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THE ENCYCLOPÆDIA BRITANNICA

A DICTIONARY OF ARTS, SCIENCES,
LITERATURE AND GENERAL INFORMATION
ELEVENTH EDITION

VOLUME XVI SLICE VIII

Logarithm to Lord Advocate

Articles in This Slice

LOGARITHM LONG, JOHN DAVIS
LOGAU, FRIEDRICH LONG BRANCH
LOGIA LONGCLOTH
LOGIC LONG EATON
LOGOCYCLIC CURVE, STROPHOID or LONGEVITY
FOLIATE
LOGOGRAPHI LONGFELLOW, HENRY
WADSWORTH
LOGOS LONG FIVES
LOGOTHETE LONGFORD (county of Ireland)
LOGROÑO (province of Spain) LONGFORD (town of Ireland)

LOGROÑO (Spanish town) LONGHI, PIETRO
LOGROSCINO, NICOLA LONGINUS, CASSIUS
LOGWOOD LONG ISLAND
LOHARU LONG ISLAND CITY
LÖHE, JOHANN KONRAD WILHELM LONGITUDE
LOHENGRIN LONGLEY, CHARLES THOMAS
LOIN LONGMANS
LOIRE (river of France) LONGOMONTANUS, CHRISTIAN
SEVERIN
LOIRE (department of France) LONGSTREET, JAMES
LOIRE-INFÉRIEURE LONGTON
LOIRET LONGUEVILLE
LOIR-ET-CHER LONGUEVILLE, ANNE
GENEVIÈVE
LOISY, ALFRED FIRMIN LONGUS
LOJA LONGWY
LOKEREN LÖNNROT, ELIAS
LOKOJA LONSDALE, EARLS OF
LOLLARDS LONSDALE, WILLIAM
LOLLIUS, MARCUS LONS-LE-SAUNIER
LOLOS LOO
LOMBARD LEAGUE LOOE
LOMBARDO LOOM (water-birds)
LOMBARDS LOOM (weaving machine)
LOMBARDY LOÓN
LOMBOK LOOP
LOMBROSO, CESARE LOOSESTRIFE
LOMÉNIE DE BRIENNE, ÉTIENNE LOOT
CHARLES DE
LOMOND, LOCH LOPES, FERNÃO
LOMONÓSOV, MIKHAIL VASILIEVICH LOPEZ, CARLOS ANTONIO
LOMZA (government of Russian) LOPEZ DE GÓMARA, FRANCISCO
LOMZA (town of Russia) LOP-NOR
LONAULI LOQUAT

LONDON (Canada) LORAIN
LONDON (capital of England) LORALAI
LONDON CLAY LORCA
LONDONDERRY, EARLS AND LORCH (Prussian town)
MARQUESSES OF
LONDONDERRY, CHARLES WILLIAM LORCH (town of kingdom of
STEWART (VANE) Württemberg)
LONDONDERRY, ROBERT STEWART LORD, JOHN
LONDONDERRY (county of Ireland) LORD
LONDONDERRY (town of Ireland) LORD ADVOCATE
LONG, GEORGE

LOGARITHM (from Gr. λόγος, word, ratio, and ἀριθμός, number),
in mathematics, a word invented by John Napier to denote a particular class of
function discovered by him, and which may be defined as follows: if a, x, m
are any three quantities satisfying the equation ax = m, then a is called the base,
and x is said to be the logarithm of m to the base a. This relation between x, a,
m, may be expressed also by the equation x = loga m.

Properties.—The principal properties of logarithms are given by the
equations

loga (mn) = loga m + loga n, loga (m/n) = loga m − loga n,
loga mr = r loga m, loga r√ m = (1/r) loga m,

which may be readily deduced from the definition of a logarithm. It follows
from these equations that the logarithm of the product of any number of
quantities is equal to the sum of the logarithms of the quantities, that the
logarithm of the quotient of two quantities is equal to the logarithm of the
numerator diminished by the logarithm of the denominator, that the logarithm
of the rth power of a quantity is equal to r times the logarithm of the quantity,
and that the logarithm of the rth root of a quantity is equal to (1/r)th of the
logarithm of the quantity.

Logarithms were originally invented for the sake of abbreviating
arithmetical calculations, as by their means the operations of multiplication and
division may be replaced by those of addition and subtraction, and the
operations of raising to powers and extraction of roots by those of
multiplication and division. For the purpose of thus simplifying the operations
of arithmetic, the base is taken to be 10, and use is made of tables of logarithms
in which the values of x, the logarithm, corresponding to values of m, the
number, are tabulated. The logarithm is also a function of frequent occurrence
in analysis, being regarded as a known and recognized function like sin x or tan
x; but in mathematical investigations the base generally employed is not 10,
but a certain quantity usually denoted by the letter e, of value 2.71828 18284....

Thus in arithmetical calculations if the base is not expressed it is understood
to be 10, so that log m denotes log10 m; but in analytical formulae it is
understood to be e.

The logarithms to base 10 of the first twelve numbers to 7 places of decimals
are

log 1 = 0.0000000 log 5 = 0.6989700 log 9 = 0.9542425
log 2 = 0.3010300 log 6 = 0.7781513 log 10 = 1.0000000
log 3 = 0.4771213 log 7 = 0.8450980 log 11 = 1.0413927
log 4 = 0.6020600 log 8 = 0.9030900 log 12 = 1.0791812

The meaning of these results is that

1 = 100, 2 = 100.3010300, 3 = 100.4771213, ...
10 = 101, 11 = 101.0413927, 12 = 101.0791812.

The integral part of a logarithm is called the index or characteristic, and the
fractional part the mantissa. When the base is 10, the logarithms of all numbers
in which the digits are the same, no matter where the decimal point may be,
have the same mantissa; thus, for example,

log 2.5613 = 0.4084604, log 25.613 = 1.4084604, log 2561300 =
6.4084604, &c.

In the case of fractional numbers (i.e. numbers in which the integral part is
0) the mantissa is still kept positive, so that, for example,

log .25613 = 1.4084604, log .0025613 = 3.4084604, &c.

the minus sign being usually written over the characteristic, and not before it,
to indicate that the characteristic only, and not the whole expression, is
negative; thus

1.4084604 stands for −1 + .4084604.

The fact that when the base is 10 the mantissa of the logarithm is independent
of the position of the decimal point in the number affords the chief reason for
the choice of 10 as base. The explanation of this property of the base 10 is
evident, for a change in the position of the decimal points amounts to
multiplication or division by some power of 10, and this corresponds to the
addition or subtraction of some integer in the case of the logarithm, the
mantissa therefore remaining intact. It should be mentioned that in most tables
of trigonometrical functions, the number 10 is added to all the logarithms in the
table in order to avoid the use of negative characteristics, so that the
characteristic 9 denotes in reality 1, 8 denotes 2, 10 denotes 0, &c. Logarithms

thus increased are frequently referred to for the sake of distinction as tabular
logarithms, so that the tabular logarithm = the true logarithm + 10.

In tables of logarithms of numbers to base 10 the mantissa only is in general
tabulated, as the characteristic of the logarithm of a number can always be
written down at sight, the rule being that, if the number is greater than unity,
the characteristic is less by unity than the number of digits in the integral
portion of it, and that if the number is less than unity the characteristic is
negative, and is greater by unity than the number of ciphers between the
decimal point and the first significant figure.

It follows very simply from the definition of a logarithm that

loga b × logb a = 1, logb m = loga m × (1/loga b).

The second of these relations is an important one, as it shows that from a
table of logarithms to base a, the corresponding table of logarithms to base b
may be deduced by multiplying all the logarithms in the former by the constant
multiplier 1/loga b, which is called the modulus of the system whose base is b
with respect to the system whose base is a.

The two systems of logarithms for which extensive tables have been
calculated are the Napierian, or hyperbolic, or natural system, of which the
base is e, and the Briggian, or decimal, or common system, of which the base is
10; and we see that the logarithms in the latter system may be deduced from
those in the former by multiplication by the constant multiplier 1/loge 10,
which is called the modulus of the common system of logarithms. The
numerical value of this modulus is 0.43429 44819 03251 82765 11289 ..., and
the value of its reciprocal, loge 10 (by multiplication by which Briggian
logarithms may be converted into Napierian logarithms) is 2.30258 50929
94045 68401 79914 ....

The quantity denoted by e is the series,

1+ 1 + 1 + 1 + 1 + ...

1 1·2 1·2·3 1·2·3·4

the numerical value of which is,

2.71828 18284 59045 23536 02874 ....

The logarithmic Function.—The mathematical function log x or loge x
is one of the small group of transcendental functions, consisting only of
the circular functions (direct and inverse) sin x, cos x, &c., arc sin x or
sin−1 x,&c., log x and ex which are universally treated in analysis as
known functions. The notation log x is generally employed in English and
American works, but on the continent of Europe writers usually denote the
function by lx or lg x. The logarithmic function is most naturally
introduced into analysis by the equation

dt
log x = ∫1x , (x > 0).
t

This equation defines log x for positive values of x; if x ≤ 0 the formula
ceases to have any meaning. Thus log x is the integral function of 1/x, and
it can be shown that log x is a genuinely new transcendent, not expressible
in finite terms by means of functions such as algebraical or circular
functions. A connexion with the circular functions, however, appears later
when the definition of log x is extended to complex values of x.

A relation which is of historical interest connects the logarithmic
function with the quadrature of the hyperbola, for, by considering the
equation of the hyperbola in the form xy = const., it is evident that the area
included between the arc of a hyperbola, its nearest asymptote, and two
ordinates drawn parallel to the other asymptote from points on the first
asymptote distant a and b from their point of intersection, is proportional
to log b/a.

The following fundamental properties of log x are readily deducible
from the definition

(i.) log xy = log x + log y.

(ii.) Limit of (xh − 1)/h = log x, when h is indefinitely diminished.

Either of these properties might be taken as itself the definition of log x.

There is no series for log x proceeding either by ascending or
descending powers of x, but there is an expansion for log (1 + x), viz.

log (1 + x) = x − 1⁄2 x2 + 1⁄3 x3 − 1⁄4 x4 + ...;

the series, however, is convergent for real values of x only when x lies
between +1 and −1. Other formulae which are deducible from this
equation are given in the portion of this article relating to the calculation
of logarithms.

The function log x as x increases from 0 towards ∞ steadily increases
from −∞ towards +∞. It has the important property that it tends to infinity
with x, but more slowly than any power of x, i.e. that x−m log x tends to
zero as x tends to ∞ for every positive value of m however small.

The exponential function, exp x, may be defined as the inverse of the
logarithm: thus x = exp y if y = log x. It is positive for all values of y and
increases steadily from 0 toward ∞ as y increases from -∞ towards +∞. As
y tends towards ∞, exp y tends towards ∞ more rapidly than any power of
y.

The exponential function possesses the properties

(i.) exp (x + y) = exp x × exp y.
(ii.) (d/dx) exp x = exp x.
(iii.) exp x = 1 + x + x2/2! + x3/3! + ...

From (i.) and (ii.) it may be deduced that

exp x = (1 + 1 + 1/2! + 1/3! + ... )x,

where the right-hand side denotes the positive xth power of the number 1
+ 1 + 1/2! + 1/3! + ... usually denoted by e. It is customary, therefore, to
denote the exponential function by ex and the result

ex = 1 + x + x2/2! + x3/3! ...

is known as the exponential theorem.

The definitions of the logarithmic and exponential functions may be
extended to complex values of x. Thus if x = ξ + iη

dt
log x = ∫1x
t

where the path of integration in the plane of the complex variable t is any
curve which does not pass through the origin; but now log x is not a
uniform function, that is to say, if x describes a closed curve it does not
follow that log x also describes a closed curve: in fact we have

log (ξ + iη) = log √(ξ2 + η2) + i(α + 2nπ),

where α is the numerically least angle whose cosine and sine are ξ/√(ξ2 +
η2) and η/√(ξ2 + η2), and n denotes any integer. Thus even when the
argument is real log x has an infinite number of values; for putting η = 0
and taking ξ positive, in which case α = 0, we obtain for log ξ the infinite
system of values log ξ + 2nπi. It follows from this property of the function
that we cannot have for log x a series which shall be convergent for all
values of x, as is the case with sin x and cos x, for such a series could only
represent a uniform function, and in fact the equation

log(1 + x) = x − 1⁄2x2 + 1⁄3x3 − 1⁄4x4 + ...

is true only when the analytical modulus of x is less than unity. The
exponential function, which may still be defined as the inverse of the

logarithmic function, is, on the other hand, a uniform function of x, and its
fundamental properties may be stated in the same form as for real values
of x. Also

exp (ξ − iη) = eξ (cos η + i sin η).

An alternative method of developing the theory of the exponential
function is to start from the definition

exp x = 1 + x + x2/2! + x3/3! + ...,

the series on the right-hand being convergent for all values of x and
therefore defining an analytical function of x which is uniform and regular
all over the plane.

Invention and Early History of Logarithms.—The invention of logarithms
has been accorded to John Napier, baron of Merchiston in Scotland, with a
unanimity which is rare with regard to important scientific discoveries: in fact,
with the exception of the tables of Justus Byrgius, which will be referred to
further on, there seems to have been no other mathematician of the time whose
mind had conceived the principle on which logarithms depend, and no partial
anticipations of the discovery are met with in previous writers.

The first announcement of the invention was made in Napier’s Mirifici
Logarithmorum Canonis Descriptio ... (Edinburgh, 1614). The work is a small
quarto containing fifty-seven pages of explanatory matter and a table of ninety
pages (see Napier, John). The nature of logarithms is explained by reference to
the motion of points in a straight line, and the principle upon which they are
based is that of the correspondence of a geometrical and an arithmetical series
of numbers. The table gives the logarithms of sines for every minute of seven
figures; it is arranged semi-quadrantally, so that the differentiae, which are the
differences of the two logarithms in the same line, are the logarithms of the
tangents. Napier’s logarithms are not the logarithms now termed Napierian or
hyperbolic, that is to say, logarithms to the base e where e = 2.7182818...; the

relation between N (a sine) and L its logarithm, as defined in the Canonis
Descriptio, being N = 107 e−L/(l07), so that (ignoring the factors 107, the effect
of which is to render sines and logarithms integral to 7 figures), the base is e−1.
Napier’s logarithms decrease as the sines increase. If l denotes the logarithm to
base e (that is, the so-called “Napierian” or hyperbolic logarithm) and L
denotes, as above, “Napier’s” logarithm, the connexion between l and L is
expressed by

L = 107 loge 107 − 107 l or el = 107 e−L/(107)

Napier’s work (which will henceforth in this article be referred to as the
Descriptio) immediately on its appearance in 1614 attracted the attention of
perhaps the two most eminent English mathematicians then living—Edward
Wright and Henry Briggs. The former translated the work into English; the
latter was concerned with Napier in the change of the logarithms from those
originally invented to decimal or common logarithms, and it is to him that the
original calculation of the logarithmic tables now in use is mainly due. Both
Napier and Wright died soon after the publication of the Descriptio, the date of
Wright’s death being 1615 and that of Napier 1617, but Briggs lived until 1631.
Edward Wright, who was a fellow of Caius College, Cambridge, occupies a
conspicuous place in the history of navigation. In 1599 he published Certaine
errors in Navigation detected and corrected, and he was the author of other
works; to him also is chiefly due the invention of the method known as
Mercator’s sailing. He at once saw the value of logarithms as an aid to
navigation, and lost no time in preparing a translation, which he submitted to
Napier himself. The preface to Wright’s edition consists of a translation of the
preface to the Descriptio, together with the addition of the following sentences
written by Napier himself: “But now some of our countreymen in this Island
well affected to these studies, and the more publique good, procured a most
learned Mathematician to translate the same into our vulgar English tongue,
who after he had finished it, sent the Coppy of it to me, to bee seene and
considered on by myselfe. I having most willingly and gladly done the same,

finde it to bee most exact and precisely conformable to my minde and the
originall. Therefore it may please you who are inclined to these studies, to
receive it from me and the Translator, with as much good will as we
recommend it unto you.” There is a short “preface to the reader” by Briggs, and
a description of a triangular diagram invented by Wright for finding the
proportional parts. The table is printed to one figure less than in the Descriptio.
Edward Wright died, as has been mentioned, in 1615, and his son, Samuel
Wright, in the preface states that his father “gave much commendation of this
work (and often in my hearing) as of very great use to mariners”; and with
respect to the translation he says that “shortly after he had it returned out of
Scotland, it pleased God to call him away afore he could publish it.” The
translation was published in 1616. It was also reissued with a new title-page in
1618.

Henry Briggs, then professor of geometry at Gresham College, London, and
afterwards Savilian professor of geometry at Oxford, welcomed the Descriptio
with enthusiasm. In a letter to Archbishop Usher, dated Gresham House, March
10, 1615, he wrote, “Napper, lord of Markinston, hath set my head and hands a
work with his new and admirable logarithms. I hope to see him this summer, if
it please God, for I never saw book which pleased me better, or made me more
wonder.1 I purpose to discourse with him concerning eclipses, for what is there
which we may not hope for at his hands,” and he also states “that he was
wholly taken up and employed about the noble invention of logarithms lately
discovered.” Briggs accordingly visited Napier in 1615, and stayed with him a
whole month.2 He brought with him some calculations he had made, and
suggested to Napier the advantages that would result from the choice of 10 as a
base, an improvement which he had explained in his lectures at Gresham
College, and on which he had written to Napier. Napier said that he had already
thought of the change, and pointed out a further improvement, viz., that the
characteristics of numbers greater than unity should be positive and not
negative, as suggested by Briggs. In 1616 Briggs again visited Napier and

showed him the work he had accomplished, and, he says, he would gladly have
paid him a third visit in 1617 had Napier’s life been spared.

Briggs’s Logarithmorum chilias prima, which contains the first published
table of decimal or common logarithms, is only a small octavo tract of sixteen
pages, and gives the logarithms of numbers from unity to 1000 to 14 places of
decimals. It was published, probably privately, in 1617, after Napier’s death,3
and there is no author’s name, place or date. The date of publication is,
however, fixed as 1617 by a letter from Sir Henry Bourchier to Usher, dated
December 6, 1617, containing the passage—“Our kind friend, Mr Briggs, hath
lately published a supplement to the most excellent tables of logarithms, which
I presume he has sent to you.” Briggs’s tract of 1617 is extremely rare, and has
generally been ignored or incorrectly described. Hutton erroneously states that
it contains the logarithms to 8 places, and his account has been followed by
most writers. There is a copy in the British Museum.

Briggs continued to labour assiduously at the calculation of logarithms, and
in 1624 published his Arithmetica logarithmica, a folio work containing the
logarithms of the numbers from l to 20,000, and from 90,000 to 100,000 (and
in some copies to 101,000) to 14 places of decimals. The table occupies 300
pages, and there is an introduction of 88 pages relating to the mode of
calculation, and the applications of logarithms.

There was thus left a gap between 20,000 and 90,000, which was filled up
by Adrian Vlacq (or Ulaccus), who published at Gouda, in Holland, in 1628, a
table containing the logarithms of the numbers from unity to 100,000 to 10
places of decimals. Having calculated 70,000 logarithms and copied only
30,000, Vlacq would have been quite entitled to have called his a new work.
He designates it, however, only a second edition of Briggs’s Arithmetica
logarithmica, the title running Arithmetica logarithmica sive Logarithmorum
Chiliades centum, ... editio secunda aucta per Adrianum Vlacq, Goudanum.
This table of Vlacq’s was published, with an English explanation prefixed, at
London in 1631 under the title Logarithmicall Arithmetike ... London, printed

by George Miller, 1631. There are also copies with the title-page and
introduction in French and in Dutch (Gouda, 1628).

Briggs had himself been engaged in filling up the gap, and in a letter to John
Pell, written after the publication of Vlacq’s work, and dated October 25, 1628,
he says:—

“My desire was to have those chiliades that are wantinge betwixt 20 and
90 calculated and printed, and I had done them all almost by my selfe, and
by some frendes whom my rules had sufficiently informed, and by
agreement the busines was conveniently parted amongst us; but I am eased
of that charge and care by one Adrian Vlacque, an Hollander, who hathe
done all the whole hundred chiliades and printed them in Latin, Dutche
and Frenche, 1000 bookes in these 3 languages, and hathe sould them
almost all. But he hathe cutt off 4 of my figures throughout; and hathe left
out my dedication, and to the reader, and two chapters the 12 and 13, in
the rest he hath not varied from me at all.”

The original calculation of the logarithms of numbers from unity to 101,000
was thus performed by Briggs and Vlacq between 1615 and 1628. Vlacq’s table
is that from which all the hundreds of tables of logarithms that have
subsequently appeared have been derived. It contains of course many errors,
which were gradually discovered and corrected in the course of the next two
hundred and fifty years.

The first calculation or publication of Briggian or common logarithms of
trigonometrical functions was made in 1620 by Edmund Gunter, who was
Briggs’s colleague as professor of astronomy in Gresham College. The title of
Gunter’s book, which is very scarce, is Canon triangulorum, and it contains
logarithmic sines and tangents for every minute of the quadrant to 7 places of
decimals.

The next publication was due to Vlacq, who appended to his logarithms of
numbers in the Arithmetica logarithmica of 1628 a table giving log sines,

tangents and secants for every minute of the quadrant to 10 places; these were
obtained by calculating the logarithms of the natural sines, &c. given in the
Thesaurus mathematicus of Pitiscus (1613).

During the last years of his life Briggs devoted himself to the calculation of
logarithmic sines, &c. and at the time of his death in 1631 he had all but
completed a logarithmic canon to every hundredth of a degree. This work was
published by Vlacq at his own expense at Gouda in 1633, under the title
Trigonometria Britannica. It contains log sines (to 14 places) and tangents (to
10 places), besides natural sines, tangents and secants, at intervals of a
hundredth of a degree. In the same year Vlacq published at Gouda his
Trigonometria artificialis, giving log sines and tangents to every 10 seconds of
the quadrant to 10 places. This work also contains the logarithms of numbers
from unity to 20,000 taken from the Arithmetica logarithmica of 1628. Briggs
appreciated clearly the advantages of a centesimal division of the quadrant, and
by dividing the degree into hundredth parts instead of into minutes, made a step
towards a reformation in this respect, and but for the appearance of Vlacq’s
work the decimal division of the degree might have become recognized, as is
now the case with the corresponding division of the second. The calculation of
the logarithms not only of numbers but also of the trigonometrical functions is
therefore due to Briggs and Vlacq; and the results contained in their four
fundamental works—Arithmetica logarithmica (Briggs), 1624; Arithmetica
logarithmica (Vlacq), 1628; Trigonometria Britannica (Briggs), 1633;
Trigonometria artificialis (Vlacq), 1633—have not been superseded by any
subsequent calculations.

In the preceding paragraphs an account has been given of the actual
announcement of the invention of logarithms and of the calculation of the
tables. It now remains to refer in more detail to the invention itself and to
examine the claims of Napier and Briggs to the capital improvement involved
in the change from Napier’s original logarithms to logarithms to the base 10.

The Descriptio contained only an explanation of the use of the logarithms
without any account of the manner in which the canon was constructed. In an
“Admonitio” on the seventh page Napier states that, although in that place the
mode of construction should be explained, he proceeds at once to the use of the
logarithms, “ut praelibatis prius usu, et rei utilitate, caetera aut magis placeant
posthac edenda, aut minus saltem displiceant silentio sepulta.” He awaits
therefore the judgment and censure of the learned “priusquam caetera in lucem
temerè prolata lividorum detrectationi exponantur”; and in an “Admonitio” on
the last page of the book he states that he will publish the mode of construction
of the canon “si huius inventi usum eruditis gratum fore intellexero.” Napier,
however, did not live to keep this promise. In 1617 he published a small work
entitled Rabdologia relating to mechanical methods of performing
multiplications and divisions, and in the same year he died.

The proposed work was published in 1619 by Robert Napier, his second son
by his second marriage, under the title Mirifici logarithmorum canonis
constructio.... It consists of two pages of preface followed by sixty-seven pages
of text. In the preface Robert Napier says that he has been assured from
undoubted authority that the new invention is much thought of by the ablest
mathematicians, and that nothing would delight them more than the publication
of the mode of construction of the canon. He therefore issues the work to
satisfy their desires, although, he states, it is manifest that it would have seen
the light in a far more perfect state if his father could have put the finishing
touches to it; and he mentions that, in the opinion of the best judges, his father
possessed, among other most excellent gifts, in the highest degree the power of
explaining the most difficult matters by a certain and easy method in the fewest
possible words.

It is important to notice that in the Constructio logarithms are called artificial
numbers; and Robert Napier states that the work was composed several years
(aliquot annos) before Napier had invented the name logarithm. The
Constructio therefore may have been written a good many years previous to the
publication of the Descriptio in 1614.

Passing now to the invention of common or decimal logarithms, that is, to
the transition from the logarithms originally invented by Napier to logarithms
to the base 10, the first allusion to a change of system occurs in the
“Admonitio” on the last page of the Descriptio (1614), the concluding
paragraph of which is “Verùm si huius inventi usum eruditis gratum fore
intellexero, dabo fortasse brevi (Deo aspirante) rationem ac methodum aut
hunc canonem emendandi, aut emendatiorem de novo condendi, ut ita plurium
Logistarum diligentia, limatior tandem et accuratior, quàm unius opera fieri
potuit, in lucem prodeat. Nihil in ortu perfectum.” In some copies, however,
this “Admonitio” is absent. In Wright’s translation of 1616 Napier has added
the sentence—“But because the addition and subtraction of these former
numbers may seeme somewhat painfull, I intend (if it shall please God) in a
second Edition, to set out such Logarithmes as shall make those numbers above
written to fall upon decimal numbers, such as 100,000,000, 200,000,000,
300,000,000, &c., which are easie to be added or abated to or from any other
number” (p. 19); and in the dedication of the Rabdologia (1617) he wrote
“Quorum quidem Logarithmorum speciem aliam multò praestantiorem nunc
etiam invenimus, & creandi methodum, unà cum eorum usu (si Deus
longiorem vitae & valetudinis usuram concesserit) evulgare statuimus; ipsam
autem novi canonis supputationem, ob infirmam corporis nostri valetudinem,
viris in hoc studii genere versatis relinquimus: imprimis verò doctissimo viro
D. Henrico Briggio Londini publico Geometriae Professori, et amico mihi
longè charissimo.”

Briggs in the short preface to his Logarithmorum chilias (1617) states that
the reason why his logarithms are different from those introduced by Napier
“sperandum, ejus librum posthumum, abunde nobis propediem satisfacturum.”
The “liber posthumus” was the Constructio (1619), in the preface to which
Robert Napier states that he has added an appendix relating to another and
more excellent species of logarithms, referred to by the inventor himself in the
Rabdologia, and in which the logarithm of unity is 0. He also mentions that he
has published some remarks upon the propositions in spherical trigonometry

and upon the new species of logarithms by Henry Briggs, “qui novi hujus
Canonis supputandi laborem gravissimum, pro singulari amicitiâ quae illi cum
Patre meo L. M. intercessit, animo libentissimo in se suscepit; creandi
methodo, et usuum explanatione Inventori relictis. Nunc autem ipso ex hâc vitâ
evocato, totius negotii onus doctissimi Briggii humeris incumbere, et Sparta
haec ornanda illi sorte quadam obtigisse videtur.”

In the address prefixed to the Arithmetica logarithmica (1625) Briggs bids
the reader not to be surprised that these logarithms are different from those
published in the Descriptio:—

“Ego enim, cum meis auditoribus Londini, publice in Collegio
Greshamensi horum doctrinam explicarem; animadverti multo futurum
commodius, si Logarithmus sinus totius servaretur 0 (ut in Canone
mirifico), Logarithmus autem partis decimae ejusdem sinus totius, nempe
sinus 5 graduum, 44, m. 21, s., esset 10000000000. atque ea de re scripsi
statim ad ipsum authorem, et quamprimum per anni tempus, et vacationem
a publico docendi munere licuit, profectus sum Edinburgum; ubi
humanissime ab eo acceptus haesi per integrum mensem. Cum autem inter
nos de horum mutatione sermo haberetur; ille se idem dudum sensisse, et
cupivisse dicebat: veruntamen istos, quos jam paraverat edendos curasse,
donec alios, si per negotia et valetudinem liceret, magis commodos
confecisset. Istam autem mutationem ita faciendam censebat, ut 0 esset
Logarithmus unitatis, et 10000000000 sinus totius: quod ego longe
commodissimum esse non potui non agnoscere. Coepi igitur, ejus hortatu,
rejectis illis quos anteà paraveram, de horum calculo serio cogitare; et
sequenti aestate iterum profectus Edinburgum, horum quos hic exhibeo
praecipuos, illi ostendi, idem etiam tertia aestate libentissime facturus, si
Deus illum nobis tamdiu superstitem esse voluisset.”

There is also a reference to the change of the logarithms on the title-page of
the work.

These extracts contain all the original statements made by Napier, Robert
Napier and Briggs which have reference to the origin of decimal logarithms. It
will be seen that they are all in perfect agreement. Briggs pointed out in his
lectures at Gresham College that it would be more convenient that 0 should
stand for the logarithm of the whole sine as in the Descriptio, but that the
logarithm of the tenth part of the whole sine should be 10,000,000,000. He
wrote also to Napier at once; and as soon as he could he went to Edinburgh to
visit him, where, as he was most hospitably received by him, he remained for a
whole month. When they conversed about the change of system, Napier said
that he had perceived and desired the same thing, but that he had published the
tables which he had already prepared, so that they might be used until he could
construct others more convenient. But he considered that the change ought to
be so made that 0 should be the logarithm of unity and 10,000,000,000 that of
the whole sine, which Briggs could not but admit was by far the most
convenient of all. Rejecting therefore, those which he had prepared already,
Briggs began, at Napier’s advice, to consider seriously the question of the
calculation of new tables. In the following summer he went to Edinburgh and
showed Napier the principal portion of the logarithms which he published in
1624. These probably included the logarithms of the first chiliad which he
published in 1617.

It has been thought necessary to give in detail the facts relating to the
conversion of the logarithms, as unfortunately Charles Hutton in his history of
logarithms, which was prefixed to the early editions of his Mathematical
Tables, and was also published as one of his Mathematical Tracts, has charged
Napier with want of candour in not telling the world of Briggs’s share in the
change of system, and he expresses the suspicion that “Napier was desirous
that the world should ascribe to him alone the merit of this very useful
improvement of the logarithms.” According to Hutton’s view, the words, “it is
to be hoped that his posthumous work” ... which occur in the preface to the
Chilias, were a modest hint that the share Briggs had had in changing the
logarithms should be mentioned, and that, as no attention was paid to it, he

himself gave the account which appears in the Arithmetica of 1624. There
seems, however, no ground whatever for supposing that Briggs meant to
express anything beyond his hope that the reason for the alteration would be
explained in the posthumous work; and in his own account, written seven years
after Napier’s death and five years after the appearance of the work itself, he
shows no injured feeling whatever, but even goes out of his way to explain that
he abandoned his own proposed alteration in favour of Napier’s, and, rejecting
the tables he had already constructed, began to consider the calculation of new
ones. The facts, as stated by Napier and Briggs, are in complete accordance,
and the friendship existing between them was perfect and unbroken to the last.
Briggs assisted Robert Napier in the editing of the “posthumous work,” the
Constructio, and in the account he gives of the alteration of the logarithms in
the Arithmetica of 1624 he seems to have been more anxious that justice
should be done to Napier than to himself; while on the other hand Napier
received Briggs most hospitably and refers to him as “amico mihi longè
charissimo.”

Hutton’s suggestions are all the more to be regretted as they occur as a
history which is the result of a good deal of investigation and which for years
was referred to as an authority by many writers. His prejudice against Napier
naturally produced retaliation, and Mark Napier in defending his ancestor has
fallen into the opposite extreme of attempting to reduce Briggs to the level of a
mere computer. In connexion with this controversy it should be noticed that the
“Admonitio” on the last page of the Descriptio, containing the reference to the
new logarithms, does not occur in all the copies. It is printed on the back of the
last page of the table itself, and so cannot have been torn out from the copies
that are without it. As there could have been no reason for omitting it after it
had once appeared, we may assume that the copies which do not have it are
those which were first issued. It is probable, therefore, that Briggs’s copy
contained no reference to the change, and it is even possible that the
“Admonitio” may have been added after Briggs had communicated with
Napier. As special attention has not been drawn to the fact that some copies

have the “Admonitio” and some have not, different writers have assumed that
Briggs did or did not know of the promise contained in the “Admonitio”
according as it was present or absent in the copies they had themselves referred
to, and this has given rise to some confusion. It may also be remarked that the
date frequently assigned to Briggs’s first visit to Napier is 1616, and not 1615
as stated above, the reason being that Napier was generally supposed to have
died in 1618 until Mark Napier showed that the true date was 1617. When the
Descriptio was published Briggs was fifty-seven years of age, and the
remaining seventeen years of his life were devoted with steady enthusiasm to
extend the utility of Napier’s great invention.

The only other mathematician besides Napier who grasped the idea on which
the use of logarithm depends and applied it to the construction of a table is
Justus Byrgius (Jobst Bürgi), whose work Arithmetische und geometrische
Progress-Tabulen ... was published at Prague in 1620, six years after the
publication of the Descriptio of Napier. This table distinctly involves the
principle of logarithms and may be described as a modified table of
antilogarithms. It consists of two series of numbers, the one being an
arithmetical and the other a geometrical progression: thus

0, 1,0000 0000
10, 1,0001 0000
20, l,0002 0001
. . . .
990, l,0099 4967
. . . .

In the arithmetical column the numbers increase by 10, in the geometrical
column each number is derived from its predecessor by multiplication by
1.0001. Thus the number 10x in the arithmetical column corresponds to 108
(1.0001)x in the geometrical column; the intermediate numbers being obtained
by interpolation. If we divide the numbers in the geometrical column by 108

the correspondence is between 10x and (1.0001)x, and the table then becomes
one of antilogarithms, the base being (1.0001)1/10, viz. for example
(l.0001)1/10·990 = 1.00994967. The table extends to 230270 in the arithmetical
column, and it is shown that 230270.022 corresponds to 9.9999 9999 or 109 in
the geometrical column; this last result showing that (1.0001)23027.022 = 10. The
first contemporary mention of Byrgius’s table occurs on page 11 of the
“Praecepta” prefixed to Kepler’s Tabulae Radolphinae (1627); his words are:
“apices logistici J. Byrgio multis annis ante editionem Neperianam viam
praeiverent ad hos ipsissimos logarithmos. Etsi homo cunctator et secretorum
suorum custos foetum in partu destituit, non ad usus publicos educavit.”
Another reference to Byrgius occurs in a work by Benjamin Bramer, the
brother-in-law and pupil of Byrgius, who, writing in 1630, says that the latter
constructed his table twenty years ago or more.4

As regards priority of publication, Napier has the advantage by six years,
and even fully accepting Bramer’s statement, there are grounds for believing
that Napier’s work dates from a still earlier period.

The power of 10, which occurs as a factor in the tables of both Napier and
Byrgius, was rendered necessary by the fact that the decimal point was not yet
in use. Omitting this factor in the case of both tables, the connexion between N
a number and L its “logarithm” is

N = (e−1)L (Napier), L =(1.0001)1⁄10N (Byrgius),

viz. Napier gives logarithms to base e-1, Byrgius gives antilogarithms to base
(1.0001)1/10.

There is indirect evidence that Napier was occupied with logarithms as early
as 1594, for in a letter to P. Crügerus from Kepler, dated September 9, 1624
(Frisch’s Kepler, vi. 47), there occurs the sentence: “Nihil autem supra
Neperianam rationem esse puto: etsi quidem Scotus quidam literis ad
Tychonem 1594 scriptis jam spem fecit Canonis illius Mirifici.” It is here

distinctly stated that some Scotsman in the year 1594, in a letter to Tycho
Brahe, gave him some hope of the logarithms; and as Kepler joined Tycho after
his expulsion from the island of Huen, and had been so closely associated with
him in his work, he would be likely to be correct in any assertion of this kind.
In connexion with Kepler’s statement the following story, told by Anthony
wood in the Athenae Oxonienses, is of some importance:—

“It must be now known, that one Dr Craig, a Scotchman ... coming out
of Denmark into his own country, called upon Joh. Neper, Baron of
Mercheston, near Edinburgh, and told him, among other discourses, of a
new invention in Denmark (by Longomontanus, as ’tis said), to save the
tedious multiplication and division in astronomical calculations. Neper
being solicitous to know farther of him concerning this matter, he could
give no other account of it than that it was by proportional numbers.
Which hint Neper taking, he desired him at his return to call upon him
again. Craig, after some weeks had passed, did so, and Neper then showed
him a rude draught of what he called Canon mirabilis logarithmorum.
which draught, with some alterations, he printing in 1614, it came
forthwith into the hands of our author Briggs, and into those of Will.
Oughtred, from whom the relation of this matter came.”

This story, though obviously untrue in some respects, gives valuable
information by connecting Dr Craig with Napier and Longomontanus, who was
Tycho Brahe’s assistant. Dr Craig was John Craig, the third son of Thomas
Craig, who was one of the colleagues of Sir Archibald Napier, John Napier’s
father, in the office of justice-depute. Between John Craig and John Napier a
friendship sprang up which may have been due to their common taste for
mathematics. There are extant three letters from Dr John Craig to Tycho Brahe,
which show that he was on the most friendly terms with him. In the first letter,
of which the date is not given, Craig says that Sir William Stuart has safely
delivered to him, “about the beginning of last winter,” the book which he sent
him. Now Mark Napier found in the library of the university of Edinburgh a
mathematical work bearing a sentence in Latin which he translates, “To Doctor

John Craig of Edinburgh, in Scotland, a most illustrious man, highly gifted
with various and excellent learning, professor of medicine, and exceedingly
skilled in the mathematics, Tycho Brahe hath sent this gift, and with his own
hand written this at Uraniburg, 2d November 1588.” As Sir William Stuart was
sent to Denmark to arrange the preliminaries of King James’s marriage, and
returned to Edinburgh on the 15th of November 1588, it would seem probable
that this was the volume referred to by Craig. It appears from Craig’s letter, to
which we may therefore assign the date 1589, that, five years before, he had
made an attempt to reach Uranienburg, but had been baffled by the storms and
rocks of Norway, and that ever since then he had been longing to visit Tycho.
Now John Craig was physician to the king, and in 1590 James VI. spent some
days at Uranienburg, before returning to Scotland from his matrimonial
expedition. It seems not unlikely therefore that Craig may have accompanied
the king in his visit to Uranienburg.5 In any case it is certain that Craig was a
friend and correspondent of Tycho’s, and it is probable that he was the “Scotus
quidam.”

We may infer therefore that as early as 1594 Napier had communicated to
some one, probably John Craig, his hope of being able to effect a simplification
in the processes of arithmetic. Everything tends to show that the invention of
logarithms was the result of many years of labour and thought,6 undertaken
with this special object, and it would seem that Napier had seen some prospect
of success nearly twenty years before the publication of the Descriptio. It is
very evident that no mere hint with regard to the use of proportional numbers
could have been of any service to him, but it is possible that the news brought
by Craig of the difficulties placed in the progress of astronomy by the labour of
the calculations may have stimulated him to persevere in his efforts.

The “new invention in Denmark” to which Anthony Wood refers as having
given the hint to Napier was probably the method of calculation called
prosthaphaeresis (often written in Greek letters προσθαφαίρεσις), which had

its origin in the solution of spherical triangles.7 The method consists in the use
of the formula

sin a sin b = 1⁄2 {cos (a − b) − cos (a + b)},

by means of which the multiplication of two sines is reduced to the addition or
subtraction of two tabular results taken from a table of sines; and, as such
products occur in the solution of spherical triangles, the method affords the
solution of spherical triangles in certain cases by addition and subtraction only.
It seems to be due to Wittich of Breslau, who was assistant for a short time to
Tycho Brahe; and it was used by them in their calculations in 1582. Wittich in
1584 made known at Cassel the calculation of one case by this
prosthaphaeresis; and Justus Byrgius proved it in such a manner that from his
proof the extension to the solution of all triangles could be deduced.8 Clavius
generalized the method in his treatise De astrolabio (1593), lib. i. lemma liii.
The lemma is enunciated as follows:—

“Quaestiones omnes, quae per sinus, tangentes, atque secantes absolvi
solent, per solam prosthaphaeresim, id est, per solam additionem,
subtractionem, sine laboriosa numerorum multiplicatione divisioneque
expedire.”

Clavius then refers to a work of Raymarus Ursus Dithmarsus as containing
an account of a particular case. The work is probably the Fundamentum
astronomicum (1588). Longomontanus, in his Astronomia Danica (1622),
gives an account of the method, stating that it is not to be found in the writings
of the Arabs or Regiomontanus. As Longomontanus is mentioned in Anthony
Wood’s anecdote, and as Wittich as well as Longomontanus were assistants of
Tycho, we may infer that Wittich’s prosthaphaeresis is the method referred to
by Wood.

It is evident that Wittich’s prosthaphaeresis could not be a good method of
practically effecting multiplications unless the quantities to be multiplied were
sines, on account of the labour of the interpolations. It satisfies the condition,

however, equally with logarithms, of enabling multiplication to be performed
by the aid of a table of single entry; and, analytically considered, it is not so
different in principle from the logarithmic method. In fact, if we put xy = φ(X
+ Y), X being a function of x only and Y a function of y only, we can show that
we must have X = Aeqx, y = Beqy; and if we put xy = φ(X + Y) − φ(X − Y), the
solutions are φ(X + Y) = 1⁄4(x + y)2, and x = sin X, y = sin Y, φ(X + Y) = −1⁄2
cos(X + Y). The former solution gives a method known as that of quarter-
squares; the latter gives the method of prosthaphaeresis.

An account has now been given of Napier’s invention and its publication,
the transition to decimal logarithms, the calculation of the tables by Briggs,
Vlacq and Gunter, as well as of the claims of Byrgius and the method of
prosthaphaeresis. To complete the early history of logarithms it is necessary to
return to Napier’s Descriptio in order to describe its reception on the continent,
and to mention the other logarithmic tables which were published while Briggs
was occupied with his calculations.

John Kepler, who has been already quoted in connexion with Craig’s visit to
Tycho Brahe, received the invention of logarithms almost as enthusiastically as
Briggs. His first mention of the subject occurs in a letter to Schikhart dated the
11th of March 1618, in which he writes-“Extitit Scotus Baro, cujus nomen mihi
excidit, qui praeclari quid praestitit, necessitate omni multiplicationum et
divisionum in meras additiones et subtractiones commutata, nec sinibus utitur;
at tamen opus est ipsi tangentium canone: et varietas, crebritas, difficultasque
additionum subtractionumque alicubi laborem multiplicandi et dividendi
superat.” This erroneous estimate was formed when he had seen the Descriptio
but had not read it; and his opinion was very different when he became
acquainted with the nature of logarithms. The dedication of his Ephemeris for
1620 consists of a letter to Napier dated the 28th of July 1619, and he there
congratulates him warmly on his invention and on the benefit he has conferred
upon astronomy generally and upon Kepler’s own Rudolphine tables. He says
that, although Napier’s book had been published five years, he first saw it at

Prague two years before; he was then unable to read it, but last year he had met
with a little work by Benjamin Ursinus9 containing the substance of the
method, and he at once recognized the importance of what had been effected.
He then explains how he verified the canon, and so found that there were no
essential errors in it, although there were a few inaccuracies near the beginning
of the quadrant, and he proceeds, “Haec te obiter scire volui, ut quibus tu
methodis incesseris, quas non dubito et plurimas et ingeniosissimas tibi in
promptu esse, eas publici juris fieri, mihi saltem (puto et caeteris) scires fore
gratissimum; eoque percepto, tua promissa folio 57, in debitum cecidisse
intelligeres.” This letter was written two years after Napier’s death (of which
Kepler was unaware), and in the same year as that in which the Constructio
was published. In the same year (1620) Napier’s Descriptio (1614) and
Constructio (1619) were reprinted by Bartholomew Vincent at Lyons and
issued together.10

Napier calculated no logarithms of numbers, and, as already stated, the
logarithms invented by him were not to base e. The first logarithms to the base
e were published by John Speidell in his New Logarithmes (London, 1619),
which contains hyperbolic log sines, tangents and secants for every minute of
the quadrant to 5 places of decimals.

In 1624 Benjamin Ursinus published at Cologne a canon of logarithms
exactly similar to Napier’s in the Descriptio of 1614, only much enlarged. The
interval of the arguments is 10″, and the results are given to 8 places; in
Napier’s canon the interval is 1′, and the number of places is 7. The logarithms
are strictly Napierian, and the arrangement is identical with that in the canon of
1614. This is the largest Napierian canon that has ever been published.

In the same year (1624) Kepler published at Marburg a table of Napierian
logarithms of sines with certain additional columns to facilitate special
calculations.

The first publication of Briggian logarithms on the continent is due to
Wingate, who published at Paris in 1625 his Arithmétique logarithmétique,
containing seven-figure logarithms of numbers up to 1000, and log sines and
tangents from Gunter’s Canon (1620). In the following year, 1626, Denis
Henrion published at Paris a Traicté des Logarithmes, containing Briggs’s
logarithms of numbers up to 20,001 to 10 places, and Gunter’s log sines and
tangents to 7 places for every minute. In the same year de Decker also
published at Gouda a work entitled Nieuwe Telkonst, inhoudende de
Logarithmi voor de Ghetallen beginnende van 1 tot 10,000, which contained
logarithms of numbers up to 10,000 to 10 places, taken from Briggs’s
Arithmetica of 1624, and Gunter’s log sines and tangents to 7 places for every
minute.11 Vlacq rendered assistance in the publication of this work, and the
privilege is made out to him.

The invention of logarithms and the calculation of the earlier tables form a
very striking episode in the history of exact science, and, with the exception of
the Principia of Newton, there is no mathematical work published in the
country which has produced such important consequences, or to which so
much interest attaches as to Napier’s Descriptio. The calculation of tables of
the natural trigonometrical functions may be said to have formed the work of
the last half of the 16th century, and the great canon of natural sines for every
10 seconds to 15 places which had been calculated by Rheticus was published
by Pitiscus only in 1613, the year before that in which the Descriptio appeared.
In the construction of the natural trigonometrical tables Great Britain had taken
no part, and it is remarkable that the discovery of the principles and the
formation of the tables that were to revolutionize or supersede all the methods
of calculation then in use should have been so rapidly effected and developed
in a country in which so little attention had been previously devoted to such
questions.

For more detailed information relating to Napier, Briggs and Vlacq, and
the invention of logarithms, the reader is referred to the life of Briggs in
Ward’s Lives of the Professors of Gresham College (London, 1740);

Thomas Smith’s Vitae quorundam eruditissimorum et illustrium virorum
(Vita Henrici Briggii) (London, 1707); Mark Napier’s Memoirs of John
Napier already referred to, and the same author’s Naperi libri qui
supersunt (1839); Hutton’s History; de Morgan’s article already referred
to; Delambre’s Histoire de l’Astronomie moderne; the report on
mathematical tables in the Report of the British Association for 1873; and
the Philosophical Magazine for October and December 1872 and May
1873. It may be remarked that the date usually assigned to Briggs’s first
visit to Napier is 1616 and not 1615 as stated above, the reason being that
Napier was generally supposed to have died in 1618; but it was shown by
Mark Napier that the true date is 1617.

In the years 1791-1807 Francis Maseres published at London, in six volumes
quarto “Scriptores Logarithmici, or a collection of several curious tracts on the
nature and construction of logarithms, mentioned in Dr Hutton’s historical
introduction to his new edition of Sherwin’s mathematical tables ...,” which
contains reprints of Napier’s Descriptio of 1614, Kepler’s writings on
logarithms (1624-1625), &c. In 1889 a translation of Napier’s Constructio of
1619 was published by Walter Rae Macdonald. Some valuable notes are added
by the translator, in one of which he shows the accuracy of the method
employed by Napier in his calculations, and explains the origin of a small error
which occurs in Napier’s table. Appended to the Catalogue is a full and careful
bibliography of all Napier’s writings, with mention of the public libraries,
British and foreign, which possess copies of each. A facsimile reproduction of
Bartholomew Vincent’s Lyons edition (1620) of the Constructio was issued in
1895 by A. Hermann at Paris (this imprint occurs on page 62 after the word
“Finis”).

It now remains to notice briefly a few of the more important events in the
history of logarithmic tables subsequent to the original calculations.

Common or Briggian Logarithms of Numbers.—Nathaniel Roe’s
Tabulae logarithmicae (1633) was the first complete seven-figure table

that was published. It contains seven-figure logarithms of numbers from 1
to 100,000, with characteristics unseparated from the mantissae, and was
formed from Vlacq’s table (1628) by leaving out the last three figures. All
the figures of the number are given at the head of the columns, except the
last two, which run down the extreme columns—1 to 50 on the left-hand
side, and 50 to 100 on the right-hand side. The first four figures of the
logarithms are printed at the top of the columns. There is thus an advance
half way towards the arrangement now universal in seven-figure tables.
The final step was made by John Newton in his Trigonometria Britannica
(1658), a work which is also noticeable as being the only extensive eight-
figure table that until recently had been published; it contains logarithms
of sines, &c., as well as logarithms of numbers.

In 1705 appeared the original edition of Sherwin’s tables, the first of the
series of ordinary seven-figure tables of logarithms of numbers and
trigonometrical functions such as are in general use now. The work went
through several editions during the 18th century, and was at length
superseded in 1785 by Hutton’s tables, which continued in successive
editions to maintain their position for a century.

In 1717 Abraham Sharp published in his Geometry Improv’d the
Briggian logarithms of numbers from 1 to 100, and of primes from 100 to
1100, to 61 places; these were copied into the later editions of Sherwin
and other works.

In 1742 a seven-figure table was published in quarto form by Gardiner,
which is celebrated on account of its accuracy and of the elegance of the
printing. A French edition, which closely resembles the original, was
published at Avignon in 1770.

In 1783 appeared at Paris the first edition of François Callet’s tables,
which correspond to those of Hutton in England. These tables, which form
perhaps the most complete and practically useful collection of logarithms

for the general computer that has been published, passed through many
editions.

In 1794 Vega published his Thesaurus logarithmorum completus, a
folio volume containing a reprint of the logarithms of numbers from
Vlacq’s Arithmetica logarithmica of 1628, and Trigonometria artificialis
of 1633. The logarithms of numbers are arranged as in an ordinary seven-
figure table. In addition to the logarithms reprinted from the
Trigonometria, there are given logarithms for every second of the first two
degrees, which were the result of an original calculation. Vega devoted
great attention to the detection and correction of the errors in Vlacq’s work
of 1628. Vega’s Thesaurus has been reproduced photographically by the
Italian government. Vega also published in 1797, in 2 vols. 8vo, a
collection of logarithmic and trigonometrical tables which has passed
through many editions, a very useful one volume stereotype edition having
been published in 1840 by Hülsse. The tables in this work may be
regarded as to some extent supplementary to those in Callet.

If we consider only the logarithms of numbers, the main line of descent
from the original calculation of Briggs and Vlacq is Roe, John Newton,
Sherwin, Gardiner; there are then two branches, viz. Hutton founded on
Sherwin and Callet on Gardiner, and the editions of Vega form a separate
offshoot from the original tables. Among the most useful and accessible of
modern ordinary seven-figure tables of logarithms of numbers and
trigonometrical functions may be mentioned those of Bremiker, Schrön
and Bruhns. For logarithms of numbers only perhaps Babbage’s table is
the most convenient.12

In 1871 Edward Sang published a seven-figure table of logarithms of
numbers from 20,000 to 200,000, the logarithms between 100,000 and
200,000 being the result of a new calculation. By beginning the table at
20,000 instead of at 10,000 the differences are halved in magnitude, while
the number of them in a page is quartered. In this table multiples of the

differences, instead of proportional parts, are given.13 John Thomson of
Greenock (1782-1855) made an independent calculation of logarithms of
numbers up to 120,000 to 12 places of decimals, and his table has been
used to verify the errata already found in Vlacq and Briggs by Lefort (see
Monthly Not. R.A.S. vol. 34, p. 447). A table of ten-figure logarithms of
numbers up to 100,009 was calculated by W. W. Duffield and published in
the Report of the U.S. Coast and Geodetic Survey for 1895-1896 as
Appendix 12, pp. 395-722. The results were compared with Vega’s
Thesaurus (1794) before publication.

Common or Briggian Logarithms of Trigonometrical Functions.—The
next great advance on the Trigonometria artificialis took place more than a
century and a half afterwards, when Michael Taylor published in 1792 his
seven-decimal table of log sines and tangents to every second of the
quadrant; it was calculated by interpolation from the Trigonometria to 10
places and then contracted to 7. On account of the great size of this table,
and for other reasons, it never came into very general use, Bagay’s
Nouvelles tables astronomiques (1829), which also contains log sines and
tangents to every second, being preferred; this latter work, which for many
years was difficult to procure, has been reprinted with the original title-
page and date unchanged. The only other logarithmic canon to every
second that has been published forms the second volume of Shortrede’s
Logarithmic Tables (1849). In 1784 the French government decided that
new tables of sines, tangents, &c., and their logarithms, should be
calculated in relation to the centesimal division of the quadrant. Prony was
charged with the direction of the work, and was expressly required “non
seulement à composer des tables qui ne laissassent rien à désirer quant à
l’exactitude, mais à en faire le monument de calcul le plus vaste et le plus
imposant qui eût jamais été exécuté ou même conçu.” Those engaged
upon the work were divided into three sections: the first consisted of five
or six mathematicians, including Legendre, who were engaged in the
purely analytical work, or the calculation of the fundamental numbers; the

second section consisted of seven or eight calculators possessing some
mathematical knowledge; and the third comprised seventy or eighty
ordinary computers. The work, which was performed wholly in duplicate,
and independently by two divisions of computers, occupied two years. As
a consequence of the double calculation, there are two manuscripts, one
deposited at the Observatory, and the other in the library of the Institute, at
Paris. Each of the two manuscripts consists essentially of seventeen large
folio volumes, the contents being as follows:—

Logarithms of numbers up to 200,000 8 vols.
Natural sines 1 ”
Logarithms of the ratios of arcs to sines from 0q.00000
to 0q.05000, and log sines throughout the quadrant 4 ”
Logarithms of the ratios of arcs to tangents from
0q.00000 to 0q.05000, and log tangents throughout
the quadrant 4 ”

The trigonometrical results are given for every hundred-thousandth of
the quadrant (10″ centesimal or 3″.24 sexagesimal). The tables were all
calculated to 14 places, with the intention that only 12 should be
published, but the twelfth figure is not to be relied upon. The tables have
never been published, and are generally known as the Tables du Cadastre,
or, in England, as the great French manuscript tables.

A very full account of these tables, with an explanation of the methods
of calculation, formulae employed, &c., was published by Lefort in vol. iv.
of the Annales de l’observatoire de Paris. The printing of the table of
natural sines was once begun, and Lefort states that he has seen six copies,
all incomplete, although including the last page. Babbage compared his
table with the Tables du Cadastre, and Lefort has given in his paper just
referred to most important lists of errors in Vlacq’s and Briggs’s

logarithms of numbers which were obtained by comparing the manuscript
tables with those contained in the Arithmetica logarithmica of 1624 and of
1628.

As the Tables du Cadastre remained unpublished, other tables appeared
in which the quadrant was divided centesimally, the most important of
these being Hobert and Ideler’s Nouvelles tables trigonométriques (1799),
and Borda and Delambre’s Tables trigonométriques décimales (1800-
1801), both of which are seven-figure tables. The latter work, which was
much used, being difficult to procure, and greater accuracy being required,
the French government in 1891 published an eight-figure centesimal table,
for every ten seconds, derived from the Tables du Cadastre.

Decimal or Briggian Antilogarithms.—In the ordinary tables of
logarithms the natural numbers are all integers, while the logarithms
tabulated are incommensurable. In an antilogarithmic table, the logarithms
are exact quantities such as .00001, .00002, &c., and the numbers are
incommensurable. The earliest and largest table of this kind that has been
constructed is Dodson’s Antilogarithmic canon (1742), which gives the
numbers to 11 places, corresponding to the logarithms from .00001 to
.99999 at intervals of .00001. Antilogarithmic tables are few in number,
the only other extensive tables of the same kind that have been published
occurring in Shortrede’s Logarithmic tables already referred to, and in
Filipowski’s Table of antilogarithms (1849). Both are similar to Dodson’s
tables, from which they were derived, but they only give numbers to 7
places.

Hyperbolic or Napierian logarithms (i.e. to base e).—The most
elaborate table of hyperbolic logarithms that exists is due to Wolfram, a
Dutch lieutenant of artillery. His table gives the logarithms of all numbers
up to 2200, and of primes (and also of a great many composite numbers)
from 2200 to 10,009, to 48 decimal places. The table appeared in
Schulze’s Neue und erweiterte Sammlung logarithmischer Tafeln (1778),

and was reprinted in Vega’s Thesaurus (1794), already referred to. Six
logarithms omitted in Schulze’s work, and which Wolfram had been
prevented from computing by a serious illness, were published
subsequently, and the table as given by Vega is complete. The largest
hyperbolic table as regards range was published by Zacharias Dase at
Vienna in 1850 under the title Tafel der natürlichen Logarithmen der
Zahlen.

Hyperbolic antilogarithms are simple exponentials, i.e. the hyperbolic
antilogarithm of x is ex. Such tables can scarcely be said to come under the
head of logarithmic tables. See Tables, Mathematical: Exponential
Functions.

Logistic or Proportional Logarithms.—The old name for what are now
called ratios or fractions are logistic numbers, so that a table of log (a/x)
where x is the argument and a a constant is called a table of logistic or
proportional logarithms; and since log (a/x) = log a − log x it is clear that
the tabular results differ from those given in an ordinary table of
logarithms only by the subtraction of a constant and a change of sign. The
first table of this kind appeared in Kepler’s work of 1624 which has been
already referred to. The object of a table of log (a/x) is to facilitate the
working out of proportions in which the third term is a constant quantity a.
In most collections of tables of logarithms, and especially those intended
for use in connexion with navigation, there occurs a small table of logistic
logarithms in which a = 3600″ (= 1° or 1h), the table giving log 3600 − log
x, and x being expressed in minutes and seconds. It is also common to find
tables in which a = 10800″ (= 3° or 3h), and x is expressed in degrees (or
hours), minutes and seconds. Such tables are generally given to 4 or 5
places. The usual practice in books seems to be to call logarithms logistic
when a is 3600″, and proportional when a has any other value.

Addition and Subtraction, or Gaussian Logarithms.—Gaussian
logarithms are intended to facilitate the finding of the logarithms of the

sum and difference of two numbers whose logarithms are known, the
numbers themselves being unknown; and on this account they are
frequently called addition and subtraction logarithms. The object of the
table is in fact to give log (a ± b) by only one entry when log a and log b
are given. The utility of such logarithms was first pointed out by Leonelli
in a book entitled Supplément logarithmique, printed at Bordeaux in the
year XI. (1802/3); he calculated a table to 14 places, but only a specimen
of it which appeared in the Supplément was printed. The first table that
was actually published is due to Gauss, and was printed in Zach’s
Monatliche Correspondenz, xxvi. 498 (1812). Corresponding to the
argument log x it gives the values of log (1 + x−1) and log (1 + x).

Dual Logarithms.—This term was used by Oliver Byrne in a series of
works published between 1860 and 1870. Dual numbers and logarithms
depend upon the expression of a number as a product of 1.1, 1.01, 1.001 ...
or of .9, .99, .999....

In the preceding résumé only those publications have been mentioned
which are of historic importance or interest.14 For fuller details with
respect to some of these works, for an account of tables published in the
latter part of the 19th century, and for those which would now be used in
actual calculation, reference should be made to the article Tables,
Mathematical.

Calculation of Logarithms.—The name logarithm is derived from the
words λόγων ἀριθμός, the number of the ratios, and the way of regarding
a logarithm which justifies the name may be explained as follows.
Suppose that the ratio of 10, or any other particular number, to 1 is
compounded of a very great number of equal ratios, as, for example,
1,000,000, then it can be shown that the ratio of 2 to 1 is very nearly equal
to a ratio compounded of 301,030 of these small ratios, or ratiunculae,
that the ratio of 3 to 1 is very nearly equal to a ratio compounded of
477,121 of them, and so on. The small ratio, or ratiuncula, is in fact that

of the millionth root of 10 to unity, and if we denote it by the ratio of a to
1, then the ratio of 2 to 1 will be nearly the same as that of a301,030 to 1,
and so on; or, in other words, if a denotes the millionth root of 10, then 2
will be nearly equal to a301,030, 3 will be nearly equal to a477,121, and so
on.

Napier’s original work, the Descriptio Canonis of 1614, contained, not
logarithms of numbers, but logarithms of sines, and the relations between
the sines and the logarithms were explained by the motions of points in
lines, in a manner not unlike that afterwards employed by Newton in the
method of fluxions. An account of the processes by which Napier
constructed his table was given in the Constructio Canonis of 1619. These
methods apply, however, specially to Napier’s own kind of logarithms,
and are different from those actually used by Briggs in the construction of
the tables in the Arithmetica Logarithmica, although some of the latter are
the same in principle as the processes described in an appendix to the
Constructio.

The processes used by Briggs are explained by him in the preface to the
Arithmetica Logarithmica (1624). His method of finding the logarithms of
the small primes, which consists in taking a great number of continued
geometric means between unity and the given primes, may be described as
follows. He first formed the table of numbers and their logarithms:—

Numbers. Logarithms,
10 1
3.162277... 0.5
1.778279... 0.25
1.333521... 0.125
1.154781... 0.0625

each quantity in the left-hand column being the square root of the one
above it, and each quantity in the right-hand column being the half of the
one above it. To construct this table Briggs, using about thirty places of
decimals, extracted the square root of 10 fifty-four times, and thus found
that the logarithm of 1.00000 00000 00000 12781 91493 20032 35 was
0.00000 00000 00000 05551 11512 31257 82702, and that for numbers of
this form (i.e. for numbers beginning with 1 followed by fifteen ciphers,
and then by seventeen or a less number of significant figures) the
logarithms were proportional to these significant figures. He then by
means of a simple proportion deduced that log (1.00000 00000 00000 1) =
0.00000 00000 00000 04342 94481 90325 1804, so that, a quantity
1.00000 00000 00000 x (where x consists of not more than seventeen
figures) having been obtained by repeated extraction of the square root of
a given number, the logarithm of 1.00000 00000 00000 x could then be
found by multiplying x by .00000 00000 00000 04342....

To find the logarithm of 2, Briggs raised it to the tenth power, viz. 1024,
and extracted the square root of 1.024 forty-seven times, the result being
1.00000 00000 00000 16851 60570 53949 77. Multiplying the significant
figures by 4342 ... he obtained the logarithm of this quantity, viz. 0.00000
00000 00000 07318 55936 90623 9336, which multiplied by 247 gave
0.01029 99566 39811 95265 277444, the logarithm of 1.024, true to 17 or
18 places. Adding the characteristic 3, and dividing by 10, he found (since
2 is the tenth root of 1024) log 2 = .30102 99956 63981 195. Briggs
calculated in a similar manner log 6, and thence deduced log 3.

It will be observed that in the first process the value of the modulus is in
fact calculated from the formula.

h 1
= ,
10h − 1 loge 10

the value of h being 1/254, and in the second process log10 2 is in effect
calculated from the formula.

1 247
log10 2 = ( 210/2 − 1 )
247 × loge 10 ×
10
.

Briggs also gave methods of forming the mean proportionals or square
roots by differences; and the general method of constructing logarithmic
tables by means of differences is due to him.

The following calculation of log 5 is given as an example of the
application of a method of mean proportionals. The process consists in
taking the geometric mean of numbers above and below 5, the object
being to at length arrive at 5.000000. To every geometric mean in the
column of numbers there corresponds the arithmetical mean in the column
of logarithms. The numbers are denoted by A, B, C, &c., in order to
indicate their mode of formation.

Numbers. Logarithms.
A= 1.000000 0.0000000
B= 10.000000 1.0000000
C = √(AB) = 3.162277 0.5000000
D = √(BC) = 5.623413 0.7500000
E = √(CD) = 4.216964 0.6250000
F = √(DE) = 4.869674 0.6875000
G = √(DF) = 5.232991 0.7187500
H = √(FG) = 5.048065 0.7031250
I = √(FH) = 4.958069 0.6953125
K = √(HI) = 5.002865 0.6992187
L = √(IK) = 4.980416 0.6972656
M = √(KL) = 4.991627 0.6982421
N = √(KM) = 4.997242 0.6987304

O = √(KN) = 5.000052 0.6989745
P = √(NO) = 4.998647 0.6988525
Q = √(OP) = 4.999350 0.6989135
R = √(OQ) = 4.999701 0.6989440
S = √(OR) = 4.999876 0.6989592
T = √(OS) = 4.999963 0.6989668
V = √(OT) = 5.000008 0.6989707
W = √(TV) = 4.999984 0.6989687
X = √(WV) = 4.999997 0.6989697
Y = √(VX) = 5.000003 0.6989702
Z = √(XY) = 5.000000 0.6989700

Great attention was devoted to the methods of calculating logarithms
during the 17th and 18th centuries. The earlier methods proposed were,
like those of Briggs, purely arithmetical, and for a long time logarithms
were regarded from the point of view indicated by their name, that is to
say, as depending on the theory of compounded ratios. The introduction of
infinite series into mathematics effected a great change in the modes of
calculation and the treatment of the subject. Besides Napier and Briggs,
special reference should be made to Kepler (Chilias, 1624) and Mercator
(Logarithmotechnia, 1668), whose methods were arithmetical, and to
Newton, Gregory, Halley and Cotes, who employed series. A full and
valuable account of these methods is given in Hutton’s “Construction of
Logarithms,” which occurs in the introduction to the early editions of his
Mathematical Tables, and also forms tract 21 of his Mathematical Tracts
(vol. i., 1812). Many of the early works on logarithms were reprinted in
the Scriptores logarithmici of Baron Maseres already referred to.

In the following account only those formulae and methods will be
referred to which would now be used in the calculation of logarithms.

Since

loge (1 + x) = x − 1⁄2x2 + 1⁄3x3 − 1⁄4x4 + &c.,

we have, by changing the sign of x,

loge (1 − x) = −x − 1⁄2x2 − 1⁄3x3 − 1⁄4x4 − &c.;

whence

1+x
loge = 2 (x + 1⁄3x3 + 1⁄5x5 + &c.),
1−x

and, therefore, replacing x by (p − q)/(p + q),

p p−q p−q p−q
loge
q =2 { p + q + 1⁄3 ( p+q )3+ ⁄ (
15
p+q ) 5 + &c. },
in which the series is always convergent, so that the formula affords a
method of deducing the logarithm of one number from that of another.

As particular cases we have, by putting q = 1,

p−1 p−1 p−1
loge p = 2 { p + 1 + ⁄ ( p + 1 ) 3 + ⁄ ( p + 1 ) 5 + &c. },
13 15

and by putting q = p + 1,

1 1 1
loge(p + 1) − loge p = 2 { 2p + 1 + ⁄ (2p + 1)3 + ⁄ (2p + 1)5 + &c. };
13 15

the former of these equations gives a convergent series for logep, and the
latter a very convergent series by means of which the logarithm of any
number may be deduced from the logarithm of the preceding number.

From the formula for loge (p/q) we may deduce the following very
convergent series for loge2, loge3 and loge5, viz.:—

loge 2 = 2 (7P + 5Q + 3R),

loge 3 = 2 (11P + 8Q + 5R),
loge 5 = 2 (16P + 12Q + 7R),

where

1 1 1
P= + 1⁄3 · + 1⁄5 · + &c.
31 (31)3 (31)5
1 1 1
Q= + 1⁄3 · + 1⁄5 · + &c.
49 (49)3 (49)5
1 1 1
R= + 1⁄3 · + 1⁄5 · + &c.
161 (161)3 (161)5

The following still more convenient formulae for the calculation of loge
2, loge 3, &c. were given by J. Couch Adams in the Proc. Roy. Soc., 1878,
27, p. 91. If

10 1 25 4
a = log
9 = −log 1 − (10 , b = )
log 24 = −log 1 − (
100 , )
81 1 50 2
c = log (
80 = log 1 + 80 , )
d = log 49 = −log 1 − 100 ,( )
126 8
e = log (
125 = log 1 + 1000 , )
then

log 2 = 7a − 2b + 3c, log 3 = 11a − 3b + 5c, log 5 = 16a − 4b + 7c,

and

log 7 = 1⁄2 (39a − 10b + 17c − d) or = 19a − 4b + 8c + e,

and we have the equation of condition,

a − 2b + c = d + 2e.

By means of these formulae Adams calculated the values of loge 2, loge 3,
loge 5, and loge 7 to 276 places of decimals, and deduced the value of loge
10 and its reciprocal M, the modulus of the Briggian system of logarithms.
The value of the modulus found by Adams is

Mo = 0.43429 44819 03251 82765 11289
18916 60508 22943 97005 80366
65661 14453 78316 58646 49208
87077 47292 24949 33843 17483
18706 10674 47663 03733 64167
92871 58963 90656 92210 64662

81226 58521 27086 56867 03295
93370 86965 88266 88331 16360
77384 90514 28443 48666 76864
65860 85135 56148 21234 87653
43543 43573 17253 83562 21868
25

which is true certainly to 272, and probably to 273, places (Proc. Roy.
Soc., 1886, 42, p. 22, where also the values of the other logarithms are
given).

If the logarithms are to be Briggian all the series in the preceding
formulae must be multiplied by M, the modulus; thus,

log10 (1 + x) = M (x − 1⁄2x2 + 1⁄3x3 − 1⁄4x4 + &c.),

and so on.

As has been stated, Abraham Sharp’s table contains 61-decimal
Briggian logarithms of primes up to 1100, so that the logarithms of all

composite numbers whose greatest prime factor does not exceed this
number may be found by simple addition; and Wolfram’s table gives 48-
decimal hyperbolic logarithms of primes up to 10,009. By means of these
tables and of a factor table we may very readily obtain the Briggian
logarithm of a number to 61 or a less number of places or of its hyperbolic
logarithm to 48 or a less number of places in the following manner.
Suppose the hyperbolic logarithm of the prime number 43,867 required.
Multiplying by 50, we have 50 × 43,867 = 2,193,350, and on looking in
Burckhardt’s Table des diviseurs for a number near to this which shall
have no prime factor greater than 10,009, it appears that

2,193,349 = 23 × 47 × 2029;

thus

43,867 = 1⁄50 (23 × 47 × 2029 + 1),

and therefore

loge 43,867 = loge 23 + loge 47 + loge 2029 − loge 50

1 1 1
+ − 1⁄2 + 1⁄3 − &c.
2,193,349 (2,193,349)2 (193,349)3

The first term of the series in the second line is

0.00000 04559 23795 07319 6286;

dividing this by 2 × 2,193,349 we obtain

0.00000 00000 00103 93325 3457,

and the third term is

0.00000 00000 00000 00003 1590,

so that the series =

0.00000 04559 23691 13997 4419;

whence, taking out the logarithms from Wolfram’s table,

loge 43,867 = 10.68891 76079 60568 10191 3661.

The principle of the method is to multiply the given prime (supposed to
consist of 4, 5 or 6 figures) by such a factor that the product may be a
number within the range of the factor tables, and such that, when it is
increased by 1 or 2, the prime factors may all be within the range of the
logarithmic tables. The logarithm is then obtained by use of the formula

d d2 d3
loge (x + d) = loge x + − 1⁄2 + 1⁄3 − &c.,
x x2 x3

in which of course the object is to render d/x as small as possible. If the
logarithm required is Briggian, the value of the series is to be multiplied
by M.

If the number is incommensurable or consists of more than seven
figures, we can take the first seven figures of it (or multiply and divide the
result by any factor, and take the first seven figures of the result) and
proceed as before. An application to the hyperbolic logarithm of π is given
by Burckhardt in the introduction to his Table des diviseurs for the second
million.

The best general method of calculating logarithms consists, in its
simplest form, in resolving the number whose logarithm is required into
factors of the form 1 − .1rn, where n is one of the nine digits; and making
use of subsidiary tables of logarithms of factors of this form. For example,
suppose the logarithm of 543839 required to twelve places. Dividing by
105 and by 5 the number becomes 1.087678, and resolving this number
into factors of the form 1 − .1rn we find that

543839 = 105 × 5(1 − .128) (1 − .146) (1 − .156) (1 − .163) (1 − .173)
× (1 − .185) (1 − .197) (1 − .1109) (1 − .1113) (1 − .1122),

where 1 − 128 denotes 1 − .08, 1 − .146 denotes 1 − .0006, &c., and so on.
All that is required therefore in order to obtain the logarithm of any
number is a table of logarithms, to the required number of places, of .n,
.9n, .99n, .999n, &c., for n = 1, 2, 3, ... 9.

The resolution of a number into factors of the above form is easily
performed. Taking, for example, the number 1.087678, the object is to
destroy the significant figure 8 in the second place of decimals; this is
effected by multiplying the number by 1-.08, that is, by subtracting from
the number eight times itself advanced two places, and we thus obtain
1.00066376. To destroy the first 6 multiply by 1 − .0006 giving
1.000063361744, and multiplying successively by 1 − .00006 and 1 −
.000003, we obtain 1.000000357932, and it is clear that these last six
significant figures represent without any further work the remaining
factors required. In the corresponding antilogarithmic process the number
is expressed as a product of factors of the form 1 + .1nx.

This method of calculating logarithms by the resolution of numbers into
factors of the form 1 − .1rn is generally known as Weddle’s method,
having been published by him in The Mathematician for November 1845,
and the corresponding method for antilogarithms by means of factors of
the form 1 + (.1)rn is known by the name of Hearn, who published it in the
same journal for 1847. In 1846 Peter Gray constructed a new table to 12
places, in which the factors were of the form 1 − (.01)rn, so that n had the
values 1, 2, ... 99; and subsequently he constructed a similar table for
factors of the form 1 + (.01)rn. He also devised a method of applying a
table of Hearn’s form (i.e. of factors of the form 1 + .1rn) to the
construction of logarithms, and calculated a table of logarithms of factors

of the form 1 + (.001)rn to 24 places. This was published in 1876 under
the title Tables for the formation of logarithms and antilogarithms to
twenty-four or any less number of places, and contains the most complete
and useful application of the method, with many improvements in points
of detail. Taking as an example the calculation of the Briggian logarithm
of the number 43,867, whose hyperbolic logarithm has been calculated
above, we multiply it by 3, giving 131,601, and find by Gray’s process
that the factors of 1.31601 are

(1) 1.316 (5) 1.(001)4002
(2) 1.000007 (6) 1.(001)5602
(3) 1.(001)2598 (7) 1.(001)6412
(4) 1.(001)3780 (8) 1.(001)7340

Taking the logarithms from Gray’s tables we obtain the required logarithm
by addition as follows:—

522 878 745 280 337 562 704 972 = colog 3
119 255 889 277 936 685 553 913 = log (1)
3 040 050 733 157 610 239 = log (2)
259 708 022 525 453 597 = log (3)
338 749 695 752 424 = log (4)
868 588 964 = log (5)
261 445 278 = log (6)
178 929 = log (7)
148 = log (8)
4.642 137 934 655 780 757 288 464 = log10 43,867

In Shortrede’s Tables there are tables of logarithms and factors of the
form 1 ± (.01)r n to 16 places and of the form 1 ± (.1)r n to 25 places; and
in his Tables de Logarithmes à 27 Décimales (Paris, 1867) Fédor Thoman
gives tables of logarithms of factors of the form 1 ± .1r n. In the
Messenger of Mathematics, vol. iii. pp. 66-92, 1873, Henry Wace gave a
simple and clear account of both the logarithmic and antilogarithmic
processes, with tables of both Briggian and hyperbolic logarithms of
factors of the form 1 ± .1rn to 20 places.

Although the method is usually known by the names of Weddle and
Hearn, it is really, in its essential features, due to Briggs, who gave in the
Arithmetica logarithmica of 1624 a table of the logarithms of 1 + .1rn up
to r = 9 to 15 places of decimals. It was first formally proposed as an
independent method, with great improvements, by Robert Flower in The
Radix, a new way of making Logarithms, which was published in 1771;
and Leonelli, in his Supplement logarithmique (1802-1803), already
noticed, referred to Flower and reproduced some of his tables. A complete
bibliography of this method has been given by A. J. Ellis in a paper “on
the potential radix as a means of calculating logarithms,” printed in the
Proceedings of the Royal Society, vol. xxxi., 1881, pp. 401-407, and vol.
xxxii., 1881, pp. 377-379. Reference should also be made to Hoppe’s
Tafeln zur dreissigstelligen logarithmischen Rechnung (Leipzig, 1876),
which give in a somewhat modified form a table of the hyperbolic
logarithm of 1 + .1rn.

The preceding methods are only appropriate for the calculation of
isolated logarithms. If a complete table had to be reconstructed, or
calculated to more places, it would undoubtedly be most convenient to
employ the method of differences. A full account of this method as applied
to the calculation of the Tables du Cadastre is given by Lefort in vol. iv. of
the Annales de l’Observatoire de Paris. (J. W. L. G.)

1 Dr Thomas Smith thus describes the ardour with which Briggs studied the Descriptio:
“Hunc in deliciis habuit, in sinu, in manibus, in pectore gestavit, oculisque avidissimis, et mente
attentissima, iterum iterumque perlegit,...” Vitae quorundam eruditissimorum et illustrium
virorum (London, 1707).

2 William Lilly’s account of the meeting of Napier and Briggs at Merchiston is quoted in the
article Napier.

3 It was certainly published after Napier’s death, as Briggs mentions his “librum
posthumum.” This liber posthumus was the Constructio referred to later in this article.

4 Frisch’s Kepleri opera omnia, ii. 834. Frisch thinks Bramer possibly relied on Kepler’s
statement quoted in the text (“Quibus forte confisus Kepleri verbis Benj. Bramer....”). See also
vol. vii. p. 298.

The claims of Byrgius are discussed in Kästner’s Geschichte der Mathematik, ii. 375, and iii.
14; Montucla’s Histoire des mathématiques, ii. 10; Delambre’s Histoire de l’astronomie
moderne, i. 560; de Morgan’s article on “Tables” in the English Cyclopaedia; Mark Napier’s
Memoirs of John Napier of Merchiston (1834), p. 392, and Cantor’s Geschichte der
Mathematik, ii. (1892), 662. See also Gieswald, Justus Byrg als Mathematiker und dessen
Einleitung in seine Logarithmen (Danzig, 1856).

5 See Mark Napier’s Memoirs of John Napier of Merchiston (1834), p. 362.

6 In the Rabdologia (1617) he speaks of the canon of logarithms as “a me longo tempore
elaboratum.”

7 A careful examination of the history of the method is given by Scheibel in his Einleitung
zur mathematischen Bücherkenntniss, Stück vii. (Breslau, 1775), pp. 13-20; and there is also an
account in Kästner’s Geschichte der Mathematik, i. 566-569 (1796); in Montucla’s Histoire des
mathématiques, i. 583-585 and 617-619; and in Klügel’s Wörterbuch (1808), article
“Prosthaphaeresis.”

8 Besides his connexion with logarithms and improvements in the method of
prosthaphaeresis, Byrgius has a share in the invention of decimal fractions. See Cantor,
Geschichte, ii. 567. Cantor attributes to him (in the use of his prosthaphaeresis) the first
introduction of a subsidiary angle into trigonometry (vol. ii. 590).

9 The title of this work is—Benjaminis Ursini ... cursus mathematici practici volumen
primum continens illustr. & generosi Dn. Dn. Johannis Neperi Baronis Merchistonij &c. Scoti
trigonometriam logarithmicam usibus discentium accommodatam ... Coloniae ... CIɔ IɔC XIX.
At the end, Napier’s table is reprinted, but to two figures less. This work forms the earliest
publication of logarithms on the continent.

10 The title is Logarithmorum canonis descriptio, seu arithmeticarum supputationum
mirabilis abbreviatio. Ejusque usus in utraque trigonometria ut etiam in omni logistica
mathematica, amplissimi, facillimi & expeditissimi explicatio. Authore ac inventore Ioanne

Nepero, Barone Merchistonii, &c. Scoto. Lugduni.... It will be seen that this title is different
from that of Napier’s work of 1614; many writers have, however, erroneously given it as the
title of the latter.

11 In describing the contents of the works referred to, the language and notation of the
present day have been adopted, so that for example a table to radius 10,000,000 is described as
a table to 7 places, and so on. Also, although logarithms have been spoken of as to the base e,
&c., it is to be noticed that neither Napier nor Briggs, nor any of their successors till long
afterwards, had any idea of connecting logarithms with exponents.

12 The smallest number of entries which are necessary in a table of logarithms in order that
the intermediate logarithms may be calculable by proportional parts has been investigated by J.
E. A. Steggall in the Proc. Edin. Math. Soc., 1892, 10, p. 35. This number is 1700 in the case of
a seven-figure table extending to 100,000.

13 Accounts of Sang’s calculations are given in the Trans. Roy. Soc. Edin., 1872, 26, p. 521,
and in subsequent papers in the Proceedings of the same society.

14 In vol. xv. (1875) of the Verhandelingen of the Amsterdam Academy of Sciences, Bierens
de Haan has given a list of 553 tables of logarithms. A previous paper of the same kind,
containing notices of some of the tables, was published by him in the Verslagen en
Mededeelingen of the same academy (Afd. Natuurkunde) deel. iv. (1862), p. 15.

LOGAU, FRIEDRICH, Freiherr von (1604-1655), German
epigrammatist, was born at Brockut, near Nimptsch, in Silesia, in June 1604.
He was educated at the gymnasium of Brieg and subsequently studied law. He
then entered the service of the duke of Brieg. In 1644 he was made “ducal
councillor.” He died at Liegnitz on the 24th of July 1655. Logau’s epigrams,
which appeared in two collections under the pseudonym “Salomon von Golaw”
(an anagram of his real name) in 1638 (Erstes Hundert Teutscher
Reimensprüche) and 1654 (Deutscher Sinngedichte drei Tausend), show a
marvellous range and variety of expression. He had suffered bitterly under the
adverse conditions of the time; but his satire is not merely the outcome of

personal feeling. In the turbulent age of the Thirty Years’ War he was one of the
few men who preserved intact his intellectual integrity and judged his
contemporaries fairly. He satirized with unsparing hand the court life, the
useless bloodshed of the war, the lack of national pride in the German people,
and their slavish imitation of the French in customs, dress and speech. He
belonged to the Fruchtbringende Gesellschaft under the name Der
Verkleinernde, and regarded himself as a follower of Martin Opitz; but he did
not allow such ties to influence his independence or originality.

Logau’s Sinngedichte were edited in 1759 by G. E. Lessing and K. W.
Ramler, who first drew attention to their merits; a second edition appeared
in 1791. A critical edition was published by G. Eitner in 1872, who also
edited a selection of Logau’s epigrams for the Deutsche Dichter des XVII.
Jahrhunderts (vol. iii., 1870); there is also a selection by H. Oesterley in
Kürschner’s Deutsche Nationalliteratur, vol. xxviii. (1885). See H.
Denker, Beiträge zur literarischen Würdigung Logaus (1889); W.
Heuschkel, Untersuchungen über Ränders und Lessings Bearbeitung
Logauscher Sinngedichte (1901).

LOGIA, a title used to describe a collection of the sayings of Jesus Christ
(λόγια Ἰησοῦ) and therefore generally applied to the “Sayings of Jesus”
discovered in Egypt by B. P. Grenfell and A. S. Hunt. There is some question
as to whether the term is rightly used for this purpose. It does not occur in the
Papyri in this sense. Each “saying” is introduced by the phrase “Jesus says”
(λέγει) and the collection is described in the introductory words of the 1903
series as λόγοι not as λόγια. Some justification for the employment of the term

is found in early Christian literature. Several writers speak of the λόγια τοῦ
κυρίου or τὰ κυριακὰ λόγια, i.e. oracles of (or concerning) the Lord.
Polycarp, for instance, speaks of “those who pervert the oracles of the Lord.”
(Philipp. 7), and Papias, as Eusebius tells us, wrote a work with the title
“Expositions of the Oracles of the Lord.” The expression has been variously
interpreted. It need mean no more (Lightfoot, Essays on Supernatural Religion,
172 seq.) than narratives of (or concerning) the Lord; on the other hand, the
phrase is capable of a much more definite meaning, and there are many
scholars who hold that it refers to a document which contained a collection of
the sayings of Jesus. Some such document, we know, must lie at the base of our
Synoptic Gospels, and it is quite possible that it may have been known to and
used by Papias. It is only on this assumption that the use of the term Logia in
the sense described above can be justified.

“The Sayings,” to which the term Logia is generally applied, consist of (a) a
papyrus leaf containing seven or eight sayings of Jesus discovered in 1897, (b)
a second leaf containing five more sayings discovered in 1903, (c) two
fragments of unknown Gospels, the former published in 1903, the latter in
1907. All these were found amongst the great mass of papyri acquired by the
Egyptian Exploration Fund from the ruins of Oxyrhynchus, one of the chief
early Christian centres in Egypt, situated some 120 m. S. of Cairo.

The eight “sayings” discovered in 1897 are as follows:—

1. ... καὶ τότε διαβλέψεις ἐκβαλεῖν τὸ κάρφος τὸ ἐν τῷ ὀφθαλμῷ
τοῦ ἀδελφοῦ σου.

2. Λέγει Ἰησοῦς ἐὰν μὴ νηστεύσητε τὸν κόσμον οὐ μὴ εὔρητε τὴν
βασίλειαν τοῦ θεοῦ. καὶ ἐὰν μὴ σαββατίσητε τὸ σάββατον οὐκ
ὄψεσθε τὸν πατέρα.

3. Λέγει Ἰησοῦς ἔ[σ]την ἐν μεσῷ τοῦ κόσμου καὶ ἐν σαρκὶ ὤφθην
αὐτοῖς, καὶ εὖρον πάντας μεθύοντας καὶ οὐδένα εὖρον διψῶντα ἐν

αὐτοῖς, καὶ πονεῖ ἡ ψυχή μου ἐπὶ τοῖς υὶοῖς τῶν ἀνθρώπων, ὅτι
τυφλοί εἰσιν τῇ καρδίᾳ αὐτῶ[ν] κ[αὶ] ο̣ὐ̣ βλ̣έ[πουσιν]....

4. [Illegible: possibly joins on to 3] ... [τ]ὴν πτωχείαν.

5. [Λέγ]ει [Ἰησοῦς ὄπ]ου ἐὰν ὧσιν [β, οὐκ] ε[ἰσὶ]ν ἄθεοι καὶ [ὅ]που
ε[ἶς] ἐστιν μόνος, [λέ]γω, ἐγώ εἰμι μετ᾽ αὐτ[οῦ] ἔγει[ρ]ον τὸν λίθον
κἀκεῖ εὑρήσεις με, σχίσον τὸ ξύλον κἀγὼ ἐκεῖ εἰμι.

6. Λέγει Ἰησοῦς ούκ ἔστιν δεκτὸς προφήτης ἐν τῇ πατρίδι αὐτ[ο]ῦ,
οὐδὲ ἰατρὸς ποιεῖ θεραπείας εἰς τοὺς γινώσκοντας αὐτόν.

7. Λέγει Ἰησοῦς πόλιςοἰ κοδομημένη ἐπ᾽ ἄκρον [ὄ]ρους ὑψηλοῦ
καὶ ἐστηριγμένη οὔτε πε[σ]εῖν δύναται οὔτε κρυ[β]ῆναι.

8. Λέγει Ἰησοῦς ἀκούεις [ε]ἰ̣ς τ̣ὸ ἓ̣ν̣ ᾠ̣τ̣ίον σοῦ τ̣ὸ [δὲ ἕτερον
συνέκλεισας].

Letters in brackets are missing in the original: letters which are dotted
beneath are doubtful.

1. “... and then shalt thou see clearly to cast out the mote that is in thy
brother’s eye.”

2. “Jesus saith, Except ye fast to the world, ye shall in no wise find the
kingdom of God; and except ye make the sabbath a real sabbath, ye shall
not see the Father.”

3. “Jesus saith, I stood in the midst of the world and in the flesh was I
seen of them, and I found all men drunken, and none found I athirst
among them, and my soul grieveth over the sons of men, because they are
blind in their heart, and see not....”

4. “... poverty....”

5. “Jesus saith, Wherever there are two, they are not without God, and
wherever there is one alone, I say, I am with him. Raise the stone and

there thou shalt find me, cleave the wood and there am I.”

6. “Jesus saith, A prophet is not acceptable in his own country, neither
doth a physician work cures upon them that know him.”

7. “Jesus saith, A city built upon the top of a high hill and stablished can
neither fall nor be hid.”

8. “Jesus saith, Thou hearest with one ear [but the other ear hast thou
closed].”

The “sayings” of 1903 were prefaced by the following introductory
statement:—

οἱ τοῖοι οἱ λόγοι οἱ [... οὓς ἐλάλησεν Ἰη(σοῦ)ς ὁ ζῶν κ[ύριος? ... καὶ
Θωμᾷ καὶ εἶπεν [αὐτοῖς· πᾶς ὅστις ἂν τῶν λόγων τούτ[ων ἀκούσῃ
θανάτου οὐ μὴ γεύσηται.

“These are the (wonderful?) words which Jesus the living (Lord) spake
to ... and Thomas and he said unto (them) every one that hearkens to these
words shall never taste of death.”

The “sayings” themselves are as follows:—

(1) [λέγει Ἰη(σοῦ)ς· μὴ παυσάσθω ὁ ζη[τῶν ...
ἕως ἄν εὕρῃ καὶ ὅταν εὔρῃ [θαμβηθήσεαι
καὶ θαμβηθεὶς βασιλεύσει κα[ὶ βασιλεύσας
ἀναπαήσεται.

(2) λέγει Ἰ[η(σοῦς ... τίνες ...
οἱ ἕλκοντες ἡμᾶς [εἰς τὴν βασιλείαν εἰ
ἡ βασιλεία ἐν οὐρα[νῷ ἐστιν;
τὰ πετεινὰ τοῦ οὐρ[ανοῦ καὶ τῶν θηρίων ὅ
τι ὑπὸ τὴν γῆν ἐστ[ιν ἤ ἐπὶ τῆς γῆς καὶ
οἱ ἰχθύες τῆς θαλά[σσης οὖτοι οἱ ἕλκον-

τες ὑμᾶς καὶ ῇ βασ[ιλεία τῶν οὐρανῶν
ἐντὸς ὐμῶν [ἐ]στι [καὶ ὅστις ἃν ἑαυτὸν
γνῷ ταύτην εὑρή[σει ...
ἑαυτοὺς γνώσεσθε [καὶ εἰδήσετε ὅτι υἱοὶ
ἔστε ὑμεῖς τοῦ πατρὸς τοῦ τ[ ...
γνώσ(εσ)θε ἐαυτοὺς ἐν[ ...
καὶ ὑ εῖς ἐστὲ ηπτο̣[

(3) [ λέγει Ἰη(σοῦ)ς
οὐκ ἀποκνήσει ἄνθ[ρωπος ...
ρων ἐπερωτῆσαι πα[ ...
ρων περὶ τοῦ τόπου τῆ[ς ...
σετε ὅτι πολλοὶ ἔσονται π[ρῶτοι ἔσχατοι καὶ
οἱ ἕσχατοι πρῶτοι καὶ [ ...
σιν.

(4) λέγει Ἰη(σοῦ)ς· [πᾶν τὸ μὴ ἔμπροσ-
θεν τῆς ὅψεως σου καὶ[τὸ κεκρυμμένον
ἀπὸ σοῦ ἀποκαλυφ(θ)ήσετ[αί σοι. οὐ γάρ ἐσ-
τιν κρυπτὸν ὃ οὐ φανε[ρὸν γενήσεται
καὶ τεθαμμένον ὃ ο[ὐκ ἐγερθήσεται.

(5) [ἐξ] ετάζουσιν αὐτὸν ο[ἱ μαθηταἱ αὐτοῦ καὶ
[λέ]γουσιν· πῶς νηστεύ[σομεν καὶ πῶς ...
[ ... ] μεθα καὶ πῶς [ ...
[ ... κ]αὶ τί παρατηρήσ[ομεν ...
[ ... ]ν; λέγει Ἰη(σοῦ)ς· [ ...
[ ... ]ειται μὴ ποεῖτ[ε ...
[ ... ]ης ἀληθείας ἀν[ ...
[ ... ]ν ἀ[π]οκεκρ[υ ...
[ ... μα] κάρι[ός] ἐστιν [ ...

[ ... ]ω ἐστ[ι ...
[ ... ]ιν [ ...

1. “Jesus saith, Let not him who seeks ... cease until he finds and
when he finds he shall be astonished; astonished he shall reach the
kingdom and having reached the kingdom he shall rest.”

2. “Jesus saith (ye ask? who are those) that draw us (to the kingdom
if) the kingdom is in Heaven? ... the fowls of the air and all beasts that
are under the earth or upon the earth and the fishes of the sea (these are
they which draw) you and the kingdom of Heaven is within you and
whosoever shall know himself shall find it. (Strive therefore?) to know
yourselves and ye shall be aware that ye are the sons of the (Almighty?)
Father; (and?) ye shall know that ye are in (the city of God?) and ye are
(the city?).”

3. “Jesus saith, A man shall not hesitate ... to ask concerning his place
(in the kingdom. Ye shall know) that many that are first shall be last and
the last first and (they shall have eternal life?).”

4. “Jesus saith, Everything that is not before thy face and that which is
hidden from thee shall be revealed to thee. For there is nothing hidden
which shall not be made manifest nor buried which shall not be raised.”

5. “His disciples question him and say, How shall we fast and how
shall we (pray?) ... and what (commandment) shall we keep ... Jesus
saith ... do not ... of truth ... blessed is he ...”

The fragment of a lost Gospel which was discovered in 1903 contained
originally about fifty lines, but many of them have perished and others are
undecipherable. The translation, as far as it can be made out, is as follows:—

1-7. “(Take no thought) from morning until even nor from evening
until morning either for your food what ye shall eat or for your raiment
what ye shall put on. 7-13. Ye are far better than the lilies which grow
but spin not. Having one garment what do ye (lack)?... 13-15. Who

could add to your stature? 15-16. He himself will give you your
garment. 17-23. His disciples say unto him, When wilt thou be manifest
unto us and when shall we see thee? He saith, When ye shall be stripped
and not be ashamed ... 41-46. He said, The key of knowledge ye hid: ye
entered not in yourselves, and to them that were entering in, ye opened
not.”

The second Gospel fragment discovered in 1907 “consists of a single
vellum leaf, practically complete except at one of the lower corners and here
most of the lacunae admit of a satisfactory solution.” The translation is as
follows:—

... before he does wrong makes all manner of subtle excuse. But give
heed lest ye also suffer the same things as they: for the evil doers among
men receive their reward not among the living only, but also await
punishment and much torment. And he took them and brought them into
the very place of purification and was walking in the temple. And a
certain Pharisee, a chief priest, whose name was Levi, met them and
said to the Saviour, Who gave thee leave to walk in this place of
purification, and to see these holy vessels when thou hast not washed
nor yet have thy disciples bathed their feet? But defiled thou hast walked
in this temple, which is a pure place, wherein no other man walks except
he has washed himself and changed his garments neither does he
venture to see these holy vessels. And the Saviour straightway stood still
with his disciples and answered him, Art thou then, being here in the
temple, clean? He saith unto him, I am clean; for I washed in the pool of
David and having descended by one staircase, I ascended by another and
I put on white and clean garments, and then I came and looked upon
these holy vessels. The Saviour answered and said unto him, Woe ye
blind, who see not. Thou hast washed in these running waters wherein
dogs and swine have been cast night and day and hast cleansed and
wiped the outside skin which also the harlots and flute-girls anoint and

wash and wipe and beautify for the lust of men; but within they are full
of scorpions and all wickedness. But I and my disciples who thou sayest
have not bathed have been dipped in the waters of eternal life which
come from.... But woe unto thee....

These documents have naturally excited considerable interest and raised
many questions. The papyri of the “sayings” date from the 3rd century and
most scholars agree that the “sayings” themselves go back to the 2nd. The
year a.d. 140 is generally assigned as the terminus ad quem. The problem as
to their origin has been keenly discussed. There are two main types of theory.
(1) Some suppose that they are excerpts from an uncanonical Gospel. (2)
Others think that they represent an independent and original collection of
sayings. The first theory has assumed three main forms. (a) Harnack
maintains that they were taken from the Gospel according to the Egyptians.
This theory, however, is based upon a hypothetical reconstruction of the
Gospel in question which has found very few supporters. (b) Others have
advocated the Gospel of the Hebrews as the source of the “sayings,” on the
ground of the resemblance between the first “saying” of the 1903 series and a
well-authenticated fragment of that Gospel. The resemblance, however, is
not sufficiently clear to support the conclusion. (c) A third view supposes that
they are extracts from the Gospel of Thomas—an apocryphal Gospel dealing
with the boyhood of Jesus. Beyond the allusion to Thomas in the
introductory paragraph to the 1903 series, there seems to be no tangible
evidence in support of this view. The second theory, which maintains that the
papyri represent an independent collection of “sayings,” seems to be the
opinion which has found greatest favour. It has won the support of W.
Sanday, H. B. Swete, Rendel Harris, W. Lock, Heinrici, &c. There is a
considerable diversity of judgment, however, with regard to the value of the
collection. (a) Some scholars maintain that the collection goes back to the 1st
century and represents one of the earliest attempts to construct an account of
the teaching of Jesus. They are therefore disposed to admit to a greater or

less extent and with widely varying degrees of confidence the presence of
genuine elements in the new matter. (b) Sanday and many others regard the
sayings as originating early in the 2nd century and think that, though not
“directly dependent on the Canonical Gospels,” they have “their origin under
conditions of thought which these Gospels had created.” The “sayings” must
be regarded as expansions of the true tradition, and little value is therefore to
be attached to the new material.

With the knowledge at our disposal, it is impossible to reach an assured
conclusion between these two views. The real problem, to which at present
no solution has been found, is to account for the new material in the
“sayings.” There seems to be no motive sufficient to explain the additions
that have been made to the text of the Gospels. It cannot be proved that the
expansions have been made in the interests of any sect or heresy. Unless new
discoveries provide the clue, or some reasonable explanation can otherwise
be found, there seems to be no reason why we should not regard the
“sayings” as containing material which ought to be taken into account in the
critical study of the teaching of Jesus.

The 1903 Gospel fragment is so mutilated in many of its parts that it is
difficult to decide upon its character and value. It appears to be earlier than
150, and to be taken from a Gospel which followed more or less closely the
version of the teaching of Jesus given by Matthew and Luke. The phrase
“when ye shall be stripped and not be ashamed” contains an idea which has
some affinity with two passages found respectively in the Gospel according
to the Egyptians and the so-called Second Epistle of Clement. The
resemblance, however, is not sufficiently close to warrant the deduction that
either the Gospel of the Egyptians or the Gospel from which the citation in 2
Clement is taken (if these two are distinct) is the source from which our
fragment is derived.

The second Gospel fragment (1907) seems to be of later origin than the
documents already mentioned. Grenfell and Hunt date the Gospel, from
which it is an excerpt, about 200. There is considerable difficulty with regard
to some of the details. The statement that an ordinary Jew was required to
wash and change his clothes before visiting the inner court of the temple is
quite unsupported by any other evidence. Nothing is known about “the place
of purification” (ἁγνευτήριον) nor “the pool of David” (λίμνη τοῦ Δαυείδ).
Nor does the statement that “the sacred vessels” were visible from the place
where Jesus was standing seem at all probable. Grenfell and Hunt conclude
therefore—“So great indeed are the divergences between this account and the
extant and no doubt well-informed authorities with regard to the topography
and ritual of the Temple that it is hardly possible to avoid the conclusion that
much of the local colour is due to the imagination of the author who was
aiming chiefly at dramatic effect and was not really well acquainted with the
Temple. But if the inaccuracy of the fragment in this important respect is
admitted the historical character of the whole episode breaks down and it is
probably to be regarded as an apocryphal elaboration of Matt. xv. 1-20 and
Mark vii. 1-23.”

See the Oxyrhynchus Papyri, part i. (1897), part iv. (1904), part v.
(1908).
(H. T. A.)

LOGIC (λογική, sc. τέχνη, the art of reasoning), the name given to one
of the four main departments of philosophy, though its sphere is very

variously delimited. The present article is divided into 1. The Problems of
Logic, II. History.

I. The Problems of Logic.

Introduction.—Logic is the science of the processes of inference, what,
then, is inference? It is that mental operation which proceeds by combining
two premises so as to cause a consequent conclusion. Some suppose that we
may infer from one premise by a so-called “immediate inference.” But one
premise can only reproduce itself in another form, e.g. all men are some
animals; therefore some animals are men. It requires the combination of at
least two premises to infer a conclusion different from both. There are as
many kinds of inference as there are different ways of combining premises,
and in the main three types:—

1. Analogical Inference, from particular to particular: e.g. border-war
between Thebes and Phocis is evil; border-war between Thebes and Athens is
similar to that between Thebes and Phocis; therefore, border-war between
Thebes and Athens is evil.

2. Inductive Inference, from particular to universal: e.g. border-war
between Thebes and Phocis is evil; all border-war is like that between
Thebes and Phocis; therefore, all border-war is evil.

3. Deductive or Syllogistic Inference, from universal to particular, e.g. all
border-war is evil; border-war between Thebes and Athens is border-war;
therefore border-war between Thebes and Athens is evil.

In each of these kinds of inference there are three mental judgments
capable of being expressed as above in three linguistic propositions; and the
two first are the premises which are combined, while the third is the
conclusion which is consequent on their combination. Each proposition

consists of two terms, the subject and its predicate, united by the copula.
Each inference contains three terms. In syllogistic inference the subject of
the conclusion is the minor term, and its predicate the major term, while
between these two extremes the term common to the two premises is the
middle term, and the premise containing the middle and major terms is the
major premise, the premise containing the middle and minor terms the minor
premise. Thus in the example of syllogism given above, “border-war
between Thebes and Athens” is the minor term, “evil” the major term, and
“border-war” the middle term. Using S for minor, P for major and M for
middle, and preserving these signs for corresponding terms in analogical and
inductive inferences, we obtain the following formula of the three inferences:
—

Analogical. Inductive. Deductive or Syllogistic.
S1 is P S is P Every M is P
S2 is similar to S1 Every M is similar to S S is M
∴ S2 is P. ∴ Every M is P. ∴ S is P.

The love of unity has often made logicians attempt to resolve these three
processes into one. But each process has a peculiarity of its own; they are
similar, not the same. Analogical and inductive inference alike begin with a
particular premise containing one or more instances; but the former adds a
particular premise to draw a particular conclusion, the latter requires a
universal premise to draw a universal conclusion. A citizen of Athens, who
had known the evils of the border-war between Thebes and Phocis, would
readily perceive the analogy of a similar war between Thebes and Athens,
and conclude analogously that it would be evil; but he would have to
generalize the similarity of all border-wars in order to draw the inductive
conclusion that all alike are evil. Induction and deduction differ still more,
and are in fact opposed, as one makes a particular premise the evidence of a

universal conclusion, the other makes a universal premise evidence of a
particular conclusion. Yet they are alike in requiring the generalization of the
universal and the belief that there are classes which are whole numbers of
similars. On this point both differ from inference by analogy, which proceeds
entirely from particular premises to a particular conclusion. Hence we may
redivide inference into particular inference by analogy and universal
inference by induction and deduction. Universal inference is what we call
reasoning; and its two species are very closely connected, because universal
conclusions of induction become universal premises of deduction. Indeed,
we often induce in order to deduce, ascending from particular to universal
and descending from universal to particular in one act as it were; so that we
may proceed either directly from particular to particular by analogical
inference, or indirectly from particular through universal to particular by an
inductive-deductive inference which might be called “perduction.” On the
whole, then, analogical, inductive and deductive inferences are not the same
but three similar and closely connected processes.

The three processes of inference, though different from one another, rest
on a common principle of similarity of which each is a different application.
Analogical inference requires that one particular is similar to another,
induction that a whole number or class is similar to its particular instances,
deduction that each particular is similar to the whole number or class. Not
that these inferences require us to believe, or assume, or premise or
formulate this principle either in general, or in its applied forms: the premises
are all that any inference needs the mind to assume. The principle of
similarity is used, not assumed by the inferring mind, which in accordance
with the similarity of things and the parity of inference spontaneously
concludes in the form that similars are similarly determined (“similia
similibus convenire”). In applying this principle of similarity, each of the
three processes in its own way has to premise both that something is
somehow determined and that something is similar, and by combining these

premises to conclude that this is similarly determined to that. Thus the very
principle of inference by similarity requires it to be a combination of
premises in order to draw a conclusion.

The three processes, as different applications of the principle of similarity,
consisting of different combinations of premises, cause different degrees of
cogency in their several conclusions. Analogy hardly requires as much
evidence as induction. Men speculate about the analogy between Mars and
the earth, and infer that it is inhabited, without troubling about all the planets.
Induction has to consider more instances, and the similarity of a whole
number or class. Even so, however, it starts from a particular premise which
only contains many instances, and leaves room to doubt the universality of
its conclusions. But deduction, starting from a premise about all the members
of a class, compels a conclusion about every and each of necessity. One
border-war may be similar to another, and the whole number may be similar,
without being similarly evil; but if all alike are evil, each is evil of necessity.
Deduction or syllogism is superior to analogy and induction in combining
premises so as to involve or contain the conclusion. For this reason it has
been elevated by some logicians above all other inferences, and for this very
same reason attacked by others as no inference at all. The truth is that,
though the premises contain the conclusion, neither premise alone contains
it, and a man who knows both but does not combine them does not draw the
conclusion; it is the synthesis of the two premises which at once contains the
conclusion and advances our knowledge; and as syllogism consists, not
indeed in the discovery, but essentially in the synthesis of two premises, it is
an inference and an advance on each premise and on both taken separately.
As again the synthesis contains or involves the conclusion, syllogism has the
advantage of compelling assent to the consequences of the premises.
Inference in general is a combination of premises to cause a conclusion;
deduction is such a combination as to compel a conclusion involved in the
combination, and following from the premises of necessity.

Nevertheless, deduction or syllogism is not independent of the other
processes of inference. It is not the primary inference of its own premises,
but constantly converts analogical and inductive conclusions into its
particular and universal premises. Of itself it causes a necessity of
consequence, but only a hypothetical necessity; if these premises are true,
then this conclusion necessarily follows. To eliminate this “if” ultimately
requires other inferences before deduction. Especially, induction to
universals is the warrant and measure of deduction from universals. So far as
it is inductively true that all border-war is evil, it is deductively true that a
given border-war is therefore evil. Now, as an inductive combination of
premises does not necessarily involve the inductive conclusion, induction
normally leads, not to a necessary, but to a probable conclusion; and
whenever its probable conclusions become deductive premises, the deduction
only involves a probable conclusion. Can we then infer any certainty at all?
In order to answer this question we must remember that there are many
degrees of probability, and that induction, and therefore deduction, draw
conclusions more or less probable, and rise to the point at which probability
becomes moral certainty, or that high degree of probability which is
sufficient to guide our lives, and even condemn murderers to death. But can
we rise still higher and infer real necessity? This is a difficult question, which
has received many answers. Some noölogists suppose a mental power of
forming necessary principles of deduction a priori; but fail to show how we
can apply principles of mind to things beyond mind. Some empiricists, on
the other hand, suppose that induction only infers probable conclusions
which are premises of probable deductions; but they give up all exact
science. Between these extremes there is room for a third theory, empirical
yet providing a knowledge of the really necessary. In some cases of induction
concerned with objects capable of abstraction and simplification, we have a
power of identification, by which, not a priori but in the act of inducing a
conclusion, we apprehend that the things signified by its subject and
predicate are one and the same thing which cannot exist apart from itself.

Thus by combined induction and identification we apprehend that one and
one are the same as two, that there is no difference between a triangle and a
three-sided rectilineal figure, that a whole must be greater than its part by
being the whole, that inter-resisting bodies necessarily force one another
apart, otherwise they would not be inter-resisting but occupy the same place
at the same moment. Necessary principles, discovered by this process of
induction and identification, become premises of deductive demonstration to
conclusions which are not only necessary consequents on the premises, but
also equally necessary in reality. Induction thus is the source of deduction, of
its truth, of its probability, of its moral certainty; and induction, combined
with identification, is the origin of the necessary principles of demonstration
or deduction to necessary conclusions.

Analogical inference in its turn is as closely allied with induction. Like
induction, it starts from a particular premise, containing one or more
examples or instances; but, as it is easier to infer a particular than a universal
conclusion, it supplies particular conclusions which in their turn become
further particular premises of induction. Its second premise is indeed merely
a particular apprehension that one particular is similar to another, whereas
the second premise of induction is a universal apprehension that a whole
number of particulars is similar to those from which the inference starts; but
at bottom these two apprehensions of similarity are so alike as to suggest that
the universal premise of induction has arisen as a generalized analogy. It
seems likely that man has arrived at the apprehension of a whole individual,
e.g. a whole animal including all its parts, and thence has inferred by analogy
a whole number, or class, e.g. of animals including all individual animals;
and accordingly that the particular analogy of one individual to another has
given rise to the general analogy of every to each individual in a class, or
whole number of individuals, contained in the second premise of induction.
In this case, analogical inference has led to induction, as induction to

deduction. Further, analogical inference from particular to particular suggests
inductive-deductive inference from particular through universal to particular.

Newton, according to Dr Pemberton, thought in 1666 that the moon moves
so like a falling body that it has a similar centripetal force to the earth, 20
years before he demonstrated this conclusion from the laws of motion in the
Principia. In fact, analogical, inductive and deductive inferences, though
different processes of combining premises to cause different conclusions, are
so similar and related, so united in principle and interdependent, so
consolidated into a system of inference, that they cannot be completely
investigated apart, but together constitute a single subject of science. This
science of inference in general is logic.

Logic, however, did not begin as a science of all inference. Rather it began
as a science of reasoning (λόγος), of syllogism (συλλογισμός), of deductive
inference. Aristotle was its founder. He was anticipated of course by many
generations of spontaneous thinking (logica naturalis). Many of the higher
animals infer by analogy: otherwise we cannot explain their thinking. Man so
infers at first: otherwise we cannot explain the actions of young children,
who before they begin to speak give no evidence of universal thinking. It is
likely that man began with particular inference and with particular language;
and that, gradually generalizing thought and language, he learnt at last to
think and say “all,” to infer universally, to induce and deduce, to reason, in
short, and raise himself above other animals. In ancient times, and especially
in Egypt, Babylon and Greece, he went on to develop reason into science or
the systematic investigation of definite subjects, e.g. arithmetic of number,
geometry of magnitude, astronomy of stars, politics of government, ethics of
goods. In Greece he became more and more reflective and conscious of
himself, of his body and soul, his manners and morals, his mental operations
and especially his reason. One of the characteristics of Greek philosophers is
their growing tendency, in investigating any subject, to turn round and ask
themselves what should be the method of investigation. In this way the

Presocratics and Sophists, and still more Socrates and Plato, threw out hints
on sense and reason, on inferential processes and scientific methods which
may be called anticipations of logic. But Aristotle was the first to conceive of
reasoning itself as a definite subject of a special science, which he called
analytics or analytic science, specially designed to analyse syllogism and
especially demonstrative syllogism, or science, and to be in fact a science of
sciences. He was therefore the founder of the science of logic.

Among the Aristotelian treatises we have the following, which
together constitute this new science of reasoning:—

1. The Categories, or names signifying things which can become
predicates;

2. The De Interpretatione, or the enumeration of conceptions and their
combinations by (1) nouns and verbs (names), (2) enunciations
(propositions);

3. The Prior Analytics, on syllogism;

4. The Posterior Analytics, on demonstrative syllogism, or science;

5. The Topics, on dialectical syllogism; or argument;

6. The Sophistical Elenchi, on sophistical or contentious syllogism, or
sophistical fallacies.

So far as we know, Aristotle had no one name for all these
investigations. “Analytics” is only applied to the Prior and Posterior
Analytics, and “logical,” which he opposed to “analytical,” only suits the
Topics and at most the Sophistical Elenchi; secondly, while he analyzed
syllogism into premises, major and minor, and premises into terms,
subject and predicate, he attempted no division of the whole science;
thirdly, he attempted no order and arrangement of the treatises into a
system of logic, but only of the Analytics, Topics and Sophistical

Elenchi into a system of syllogisms. Nevertheless, when his followers
had arranged the treatises into the Organon, as they called it to express
that it is an instrument of science, then there gradually emerged a system
of syllogistic logic, arranged in the triple division—terms, propositions
and syllogisms—which has survived to this day as technical logic, and
has been the foundation of all other logics, even of those which aim at
its destruction.

The main problem which Aristotle set before him was the analysis of
syllogism, which he defined as “reasoning in which certain things having
been posited something different from them of necessity follows by their
being those things” (Prior Analytics, i. 1). What then did he mean by
reasoning, or rather by the Greek word λόγος of which “reasoning” is an
approximate rendering? It was meant (cf. Post. An. i. 10) to be both internal,
in the soul (ὁ ἔσω λόγος, ἐν τῇ ψυχῇ), and external, in language (ὁ ἔξω
λόγος): hence after Aristotle the Stoics distinguished λόγος ἐνδιάθετος and
προφορικός. It meant, then, both reason and discourse of reason (cf.
Shakespeare, Hamlet, i. 2). On its mental side, as reason it meant
combination of thoughts. On its linguistic side, as discourse it was used for
any combination of names to form a phrase, such as the definition “rational
animal,” or a book, such as the Iliad. It had also the mathematical meaning of
ratio; and in its use for definition it is sometimes transferred to essence as the
object of definition, and has a mixed meaning, which may be expressed by
“account.” In all its uses, however, the common meaning is combination.
When Aristotle called syllogism λόγος, he meant that it is a combination of
premises involving a conclusion of necessity. Moreover, he tended to confine
the term λόγος to syllogistic inference. Not that he omitted other inferences
(πίστεις). On the contrary, to him (cf. Prior Analytics, ii. 24) we owe the
triple distinction into inference from particular to particular (παράδειγμα,
example, or what we call “analogy”), inference from particular to universal
(ἐπαγωγή, induction), and inference from universal to particular

(συλλογισμός, syllogism, or deduction). But he thought that inferences
other than syllogism are imperfect; that analogical inference is rhetorical
induction; and that induction, through the necessary preliminary of syllogism
and the sole process of ascent from sense, memory and experience to the
principles of science, is itself neither reasoning nor science. To be perfect he
thought that all inference must be reduced to syllogism of the first figure,
which he regarded as the specially scientific inference. Accordingly, the
syllogism appeared to him to be the rational process (μετὰ λόγου), and the
demonstrative syllogism from inductively discovered principles to be science
(ἐπιστήμη). Hence, without his saying it in so many words, Aristotle’s logic
perforce became a logic of deductive reasoning, or syllogism. As it happened
this deductive tendency helped the development of logic. The obscurer
premises of analogy and induction, together with the paucity of experience
and the backward state of physical science in Aristotle’s time would have
baffled even his analytical genius. On the other hand, the demonstrations of
mathematical sciences of his time, and the logical forms of deduction
evinced in Plato’s dialogues, provided him with admirable examples of
deduction, which is also the inference most capable of analysis. Aristotle’s
analysis of the syllogism showed man how to advance by combining his
thoughts in trains of deductive reasoning. Nevertheless, the wider question
remained for logic: what is the nature of all inference, and the special form of
each of its three main processes?

As then the reasoning of the syllogism was the main problem of
Aristotle’s logic, what was his analysis of it? In distinguishing inner and
outer reason, or reasoning and discourse, he added that it is not to outer
reason but to inner reason in the soul that demonstration and syllogism
are directed (Post. An. i. 10). One would expect, then, an analysis of
mental reasoning into mental judgments (κρίσεις) as premises and
conclusion. In point of fact, he analysed it into premises, but then
analysed a premise into terms, which he divided into subject and

predicate, with the addition of the copula “is” or “is not.” This analysis,
regarded as a whole and as it is applied in the Analytics and in the other
logical treatises, was evidently intended as a linguistic analysis. So in
the Categories, he first divided things said (τὰ λεγόμενα) into
uncombined and combined, or names and propositions, and then divided
the former into categories; and in the De interpretatione he expressly
excluded mental conceptions and their combinations, and confined
himself to nouns and verbs and enunciations, or, as we should say, to
names and propositions. Aristotle apparently intended, or at all events
has given logicians in general the impression, that he intended to analyse
syllogism into propositions as premises, and premise into names as
terms. His logic therefore exhibits the curious paradox of being an
analysis of mental reasoning into linguistic elements. The explanation is
that outer speech is more obvious than inner thought, and that grammar
and poetic criticism, rhetoric and dialectic preceded logic, and that out
of those arts of language arose the science of reasoning. The sophist
Protagoras had distinguished various kinds of sentences, and Plato had
divided the sentence into noun and verb, signifying a thing and the
action of a thing. Rhetoricians had enumerated various means of
persuasion, some of which are logical forms, e.g. probability and sign,
example and enthymeme. Among the dialecticians, Socrates had used
inductive arguments to obtain definitions as data of deductive arguments
against his opponents, and Plato had insisted on the processes of
ascending to and descending from an unconditional principle by the
power of giving and receiving argument. All these points about speech,
eloquence and argument between man and man were absorbed into
Aristotle’s theory of reasoning, and in particular the grammar of the
sentence consisting of noun and verb caused the logic of the proposition
consisting of subject and predicate. At the same time, Aristotle was well
aware that the science of reasoning is no art of language and must take
up a different position towards speech as the expression of thought. In

the Categories he classified names, not, however, as a grammarian by
their structure, but as a logician by their signification. In the De
interpretatione, having distinguished the enunciation, or proposition,
from other sentences as that in which there is truth or falsity, he
relegated the rest to rhetoric or poetry, and founded the logic of the
proposition, in which, however, he retained the grammatical analysis
into noun and verb. In the Analytics he took the final step of originating
the logical analysis of the proposition as premise into subject and
predicate as terms mediated by the copula, and analysed the syllogism
into these elements. Thus did he become the founder of the logical but
linguistic analysis of reasoning as discourse (ὁ ἔξω λόγος) into
propositions and terms. Nevertheless, the deeper question remained,
what is the logical but mental analysis of reasoning itself (ὁ ἔσω
λόγος) into its mental premises and conclusion?

Aristotle thus was the founder of logic as a science. But he laid too much
stress on reasoning as syllogism or deduction, and on deductive science; and
he laid too much stress on the linguistic analysis of rational discourse into
proposition and terms. These two defects remain ingrained in technical logic
to this day. But in the course of the development of the science, logicians
have endeavoured to correct those defects, and have diverged into two
schools. Some have devoted themselves to induction from sense and
experience and widened logic till it has become a general science of
inference and scientific method. Others have devoted themselves to the
mental analysis of reasoning, and have narrowed logic into a science of
conception, judgment and reasoning. The former belong to the school of
empirical logic, the latter to the school of conceptual and formal logic. Both
have started from points which Aristotle indicated without developing them.
But we shall find that his true descendants are the empirical logicians.

Aristotle was the first of the empiricists. He consistently maintained that
sense is knowledge of particulars and the origin of scientific knowledge of

universals. In his view, sense is a congenital form of judgment (δύναμις
σύμφυτος κριτική, Post. An. ii. 19); a sensation of each of the five senses is
always true of its proper object; without sense there is no science; sense is
the origin of induction, which is the origin of deduction and science. The
Analytics end (Post. An. ii. 19) with a detailed system of empiricism,
according to which sense is the primary knowledge of particulars, memory is
the retention of a sensation, experience is the sum of many memories,
induction infers universals, and intelligence is the true apprehension of the
universal principles of science, which is rational, deductive, demonstrative,
from empirical principles.

This empirical groundwork of Aristotle’s logic was accepted by the
Epicureans, who enunciated most distinctly the fundamental doctrine
that all sensations are true of their immediate objects, and falsity begins
with subsequent opinions, or what the moderns call “interpretation.”
Beneath deductive logic, in the logic of Aristotle and the canonic of the
Epicureans, there already lay the basis of empirical logic: sensory
experience is the origin of all inference and science. It remained for
Francis Bacon to develop these beginnings into a new logic of induction.
He did not indeed accept the infallibility of sense or of any other
operation unaided. He thought, rather, that every operation becomes
infallible by method. Following Aristotle in this order—sense, memory,
intellect—he resolved the whole process of induction into three
ministrations:—

1. The ministration to sense, aided by observation and experiment.

2. The ministration to memory, aided by registering and arranging the
data, of observation and experiment in tables of instances of agreement,
difference and concomitant variations.

3. The ministration to intellect or reason, aided by the negative
elimination by means of contradictory instances of whatever in the

instances is not always present, absent and varying with the given
subject investigated, and finally by the positive inference that whatever
in the instances is always present, absent and varying with the subject is
its essential cause.

Bacon, like Aristotle, was anticipated in this or that point; but, as
Aristotle was the first to construct a system of deduction in the
syllogism and its three figures, so Bacon was the first to construct a
system of induction in three ministrations, in which the requisites of
induction, hitherto recognized only in sporadic hints, were combined for
the first time in one logic of induction. Bacon taught men to labour in
inferring from particular to universal, to lay as much stress on induction
as on deduction, and to think and speak of inductive reasoning,
inductive science, inductive logic. Moreover, while Aristotle had the
merit of discerning the triplicity of inference, to Bacon we owe the merit
of distinguishing the three processes without reduction:—

1. Inference from particular to particular by Experientia Literata, in
plano;

2. Inference from particular to universal by Inductio, ascendendo;

3. Inference from universal to particular by Syllogism,
descendendo.

In short, the comprehensive genius of Bacon widened logic into a
general science of inference.

On the other hand, as Aristotle over-emphasized deduction so Bacon
over-emphasized induction by contending that it is the only process of
discovering universals (axiomata), which deduction only applies to
particulars. J. S. Mill in his Logic pointed out this defect, and without
departing from Baconian principles remedied it by quoting scientific
examples, in which deduction, starting from inductive principles, applies

more general to less general universals, e.g. when the more general law
of gravitation is shown to include the less general laws of planetary
gravitation. Mill’s logic has the great merit of copiously exemplifying
the principles of the variety of method according to subject-matter. It
teaches us that scientific method is sometimes induction, sometimes
deduction, and sometimes the consilience of both, either by the
inductive verification of previous deductions, or by the deductive
explanation of previous inductions.

It is also most interesting to notice that Aristotle saw further than
Bacon in this direction. The founder of logic anticipated the latest logic
of science, when he recognized, not only the deduction of mathematics,
but also the experience of facts followed by deductive explanations of
their causes in physics.

The consilience of empirical and deductive processes was an Aristotelian
discovery, elaborated by Mill against Bacon. On the whole, however,
Aristotle, Bacon and Mill, purged from their errors, form one empirical
school, gradually growing by adapting itself to the advance of science; a
school in which Aristotle was most influenced by Greek deductive
Mathematics, Bacon by the rise of empirical physics at the Renaissance, and
Mill by the Newtonian combination of empirical facts and mathematical
principles in the Principia. From studying this succession of empirical
logicians, we cannot doubt that sense, memory and experience are the real
origin of inference, analogical, inductive and deductive. The deepest problem
of logic is the relation of sense and inference. But we must first consider the
mental analysis of inference, and this brings us to conceptual and formal
logic.

Aristotle’s logic has often been called formal logic; it was really a
technical logic of syllogism analysed into linguistic elements, and of science
rested on an empirical basis. At the same time his psychology, though

maintaining his empiricism, contained some seeds of conceptual logic, and
indirectly of formal logic. Intellectual development, which according to the
logic of the Analytics consists of sense, memory, experience, induction and
intellect, according to the psychology of the De Anima consists of sense,
imagination and intellect, and one division of intellect is into conception of
the undivided and combination of conceptions as one (De An. iii. 6). The De
Interpretatione opens with a reference to this psychological distinction,
implying that names represent conceptions, propositions represent
combinations of conceptions. But the same passage relegates conceptions
and their combinations to the De Anima, and confines the De Interpretatione
to names and propositions in conformity with the linguistic analysis which
pervades the logical treatises of Aristotle, who neither brought his
psychological distinction between conceptions and their combinations into
his logic, nor advanced the combinations of conceptions as a definition of
judgment (κρίσις), nor employed the mental distinction between conceptions
and judgments as an analysis of inference, or reasoning, or syllogism: he was
no conceptual logician. The history of logic shows that the linguistic
distinction between terms and propositions was the sole analysis of reasoning
in the logical treatises of Aristotle; that the mental distinction between
conceptions (ἔννοιαι) and judgments (ἀξιώματα in a wide sense) was
imported into logic by the Stoics; and that this mental distinction became the
logical analysis of reasoning under the authority of St Thomas Aquinas. In
his commentary on the De Interpretatione, St Thomas, after citing from the
De Anima Aristotle’s “duplex operatio intellectus,” said, “Additur autem et
tertia operatio, scilicet ratiocinandi,” and concluded that, since logic is a
rational science (rationalis scientia), its consideration must be directed to all
these operations of reason. Hence arose conceptual logic; according to which
conception is a simple apprehension of an idea without belief in being or not
being, e.g. the idea of man or of running; judgment is a combination of
conceptions, adding being or not being, e.g. man is running or not running;
and reasoning is a combination of judgments: conversely, there is a mental

analysis of reasoning into judgments, and judgment into conceptions,
beneath the linguistic analysis of rational discourse into propositions, and
propositions into terms. Logic, according to this new school, which has by
our time become an old school, has to co-ordinate these three operations,
direct them, and, beginning with conceptions, combine conceptions into
judgments, and judgments into inference, which thus becomes a complex
combination of conceptions, or, in modern parlance, an extension of our
ideas. Conceptual logicians were, indeed, from the first aware that sense
supplies the data, and that judgment and therefore inference contains belief
that things are or are not. But they held, and still hold that sensation and
conception are alike mere apprehensions, and that the belief that things are or
are not arises somehow after sensation and conception in judgment, from
which it passes into inference. At first, they were more sanguine of extracting
from these unpromising beginnings some knowledge of things beyond ideas.
But at length many of them became formal logicians, who held that logic is
the investigation of formal thinking, or consistent conception, judgment and
reasoning; that it shows how we infer formal truths of consistency without
material truth of signifying things; that, as the science of the form or process,
it must entirely abstract from the matter, or objects, of thought; and that it
does not tell us how we infer from experience. Thus has logic drifted further
and further from the real and empirical logic of Aristotle the founder and
Bacon the reformer of the science.

The great merit of conceptual logic was the demand for a mental analysis
of mental reasoning, and the direct analysis of reasoning into judgments
which are the sole premises and conclusions of reasoning and of all mental
inferences. Aristotle had fallen into the paradox of resolving a mental act into
verbal elements. The Schoolmen, however, gradually came to realize that the
result to their logic was to make it a sermocionalis scientia, and to their
metaphysics the danger of nominalism. St Thomas made a great advance by
making logic throughout a rationalis scientia; and logicians are now agreed

that reasoning consists of judgments, discourse of propositions. This
distinction is, moreover, vital to the whole logic of inference, because we
always think all the judgments of which our inference consists, but seldom
state all the propositions by which it is expressed. We omit propositions,
curtail them, and even express a judgment by a single term, e.g. “Good!”
“Fire!”. Hence the linguistic expression is not a true measure of inference;
and to say that an inference consists of two propositions causing a third is not
strictly true. But to say that it is two judgments causing a third is always true,
and the very essence of inference, because we must think the two to conclude
the third in “the sessions of sweet silent thought.” Inference, in short,
consists of actual judgments capable of being expressed in propositions.

Inference always consists of judgments. But judgment does not
always consist of conceptions. It is not a combination of conceptions; it
does not arise from conceptions, nor even at first require conception.
Sense is the origin of judgment. One who feels pained or pleased, who
feels hot or cold or resisting in touch, who tastes the flavoured, who
smells the odorous, who hears the sounding, who sees the coloured, or is
conscious, already believes that something sensible exists before
conception, before inference, and before language; and his belief is true
of the immediate object of sense, the sensible thing, e.g. the hot felt in
touch. But a belief in the existence of something is a judgment and a
categorical judgment of existence. Sense, then, outer and inner, or
sensation and consciousness, is the origin of sensory judgments which
are true categorical beliefs in the existence of sensible things; and
primary judgments are such true categorical sensory beliefs that things
exist, and neither require conception nor are combinations of
conceptions. Again, since sense is the origin of memory and experience,
memorial and experiential judgments are categorical and existential
judgments, which so far as they report sensory judgments are always
true. Finally, since sense, memory and experience are the origin of

inference, primary inference is categorical and existential, starting from
sensory, memorial and experiential judgments as premises, and
proceeding to inferential judgments as conclusions, which are
categorical and existential, and are true, so far as they depend on sense,
memory and experience.

Sense, then, is the origin of judgment; and the consequence is that
primary judgments are true, categorical and existential judgments of
sense, and primary inferences are inferences from categorical and
existential premises to categorical and existential conclusions, which are
true so far as they arise from outer and inner sense, and proceed to
things similar to sensible things. All other judgments and inferences
about existing things, or ideas, or names, whether categorical or
hypothetical, are afterthoughts, partly true and partly false.

Sense, then, because it involves a true belief in existence is fitted to
be the origin of judgment. Conception on the other hand is the simple
apprehension of an idea, particular or universal, but without belief that
anything is or is not, and therefore is unfitted to beget judgment. Nor
could a combination of conceptions make a difference so fundamental as
that between conceiving and believing. The most that it could do would
be to cause an ideal judgment, e.g. that the idea of a centaur is the idea
of a man-horse; and even here some further origin is needed for the
addition of the copula “is.”

So far from being a cause, conception is not even a condition of all
judgments; a sensation of hot is sufficient evidence that hot exists,
before the idea of hot is either present or wanted. Conception is,
however, a condition of a memorial judgment: in order to remember
being hot, we require an idea of hot. Memory, however, is not that idea,
but involves a judgment that there previously existed the hot now
represented by the idea, which is about the sensible thing beyond the

conceived idea; and the cause of this memorial judgment is past sense
and present memory. So sense, memory and experience, the sum of
sense and memory, though requiring conception, are the causes of the
experiential judgment that there exist and have existed many similar,
sensible things, and these sensory, memorial and experiential judgments
about the existence of past and present sensible things beyond conceived
ideas become the particular premises of primary inference. Starting from
them, inference is enabled to draw conclusions which are inferential
judgments about the existence of things similar to sensible things
beyond conceived ideas. In rising, however, from particular to universal
inference, induction, as we have seen, adds to its particular premise, S is
P, a universal premise, every M is similar to S, in order to infer the
universal conclusion, every M is P. This universal premise requires a
universal conception of a class or whole number of similar particulars,
as a condition. But the premise is not that conception; it is a belief that
there is a whole number of particulars similar to those already
experienced. The generalization of a class is not, as the conceptual logic
assumes, the abstraction of a general idea, but an inference from the
analogy of a whole individual thing, e.g. a whole man, to a whole
number of similar individuals, e.g. the whole of men. The general idea
of all men or the combination that the idea of all men is similar to the
idea of particular men would not be enough; the universal premise that
all men in fact are similar to those who have died is required to induce
the universal conclusion that all men in fact die. Universal inference
thus requires particular and universal conceptions as its condition; but,
so far as it arises from sense, memory, experience, and involves
generalization, it consists of judgments which do not consist of
conceptions, but are beliefs in things existing beyond conception.
Inference then, so far as it starts from categorical and existential
premises, causes conclusions, or inferential judgments, which require
conceptions, but are categorical and existential judgments beyond

conception. Moreover, as it becomes more deductive, and causes
conclusions further from sensory experience, these inferential judgments
become causes of inferential conceptions. For example, from the
evidence of molar changes due to the obvious parts of bodies, science
first comes to believe in molecular changes due to imperceptible
particles, and then tries to conceive the ideas of particles, molecules,
atoms, electrons. The conceptual logic supposes that conception always
precedes judgment; but the truth is that sensory judgment begins and
inferential judgment ends by preceding conception. The supposed triple
order—conception, judgment, reasoning—is defective and false. The
real order is sensation and sensory judgment, conception, memory and
memorial judgment, experience and experiential judgment, inference,
inferential judgment, inferential conception. This is not all: inferential
conceptions are inadequate, and finally fail. They are often symbolical;
that is, we conceive one thing only by another like it, e.g. atoms by
minute bodies not nearly small enough. Often the symbol is not like.
What idea can the physicist form of intraspatial ether? What believer in
God pretends to conceive Him as He really is? We believe many things
that we cannot conceive; as Mill said, the inconceivable is not the
incredible; and the point of science is not what we can conceive but
what we should believe on evidence. Conception is the weakest,
judgment the strongest power of man’s mind. Sense before conception is
the original cause of judgment; and inference from sense enables
judgment to continue after conception ceases. Finally, as there is
judgment without conception, so there is conception without judgment.
We often say “I understand, but do not decide.” But this suspension of
judgment is a highly refined act, unfitted to the beginning of thought.
Conception begins as a condition of memory, and after a long
continuous process of inference ends in mere ideation. The conceptual
logic has made the mistake of making ideation a stage in thought prior
to judgment.

It was natural enough that the originators of conceptual logic, seeing
that judgments can be expressed by propositions, and conceptions by
terms, should fall into the error of supposing that, as propositions consist
of terms, so judgments consist of conceptions, and that there is a triple
mental order—conception, judgment, reasoning—parallel to the triple
linguistic order—term, proposition, discourse. They overlooked the fact
that man thinks long before he speaks, makes judgments which he does
not express at all, or expresses them by interjections, names and phrases,
before he uses regular propositions, and that he does not begin by
conceiving and naming, and then proceed to believing and proposing.
Feeling and sensation, involving believing or judging, come before
conception and language. As conceptions are not always present in
judgment, as they are only occasional conditions, and as they are
unfitted to cause beliefs or judgments, and especially judgments of
existence, and as judgments both precede conceptions in sense and
continue after them in inference, it follows that conceptions are not the
constituents of judgment, and judgment is not a combination of
conceptions. Is there then any analysis of judgment? Paradoxical as it
may sound, the truth seems to be that primary judgment, beginning as it
does with the simplest feeling and sensation, is not a combination of two
mental elements into one, but is a division of one sensible thing into the
thing itself and its existence and the belief that it is determined as
existing, e.g. that hot exists, cold exists, the pained exists, the pleased
exists. Such a judgment has a cause, namely sense, but no mental
elements. Afterwards come judgments of complex sense, e.g. that the
existing hot is burning or becoming more or less hot, &c. Thus there is a
combination of sensations causing the judgment; but the judgment is
still a division of the sensible thing into itself and its being, and a belief
that it is so determined. Afterwards follow judgments arising from more
complex causes, e.g. memory, experience, inference. But however
complicated these mental causes, there still remain these points common

to all judgment:—(1) The mental causes of judgment are sense, memory,
experience and inference; while conception is a condition of some
judgments. (2) A judgment is not a combination either of its causes or of
its conditions, e.g. it is not a combination of sensations any more than of
ideas. (3) A judgment is a unitary mental act, dividing not itself but its
object into the object itself and itself as determined, and signifying that
it is so determined. (4) A primary judgment is a judgment that a sensible
thing is determined as existing; but later judgments are concerned with
either existing things, or with ideas, or with words, and signify that they
are determined in all sorts of ways. (5) When a judgment is expressed by
a proposition, the proposition expresses the results of the division by
two terms, subject and predicate, and by the copula that what is signified
by the subject is what is signified by the predicate; and the proposition is
a combination of the two terms; e.g. border war is evil. (6) A complex
judgment is a combination of two judgments, and may be copulative,
e.g. you and I are men, or hypothetical, or disjunctive, &c.

Empirical logic, the logic of Aristotle and Bacon, is on the right way. It is
the business of the logician to find the causes of the judgments which form
the premises and the conclusions of inference, reasoning and science. What
knowledge do we get by sense, memory and experience, the first mental
causes of judgment? What is judgment, and what its various kinds? What is
inference, how does it proceed by combining judgments as premises to cause
judgments as conclusions, and what are its various kinds? How does
inference draw conclusions more or less probable up to moral certainty?
How does it by the aid of identification convert probable into necessary
conclusions, which become necessary principles of demonstration? How is
categorical succeeded by conditional inference? What is scientific method as
a system of inferences about definite subjects? How does inference become
the source of error and fallacy? How does the whole process from sense to
inference discover the real truth of judgments, which are true so far as they

signify things known by sense, memory, experience and inference? These are
the fundamental questions of the science of inference. Conceptual logic, on
the other hand, is false from the start. It is not the first business of logic to
direct us how to form conceptions signified by terms, because sense is a prior
cause of judgment and inference. It is not the second business of logic to
direct us how out of conceptions to form judgments signified by
propositions, because the real causes of judgments are sense, memory,
experience and inference. It is, however, the main business of logic to direct
us how out of judgments to form inferences signified by discourse; and this
is the one point which conceptual logic has contributed to the science of
inference. But why spoil the further mental analysis of inference by
supposing that conceptions are constituents of judgment and therefore of
inference, which thus becomes merely a complex combination of
conceptions, an extension of ideas? The mistake has been to convert three
operations of mind into three processes in a fixed order—conception,
judgment, inference. Conception and judgment are decisions: inference alone
is a process, from decisions to decision, from judgments to judgment. Sense,
not conception, is the origin of judgment. Inference is the process which
from judgments about sensible things proceeds to judgments about things
similar to sensible things. Though some conceptions are its conditions and
some judgments its causes, inference itself in its conclusions causes many
more judgments and conceptions. Finally, inference is an extension, not of
ideas, but of beliefs, at first about existing things, afterwards about ideas, and
even about words; about anything in short about which we think, in what is
too fancifully called “the universe of discourse.”

Formal logic has arisen out of the narrowness of conceptual logic. The
science of inference no doubt has to deal primarily with formal truth or the
consistency of premises and conclusion. But as all truth, real as well as
formal, is consistent, formal rules of consistency become real rules of truth,
when the premises are true and the consistent conclusion is therefore true.

The science of inference again rightly emphasizes the formal thinking of the
syllogism in which the combination of premises involves the conclusion. But
the combinations of premises in analogical and inductive inference, although
the combination does not involve the conclusion, yet causes us to infer it, and
in so similar a way that the science of inference is not complete without
investigating all the combinations which characterize different kinds of
inference. The question of logic is how we infer in fact, as well as perfectly;
and we cannot understand inference unless we consider inferences of
probability of all kinds. Moreover, the study of analogical and inductive
inference is necessary to that of the syllogism itself, because they discover
the premises of syllogism. The formal thinking of syllogism alone is merely
necessary consequence; but when its premises are necessary principles, its
conclusions are not only necessary consequents but also necessary truths.
Hence the manner in which induction aided by identification discovers
necessary principles must be studied by the logician in order to decide when
the syllogism can really arrive at necessary conclusions. Again, the science
of inference has for its subject the form, or processes, of thought, but not its
matter or objects. But it does not follow that it can investigate the former
without the latter. Formal logicians say that, if they had to consider the
matter, they must either consider all things, which would be impossible, or
select some, which would be arbitrary. But there is an intermediate
alternative, which is neither impossible nor arbitrary; namely, to consider the
general distinctions and principles of all things; and without this general
consideration of the matter the logician cannot know the form of thought,
which consists in drawing inferences about things on these general
principles. Lastly, the science of inference is not indeed the science of
sensation, memory and experience, but at the same time it is the science of
using those mental operations as data of inference; and, if logic does not
show how analogical and inductive inferences directly, and deductive
inferences indirectly, arise from experience, it becomes a science of mere
thinking without knowledge.

Logic is related to all the sciences, because it considers the common
inferences and varying methods used in investigating different subjects. But
it is most closely related to the sciences of metaphysics and psychology,
which form with it a triad of sciences. Metaphysics is the science of being in
general, and therefore of the things which become objects apprehended by
our minds. Psychology is the science of mind in general, and therefore of the
mental operations, of which inference is one. Logic is the science of the
processes of inference. These three sciences, of the objects of mind, of the
operations of mind, of the processes used in the inferences of mind, are
differently, but closely related, so that they are constantly confused. The real
point is their interdependence, which is so intimate that one sign of great
philosophy is a consistent metaphysics, psychology and logic. If the world of
things is known to be partly material and partly mental, then the mind must
have powers of sense and inference enabling it to know these things, and
there must be processes of inference carrying us from and beyond the
sensible to the insensible world of matter and mind. If the whole world of
things is matter, operations and processes of mind are themselves material. If
the whole world of things is mind, operations and processes of mind have
only to recognize their like all the world over. It is clear then that a man’s
metaphysics and psychology must colour his logic. It is accordingly
necessary to the logician to know beforehand the general distinctions and
principles of things in metaphysics, and the mental operations of sense,
conception, memory and experience in psychology, so as to discover the
processes of inference from experience about things in logic.

The interdependence of this triad of sciences has sometimes led to their
confusion. Hegel, having identified being with thought, merged metaphysics
in logic. But he divided logic into objective and subjective, and thus
practically confessed that there is one science of the objects and another of
the processes of thought. Psychologists, seeing that inference is a mental
operation, often extemporize a theory of inference to the neglect of logic. But

we have a double consciousness of inference. We are conscious of it as one
operation among many, and of its omnipresence, so to speak, to all the rest.
But we are also conscious of the processes of the operation of inference. To a
certain extent this second consciousness applies to other operations: for
example, we are conscious of the process of association by which various
mental causes recall ideas in the imagination. But how little does the
psychologist know about the association of ideas, compared with what the
logician has discovered about the processes of inference! The fact is that our
primary consciousness of all mental operations is hardly equal to our
secondary consciousness of the processes of the one operation of inference
from premises to conclusions permeating long trains and pervading whole
sciences. This elaborate consciousness of inferential process is the
justification of logic as a distinct science, and is the first step in its method.
But it is not the whole method of logic, which also and rightly considers the
mental process necessary to language, without substituting linguistic for
mental distinctions.

Nor are consciousness and linguistic analysis all the instruments of the
logician. Logic has to consider the things we know, the minds by which we
know them from sense, memory and experience to inference, and the
sciences which systematize and extend our knowledge of things; and having
considered these facts, the logician must make such a science of inference as
will explain the power and the poverty of human knowledge.

General Tendencies of Modern Logic

There are several grounds for hope in the logic of our day. In the first
place, it tends to take up an intermediate position between the extremes of
Kant and Hegel. It does not, with the former, regard logic as purely formal in
the sense of abstracting thought from being, nor does it follow the latter in
amalgamating metaphysics with logic by identifying being with thought.
Secondly, it does not content itself with the mere formulae of thinking, but

pushes forward to theories of method, knowledge and science; and it is a
hopeful sign to find this epistemological spirit, to which England was
accustomed by Mill, animating German logicians such as Lotze, Dühring,
Schuppe, Sigwart and Wundt. Thirdly, there is a determination to reveal the
psychological basis of logical processes, and not merely to describe them as
they are in adult reasoning, but to explain also how they arise from simpler
mental operations and primarily from sense. This attempt is connected with
the psychological turn given to recent philosophy by Wundt and others, and
is dangerous only so far as psychology itself is hypothetical. Unfortunately,
however, these merits are usually connected with a less admirable
characteristic—contempt for tradition, Writing his preface to his second
edition in 1888, Sigwart says: “Important works have appeared by Lotze,
Schuppe, Wundt and Bradley, to name only the most eminent; and all start
from the conception which has guided this attempt. That is, logic is grounded
by them, not upon an effete tradition but upon a new investigation of thought
as it actually is in its psychological foundations, in its significance for
knowledge, and its actual operation in scientific methods.” How strange! The
spirit of every one of the three reforms above enumerated is an unconscious
return to Aristotle’s Organon. Aristotle’s was a logic which steered, as
Trendelenburg has shown, between Kantian formalism and Hegelian
metaphysics; it was a logic which in the Analytics investigated the syllogism
as a means to understanding knowledge and science: it was a logic which,
starting from the psychological foundations of sense, memory and
experience, built up the logical structure of induction and deduction on the
profoundly Aristotelian principle that “there is no process from universals
without induction, and none by induction without sense.” Wundt’s
comprehensive view that logic looks backwards to psychology and forward
to epistemology was hundreds of years ago one of the many discoveries of
Aristotle.

Judgment

1. Judgment and Conception.—The emphasis now laid on judgment, the
recovery from Hume’s confusion of beliefs with ideas and the association of
ideas, and the distinction of the mental act of judging from its verbal
expression in a proposition, are all healthy signs in recent logic. The most
fundamental question, before proceeding to the investigation of inference, is
not what we say but what we think in making the judgments which, whether
we express them in propositions or not, are both the premises and the
conclusion of inference; and, as this question has been diligently studied of
late, but has been variously answered, it will be well to give a list of the more
important theories of judgment as follows:—

a. It expresses a relation between the content of two ideas, not a
relation of these ideas (Lotze).

b. It is consciousness concerning the objective validity of a subjective
combination of ideas, i.e. whether between the corresponding objective
elements an analogous combination exists (Ueberweg).

c. It is the synthesis of ideas into unity and consciousness of their
objective validity, not in the sense of agreement with external reality but
in the sense of the logical necessity of their synthesis (Sigwart).

d. It is the analysis of an aggregate idea (Gesammtvorstellung) into
subject and predicate; based on a previous association of ideas, on
relating and comparing, and on the apperceptive synthesis of an
aggregate idea in consequence; but itself consisting in an apperceptive
analysis of that aggregate idea; and requiring will in the form of
apperception or attention (Wundt).

e. It requires an idea, because every object is conceived as well as
recognized or denied; but it is itself an assertion of actual fact, every
perception counts for a judgment, and every categorical is changeable
into an existential judgment without change of sense (Brentano, who

derives his theory from Mill except that he denies the necessity of a
combination of ideas, and reduces a categorical to an existential
judgment).

f. It is a decision of the validity of an idea requiring will (Bergmann,
following Brentano).

g. Judgment (Urtheil) expresses that two ideas belong together: “by-
judgment” (Beurtheilung) is the reaction of will expressing the validity
or invalidity of the combination of ideas (Windelband, following
Bergmann, but distinguishing the decision of validity from the
judgment).

h. Judgment is consciousness of the identity or difference and of the
causal relations of the given; naming the actual combinations of the
data, but also requiring a priori categories of the understanding, the
notions of identity, difference and causality, as principles of thought or
laws, to combine the plurality of the given into a unity (Schuppe).

i. Judgment is the act which refers an ideal content recognized as such
to a reality beyond the act, predicating an idea of a reality, a what of a
that; so that the subject is reality and the predicate the meaning of an
idea, while the judgment refers the idea to reality by an identity of
content (Bradley and Bosanquet).

k. Judgment is an assertion of reality, requiring comparison and ideas
which render it directly expressible in words (Hobhouse, mainly
following Bradley).

These theories are of varying value in proportion to their proximity to
Aristotle’s point that predication is about things, and to Mill’s point that
judgments and propositions are about things, not about ideas. The essence of
judgment is belief that something is (or is not) determined, either as existing
(e.g. “I am,” “A centaur is not”) or as something in particular (e.g. “I am a

man,” “I am not a monkey”). Neither Mill, however, nor any of the later
logicians whose theories we have quoted, has been able quite to detach
judgment from conception; they all suppose that an idea, or ideas, is a
condition of all judgment. But judgment starts from sensation (Empfindung)
and feeling (Gefühl), and not from idea (Vorstellung). When I feel pleased or
pained, or when I use my senses to perceive a pressure, a temperature, a
flavour, an odour, a colour, a sound, or when I am conscious of feeling and
perceiving, I cannot resist the belief that something sensible is present; and
this belief that something exists is already a judgment, a judgment of
existence, and, so far as it is limited to sense without inference, a true
judgment. It is a matter of words whether or not we should call this sensory
belief a judgment; but it is no matter of choice to the logician, who regards
all the constituents of inference as judgments; for the fundamental
constituents are sensory beliefs, which are therefore judgments in the logical
sense. Sense is the evidence of inference; directly of analogical and
inductive, directly or indirectly of deductive, inference; and therefore, if
logic refuses to include sensory beliefs among judgments, it will omit the
fundamental constituents of inference, inference will no longer consist of
judgments but of sensory beliefs plus judgments, and the second part of
logic, the logic of judgment, the purpose of which is to investigate the
constituents of inference, will be like Hamlet without the prince of Denmark.
If, on the other hand, all the constituents of inference are judgments, there
are judgments of sense; and the evidence of the senses means that a judgment
of sense is true, while a judgment of inference is true so far as it is directly or
indirectly concluded from judgments of sense. Now a sensory judgment, e.g.
that a sensible pressure is existing, is explained by none of the foregoing
theories, because it requires nothing but sensation and belief. It requires no
will, but is usually involuntary, for the stimulus forces one’s attention, which
is not always voluntary; not all judgment then requires will, as Wundt
supposes. It requires no reference to reality beyond the sensible pressure,
because it is merely a belief that this exists without inference of the external

stimulus or any inference at all: not all judgment then requires the reference
of subjective to objective supposed by Ueberweg, or the consciousness of
logical necessity supposed by Sigwart. It requires in addition to the belief
that something exists, no consideration as to whether the belief itself be true,
because a man who feels pressure believes in the thing without further
question about the belief: not all judgment then requires a decision of
validity, as Bergmann supposes. It requires nothing beyond the sensation and
belief in the given existence of the given pressure: not all judgment then
requires categories of understanding, or notions of identity, difference and
causality, or even of existence, such as Schuppe supposes. It requires no
comparison in order to express it in words, for a judgment need not be
expressed, and a sensory judgment of pressure is an irresistible belief that a
real pressure exists, without waiting for words, or for a comparison which is
wanted not to make a sensation a judgment, but to turn a judgment into
language: not all judgment then requires comparison with a view to its
expression, as supposed by Hobhouse. Lastly, all the authors of the above-
quoted theories err in supposing that all judgment requires conception; for
even Mill thinks a combination of ideas necessary, and Brentano, who comes
still nearer to the nature of sensory judgment when he says, “Every
perception counts for a judgment,” yet thinks that an idea is necessary at the
same time in order to understand the thing judged. In reality, the sensation
and the belief are sufficient; when I feel a sensible pressure, I cannot help
believing in its reality, and therefore judging that it is real, without any
tertium quid—an idea of pressure, or of existence or of pressure existing—
intervening between the sensation and the belief. Only after sensation has
ceased does an idea, or representation of what is not presented, become
necessary as a substitute for a sensation and as a condition not of the first
judgment that there is, but of a second judgment that there was, something
sensible. Otherwise there would be no judgment of sensible fact, for the first
sensation would not give it, and the idea following the sensation would be
still farther off. The sensory judgment then, which is nothing but a belief that

at the moment of sense something sensible exists, is a proof that not all
judgment requires conception, or synthesis or analysis of ideas, or decision
about the content, or about the validity, of ideas, or reference of an ideal
content to reality, as commonly, though variously, supposed in the logic of
our day.

Not, however, that all judgment is sensory: after the first judgments of
sense follow judgments of memory, and memory requires ideas. Yet memory
is not mere conception, as Aristotle, and Mill after him, have perceived. To
remember, we must have a present idea; but we must also have a belief that
the thing, of which the idea is a representation, was (or was not) determined;
and this belief is the memorial judgment. Originally such judgments arise
from sensory judgments followed by ideas, and are judgments of memory
after sense that something sensible existed, e.g. pressure existed: afterwards
come judgments of memory after inference, e.g. Caesar was murdered.
Finally, most judgments are inferential. These are conclusions which
primarily are inferred from sensory and memorial judgments; and so far as
inference starts from sense of something sensible in the present, and from
memory after sense of something sensible in the past, and concludes similar
things, inferential judgments are indirect beliefs in being and in existence
beyond ideas. When from the sensible pressures between the parts of my
mouth, which I feel and remember and judge that they exist and have existed,
I infer another similar pressure (e.g. of the food which presses and is pressed
by my mouth in eating), the inferential judgment with which I conclude is a
belief that the latter exists as well as the former (e.g. the pressure of food
without as well as the sensible pressures within). Inference, no doubt, is
closely involved with conception. So far as it depends on memory, an
inferential judgment presupposes memorial ideas in its data; and so far as it
infers universal classes and laws, it produces general ideas. But even so the
part played by conception is quite subordinate to that of belief. In the first
place, the remembered datum, from which an inference of pressure starts, is

not the conceived idea, but the belief that the sensible pressure existed.
Secondly, the conclusion in which it ends is not the general idea of a class,
but the belief that a class, represented by a general idea, exists, and is (or is
not) otherwise determined (e.g. that things pressing and pressed exist and
move). Two things are certain about inferential judgment: one, that when
inference is based on sense and memory, inferential judgment starts from a
combination of sensory and memorial judgment, both of which are beliefs
that things exist; the other, that in consequence inferential judgment is a
belief that similar things exist. There are thus three primary judgments:
judgments of sense, of memory after sense, and of inference from sense. All
these are beliefs in being and existence, and this existential belief is first in
sense, and afterwards transferred to memory and inference. Moreover, it is
transferred in the same irresistible way: frequently we cannot help either
feeling pressure, or remembering it, or inferring it; and as there are
involuntary sensation and attention, so there are involuntary memory and
inference. Again, in a primary judgment existence need not be expressed; but
if expressed, it may be expressed either by the predicate, e.g. “I exist,” or by
the subject, e.g. “I who exist think.” There are indeed differences between
primary judgments, in that the sensory is a belief in present, the memorial in
past, and the inferential in present, past and future existence. But these
differences in detail do not alter the main point that all these are beliefs in the
existing, in the real as opposed to the ideal, in actual things which are not
ideas. In short, a primary judgment is a belief in something existing apart
from our idea of it; and not because we have an idea of it, or by comparing
an idea with, or referring an idea to, reality; but because we have a sensation
of it, or a memory of it or an inference of it. Sensation, not conception, is the
origin of judgment.

2. Different Significations of Being in different Kinds of Judgment.—As
Aristotle remarked both in the De Interpretatione and in the Sophistici
Elenchi, “not-being is thinkable” does not mean “not-being exists.” In the

latter treatise he added that it is a fallacia a dicto secundum quid ad dictum
simpliciter to argue from the former to the latter; “for,” as he says, “it is not
the same thing to be something and to exist absolutely.” Without realizing
their debt to tradition, Herbart, Mill and recently Sigwart, have repeated
Aristotle’s separation of the copula from the verb of existence, as if it were a
modern discovery that “is” is not the same as “exists.” It may be added that
they do not quite realize what the copula exactly signifies: it does not signify
existence, but it does signify a fact, namely, that something is (or is not)
determined, either absolutely in a categorical judgment, or conditionally in a
conditional judgment. Now we have seen that all primary judgments signify
more than this fact; they are also beliefs in the existence of the thing
signified by the subject. But, in the first place, primary judgments signify this
existence never by the copula, but sometimes by the predicate, and
sometimes by the subject; and, secondly, it does not follow that all judgments
whatever signify existence. Besides inference of existence there is inference
of non-existence, of things inconsistent with the objects of primary
judgments. Hence secondary judgments, which no longer contain a belief
that the thing exists, e.g. the judgment, “not-being is thinkable,” cited by
Aristotle; the judgment, “A square circle is impossible,” cited by Herbart; the
judgment, “A centaur is a fiction of the poets,” cited by Mill. These
secondary judgments of non-existence are partly like and partly unlike
primary judgments of existence. They resemble them in that they are beliefs
in being signified by the copula. They are beliefs in things of a sort; for, after
all, ideas and names are things; their objects, even though non-existent, are at
all events things conceivable or nameable; and therefore we are able to make
judgments that things, non-existent but conceivable or nameable, are (or are
not) determined in a particular manner. Thus the judgment about a centaur is
the belief, “A conceivable centaur is a fiction of the poets,” and the judgment
about a square circle is the belief, “A so-called square circle is an
impossibility.” But, though beliefs that things of some sort are (or are not)
determined, these secondary judgments fall short of primary judgments of

existence. Whereas in a primary judgment there is a further belief, signified
by subject or predicate, that the thing is an existing thing in the sense of
being a real thing (e.g. a man), different from the idea of it as well as from
the name for it; in a secondary judgment there is no further belief that the
thing has any existence beyond the idea (e.g. a centaur), or even beyond the
name (e.g. a square circle): though the idea or name exists, there is no belief
that anything represented by idea or name exists. Starting, then, from this
fundamental distinction between judgments of existence and judgments of
non-existence, we may hope to steer our way between two extreme views
which emanate from two important thinkers, each of whom has produced a
flourishing school of psychological logic.

On the one hand, early in the 19th century Herbart started the view that a
categorical judgment is never a judgment of existence, but always
hypothetical; on the other hand, in the latter part of the century Brentano
started the view that all categorical judgments are existential. The truth lies
between these contraries. The view of Herbart and his school is contradicted
by our primary judgments of and from sense, in which we cannot help
believing existence; and it gives an inadequate account even of our
secondary judgments in which we no longer indeed believe existence, but do
frequently believe that a non-existent thing is (or is not) somehow
determined unconditionally. It is true, as Herbart says, that the judgment, “A
square circle is an impossibility,” does not contain the belief, “A square
circle is existent”; but when he goes on to argue that it means, “If a square
circle is thought, the conception of impossibility must be added in thought,”
he falls into a non-sequitur. To be categorical, a judgment does not require a
belief in existence, but only that something, existent or not, is (or is not)
determined; and there are two quite different attitudes of mind even to a non-
existent thing, such as a square circle, namely, unconditional and conditional
belief. The judgment, “A non-existent but so-called square circle is an
impossibility,” is an unconditional, or categorical judgment of non-existence,

quite different from any hypothetical judgment, which depends on the
conditions “if it is thought,” or “if it exists,” or any other “if.” On the other
hand, the view of Brentano and his school is contradicted by these very
categorical judgments of non-existence; and while it applies only to
categorical judgments of existence, it does so inadequately. To begin with the
latter objection, Brentano proposed to change the four Aristotelian forms of
judgment, A, E, I, O, into the following existential forms:—

A. “There is not an immortal man.”

E. “There is not a live stone.”

I. “There is a sick man.”

O. “There is an unlearned man.”

This reconstruction, which merges subject and predicate in one expression, in
order to combine it with the verb of existence, is repeated in similar
proposals of recent English logicians. Venn, in his Symbolic Logic, proposes
the four forms, xy = 0, xy = 0, xy > 0, xy > 0 (where y means “not-y”), but
only as alternative to the ordinary forms. Bradley says that “‘S-P is real’
attributes S-P, directly or indirectly, to the ultimate reality,” and agrees with
Brentano that “’is’ never stands for anything but ‘exists’”; while Bosanquet,
who follows Bradley, goes so far as to define a categorical judgment as “that
which affirms the existence of its subject, or, in other words, asserts a fact.”
Now it is true that our primary judgments do contain a belief in existence;
but they do not all contain it in the same way, but are beliefs sometimes that
something is determined as existing, and sometimes that something existing
is particularly determined. Brentano’s forms do not express such a judgment
of existence, as “All existing men are mortal”: nor does Bradley’s form,
“Reality includes S-P.” Metaphysically, all realities are parts of one ultimate
reality; but logically, even philosophers think more often only of finite
realities, existing men, dogs, horses, &c.; and children know that their

parents exist long before they apprehend ultimate reality. The normal form,
then, of a judgment of existence is either “S is a real P,” or “A real S is P.”
Hence the reconstruction of all categorical judgments by merging subject and
predicate, either on Brentano’s or on Bradley’s plan, is a misrepresentation
even of normal categorical judgments of existence. Secondly, it is much
more a misrepresentation of categorical judgments of non-existence. No
existential form suits a judgment such as “A centaur is a fiction,” when we
do not believe that there is a centaur, or that reality includes a centaur. As
Mill pointed out, it cannot be implied that a centaur exists, since the very
thing asserted is that the thing has no real existence. In a correspondence
with Mill, Brentano rejoined that the centaur exists in imagination; Bradley
says, “inside our heads.” According to one, then, the judgment becomes
“There is an imaginary centaur”; according to the other “Reality includes an
imaginary centaur.” The rejoinder, however, though partly true, is not to the
point. The idea of the centaur does exist in our imagination, and inside our
heads, and the name of it in our mouths. But the point is that the centaur
conceived and named does not exist beyond the idea of it and the name for it;
it is not, like a man, a real thing which is neither the idea of it nor the name
for it. No amount of subtlety will remove the difference between a
categorical judgment of existence, e.g. “An existing man is mortal,” and a
categorical judgment of non-existence, e.g. “A conceivable centaur is a
fiction,” because in the former we believe and mean that the thing exists
beyond the idea, and in the latter we do not. If, contrary to usage, we choose
to call the latter a judgment of existence, there is no use in quarrelling about
words; but we must insist that new terms must in that case be invented to
express so fundamental a difference as that between judgments about real
men and judgments about ideal centaurs. So long, however, as we use words
in the natural sense, and call the former judgments of existence, and the latter
judgments of non-existence, then “is” will not be, as Bradley supposes, the
same as “exists,” for we use “is” in both judgments, but “exists” only in the
first kind. Bosanquet’s definition of a categorical judgment contains a similar

confusion. To assert a fact and to affirm the existence of a subject are not, as
he makes out, the same thing: a judgment often asserts a fact and denies
existence in the same breath, e.g. “Jupiter is non-existent.” Here, as usual in
logic, tradition is better than innovation. All categorical judgment is an
unconditional belief in the fact, signified by the copula, that a thing of some
sort is (or is not) determined; but some categorical judgments are also beliefs
that the thing is an existing thing, signified either by the subject or by the
predicate, while others are not beliefs that the thing exists at all, but are only
beliefs in something conceivable, or nameable, or in something or other,
without particularizing what. Judgment then always signifies being, but not
always existence.

3. Particular and Universal Judgments.—Aristotle, by distinguishing
affirmative and negative, particular and universal, made the fourfold
classification of judgments, A, E, I and O, the foundation both of opposition
and of inference. With regard to inference, he remarked that a universal
judgment means by “all,” not every individual we know, but every individual
absolutely, so that, when it becomes a major premise, we know therein every
individual universally, not individually, and often do not know a given
individual individually until we add a minor premise in a syllogism.
Whereas, then, a particular judgment is a belief that some, a universal
judgment is a belief that all, the individuals of a kind or total of similar
individuals, are similarly determined, whether they are known or unknown
individuals. Now, as we have already seen, what is signified by the subject
may be existing or not, and in either case a judgment remains categorical so
long as it is a belief without conditions. Thus, “Some existing men are
poets,” “All existing men are mortal,” “Some conceivable centaurs are
human in their forequarters,” “All conceivable centaurs are equine in their
hindquarters,” are all categorical judgments, while the two first are also
categorical judgments of existence. Nevertheless these obvious applications
of Aristotelian traditions have been recently challenged, especially by

Sigwart, who holds in his Logic (secs. 27, 36) that, while a particular is a
categorical judgment of existence, a universal is hypothetical, on the ground
that it does not refer to a definite number of individuals, or to individuals at
all, but rather to general ideas, and that the appropriate form of “all M is P”
is “if anything is M it is P.” This view, which has influenced not only
German but also English logicians, such as Venn, Bradley and Bosanquet,
destroys the fabric of inference, and reduces scientific laws to mere
hypotheses. In reality, however, particular and universal judgments are too
closely connected to have such different imports. In opposition, a categorical
particular is the contradictory of a universal, which is also categorical, not
hypothetical, e.g., “not all M is P” is the contradictory of “all M is P,” not of
“if anything is M it is P.” In inference, a particular is an example of a
universal which in its turn may become a particular example of a higher
universal. For instance, in the history of mechanics it was first inferred from
some that all terrestrial bodies gravitate, and then from these as some that all
ponderable bodies, terrestrial and celestial, gravitate. How absurd to suppose
that here we pass from a particular categorical to a universal hypothetical,
and then treat this very conclusion as a particular categorical to pass to a
higher universal hypothetical! Sigwart, indeed, is deceived both about
particulars and universals. On the one hand, some particulars are not
judgments of existence, e.g. “some imaginary deities are goddesses”; on the
other hand, some universals are not judgments of non-existence, e.g. “every
existing man is mortal.” Neither kind is always a judgment of existence, but
each is sometimes the one and sometimes the other. In no case is a universal
hypothetical, unless we think it under a condition; for in a universal
judgment about the non-existing, e.g. about all conceivable centaurs, we do
not think, “If anything is a centaur,” because we do not believe that there are
any; and in a universal judgment about the existent, e.g. about all existing
men, we do not think, “If anything is a man,” because we believe that there is
a whole class of men existing at different times and places. The cause of
Sigwart’s error is his misconception of “all.” So far as he follows Aristotle in

saying that “all” does not mean a definite number of individuals he is right;
but when he says that we mean no individuals at all he deserts Aristotle and
goes wrong. By “all” we mean every individual whatever of a kind; and
when from the experience of sense and memory we start with particular
judgments of existence, and infer universal judgments of existence and
scientific laws, we further mean those existing individuals which we have
experienced, and every individual whatever of the kind which exists. We
mean neither a definite number of individuals, nor yet an infinite number, but
an incalculable number, whether experienced or inferred to exist. We do not
mean existing here and now, nor yet out of time and place, but at any time
and place (semper et ubique)—past, present and future being treated as
simply existing, by what logicians used to call suppositio naturalis. We mean
then by “all existing” every similar individual whatever, whenever, and
wherever existing. Hence Sigwart is right in saying that “All bodies are
extended” means “Whatever is a body is extended,” but wrong in identifying
this form with “If anything is a body it is extended.” “Whatever” is not “if
anything.” For the same reason it is erroneous to confuse “all existing” with
a general idea. Nor does the use of abstract ideas and terms make any
difference. When Bosanquet says that in “Heat is a mode of motion” there is
no reference to individual objects, but “a pure hypothetical form which
absolutely neglects the existence of objects,” he falls far short of expressing
the nature of this scientific judgment, for in his Theory of Heat Clerk
Maxwell describes it as “believing heat as it exists in a hot body to be in the
form of kinetic energy.” As Bacon would say, it is a belief that all individual
bodies qua hot are individually but similarly moving in their particles. When,
again, Bradley and Bosanquet speak of the universal as if it always meant
one ideal content referred to reality, they forget that in universal judgments
of existence, such as “All men existing are mortal,” we believe that every
individually existing man dies his own death individually, though similarly to
other men; and that we are thinking neither of ideas nor of reality; but of all
existent individual men being individually but similarly determined. A

universal is indeed one whole; but it is one whole of many similars, which
are not the same with one another. This is indeed the very essence of
distribution, that a universal is predicable, not singly or collectively, but
severally and similarly of each and every individual of a kind, or total of
similar individuals. So also the essence of a universal judgment is that every
individual of the kind is severally but similarly determined. Finally, a
universal judgment is often existential; but whether it is so or not it remains
categorical, so long as it introduces no hypothetical antecedent about the
existence of the thing signified by the subject. It is true that even in universal
judgments of existence there is often a hypothetical element; for example,
“All men are mortal” contains a doubt whether every man whatever,
whenever and wherever existing, must die. But this is only a doubt whether
all the things signified by the subject are similarly determined as signified by
the predicate, and not a doubt whether there are such things at all. Hence the
hypothetical element is not a hypothetical antecedent “If anything is a man,”
but an uncertain conclusion that “All existing men are mortal.” In other
words, a categorical universal is often problematic, but a problematic is not
the same as a hypothetical judgment.

4. The Judgment and the Proposition.—Judgment in general is the mental
act of believing that something is (or is not) determined. A proposition is the
consequent verbal expression of such a belief, and consists in asserting that
the thing as signified by the subject is (or is not) determined as signified by
the predicate. But the expression is not necessary. Sensation irresistibly
produces a judgment of existence without needing language. Children think
long before they speak; and indeed, as mere vocal sounds are not speech, and
as the apprehension that a word signifies a thing is a judgment, judgment is
originally not an effect, but a cause of significant language. At any rate, even
when we have learnt to speak, we do not express all we think, as we may see
not only from the fewness of words known to a child, but also from our own
adult consciousness. The principle of thought is to judge enough to conclude.

The principle of language is to speak only so far as to understand and be
understood. Hence speech is only a curtailed expression of thought.
Sometimes we express a whole judgment by one word, e.g. “Fire!” or by a
phrase, e.g. “What a fire!” and only usually by a proposition. But even the
normal proposition in the syllogistic form tertii adjacentis, with subject,
predicate and copula, is seldom a complete expression of the judgment. The
consequence is that the proposition, being different from a judgment arising
after a judgment, and remaining an imperfect copy of judgment, is only a
superficial evidence of its real nature. Fortunately, we have more profound
evidences, and at least three evidences in all: the linguistic expression of
belief in the proposition; the consciousness of what we mentally believe; and
the analysis of reasoning, which shows what we must believe, and have
believed, as data for inference. In these ways we find that a judgment is both
different from, and more than, a proposition. But recent logicians, although
they perceive the difference, nevertheless tend to make the proposition the
measure of the judgment. This makes them omit sensory judgments, and
count only those which require ideas, and even general ideas expressed in
general terms. Sigwart, for example, gives as instances of our most
elementary judgments, “This is Socrates,” “This is snow”—beliefs in things
existing beyond ourselves which require considerable inferences from many
previous judgments of sense and memory. Worse still, logicians seem unable
to keep the judgment apart from the proposition. Herbart says that the
judgment “A is B” does not contain the usually added thought that A is,
because there is no statement of A’s existence; as if the statement mattered to
the thought. So Sigwart, in order to reduce universals to hypotheticals, while
admitting that existence is usually thought, argues that it is not stated in the
universal judgment; so also Bosanquet. But in the judgment the point is not
what we state, but what we think; and so long as the existence of A is added
in thought, the judgment in question must contain the thought that A exists as
well as that A is B, and therefore is a judgment that something is determined
both as existing and in a particular manner. The statement only affects the

proposition; and whenever we believe the existence of the thing, the belief in
existence is part of the judgment thought, whether it is part of the proposition
stated or not.

Here Sir William Hamilton did a real service to logic in pointing out
that “Logic postulates to be allowed to state explicitly in language all
that is implicitly contained in the thought.” Not that men should or can
carry this logical postulate out in ordinary life; but it is necessary in the
logical analysis of judgments, and yet logicians neglect it. This is why
they confuse the categorical and the universal with the hypothetical.
Taking the carelessly expressed propositions of ordinary life, they do not
perceive that similar judgments are often differently expressed, e.g. “I,
being a man, am mortal,” and “If I am a man, I am mortal”; and
conversely, that different judgments are often similarly expressed. In
ordinary life we may say, “All men are mortal,” “All centaurs are
figments,” “All square circles are impossibilities,” “All candidates
arriving five minutes late are fined” (the last proposition being an
example of the identification of categorical with hypothetical in
Keynes’s Formal Logic). But of these universal propositions the first
imperfectly expresses a categorical belief in existing things, the second
in thinkable things, and the third in nameable things, while the fourth is
a slipshod categorical expression of the hypothetical belief, “If any
candidates arrive late they are fined.” The four judgments are different,
and therefore logically the propositions fully expressing them are also
different. The judgment, then, is the measure of the proposition, not the
proposition the measure of the judgment. On the other hand, we may go
too far in the opposite direction, as Hamilton did in proposing the
universal quantification of the predicate. If the quantity of the predicate
were always thought, it ought logically to be always stated. But we only
sometimes think it. Usually we leave the predicate indefinite, because,
as long as the thing in question is (or is not) determined, it does not

matter about other things, and it is vain for us to try to think all things at
once. It is remarkable that in Barbara, and therefore in many scientific
deductions, to think the quantity of the predicate is not to the point either
in the premises or in the conclusion; so that to quantify the propositions,
as Hamilton proposes, would be to express more than a rational man
thinks and judges. In judgments, and therefore in propositions, indefinite
predicates are the rule, quantified predicates the exception.
Consequently, A E I O are the normal propositions with indefinite
predicates; whereas propositions with quantified predicates are only
occasional forms, which we should use whenever we require to think the
quantity of the predicate, e.g. (1) in conversion, when we must think that
all men are some animals, in order to judge that some animals are men;
(2) in syllogisms of the 3rd figure, when the predicate of the minor
premise must be particularly quantified in thought in order to become
the particularly quantified subject of the conclusion; (3) in identical
propositions including definitions, where we must think both that 1 + 1
are 2 and 2 are 1 + 1. But the normal judgment, and therefore the normal
proposition, do not require the quantity of the predicate. It follows also
that the normal judgment is not an equation. The symbol of equality (=)
is not the same as the copula (is); it means “is equal to,” where “equal
to” is part of the predicate, leaving “is” as the copula. Now, in all
judgment we think “is,” but in few judgments predicate “equal to.” In
quantitative judgments we may think x = y, or, as Boole proposes, x =
vy = (0/0)y or, as Jevons proposes, x = xy, or, as Venn proposes, x which
is not y = 0; and equational symbolic logic is useful whenever we think
in this quantitative way. But it is a byway of thought. In most judgments
all we believe is that x is (or is not) y, that a thing is (or is not)
determined, and that the thing signified by the subject is a thing
signified by the predicate, but not that it is the only thing, or equal to
everything signified by the predicate. The symbolic logic, which
confuses “is” with “is equal to,” having introduced a particular kind of

predicate into the copula, falls into the mistake of reducing all
predication to the one category of the quantitative; whereas it is more
often in the substantial, e.g. “I am a man,” not “I am equal to a man,” or
in the qualitative, e.g. “I am white,” not “I am equal to white,” or in the
relative, e.g. “I am born in sin,” not “I am equal to born in sin.”
Predication, as Aristotle saw, is as various as the categories of being.
Finally, the great difficulty of the logic of judgment is to find the mental
act behind the linguistic expression, to ascribe to it exactly what is
thought, neither more nor less, and to apply the judgment thought to the
logical proposition, without expecting to find it in ordinary propositions.
Beneath Hamilton’s postulate there is a deeper principle of logic—A
rational being thinks only to the point, and speaks only to understand
and be understood.

Inference

The nature and analysis of inference have been so fully treated in the
Introduction that here we may content ourselves with some points of detail.

1. False Views of Syllogism arising from False Views of Judgment.—The
false views of judgment, which we have been examining, have led to false
views of inference. On the one hand, having reduced categorical judgments
to an existential form, Brentano proposes to reform the syllogism, with the
results that it must contain four terms, of which two are opposed and two
appear twice; that, when it is negative, both premises are negative; and that,
when it is affirmative, one premise, at least, is negative. In order to infer the
universal affirmative that every professor is mortal because he is a man,
Brentano’s existential syllogism would run as follows:—

There is not a not-mortal man.
There is not a not-human professor.
∴ There is not a non-mortal professor.

On the other hand, if on the plan of Sigwart categorical universals were
reducible to hypothetical, the same inference would be a pure hypothetical
syllogism, thus:—

If anything is a man it is mortal.
If anything is a professor it is a man.
∴ If anything is a professor it is mortal.

But both these unnatural forms, which are certainly not analyses of any
conscious process of categorical reasoning, break down at once, because they
cannot explain those moods in the third figure, e.g. Darapti, which reason
from universal premises to a particular conclusion. Thus, in order to infer
that some wise men are good from the example of professors, Brentano’s
syllogism would be the following non-sequitur:—

There is not a not-good professor.
There is not a not-wise professor.
There is a wise good (non-sequitur).

So Sigwart’s syllogism would be the following non-sequitur:—

If anything is a professor, it is good.
If anything is a professor, it is wise.
Something wise is good (non-sequitur).

But as by the admission of both logicians these reconstructions of Darapti
are illogical, it follows that their respective reductions of categorical
universals to existentials and hypotheticals are false, because they do not
explain an actual inference. Sigwart does not indeed shrink from this and
greater absurdities; he reduces the first figure to the modus ponens and the
second to the modus tollens of the hypothetical syllogism, and then, finding
no place for the third figure, denies that it can infer necessity; whereas it
really infers the necessary consequence of particular conclusions. But the

crowning absurdity is that, if all universals were hypothetical, Barbara in the
first figure would become a purely hypothetical syllogism—a consequence
which seems innocent enough until we remember that all universal
affirmative conclusions in all sciences would with their premises dissolve
into mere hypothesis. No logic can be sound which leads to the following
analysis:—

If anything is a body it is extended.
If anything is a planet it is a body.
∴ If anything is a planet it is extended.

Sigwart, indeed, has missed the essential difference between the categorical
and the hypothetical construction of syllogisms. In a categorical syllogism of
the first figure, the major premise, “Every M whatever is P,” is a universal,
which we believe on account of previous evidence without any condition
about the thing signified by the subject M, which we simply believe
sometimes to be existent (e.g. “Every man existent”), and sometimes not
(e.g., “Every centaur conceivable”); and the minor premise, “S is M,”
establishes no part of the major, but adds the evidence of a particular not
thought of in the major at all. But in a hypothetical syllogism of the ordinary
mixed type, the first or hypothetical premise is a conditional belief, e.g. “If
anything is M it is P,” containing a hypothetical antecedent, “If anything is
M,” which is sometimes a hypothesis of existence (e.g. “If anything is an
angel”), and sometimes a hypothesis of fact (e.g. “If an existing man is
wise”); and the second premise or assumption, “Something is M,” establishes
part of the first, namely, the hypothetical antecedent, whether as regards
existence (e.g. “Something is an angel”), or as regards fact (e.g. “This
existing man is wise”). These very different relations of premises are
obliterated by Sigwart’s false reduction of categorical universals to
hypotheticals. But even Sigwart’s errors are outdone by Lotze, who not only
reduces “Every M is P” so “If S is M, S is P,” but proceeds to reduce this

hypothetical to the disjunctive, “If S is M, S is P1 or P2 or P3,” and finds fault
with the Aristotelian syllogism because it contents itself with inferring “S is
P” without showing what P. Now there are occasions when we want to
reason in this disjunctive manner, to consider whether S is P1 or P2 or P3, and
to conclude that “S is a particular P”; but ordinarily all we want to know is
that “S is P”; e.g. in arithmetic, that 2 + 2 are 4, not any particular 4, and in
life that all our contemporaries must die, without enumerating all their
particular sorts of deaths. Lotze’s mistake is the same as that of Hamilton
about the quantification of the predicate, and that of those symbolists who
held that reasoning ought always to exhaust all alternatives by equations. It is
the mistake of exaggerating exceptional into normal forms of thought, and
ignoring the principle that a rational being thinks only to the point.

2. Quasi-syllogisms.—Besides reconstructions of the syllogistic fabric, we
find in recent logic attempts to extend the figures of the syllogism beyond the
syllogistic rules. An old error that we may have a valid syllogism from
merely negative premises (ex omnibus negativis), long ago answered by
Alexander and Boethius, is now revived by Lotze, Jevons and Bradley, who
do not perceive that the supposed second negative is really an affirmative
containing a “not” which can only be carried through the syllogism by
separating it from the copula and attaching it to one of the extremes, thus:—

The just are not unhappy (negative).
The just are not-recognized (affirmative).
∴ Some not-recognized are not unhappy (negative).

Here the minor being the infinite term “not-recognized” in the conclusion,
must be the same term also in the minor premise. Schuppe, however, who is
a fertile creator of quasi-syllogisms, has managed to invent some examples
from two negative premises of a different kind:—

(1) (2) (3)

No M is P. No M is P. No P is M.
S is not P. S is not M. S is not M.
∴ Neither S nor M is P. ∴ S may be P. ∴ S may be P.

But (1) concludes with a mere repetition, (2) and (3) with a contingent “may
be,” which, as Aristotle says, also “may not be,” and therefore nihil certo
colligitur. The same answer applies to Schuppe’s supposed syllogisms from
two particular premises:—

(1) (2)
Some M is P. Some M is P.
Some S is M. Some M is S.
∴ Some S may be P. ∴ Some S may be P.

The only difference between these and the previous examples (2) and (3) is
that, while those break the rule against two negative premises, these break
that against undistributed middle. Equally fallacious are two other attempts
of Schuppe to produce syllogisms from invalid moods:—

(1) 1st Fig. (2) 2nd Fig.
All M is P. P is M.
No S is M. S is M.
∴ S may be P. ∴ S is partially identical with P.

In the first the fallacy is the indifferent contingency of the conclusion caused
by the non-sequitur from a negative premise to an affirmative conclusion;
while the second is either a mere repetition of the premises if the conclusion
means “S is like P in being M,” or, if it means “S is P,” a non-sequitur on
account of the undistributed middle. It must not be thought that this trifling
with logical rules has no effect. The last supposed syllogism, namely, that
having two affirmative premises and entailing an undistributed middle in the
second figure, is accepted by Wundt under the title “Inference by
Comparison” (Vergleichungsschluss), and is supposed by him to be useful for

abstraction and subsidiary to induction, and by Bosanquet to be useful for
analogy. Wundt, for example, proposes the following premises:—

Gold is a shining, fusible, ductile, simple body.
Metals are shining, fusible, ductile, simple bodies.

But to say from these premises, “Gold and metal are similar in what is
signified by the middle term,” is a mere repetition of the premises; to say,
further, that “Gold may be a metal” is a non-sequitur, because, the middle
being undistributed, the logical conclusion is the contingent “Gold may or
may not be a metal,” which leaves the question quite open, and therefore
there is no syllogism. Wundt, who is again followed by Bosanquet, also
supposes another syllogism in the third figure, under the title of “Inference
by Connexion” (Verbindungsschluss), to be useful for induction. He
proposes, for example, the following premises:—

Gold, silver, copper, lead, are fusible.
Gold, silver, copper, lead, are metals.

Here there is no syllogistic fallacy in the premises; but the question is what
syllogistic conclusion can be drawn, and there is only one which follows
without an illicit process of the minor, namely, “Some metals are fusible.”
The moment we stir a step further with Wundt m the direction of a more
general conclusion (ein allgemeinerer Satz), we cannot infer from the
premises the conclusion desired by Wundt, “Metals and fusible are
connected”; nor can we infer “All metals are fusible,” nor “Metals are
fusible,” nor “Metals may be fusible,” nor “All metals may be fusible,” nor
any assertory conclusion, determinate or indeterminate, but the indifferent
contingent, “All metals may or may not be fusible,” which leaves the
question undecided, so that there is no syllogism. We do not mean that in
Wundt’s supposed “inferences of relation by comparison and connexion” the
premises are of no further use; but those of the first kind are of no syllogistic

use in the second figure, and those of the second kind of no syllogistic use
beyond particular conclusions in the third figure. What they really are in the
inferences proposed by Wundt is not premises for syllogism, but data for
induction parading as syllogism. We must pass the same sentence on Lotze’s
attempt to extend the second figure of the syllogism for inductive purposes,
thus:—

S is M.
Q is M.
R is M.
∴ Every Σ, which is common to S, Q, R, is M.

We could not have a more flagrant abuse of the rule Ne esto plus minusque in
conclusione quam in praemissis. As we see from Lotze’s own defence, the
conclusion cannot be drawn without another premise or premises to the
effect that “S, Q, R, are Σ, and Σ is the one real subject of M.” But how is all
this to be got into the second figure? Again, Wundt and B. Erdmann propose
new moods of syllogism with convertible premises, containing definitions
and equations. Wundt’s Logic has the following forms:—

(1) 1st Fig. (2) 2nd Fig. (3) 3rd Fig.
Only M is P. x = y. y = x.
No S is M. z = y. y = z.
∴ No S is P. ∴ x = z. ∴ x = z.

Now, there is no doubt that, especially in mathematical equations, universal
conclusions are obtainable from convertible premises expressed in these
ways. But the question is how the premises must be thought, and they must
be thought in the converse way to produce a logical conclusion. Thus, we
must think in (1) “All P is M” to avoid illicit process of the major, in (2) “All
y is z” to avoid undistributed middle, in (3) “All x is y” to avoid illicit
process of the minor. Indeed, it is the very essence of a convertible judgment
to think it in both orders, and especially to think it in the order necessary to

an inference from it. Accordingly, however expressed, the syllogisms quoted
above are, as thought, ordinary syllogisms, (1) being Camestres in the second
figure, (2) and (3) Barbara in the first figure. Aristotle, indeed, was as well
aware as German logicians of the force of convertible premises; but he was
also aware that they require no special syllogisms, and made it a point that,
in a syllogism from a definition, the definition is the middle, and the
definitum the major in a convertible major premise of Barbara in the first
figure, e.g.:—

The interposition of an opaque body is (essentially) deprivation of light.
The moon suffers the interposition of the opaque earth.
∴ The moon suffers deprivation of light.

It is the same with all the recent attempts to extend the syllogism beyond
its rules, which are not liable to exceptions, because they follow from the
nature of syllogistic inference from universal to particular. To give the name
of syllogism to inferences which infringe the general rules against
undistributed middle, illicit process, two negative premises, non-sequitur
from negative to affirmative, and the introduction of what is not in the
premises into the conclusion, and which consequently infringe the special
rules against affirmative conclusions in the second figure, and against
universal conclusions in the third figure, is to open the door to fallacy, and at
best to confuse the syllogism with other kinds of inference, without enabling
us to understand any one kind.

3. Analytic and Synthetic Deduction.—Alexander the Commentator
defined synthesis as a progress from principles to consequences, analysis as a
regress from consequences to principles; and Latin logicians preserved the
same distinction between the progressus a principiis ad principiata, and the
regressus a principiatis ad principia. No distinction is more vital in the logic
of inference in general and of scientific inference in particular; and yet none
has been so little understood, because, though analysis is the more usual

order of discovery, synthesis is that of instruction, and therefore, by
becoming more familiar, tends to replace and obscure the previous analysis.
The distinction, however, did not escape Aristotle, who saw that a
progressive syllogism can be reversed thus:—

1. Progression. 2. Regression.
(1) (2)
All M is P. All P is M. All S is P.
All S is M. All S is P. All M is S.
∴ All S is P. ∴ All S is M. ∴ All M is P.

Proceeding from one order to the other, by converting one of the premises,
and substituting the conclusion as premise for the other premise, so as to
deduce the latter as conclusion, is what he calls circular inference; and he
remarked that the process is fallacious unless it contains propositions which
are convertible, as in mathematical equations. Further, he perceived that the
difference between the progressive and regressive orders extends from
mathematics to physics, and that there are two kinds of syllogism: one
progressing a priori from real ground to consequent fact (ὁ τοῦ διότι
συλλογισμός), and the other regressing a posteriori from consequent fact to
real ground (ὁ τοῦ ὄτι συλλογισμός). For example, as he says, the
sphericity of the moon is the real ground of the fact of its light waxing; but
we can deduce either from the other, as follows:—

1. Progression. 2. Regression.
What is spherical waxes. What waxes is spherical.
The moon is spherical. The moon waxes.
∴ The moon waxes. ∴ The moon is spherical.

These two kinds of syllogism are synthesis and analysis in the ancient sense.
Deduction is analysis when it is regressive from consequence to real ground,
as when we start from the proposition that the angles of a triangle are equal
to two right angles and deduce analytically that therefore (1) they are equal

to equal angles made by a straight line standing on another straight line, and
(2) such equal angles are two right angles. Deduction is synthesis when it is
progressive from real ground to consequence, as when we start from these
two results of analysis as principles and deduce synthetically the proposition
that therefore the angles of a triangle are equal to two right angles, in the
order familiar to the student of Euclid. But the full value of the ancient
theory of these processes cannot be appreciated until we recognize that as
Aristotle planned them Newton used them. Much of the Principia consists of
synthetical deductions from definitions and axioms. But the discovery of the
centripetal force of the planets to the sun is an analytic deduction from the
facts of their motion discovered by Kepler to their real ground, and is so
stated by Newton in the first regressive order of Aristotle—P-M, S-P, S-M.
Newton did indeed first show synthetically what kind of motions by
mechanical laws have their ground in a centripetal force varying inversely as
the square of the distance (all P is M); but his next step was, not to deduce
synthetically the planetary motions, but to make a new start from the
planetary motions as facts established by Kepler’s laws and as examples of
the kind of motions in question (all S is P); and then, by combining these two
premises, one mechanical and the other astronomical, he analytically
deduced that these facts of planetary motion have their ground in a
centripetal force varying inversely as the squares of the distances of the
planets from the sun (all S is M). (See Principia I. prop. 2; 4 coroll. 6; III.
Phaenomena, 4-5; prop. 2.) What Newton did, in short, was to prove by
analysis that the planets, revolving by Kepler’s astronomical laws round the
sun, have motions such as by mechanical laws are consequences of a
centripetal force to the sun. This done, as the major is convertible, the
analytic order—P-M, S-P, S-M—was easily inverted into the synthetic order
—M-P, S-M, S-P; and in this progressive order the deduction as now taught
begins with the centripetal force of the sun as real ground, and deduces the
facts of planetary motion as consequences. Thereupon the Newtonian
analysis which preceded this synthesis, became forgotten; until at last Mill in

his Logic, neglecting the Principia, had the temerity to distort Newton’s
discovery, which was really a pure example of analytic deduction, into a
mere hypothetical deduction; as if the author of the saying “Hypotheses non
fingo” started from the hypothesis of a centripetal force to the sun, and
thence deductively explained the facts of planetary motion, which
reciprocally verified the hypothesis. This gross misrepresentation has made
hypothesis a kind of logical fashion. Worse still, Jevons proceeded to confuse
analytic deduction from consequence to ground with hypothetical deduction
from ground to consequence under the common term “inverse deduction.”
Wundt attempts, but in vain, to make a compromise between the old and the
new. He re-defines analysis in the very opposite way to the ancients; whereas
they defined it as a regressive process from consequence to ground,
according to Wundt it is a progressive process of taking for granted a
proposition and deducing a consequence, which being true verifies the
proposition. He then divides it into two species: one categorical, the other
hypothetical. By the categorical he means the ancient analysis from a given
proposition to more general propositions. By the hypothetical he means the
new-fangled analysis from a given proposition to more particular
propositions, i.e. from a hypothesis to consequent facts. But his account of
the first is imperfect, because in ancient analysis the more general
propositions, with which it concludes, are not mere consequences, but the
real grounds of the given proposition; while his addition of the second
reduces the nature of analysis to the utmost confusion, because hypothetical
deduction is progressive from hypothesis to consequent facts whereas
analysis is regressive from consequent facts to real ground. There is indeed a
sense in which all inference is from ground to consequence, because it is
from logical ground (principium cognoscendi) to logical consequence. But in
the sense in which deductive analysis is opposed to deductive synthesis,
analysis is deduction from real consequence as logical ground (principiatum
as principium cognoscendi) to real ground (principium essendi), e.g. from the
consequential facts of planetary motion to their real ground, i.e. centripetal

force to the sun. Hence Sigwart is undoubtedly right in distinguishing
analysis from hypothetical deduction, for which he proposes the name
“reduction.” We have only further to add that many scientific discoveries
about sound, heat, light, colour and so forth, which it is the fashion to
represent as hypotheses to explain facts, are really analytical deductions from
the facts to their real grounds in accordance with mechanical laws. Recent
logic does scant justice to scientific analysis.

4. Induction.—As induction is the process from particulars to universals, it
might have been thought that it would always have been opposed to
syllogism, in which one of the rules is against using particular premises to
draw universal conclusions. Yet such is the passion for one type that from
Aristotle’s time till now constant attempts have been made to reduce
induction to syllogism. Aristotle himself invented an inductive syllogism in
which the major (P) is to be referred to the middle (M) by means of the
minor (S), thus:—

A, B, C magnets (S) attract iron (P).
A, B, C magnets (S) are all magnets whatever (M).
∴ All magnets whatever (M) attract iron (P).

As the second premise is supposed to be convertible, he reduced the
inductive to a deductive syllogism as follows:—

Every S is P. Every S is P.
Every S is M (convertibly). Every M is S.
∴ Every M is P. ∴ Every M is P.

In the reduced form the inductive syllogism was described by Aldrich as
“Syllogismus in Barbara cujus minor (i.e. every M is S) reticetur.” Whately,
on the other hand, proposed an inductive syllogism with the major
suppressed, that is, instead of the minor premise above, he supposed a major
premise, “Whatever belongs to A, B, C magnets belongs to all.” Mill

thereupon supposed a still more general premise, an assumption of the
uniformity of nature. Since Mill’s time, however, the logic of induction tends
to revert towards syllogisms more like that of Aristotle. Jevons supposed
induction to be inverse deduction, distinguished from direct deduction as
analysis from synthesis, e.g. as division from multiplication; but he really
meant that it is a deduction from a hypothesis of the law of a cause to
particular effects which, being true, verify the hypothesis. Sigwart declares
himself in agreement with Jevons; except that, being aware of the difference
between hypothetical deduction and mathematical analysis, and seeing that,
whereas analysis (e.g. in division) leads to certain conclusions, hypothetical
deduction is not certain of the hypothesis, he arrives at the more definite
view that induction is not analysis proper but hypothetical deduction, or
“reduction,” as he proposes to call it. Reduction he defines as “the framing of
possible premises for given propositions, or the construction of a syllogism
when the conclusion and one premise is given.” On this view induction
becomes a reduction in the form: all M is P (hypothesis), S is M (given), ∴ S
is P (given). The views of Jevons and Sigwart are in agreement in two main
points. According to both, induction, instead of inferring from A, B, C
magnets the conclusion “Therefore all magnets attract iron,” infers from the
hypothesis, “Let every magnet attract iron,” to A, B, C magnets, whose given
attraction verifies the hypothesis. According to both, again, the hypothesis of
a law with which the process starts contains more than is present in the
particular data: according to Jevons, it is the hypothesis of a law of a cause
from which induction deduces particular effects; and according to Sigwart, it
is a hypothesis of the ground from which the particular data necessarily
follow according to universal laws. Lastly, Wundt’s view is an interesting
piece of eclecticism, for he supposes that induction begins in the form of
Aristotle’s inductive syllogism, S-P, S-M, M-P, and becomes an inductive
method in the form of Jevons’s inverse deduction, or hypothetical deduction,
or analysis, M-P, S-M, S-P. In detail, he supposes that, while an “inference by
comparison,” which he erroneously calls an affirmative syllogism in the

second figure, is preliminary to induction, a second “inference by
connexion,” which he erroneously calls a syllogism in the third figure with
an indeterminate conclusion, is the inductive syllogism itself. This is like
Aristotle’s inductive syllogism in the arrangement of terms; but, while on the
one hand Aristotle did not, like Wundt, confuse it with the third figure, on the
other hand Wundt does not, like Aristotle, suppose it to be practicable to get
inductive data so wide as the convertible premise, “All S is M, and all M is
S,” which would at once establish the conclusion, “All M is P.” Wundt’s
point is that the conclusion of the inductive syllogism is neither so much as
all, nor so little as some, but rather the indeterminate “M and P are
connected.” The question therefore arises, how we are to discover “All M is
P,” and this question Wundt answers by adding an inductive method, which
involves inverting the inductive syllogism in the style of Aristotle into a
deductive syllogism from a hypothesis in the style of Jevons, thus:—

(1) (2)
S is P. Every M is P.
S is M. S is M.
∴ M and P are connected. ∴ S is P.

He agrees with Jevons in calling this second syllogism analytical deduction,
and with Jevons and Sigwart in calling it hypothetical deduction. It is, in fact,
a common point of Jevons, Sigwart and Wundt that the universal is not really
a conclusion inferred from given particulars, but a hypothetical major
premise from which given particulars are inferred, and that this major
contains presuppositions of causation not contained in the particulars.

It is noticeable that Wundt quotes Newton’s discovery of the centripetal
force of the planets to the sun as an instance of this supposed hypothetical,
analytic, inductive method; as if Newton’s analysis were a hypothesis of the
centripetal force to the sun, a deduction of the given facts of planetary
motion, and a verification of the hypothesis by the given facts, and as if such

a process of hypothetical deduction could be identical with either analysis or
induction. The abuse of this instance of Newtonian analysis betrays the
whole origin of the current confusion of induction with deduction. One
confusion has led to another. Mill confused Newton’s analytical deduction
with hypothetical deduction; and thereupon Jevons confused induction with
both. The result is that both Sigwart and Wundt transform the inductive
process of adducing particular examples to induce a universal law into a
deductive process of presupposing a universal law as a ground to deduce
particular consequences. But we can easily extricate ourselves from these
confusions by comparing induction with different kinds of deduction. The
point about induction is that it starts from experience, and that, though in
most classes we can experience only some particulars individually, yet we
infer all. Hence induction cannot be reduced to Aristotle’s inductive
syllogism, because experience cannot give the convertible premise, “Every S
is M, and every M is S”; that “All A, B, C are magnets” is, but that “All
magnets are A, B, C” is not, a fact of experience. For the same reason
induction cannot be reduced to analytical deduction of the second kind in the
form, S-P, M-S, ∴ M-P; because, though both end in a universal conclusion,
the limits of experience prevent induction from such inference as:—

Every experienced magnet attracts iron.
Every magnet whatever is every experienced magnet.
∴ Every magnet whatever attracts iron.

Still less can induction be reduced to analytical deduction of the first kind in
the form—P-M, S-P, ∴ S-M, of which Newton has left so conspicuous an
example in his Principia. As the example shows, that analytic process starts
from the scientific knowledge of a universal and convertible law (every M is
P, and every P is M), e.g. a mechanical law of all centripetal force, and ends
in a particular application, e.g. this centripetal force of planets to the sun. But
induction cannot start from a known law. Hence it is that Jevons, followed by
Sigwart and Wundt, reduces it to deduction from a hypothesis in the form

“Let every M be P, S is M, ∴ S is P.” There is a superficial resemblance
between induction and this hypothetical deduction. Both in a way use given
particulars as evidence. But in induction the given particulars are the
evidence by which we discover the universal, e.g. particular magnets
attracting iron are the origin of an inference that all do; in hypothetical
deduction, the universal is the evidence by which we explain the given
particulars, as when we suppose undulating aether to explain the facts of heat
and light. In the former process, the given particulars are the data from which
we infer the universal; in the latter, they are only the consequent facts by
which we verify it. Or rather, there are two uses of induction: inductive
discovery before deduction, and inductive verification after deduction. But
neither use of induction is the same as the deduction itself: the former
precedes, the latter follows it. Lastly, the theory of Mill, though frequently
adopted, e.g. by B. Erdmann, need not detain us long. Most inductions are
made without any assumption of the uniformity of nature; for, whether it is
itself induced, or a priori or postulated, this like every assumption is a
judgment, and most men are incapable of judgment on so universal a scale,
when they are quite capable of induction. The fact is that the uniformity of
nature stands to induction as the axioms of syllogism do to syllogism; they
are not premises, but conditions of inference, which ordinary men use
spontaneously, as was pointed out in Physical Realism, and afterwards in
Venn’s Empirical Logic. The axiom of contradiction is not a major premise
of a judgment: the dictum de omni et nullo is not a major premise of a
syllogism: the principle of uniformity is not a major premise of an induction.
Induction, in fact, is no species of deduction; they are opposite processes, as
Aristotle regarded them except in the one passage where he was reducing the
former to the latter, and as Bacon always regarded them. But it is easy to
confuse them by mistaking examples of deduction for inductions. Thus
Whewell mistook Kepler’s inference that Mars moves in an ellipse for an
induction, though it required the combination of Tycho’s and Kepler’s
observations, as a minor, with the laws of conic sections discovered by the

Greeks, as a major, premise. Jevons, in his Principles of Science, constantly
makes the same sort of mistake. For example, the inference from the
similarity between solar spectra and the spectra of various gases on the earth
to the existence of similar gases in the sun, is called by him an induction; but
it really is an analytical deduction from effect to cause, thus:—

Such and such spectra are effects of various gases.
Solar spectra are such spectra.
∴ Solar spectra are effects of those gases.

In the same way, to infer a machine from hearing the regular tick of a clock,
to infer a player from finding a pack of cards arranged in suits, to infer a
human origin of stone implements, and all such inferences from patent
effects to latent causes, though they appear to Jevons to be typical inductions,
are really deductions which, besides the minor premise stating the particular
effects, require a major premise discovered by a previous induction and
stating the general kind of effects of a general kind of cause. B. Erdmann,
again, has invented an induction from particular predicates to a totality of
predicates which he calls “ergänzende Induction,” giving as an example,
“This body has the colour, extensibility and specific gravity of magnesium;
therefore it is magnesium.” But this inference contains the tacit major, “What
has a given colour, &c., is magnesium,” and is a syllogism of recognition. A
deduction is often like an induction, in inferring from particulars; the
difference is that deduction combines a law in the major with the particulars
in the minor premise, and infers syllogistically that the particulars of the
minor have the predicate of the major premise, whereas induction uses the
particulars simply as instances to generalize a law. An infallible sign of an
induction is that the subject and predicate of the universal conclusion are
merely those of the particular instances generalized; e.g. “These magnets
attract iron, ∴ all do.”

This brings us to another source of error. As we have seen, Jevons, Sigwart
and Wundt all think that induction contains a belief in causation, in a cause,
or ground, which is not present in the particular facts of experience, but is
contributed by a hypothesis added as a major premise to the particulars in
order to explain them by the cause or ground. Not so; when an induction is
causal, the particular instances are already beliefs in particular causes, e.g.
“My right hand is exerting pressure reciprocally with my left,” “A, B, C
magnets attract iron”; and the problem is to generalize these causes, not to
introduce them. Induction is not introduction. It would make no difference to
the form of induction, if, as Kant thought, the notion of causality is a priori;
for even Kant thought that it is already contained in experience. But whether
Kant be right or wrong, Wundt and his school are decidedly wrong in
supposing “supplementary notions which are not contained in experience
itself, but are gained by a process of logical treatment of this experience”; as
if our behalf in causality could be neither a posteriori nor a priori, but beyond
experience wake up in a hypothetical major premise of induction. Really, we
first experience that particular causes have particular effects; then induce that
causes similar to those have effects similar to these; finally, deduce that when
a particular cause of the kind occurs it has a particular effect of the kind by
synthetic deduction, and that when a particular effect of the kind occurs it has
a particular cause of the kind by analytic deduction with a convertible
premise, as when Newton from planetary motions, like terrestrial motions,
analytically deduced a centripetal force to the sun like centripetal forces to
the earth. Moreover, causal induction is itself both synthetic and analytic:
according as experiment combines elements into a compound, or resolves a
compound into elements, it is the origin of a synthetic or an analytic
generalization. Not, however, that all induction is causal; but where it is not,
there is still less reason for making it a deduction from hypothesis. When
from the fact that the many crows in our experience are black, we induce the
probability that all crows whatever are black, the belief in the particulars is
quite independent of this universal. How then can this universal be called, as

Sigwart, for example, calls it, the ground from which these particulars
follow? I do not believe that the crows I have seen are black because all
crows are black, but vice versa. Sigwart simply inverts the order of our
knowledge. In all induction, as Aristotle said, the particulars are the
evidence, or ground of our knowledge (principium cognoscendi), of the
universal. In causal induction, the particulars further contain the cause, or
ground of the being (principium essendi), of the effect, as well as the ground
of our inducing the law. In all induction the universal is the conclusion, in
none a major premise, and in none the ground of either the being or the
knowing of the particulars. Induction is generalization. It is not syllogism in
the form of Aristotle’s or Wundt’s inductive syllogism, because, though
starting only from some particulars, it concludes with a universal; it is not
syllogism in the form called inverse deduction by Jevons, reduction by
Sigwart, inductive method by Wundt, because it often uses particular facts of
causation to infer universal laws of causation; it is not syllogism in the form
of Mill’s syllogism from a belief in uniformity of nature, because few men
have believed in uniformity, but all have induced from particulars to
universals. Bacon alone was right in altogether opposing induction to
syllogism, and in finding inductive rules for the inductive process from
particular instances of presence, absence in similar circumstances, and
comparison.

5. Inference in General.—There are, as we have seen (ad init.), three types
—syllogism, induction and analogy. Different as they are, the three kinds
have something in common: first, they are all processes from similar to
similar; secondly, they all consist in combining two judgments so as to cause
a third, whether expressed in so many propositions or not; thirdly, as a
judgment is a belief in being, they all proceed from premises which are
beliefs in being to a conclusion which is a belief in being. Nevertheless,
simple as this account appears, it is opposed in every point to recent logic. In
the first place, the point of Bradley’s logic is that “similarity is not a principle

which works. What operates is identity, and that identity is a universal.” This
view makes inference easy: induction is all over before it begins; for,
according to Bradley, “every one of the instances is already a universal
proposition; and it is not a particular fact or phenomenon at all,” so that the
moment you observe that this magnet attracts iron, you ipso facto know that
every magnet does so, and all that remains for deduction is to identify a
second magnet as the same with the first, and conclude that it attracts iron. In
dealing with Bradley’s works we feel inclined to repeat what Aristotle says
of the discourses of Socrates: they all exhibit excellence, cleverness, novelty
and inquiry, but their truth is a difficult matter; and the Socratic paradox that
virtue is knowledge is not more difficult than the Bradleian paradox that as
two different things are the same, inference is identification. The basis of
Bradley’s logic is the fallacious dialectic of Hegel’s metaphysics, founded on
the supposition that two things, which are different, but have something in
common, are the same. For example, according to Hegel, being and not-
being are both indeterminate and therefore the same. “If,” says Bradley, “A
and B, for instance, both have lungs or gills, they are so far the same.” The
answer to Hegel is that being and not-being are at most similarly
indeterminate, and to Bradley that each animal has its own different lungs,
whereby they are only similar. If they were the same, then in descending, two
things, one of which has healthy and the other diseased lungs, would be the
same; and in ascending, two things, one of which has lungs and the other has
not, but both of which have life, e.g. plants and animals, would be so far the
same. There would be no limit to identity either downwards or upwards; so
that a man would be the same as a man-of-war, and all things would be the
same thing, and not different parts of one universe. But a thing which has
healthy lungs and a thing which has diseased lungs are only similar
individuals numerically different. Each individual thing is the same only with
itself, although related to other things; and each individual of a class has its
own individual, though similar, attributes. The consequence of this true
metaphysics to logic is twofold: on the one hand, one singular or particular

judgment, e.g. “this magnet attracts iron,” is not another, e.g. “that magnet
attracts iron,” and neither is universal; on the other hand, a universal
judgment, e.g. “every magnet attracts iron,” means, distributively, that each
individual magnet exerts its individual attraction, though it is similar to other
magnets exerting similar attractions. A universal is not “one identical point,”
but one distributive whole. Hence in a syllogism, a middle term, e.g.
magnets, is “absolutely the same,” not in the sense of “one identical point”
making each individual the same as any other, as Bradley supposes, but only
in the sense of one whole class, or total of many similar individuals, e.g.
magnets, each of which is separately though similarly a magnet, not magnet
in general. Hence also induction is a real process, because, when we know
that this individual magnet attracts iron, we are very far from knowing that
all alike do so similarly; and the question of inductive logic, how we get
from some similars to all similars, remains, as before, a difficulty, but not to
be solved by the fallacy that inference is identification.

Secondly, a subordinate point in Bradley’s logic is that there are inferences
which are not syllogisms; and this is true. But when he goes on to propose, as
a complete independent inference, “A is to the right of B, B is to the right of
C, therefore A is to the right of C,” he confuses two different operations.
When A, B and C are objects of sense, their relative positions are matters, not
of inference, but of observation; when they are not, there is an inference, but
a syllogistic inference with a major premise induced from previous
observations, “whenever of three things the first is to the right of the second,
and the second to the right of the third, the first is to the right of the third.”
To reply that this universal judgment is not expressed, or that its expression
is cumbrous, is no answer, because, whether expressed or not, it is required
for the thought. As Aristotle puts it, the syllogism is directed “not to the
outer, but to the inner discourse,” or as we should say, not to the expression
but to the thought, not to the proposition but to the judgment, and to the
inference not verbally but mentally. Bradley seems to suppose that the major

premise of a syllogism must be explicit, or else is nothing at all. But it is
often thought without being expressed, and to judge the syllogism by its
mere explicit expression is to commit an ignoratio elenchi; for it has been
known all along that we express less than we think, and the very purpose of
syllogistic logic is to analyse the whole thought necessary to the conclusion.
In this syllogistic analysis two points must always be considered: one, that
we usually use premises in thought which we do not express; and the other,
that we sometimes use them unconsciously, and therefore infer and reason
unconsciously, in the manner excellently described by Zeller in his Vorträge,
iii. pp. 249-255. Inference is a deeper thinking process from judgments to
judgment, which only occasionally and partially emerges in the linguistic
process from propositions to proposition. We may now then reassert two
points about inference against Bradley’s logic: the first, that it is a process
from similar to similar, and not a process of identification, because two
different things are not at all the same thing; the second, that it is the mental
process from judgments to judgment rather than the linguistic process from
propositions to proposition, because, besides the judgments expressed in
propositions, it requires judgments which are not always expressed, and are
sometimes even unconscious.

Our third point is that, as a process of judgments, inference is a process of
concluding from two beliefs in being to another belief in being, and not an
ideal construction, because a judgment does not always require ideas, but is
always a belief about things, existing or not. This point is challenged by all
the many ideal theories of judgment already quoted. If, for example,
judgment were an analysis of an aggregate idea as Wundt supposes, it would
certainly be true with him to conclude that “as judgment is an immediate,
inference is a mediate, reference of the members of an aggregate of ideas to
one another.” But really a judgment is a belief that something, existing, or
thinkable, or nameable or what not, is (or is not) determined; and inference is
a process from and to such beliefs in being. Hence the fallacy of those who,

like Bosanquet, or like Paulsen in his Einleitung in die Philosophie, represent
the realistic theory of inference as if it meant that knowledge starts from
ideas and then infers that ideas are copies of things, and who then object,
rightly enough, that we could not in that case compare the copy with the
original, but only be able to infer from idea to idea. But there is another
realism which holds that inference is a process neither from ideas to ideas,
nor from ideas to things, but from beliefs to beliefs, from judgments about
things in the premises to judgments about similar things in the conclusion.
Logical inference never goes through the impossible process of premising
nothing but ideas, and concluding that ideas are copies of things. Moreover,
as we have shown, our primary judgments of sense are beliefs founded on
sensations without requiring ideas, and are beliefs, not merely that something
is determined, but that it is determined as existing; and, accordingly, our
primary inferences from these sensory judgments of existence are inferences
that other things beyond sense are similarly determined as existing. First
press your lips together and then press a pen between them: you will not be
conscious of perceiving any ideas: you will be conscious first of perceiving
one existing lip exerting pressure reciprocally with the other existing lip;
then, on putting the pen between your lips, of perceiving each lip similarly
exerting pressure, but not with the other; and consequently of inferring that
each existing lip is exerting pressure reciprocally with another existing body,
the pen. Inference then, though it is accompanied by ideas, is not an ideal
construction, nor a process from idea to idea, nor a process from idea to
thing, but a process from direct to indirect beliefs in things, and originally in
existing things. Logic cannot, it is true, decide what these things are, nor
what the senses know about them, without appealing to metaphysics and
psychology. But, as the science of inference, it can make sure that inference,
on the one hand, starts from sensory judgments about sensible things and
logically proceeds to inferential judgments about similar things beyond
sense, and, on the other hand, cannot logically go beyond the similar. These
are the limits within which logical inference works, because its nature

essentially consists in proceeding from two judgments to another about
similar things, existing or not.

6. Truth.—Finally, though sensory judgment is always true of its sensible
object, inferential judgments are not always true, but are true so far as they
are logically inferred, however indirectly, from sense; and knowledge
consists of sense, memory after sense and logical inference from sense,
which, we must remember, is not merely the outer sense of our five senses,
but also the inner sense of ourselves as conscious thinking persons. We come
then at last to the old question—what is truth? Truth proper, as Aristotle said
in the Metaphysics, is in the mind: it is not being, but one’s signification of
being. Its requisites are that there are things to be known and powers of
knowing things. It is an attribute of judgments and derivatively of
propositions. That judgment is true which apprehends a thing as it is capable
of being known to be; and that proposition is true which so asserts the thing
to be. Or, to combine truth in thought and in speech, the true is what signifies
a thing as it is capable of being known. Secondarily, the thing itself is
ambiguously said to be true in the sense of being signified as it is. For
example, as I am weary and am conscious of being weary, my judgment and
proposition that I am weary are true because they signify what I am and
know myself to be by direct consciousness; and my being weary is
ambiguously said to be true because it is so signified. But it will be said that
Kant has proved that real truth, in the sense of the “agreement of knowledge
with the object,” is unattainable, because we could compare knowledge with
the object only by knowing both. Sigwart, indeed, adopting Kant’s argument,
concludes that we must be satisfied with consistency among the thoughts
which presuppose an existent; this, too, is the reason why he thinks that
induction is reduction, on the theory that we can show the necessary
consequence of the given particular, but that truth of fact is unattainable. But
Kant’s criticism and Sigwart’s corollary only derive plausibility from a false
definition of truth. Truth is not the agreement of knowledge with an object

beyond itself, and therefore ex hypothesi unknowable, but the agreement of
our judgments with the objects of our knowledge. A judgment is true
whenever it is a belief that a thing is determined as it is known to be by
sense, or by memory after sense, or by inference from sense, however
indirect the inference may be, and even when in the form of inference of
non-existence it extends consequently from primary to secondary judgments.
Thus the judgments “this sensible pressure exists,” “that sensible pressure
existed,” “other similar pressures exist,” “a conceivable centaur does not
exist but is a figment,” are all equally true, because they are in accordance
with one or other of these kinds of knowledge. Consequently, as knowledge
is attainable by sense, memory and inference, truth is also attainable,
because, though we cannot test what we know by something else, we can test
what we judge and assert by what we know. Not that all inference is
knowledge, but it is sometimes. The aim of logic in general is to find the
laws of all inference, which, so far as it obeys those laws, is always
consistent, but is true or false according to its data as well as its consistency;
and the aim of the special logic of knowledge is to find the laws of direct and
indirect inferences from sense, because as sense produces sensory judgments
which are always true of the sensible things actually perceived, inference
from sense produces inferential judgments which, so far as they are
consequent on sensory judgments, are always true of things similar to
sensible things, by the very consistency of inference, or, as we say, by parity
of reasoning. We return then to the old view of Aristotle, that truth is
believing in being; that sense is true of its immediate objects, and reasoning
from sense true of its mediate objects; and that logic is the science of
reasoning with a view to truth, or Logica est ars ratiocinandi, ut discernatur
verum a falso. All we aspire to add is that, in order to attain to real truth, we
must proceed gradually from sense, memory and experience through
analogical particular inference, to inductive and deductive universal
inference or reasoning. Logic is the science of all inference, beginning from
sense and ending in reason.

In conclusion, the logic of the last quarter of the 19th century may be said
to be animated by a spirit of inquiry, marred by a love of paradox and a
corresponding hatred of tradition. But we have found, on the whole, that
logical tradition rises superior to logical innovation. There are two old logics
which still remain indispensable, Aristotle’s Organon and Bacon’s Novum
Organum. If, and only if, the study of deductive logic begins with Aristotle,
and the study of inductive logic with Aristotle and Bacon, it will be
profitable to add the works of the following recent German and English
authors:—

Authorities.—J. Bergmann, Reine Logik (Berlin, 1879); Die
Grundprobleme der Logik (2nd ed., Berlin, 1895); B. Bosanquet, Logic
(Oxford, 1888); The Essentials of Logic (London, 1895); F. H. Bradley,
The Principles of Logic (London, 1883); F. Brentano, Psychologie vom
empirischen Standpunkte (Vienna, 1874); R. F. Clarke, Logic (London,
1889); W. L. Davidson, The Logic of Definition (London, 1885); E.
Dühring, Logik und Wissenschaftstheorie (Leipzig, 1878); B. Erdmann,
Logik (Halle, 1892); T. Fowler, Bacon’s Novum Organum, edited, with
introduction, notes, &c. (2nd ed., Oxford, 1889); T. H. Green, Lectures
on Logic, in Works, vol. iii. (London, 1886); J. G. Hibben, Inductive
Logic (Edinburgh and London, 1896); F. Hillebrand, Die neuen
Theorien der kategorischen Schlüsse (Vienna, 1891); L. T. Hobhouse,
The Theory of Knowledge (London, 1896); H. Hughes, The Theory of
Inference (London, 1894); E. Husserl, Logische Untersuchungen
(Halle, 1891, 1901); W. Jerusalem, Die Urtheilsfunction (Vienna and
Leipzig, 1895); W. Stanley Jevons, The Principles of Science (3rd ed.,
London, 1879); Studies in Deductive Logic (London, 1880); H. W. B.
Joseph, Introduction to Logic (1906); E. E. Constance Jones, Elements
of Logic (Edinburgh, 1890); G. H. Joyce, Principles of Logic (1908); J.
N. Keynes, Studies and Exercises in Formal Logic (2nd ed., London,
1887); F. A. Lange, Logische Studien (2nd ed., Leipzig, 1894); T.
Lipps, Grundzüge der Logik (Hamburg and Leipzig, 1893); R. H.
Lotze, Logik (2nd ed., Leipzig, 1881, English translation edited by B.
Bosanquet, Oxford, 1884); Grundzüge der Logik (Diktate) (3rd ed.,
Leipzig, 1891, English translation by G. T. Ladd, Boston, 1887);
Werner Luthe, Beiträge zur Logik (Berlin, 1872, 1877); Members of
Johns Hopkins University, Studies in Logic (edited by C. S. Peirce,
Boston, 1883); J. B. Meyer, Ueberweg’s System der Logik, fünfte
vermehrte Auflage (Bonn, 1882); Max Müller, Science of Thought
(London, 1887); Carveth Read, On the Theory of Logic (London,

1878); Logic, Deductive and Inductive (2nd ed., London, 1901); E.
Schröder, Vorlesungen über die Algebra der Logik (Leipzig, 1890,
1891, 1895); W. Schuppe, Erkenntnistheoretische Logik (Bonn, 1878);
Grundriss der Erkenntnistheorie und Logik (Berlin, 1894); R. Shute, A
Discourse on Truth (London, 1877); Alfred Sidgwick, Fallacies
(London, 1883); The Use of Words in Reasoning (London, 1901); C.
Sigwart, Logik (2nd ed., Freiburg-i.-Br. and Leipzig, 1889-1893,
English translation by Helen Dendy, London, 1895); K. Uphues,
Grundlehren der Logik (Breslau, 1883); J. Veitch, Institutes of Logic
(Edinburgh and London, 1885); J. Venn, Symbolic Logic (2nd ed.,
London, 1894); The Principles of Empirical or Inductive Logic
(London, 1889); J. Volkelt, Erfahren und Denken (Hamburg and
Leipzig, 1886); T. Welton, A Manual of Logic (London, 1891, 1896);
W. Windelband, Präludien (Freiburg-i.-Br., 1884); W. Wundt, Logik
(2nd ed., Stuttgart, 1893-1895). Text-books are not comprised in this
list. (T. Ca.)

II. History

Logic cannot dispense with the light afforded by its history so long as
counter-solutions of the same fundamental problems continue to hold the
field. A critical review of some of the chief types of logical theory, with a
view to determine development, needs no further justification.

Logic arose, at least for the Western world, in the golden age of Greek
speculation which culminated in Plato and Aristotle. There is an Indian
logic, it is true, but its priority is more than disputable. In any case no
influence upon Greek thought can be shown. The movement which ends in
the logic of Aristotle is demonstrably self-contained. When we have shaken
ourselves free of the prejudice that all stars are first seen in the East,

Oriental attempts at analysis of the structure of thought may be treated as
negligible.

It is with Aristotle that the bookish tradition begins to dominate the
evolution of logic. The technical perfection of the analysis which he offers
is, granted the circle of presuppositions within which it works, so decisive,
that what precedes, even Plato’s logic, is not unnaturally regarded as merely
preliminary and subsidiary to it. What follows is inevitably, whether
directly or indirectly, by sympathy or by antagonism, affected by the
Aristotelian tradition.

A. Greek Logic

i. Before Aristotle

Logic needs as its presuppositions that thought should distinguish itself
from things and from sense, that the problem of validity should be seen to
be raised in the field of thought itself, and that analysis of the structure of
thought should be recognized as the one way of solution.
The physical Thought is somewhat late in coming to self-
philosophers. consciousness. Implied in every contrast of principle and
fact, of rule and application, involved as we see after the
event, most decisively when we react correctly upon a world incorrectly
perceived, thought is yet not reflected on in the common experience. Its so-
called natural logic is only the potentiality of logic. The same thing is true
of the first stage of Greek philosophy. In seeking for a single material
principle underlying the multiplicity of phenomena, the first nature-
philosophers, Thales and the rest, did indeed raise the problem of the one
and the many, the endeavour to answer which must at last lead to logic. But
it is only from a point of view won by later speculation that it can be said
that they sought to determine the predicates of the single subject-reality, or

to establish the permanent subject of varied and varying predicates.1 The
direction of their inquiry is persistently outward. They hope to explain the
opposed appearance and reality wholly within the world of things, and
irrespective of the thought that thinks things. Their universal is still a
material one. The level of thought on which they move is still clearly pre-
logical. It is an advance on this when Heraclitus2 opposes to the eyes and
ears which are bad witnesses “for such as understand not their language” a
common something which we would do well to follow; or again when in
the incommensurability of the diagonal and side of a square the
Pythagoreans stumbled upon what was clearly neither thing nor image of
sense, but yet was endowed with meaning, and henceforth were
increasingly at home with symbol and formula. So far, however, it might
well be that thought, contradistinguished from sense with its illusions, was
itself infallible. A further step, then, was necessary, and it was taken at any
rate by the Eleatics, when they opposed their thought to the thought of
others, as the way of truth in contrast to the way of opinion. If Eleatic
thought stands over against Pythagorean thought as what is valid or
grounded against what is ungrounded or invalid, we are embarked upon
dialectic, or the debate in which thought is countered by thought. Claims to
a favourable verdict must now be substantiated in this field and in this field
alone. It was Zeno, the controversialist of the Eleatic school, who was
regarded in after times as the “discoverer” of dialectic.3

Zeno’s amazing skill in argumentation and his paradoxical
conclusions, particular and general, inaugurate a new era. “The
philosophical mind,” says waiter Pater,4 “will perhaps never be quite in
health, quite sane or natural again.” The give and take of thought had
by a swift transformation of values come by something more than its
own. Zeno’s paradoxes, notably, for example, the puzzle of Achilles
and the Tortoise, are still capable of amusing the modern world. In his

own age they found him imitators. And there follows the sophistic
movement.

The sophists have other claims to consideration than their service to
the development of logic. In the history of the origins of logic the
sophistic age is simply the age of the free play of thought in which men
were aware that in a sense anything can be debated
The Sophists. and not yet aware of the sense in which all things
cannot be so. It is the age of discussion used as a
universal solvent, before it has been brought to book by a deliberate
unfolding of the principles of the structure of thought determining and
limiting the movement of thought itself. The sophists furthered the
transition from dialectic to logic in two ways. In the first place they
made it possible. Incessant questioning leads to answers. Hair-splitting,
even when mischievous in intent, leads to distinctions of value.
Paradoxical insistence on the accidents of speech-forms and thought-
forms leads in the end to perception of the essentials. Secondly they
made it necessary. The spirit of debate run riot evokes a counter-spirit
to order and control it. The result is a self-limiting dialectic. This
higher dialectic is a logic. It is no accident that the first of the
philosophical sophists, Gorgias, on the one hand, is Eleatic in his
affinities, and on the other raises in the characteristic formula of his
intellectual nihilism5 issues which are as much logical and
epistemological as ontological. The meaning of the copula and the
relation of thoughts to the objects of which they are the thoughts are as
much involved as the nature of being. It is equally no accident that the
name of Protagoras is to be connected, in Plato’s view at least, with the
rival school of Heracliteans. The problems raised by the relativism of
Protagoras are no less fundamentally problems of the nature of
knowledge and of the structure of thought. The Theaetetus indeed, in
which Plato essays to deal with them, is in the broad sense of the word

logical, the first distinctively logical treatise that has come down to us.
Other sophists, of course, with more practical interests, or of humbler
attainments, were content to move on a lower plane of philosophical
speculation. As presented to us, for example, in Plato’s surely not
altogether hostile caricature in the Euthydemus, they mark the
intellectual preparation for, and the moral need for, the advance of the
next generation.

Among the pioneers of the sophistic age Socrates stands apart. He
has no other instrument than the dialectic of his compeers, and he is as
far off as the rest from a criticism of the instrument, but he uses it
differently and with a difference of aim. He
Socrates. construes the give and take of the debate-game with
extreme rigour. The rhetorical element must be
exorcised. The set harangue of teacher to pupil, in which steps in
argument are slurred and the semblance of co-inquiry is rendered
nugatory, must be eliminated. The interlocutors must in truth render an
account under the stimulus of organized heckling from their equals or
superiors in debating ability. And the aim is heuristic, though often
enough the search ends in no overt positive conclusion. Something can
be found and something is found. Common names are fitted for use by
the would-be users being first delivered from abortive conceptions, and
thereupon enabled to bring to the birth living and organic notions.

Aristotle would assign to Socrates the elaboration of two logical
functions:—general definition and inductive method.6 Rightly, if we
add that he gives no theory of either, and that his practical use of the
latter depends for its value on selection.7 It is rather in virtue of his
general faith in the possibility of construction, which he still does not
undertake, and because of his consequent insistence on the elucidation
of general concepts, which in common with some of his

contemporaries, he may have thought of as endued with a certain
objectivity, that he induces the controversies of what are called the
Socratic schools as to the nature of predication. These result in the
formulation of a new dialectic or logic by Plato. Manifestly Socrates’
use of certain forms of argumentation, like their abuse by the sophists,
tended to evoke their logical analysis. The use and abuse, confronted
one with the other, could not but evoke it.

The one in the many, the formula which lies at the base of the
possibility of predication, is involved in the Socratic doctrine of
general concepts or ideas. The nihilism of Gorgias from the Eleatic
point of view of bare identity, and the speechlessness of Cratylus from
the Heraclitean ground of absolute difference, are alike disowned. But
the one in the many, the identity in difference, is so far only postulated,
not established. When the personality of Socrates is removed, the
difficulty as to the nature of the Socratic universal, developed in the
medium of the individual processes of individual minds, carries
disciples of diverse general sympathies, united only through the
practical inspiration of the master’s life, towards the identity-formula
or the difference-formula of other teachers. The paradox of predication,
that it seems to deny identity, or to deny difference, becomes a pons
asinorum. Knowledge involves synthesis or nexus. Yet from the points
of view alike of an absolute pluralism, of a flux, and of a formula of
bare identity—and a fortiori with any blending of these principles
sufficiently within the bounds of plausibility to find an exponent—all
knowledge, because all predication of unity, in difference, must be held
to be impossible. Plato’s problem was to find a way of escape from this
impasse, and among his Socratic contemporaries he seems to have
singled out Antisthenes8 as most in need of refutation. Antisthenes,
starting with the doctrine of identity without difference, recognizes as
the only expression proper to anything its own peculiar sign, its name.

This extreme of nominalism for which predication is
Antisthenes. impossible is, however, compromised by two
concessions. A thing can be described as like
something else. And a compound can have a λόγος or account given of
it by the (literally) adequate enumeration of the names of its simple
elements or πρῶτα.9 This analytical λόγος he offers as his substitute
for knowledge.10 The simple elements still remain, sensed and named
but not known. The expressions of them are simply the speech-signs
for them. The account of the compound simply sets itself taken
piecemeal as equivalent to itself taken as aggregate. The subject-
predicate relation fails really to arise. Euclides11 found no difficulty in
fixing Antisthenes’ mode of illustrating his simple elements by
comparison, and therewith perhaps the “induction” of Socrates, with
the dilemma; so far as the example is dissimilar, the comparison is
invalid; so far as it is similar, it is useless. It is better to say what the
thing is. Between Euclides and Antisthenes the Socratic induction and
universal definition were alike discredited from the point of view of the
Eleatic logic. It is with the other point of doctrine that Plato comes to
grips, that which allows of a certainty or knowledge consisting in an
analysis of a compound into simple elements themselves not known.
The syllable or combination is, he shows, not known by resolution of it
into letters or elements themselves not known. An aggregate analysed
into its mechanical parts is as much and as little known as they. A
whole which is more than its parts is from Antisthenes’ point of view
inconceivable. Propositions analytical of a combination in the sense
alleged do not give knowledge. Yet knowledge is possible. The
development of a positive theory of predication has become quite
crucial.

Plato’s logic supplies a theory of universals in the doctrine of ideas. Upon
this it bases a theory of predication, which, however, is compatible with

more than one reading of the metaphysical import of the ideas. And it sets
forth a dialectic with a twofold movement, towards
Plato. differentiation and integration severally, which amounts
to a formulation of inference. The more fully analysed
movement, that which proceeds downward from less determinate to more
determinate universals, is named Division. Its associations, accordingly, are
to the modern ear almost inevitably those of a doctrine of classification
only. Aristotle, however, treats it as a dialectical rival to syllogism, and it
influenced Galilei and Bacon in their views of inference after the
Renaissance. If we add to this logic of “idea,” judgment and inference, a
doctrine of categories in the modern sense of the word which makes the
Theaetetus, in which it first occurs, a forerunner of Kant’s Critique of Pure
Reason, we have clearly a very significant contribution to logic even in
technical regard. Its general philosophical setting may be said to enhance its
value even as logic.

(a) Of the idea we may say that whatever else it is, and apart from all
puzzles as to ideas of relations such as smallness, of negative qualities such
as injustice, or of human inventions such as beds, it is opposed to that of
which it is the idea as its intelligible formula or law, the
The “Idea.” truth or validity—Herbart’s word—of the phenomenon
from the point of view of nexus or system. The thing of
sense in its relative isolation is unstable. It is and is not. What gives stability
is the insensible principle or principles which it holds, as it were, in
solution. These are the ideas, and their mode of being is naturally quite
other than that of the sensible phenomena which they order. The formula for
an indefinite number of particular things in particular places at particular
times, and all of them presentable in sensuous imagery of a given time and
place, is not itself presentable in sensuous imagery side by side with the
individual members of the group it orders. The law, e.g., of the equality of
the radii of a circle cannot be exhibited to sense, even if equal radii may be

so exhibited. It is the wealth of illustration with which Plato expresses his
meaning, and the range of application which he gives the idea—to the class-
concepts of natural groups objectively regarded, to categories, to aesthetic
and ethical ideals, to the concrete aims of the craftsman as well as to
scientific laws—that have obscured his doctrine, viz. that wherever there is
law, there is an idea.

(b) The paradox of the one in the many is none, if the idea may be
regarded as supplying a principle of nexus or organization to an indefinite
multiplicity of particulars. But if Antisthenes is to be answered, a further
step must be taken. The principle of difference must be
The one in the carried into the field of the ideas. Not only sense is a
many. principle of difference. The ideas are many. The
multiplicity in unity must be established within thought
itself. Otherwise the objection stands: man is man and good is good, but to
say that man is good is clearly to say the thing that is not. Plato replies with
the doctrine of the interpenetration of ideas, obviously not of all with all,
but of some with some, the formula of identity in difference within thought
itself. Nor can the opponent fairly refuse to admit it, if he affirms the
participation of the identical with being, and denies the participation of
difference with being, or affirms it with not-being. The Sophistes shows
among other things that an identity-philosophy breaks down into a dualism
of thought and expression, when it applies the predicate of unity to the real,
just as the absolute pluralism on the other hand collapses into unity if it
affirms or admits any form of relation whatsoever. Identity and difference
are all-pervasive categories, and the speech-form and the corresponding
thought-form involve both. For proposition and judgment involve subject
and predicate and exhibit what a modern writer calls “identity of reference
with diversity of characterization.” Plato proceeds to explain by his
principle of difference both privative and negative predicates, and also the
possibility of false predication. It is obvious that without the principle of

difference error is inexplicable. Even Plato, however, perhaps scarcely
shows that with it, and nothing else but it, error is explained.

(c) Plato’s Division, or the articulation of a relatively indeterminate and
generic concept into species and sub-species with resultant determinate
judgments, presumes of course the doctrine of the interpenetration of ideas
laid down in the Sophistes as the basis of predication, but
Division. its use precedes the positive development of that
formula, though not, save very vaguely, the exhibition of
it, negatively, in the antinomies of the one and the many in the Parmenides.
It is its use, however, not the theory of it, that precedes. The latter is
expounded in the Politicus (260 sqq.) and Philebus (16c sqq.). The ideal is
progressively to determine a universe of discourse till true infimae species
are reached, when no further distinction in the determinate many is
possible, though there is still the numerical difference of the indefinite
plurality of particulars. The process is to take as far as possible the form of
a continuous disjunction of contraries. We must bisect as far as may be, but
the division is after all to be into limbs, not parts. The later examples of the
Politicus show that the permission of three or more co-ordinate species is
not nugatory, and that the precept of dichotomy is merely in order to secure
as little of a saltus as possible; to avoid e.g. the division of the animal world
into men and brutes. It is the middle range of the μέσα of Philebus 17a that
appeals to Bacon, not only this but their mediating quality that appeals to
Aristotle. The media axiomata of the one and the middle term of the other
lie in the phrase. Plato’s division is nevertheless neither syllogism nor
exclusiva. It is not syllogism because it is based on the disjunctive, not on
the hypothetical relation, and so extends horizontally where syllogism
strikes vertically downward. Again it is not syllogism because it is
necessarily and finally dialectical. It brings in the choice of an interlocutor
at each stage, and so depends on a concession for what it should prove.12
Nor is it Bacon’s method of exclusions, which escapes the imputation of

being dialectical, if not that of being unduly cumbrous, in virtue of the
cogency of the negative instance. The Platonic division was, however,
offered as the scientific method of the school. A fragment of the comic poet
Epicrates gives a picture of it at work.13 And the movement of disjunction
as truly has a place in the scientific specification of a concept in all its
differences as the linking of lower to higher in syllogism. The two are
complementary, and the reinstatement of the disjunctive judgment to the
more honourable rôle in inference has been made by so notable a modern
logician as Lotze.

(d) The correlative process of Combination is less elaborately sketched,
but in a luminous passage in the Politicus (§ 278), in explaining by means
of an example the nature and use of examples, Plato represents it as the
bringing of one and the same element seen in diverse
Combination. settings to conscious realization, with the result that it is
viewed as a single truth of which the terms compared are
now accepted as the differences. The learner is to be led forward to the
unknown by being made to hark back to more familiar groupings of the
alphabet of nature which he is coming to recognize with some certainty. To
lead on, ἐπάγειν, is to refer back, ἀνάγειν,14 to what has been correctly
divined of the same elements in clearer cases. Introduction to unfamiliar
collocations follows upon this, and, only so, is it possible finally to gather
scattered examples into a conspectus as instances of one idea or law. This is
not only of importance in the history of the terminology of logic, but
supplies a philosophy of induction.

(e) Back of Plato’s illustration and explanation of predication and
dialectical inference there lies not only the question of their metaphysical
grounding in the interconnexion of ideas, but that of their epistemological
presuppositions. This is dealt with in the Theaetetus
Mental synthesis. (184b sqq.). The manifold affections of sense are not

simply aggregated in the individual, like the heroes in the Trojan horse.
There must be convergence in a unitary principle, soul or consciousness,
which is that which really functions in perception, the senses and their
organs being merely its instruments. It is this unity of apperception which
enables us to combine the data of more than one sense, to affirm reality,
unreality, identity, difference, unity, plurality and so forth, as also the good,
the beautiful and their contraries. Plato calls these pervasive factors in
knowledge κοινὰ, and describes them as developed by the soul in virtue of
its own activity. They are objects of its reflection and made explicit in the
few with pains and gradually.15 That they are not, however, psychological
or acquired categories, due to “the workmanship of the mind” as conceived
by Locke, is obvious from their attribution to the structure of mind16 and
from their correlation with immanent principles of the objective order.
Considered from the epistemological point of view, they are the implicit
presuppositions of the construction or συλλογισμός17 in which knowledge
consists. But as ideas,18 though of a type quite apart,19 they have also a
constitutive application to reality. Accordingly, of the selected “kinds” by
means of which the interpenetration of ideas is expounded in the Sophistes,
only motion and rest, the ultimate “kinds” in the physical world, have no
counterparts in the “categories” of the Theaetetus. In his doctrine as to ἕν
τὸ ποιοῦν or κρῖνον, as generally in that of the activity of the νοῦς
ἀπαθής, Aristotle in the de Anima20 is in the main but echoing the teaching
of Plato.21

ii. Aristotle.

Plato’s episodic use of logical distinctions22 is frequent. His recourse to
such logical analysis as would meet the requirements of the problem in
hand23 is not rare. In the “dialectical” dialogues the question of method and
of the justification of its postulates attains at least a like prominence with

the ostensible subject matter. There is even formal recognition of the fact
that to advance in dialectic is a greater thing than to bring any special
inquiry to a successful issue.24 But to the end there is a lack of interest in,
and therefore a relative immaturity of, technique as such. In the forcing
atmosphere, however, of that age of controversy, seed such as that sown in
the master’s treatment of the uttered λόγος25 quickly germinated. Plato’s
successors in the Academy must have developed a system of grammatico-
logical categories which Aristotle could make his own. Else much of his
criticism of Platonic doctrine26 does, indeed, miss fire. The gulf too, which
the Philebus27 apparently left unbridged between the sensuous
apprehension of particulars and the knowledge of universals of even
minimum generality led with Speusippus to a formula of knowledge in
perception (ἐπιστημονικὴ αἴσθησις). These and like developments, which
are to be divined from references in the Aristotelian writings, jejune, and,
for the most part, of probable interpretation only, complete the material
which Aristotle could utilize when he seceded from the Platonic school and
embarked upon his own course of logical inquiry.

This is embodied in the group of treatises later known as the Organon28
and culminates in the theory of syllogism and of demonstrative knowledge
in the Analytics. All else is finally subsidiary. In the well-known sentences
with which the Organon closes29 Aristotle has been
Syllogism. supposed to lay claim to the discovery of the principle of
syllogism. He at least claims to have been the first to
dissect the procedure of the debate-game, and the larger claim may be
thought to follow. In the course of inquiry into the formal consequences
from probable premises, the principle of mediation or linking was so laid
bare that the advance to the analytic determination of the species and
varieties of syllogism was natural. Once embarked upon such an analysis,
where valid process from assured principles gave truth, Aristotle could find

little difficulty in determining the formula of demonstrative knowledge or
science. It must be grounded in principles of assured certainty and must
demonstrate its conclusions with the use of such middle or linking terms
only as it is possible to equate with the real ground or cause in the object of
knowledge. Hence the account of axioms and of definitions, both of
substances and of derivative attributes. Hence the importance of
determining how first principles are established. It is, then, a fair working
hypothesis as to the structure of the Organon to place the Topics, which deal
with dialectical reasoning, before the Analytics.30 Of the remaining treatises
nothing of fundamental import depends on their order. One, however, the
Categories, may be regarded with an ancient commentator,31 as preliminary
to the dialectical inquiry in the Topics. The other, on thought as expressed in
language (Περὶ ἐρμηνείας) is possibly spurious, though in any case a
compilation of the Aristotelian school. If genuine, its naïve theory that
thought copies things and other features of its contents would tend to place
it among the earliest works of the philosopher.

Production in the form of a series of relatively self-contained treatises
accounts for the absence of a name and general definition of their common
field of inquiry. A more important lack which results is that of any clear
intimation as to the relation in which Aristotle supposed
The logical it to stand to other disciplines. In his definite
treatises. classification of the sciences,32 into First Philosophy,
Mathematics and Physics, it has no place. Its axioms,
such as the law of contradiction, belong to first philosophy, but the doctrine
as a whole falls neither under this head nor yet, though the thought has been
entertained, under that of mathematics, since logic orders mathematical
reasoning as well as all other. The speculative sciences, indeed, are
classified according to their relation to form, pure, abstract or concrete, i.e.
according to their objects. The logical inquiry seems to be conceived as
dealing with the thought of which the objects are objects. It is to be

regarded as a propaedeutic,33 which, although it is in contact with reality in
and through the metaphysical import of the axioms, or again in the fact that
the categories, though primarily taken as forms of predication, must also be
regarded as kinds of being, is not directly concerned with object-reality, but
with the determination for the thinking subject of what constitutes the
knowledge correlative to being. Logic, therefore, is not classed as one, still
less as a branch of one, among the ’ologies, ontology not excepted.

The way in which logical doctrine is developed in the Aristotelian
treatises fits in with this view. Doubtless what we have is in the main a
reflex of the heuristic character of Aristotle’s own work as pioneer. But it at
least satisfies the requirement that the inquiry shall carry the plain man
along with it. Actual modes of expression are shown to embody distinctions
which average intelligence can easily recognize and will readily
acknowledge, though they may tend by progressive rectification
fundamentally to modify the assumption natural to the level of thought from
which he begins. Thus we start34 from the point of view of a world of
separate persons and things, in which thought mirrors these concrete
realities, taken as ultimate subjects of predicates. It is a world of
communication of thought, where persons as thinkers need to utter in
language truths objectively valid for the mundus communis. In these truths
predicates are accepted or rejected by subjects, and therefore depend on the
reflection of fact in λόγοι (propositions). These are combinatory of parts,
attaching or detaching predicates, and so involving subject, predicate and
copula.35 At this stage we are as much concerned with speech-forms as the
thought-forms of which they are conventional symbols, with Plato’s
analysis, for instance, into a noun and a verb, whose connotation of time is
as yet a difficulty. The universal of this stage is the universal of fact, what is
recognized as predicable of a plurality of subjects. The dialectical doctrine
of judgment as the declaration of one member of a disjunction by
contradiction, which is later so important, is struggling with one of its initial

difficulties,36 viz. the contingency of particular events future, the solution of
which remains imperfect.37

The doctrine of the Categories is still on the same level of thought,38
though its grammatico-logical analysis is the more advanced one which had
probably been developed by the Academy before Aristotle came to think of
his friends there as “them” rather than “us.” It is what in
The Categories. one direction gave the now familiar classification of
parts of speech, in the other that of thought-categories
underlying them. If we abstract from any actual combination of subject and
predicate and proceed to determine the types of predicate asserted in simple
propositions of fact, we have on the one hand a subject which is never
object, a “first substance” or concrete thing, of which may be predicated in
the first place “second substance” expressing that it is a member of a
concrete class, and in the second place quantity, quality, correlation, action
and the like. The list follows the forms of the Greek language so closely
that a category emerges appropriated to the use of the perfect tense of the
middle voice to express the relation of the subject to a garb that it dons. In
all this the individual is the sole self-subsistent reality. Truth and error are
about the individual and attach or detach predicates correctly and
incorrectly. There is no committal to the metaphysics in the light of which
the logical inquiry is at last to find its complete justification. The point of
view is to be modified profoundly by what follows—by the doctrine of the
class-concept behind the class, of the form or idea as the constitutive
formula of a substance, or, again, by the requirement that an essential
attribute must be grounded in the nature or essence of the substance of
which it is predicated, and that such attributes alone are admissible
predicates from the point of view of the strict ideal of science. But we are
still on the ground of common opinion, and these doctrines are not yet laid
down as fundamental to the development.

Dialectic then, though it may prove to be the ultimate method of
establishing principles in philosophy,39 starts from probable and conceded
premises,40 and deals with them only in the light of common principles
such as may be reasonably appealed to or easily
The Topics. established against challenge. To the expert, in any study
which involves contingent matter, i.e. an irreducible
element of indetermination, e.g. to the physician, there is a specific form of
this, but the reflection that this is so is something of an afterthought. We
start with what is prima facie given, to return upon it from the ground of
principles clarified by the sifting process of dialectic41 and certified by
νοῦς. The Topics deal with dialectic and constitute an anatomy of
argumentation, or, according to what seems to be Aristotle’s own metaphor,
a survey of the tactical vantage-points (τόποι) for the conflict of wits in
which the prize is primarily victory, though it is a barren victory unless it is
also knowledge. It is in this treatise that what have been called “the
conceptual categories”42 emerge, viz. the predicables, or heads of
predication as it is analysed in relation to the provisional theory of
definition that dialectic allows and requires. A predicate either is expressive
of the essence or part of the essence of the subject, viz. that original group
of mutually underivable attributes of which the absence of any one destroys
its right to the class-name, or it is not. Either it is convertible with the
subject or it is not. Here then judgment, though still viewed as combinatory,
has the types which belong to coherent systems of implication
discriminated from those that predicate coincidence or accident, i.e. any
happening not even derivatively essential from the point of view of the
grouping in which the subject has found a place. In the theory of dialectic
any predicate may be suggested for a subject, and if not affirmed of it, must
be denied of it, if not denied must be affirmed. The development of a theory
of the ground on which subjects claim their predicates and disown alien
predicates could not be long postponed. In practical dialectic the unlimited

possibility was reduced to manageable proportions in virtue of the
groundwork of received opinion upon which the operation proceeded. It is
in the Topics, further, that we clearly have a first treatment of syllogism as
formal implication, with the suggestion that advance must be made to a
view of its use for material implication from true and necessary principles.
It is in the Topics,43 again, that we have hints at the devices of an inductive
process, which, as dialectical, throw the burden of producing contradictory
instances upon the other party to the discussion. In virtue of the common-
stock of opinion among the interlocutors and their potentially controlling
audience, this process was more valuable than appears on the face of things.
Obviously tentative, and with limits and ultimate interpretation to be
determined elsewhere, it failed to bear fruit till the Renaissance, and then by
the irony of fate to the discrediting of Aristotle. In any case, however,
definition, syllogism, induction all invited further determination, especially
if they were to take their place in a doctrine of truth or knowledge. The
problem of analytic, i.e. of the resolution of the various forms of inference
into their equivalents in that grouping of terms or premises which was most
obviously cogent, was a legacy of the Topics. The debate-game had sought
for diversion and found truth, and truth raised the logical problem on a
different plane.

At first the problem of formal analysis only. We proceed with the talk of
instances and concern ourselves first with relations of inclusion and
exclusion. The question is as to membership of a class, and the dominant
formula is the dictum de omni et nullo. Until the view of
Class concept. the individual units with which we are so far familiar has
undergone radical revision, the primary inquiry must be
into the forms of a class-calculus. Individuals fall into groups in virtue of
the possession of certain predicates. Does one group include, or exclude, or
intersect another with which it is compared? We are clearly in the field of
the diagrams of the text-books, and much of the phraseology is based upon

an original graphic representation in extension. The middle term, though
conceived as an intermediary or linking term, gets its name as intermediate
in a homogeneous scheme of quantity, where it cannot be of narrower
extension than the subject nor wider than the predicate of the conclusion.44
It is also, as Aristotle adds,45 middle in position in the syllogism that
concludes to a universal affirmative.45 Again, so long as we keep to the
syllogism as complete in itself and without reference to its place in the great
structure of knowledge, the nerve of proof cannot be conceived in other
than a formal manner. In analytic we work with an ethos different from that
of dialectic. We presume truth and not probability or concession, but a true
conclusion can follow from false premises, and it is only in the attempt to
derive the premises in turn from their grounds that we unmask the
deception. The passage to the conception of system is still required. The
Prior Analytics then are concerned with a formal logic to
The Prior be knit into a system of knowledge of the real only in
Analytics. virtue of a formula which is at this stage still to seek. The
forms of syllogism, however, are tracked successfully
through their figures, i.e. through the positions of the middle term that
Aristotle recognizes as of actual employment, and all their moods, i.e. all
differences of affirmative and negative, universal and particular within the
figures, the cogent or legitimate forms are alone left standing, and the
formal doctrine of syllogism is complete. Syllogism already defined46
becomes through exhibition in its valid forms clear in its principle. It is a
speech-and-thought-form (λόγος) in which certain matters being posited
something other than the matters posited necessarily results because of
them, and, though it still needs to receive a deeper meaning when presumed
truth gives way to necessary truth of premises, the notion of the class to that
of the class-concept, collective fact to universal law, its formal claim is
manifest. “Certain matters being posited.” Subject and predicate not already
seen to be conjoined must be severally known to be in relation with that

which joins them, so that more than one direct conjunction must be given.
“Of necessity.” If what are to be conjoined are severally in relation to a
common third it does perforce relate or conjoin them. “Something other.”
The conjunction was by hypothesis not given, and is a new result by no
means to be reached, apart from direct perception save by use of at least
two given conjunctions. “Because of them,” therefore. Yet so long as the
class-view is prominent, there is a suggestion of a begging of the question.
The class is either constituted by enumeration of its members, and, passing
by the difficulty involved in the thought of “its” members, is an empirical
universal of fact merely, or it is grounded in the class-concept. In the first
case it is a formal scheme which helps knowledge and the theory of
knowledge not at all. We need then to develop the alternative, and to pass
from the external aspect of all-ness to the intrinsic ground of it in the
universal καθ᾽ αὑτὸ καὶ ᾗ αὐτό, which, whatsoever the assistance it
receives from induction in some sense of the word, in the course of its
development for the individual mind, is secured against dependence on
instances by the decisive fiat or guarantee of νοῦς, insight into the
systematic nexus of things. The conception of linkage needs to be deepened
by the realization of the middle term as the ground of nexus in a real order
which is also rational.

Aristotle’s solution of the paradox of inference, viz. of the fact that in one
sense to go beyond what is in the premises is fallacy,
Problem of while in another sense not to go beyond them is futility,
inference. lies in his formula of implicit and explicit, potential and
actual.47 The real nexus underlying the thought-process
is to be articulated in the light of the voucher by intelligence as to the truth
of the principles of the various departments of knowledge which we call
sciences, and at the ideal limit it is possible to transform syllogism into
systematic presentation, so that, differently written down, it is definition.
But for human thought sense, with its accidental setting in matter itself

incognizable is always with us. The activity of νοῦς is
Nous. never so perfectly realized as to merge implication in
intuition. Syllogism must indeed be objective, i.e. valid
for any thinker, but it is also a process in the medium of individual thinking,
whereby new truth is reached. A man may know that mules are sterile and
that the beast before him is a mule, and yet believe her to be in foal “not
viewing the several truths in connexion.”48 The doctrine, then, that the
universal premise contains the conclusion not otherwise than potentially is
with Aristotle cardinal. The datum of sense is only retained through the
universal.49 It is possible to take a universal view with some at least of the
particular instances left uninvestigated.50 Recognition that the class-concept
is applicable may be independent of knowledge of much that it involves.
Knowledge of the implications of it does not depend on observation of all
members of the class. Syllogism as formula for the exhibition of truth
attained, and construction or what not as the instrumental process by which
we reach the truth, have with writers since Hegel and Herbart tended to fall
apart. Aristotle’s view is other. Both are syllogisms, though in different
points of view. For this reason, if for no other, the conception of movement
from the potential possession of knowledge to its actualization remains
indispensable. Whether this is explanation or description, a problem or its
solution, is of course another matter.

In the Posterior Analytics the syllogism is brought into decisive
connexion with the real by being set within a system in which its function is
that of material implication from principles which are primary, immediate
and necessary truths. Hitherto the assumption of the
Posterior probable as true rather than as what will be conceded in
Analytics. debate51 has been the main distinction of the standpoint
of analytic from that of dialectic. But the true is true only
in reference to a coherent system in which it is an immediate ascertainment
of νοῦς, or to be deduced from a ground which is such. The ideal of science

or demonstrative knowledge is to exhibit as flowing from the definitions
and postulates of a science, from its special principles, by the help only of
axioms or principles common to all knowledge, and these not as premises
but as guiding rules, all the properties of the subject-matter, i.e. all the
predicates that belong to it in its own nature. In the case of any subject-
kind, its definition and its existence being avouched by νοῦς, “heavenly
body” for example, the problem is, given the fact of a non-self-subsistent
characteristic of it, such as the eclipse of the said body, to find a ground, a
μέσον which expressed the αἴτιον, in virtue of which the adjectival concept
can be exhibited as belonging to the subject-concept καθ᾽ αὑτὸ in the
strictly adequate sense of the phrase in which it means also ᾖ αὑτὸ.52 We
are under the necessity then of revising the point of view of the syllogism of
all-ness. We discard the conception of the universal as a predicate
applicable to a plurality, or even to all, of the members of a group. To know
merely κατὰ παντὸς is not to know, save accidentally. The exhaustive
judgment, if attainable, could not be known to be exhaustive. The universal
is the ground of the empirical “all” and not conversely. A formula such as
the equality of the interior angles of a triangle to two right angles is only
scientifically known when it is not of isosceles or scalene triangle that it is
known, nor even of all the several types of triangle collectively, but as a
predicate of triangle recognized as the widest class-concept of which it is
true, the first stage in the progressive differentiation of figure at which it
can be asserted.53

Three points obviously need development, the nature of definition, its
connexion with the syllogism in which the middle term is cause or ground,
and the way in which we have assurance of our principles.

Definition is either of the subject-kind or of the property that is grounded
in it. Of the self-subsistent definition is οὐσίας τις γνωρισμός54 by
exposition of genus and differentia.55 It is indemonstrable. It presumes the

reality of its subject in a postulate of existence. It belongs
Definition. to the principles of demonstration. Summa genera and
groups below infimae species are indefinable. The former
are susceptible of elucidation by indication of what falls under them. The
latter are only describable by their accidents. There can here be no true
differentia. The artificiality of the limit to the articulation of species was
one of the points to which the downfall of Aristotle’s influence was largely
due. Of a non-self-subsistent or attributive conception definition in its
highest attainable form is a recasting of the syllogism, in which it was
shown that the attribute was grounded in the substance or self-subsistent
subject of which it is. Eclipse of the moon, e.g. is privation of light from the
moon by the interposition of the earth between it and the sun. In the
scientific syllogism the interposition of the earth is the middle term, the
cause or “because” (διότι), the residue of the definition is conclusion. The
difference then is in verbal expression, way of putting, inflexion.56 If we
pluck the fruit of the conclusion, severing its nexus with the stock from
which it springs, we have an imperfect form of definition, while, if further
we abandon all idea of making it adequate by exhibition of its ground, we
have, with still the same form of words, a definition merely nominal or
lexicographical. In the aporematic treatment of the relation of definition and
syllogism identical as to one form and in one view, distinct as to another
form and in another view, much of Aristotle’s discussion consists. The rest
is a consideration of scientific inquiry as converging in
The middle term. μέσου ζήτησις, the investigation of the link or
“because” as ground in the nature of things. Τὸ γὰρ
αἴτιον τὸ μέσον57 real ground and thought link fall together. The advance
from syllogism as formal implication is a notable one. It is not enough to
have for middle term a causa cognoscendi merely. We must have a causa
essendi. The planets are near, and we know it by their not twinkling,58 but
science must conceive their nearness as the cause of their not twinkling and

make the prius in the real order the middle term of its syllogism. In this
irreversible catena proceeding from ground to consequent, we have left far
behind such things as the formal parity of genus and differentia considered
as falling under the same predicable,59 and hence justified in part
Porphyry’s divergence from the scheme of predicables. We need devices,
indeed, to determine priority or superior claim to be “better known
absolutely or in the order of nature,” but on the whole the problem is fairly
faced.60

Of science Aristotle takes for his examples sometimes celestial physics,
more often geometry or arithmetic, sometimes a concrete science, e.g.
botany.61 In the field of pure form, free from the disconcerting surprises of
sensible matter and so of absolute necessity, no difficulty arises as to the
deducibility of the whole body of a science from its first principles. In the
sphere of abstract form, mathematics, the like may be allowed, abstraction
being treated as an elimination of matter from the σύνολον by one act.
When we take into account relative matter, however, and traces of a
conception of abstraction as admitting of degree,62 the question is not free
from difficulty. In the sphere of the concrete sciences where law obtains
only ὡς ἐπὶ τὸ πολὺ this ideal of science can clearly find only a relative
satisfaction with large reserves. In any case, however, the problem as to first
principles remains fundamental.

If we reject the infinite regress and the circle in proof (circulus in
probando) which resolves itself ultimately into proving A by B and B by
A,63 we are confronted by the need for principles of two kinds, those which
condition all search for truth, and those which are the
Formal and peculiar or proper principles of special sciences, their
scientific “positions,” viz. the definitions of their subjects and the
principles. postulates of the existence of these. All are
indemonstrable and cannot be less sure than the body of

doctrine that flows from them. They must indeed be recognized as true,
primary, causative and the like. But64 they are not congenitally present in
the individual in a determinate shape. The doctrine of latency is mystical
and savours of Plato’s reminiscence (anamnesis). Yet they must have
something to develop from, and thereupon Aristotle gives an account of a
process in the psychological mechanism which he illustrates by
comparative psychology, wherein a λόγος or meaning emerges, a “first”
universal recognized by induction. Yet νοῦς,
Induction and intelligence, is the principle of first principles. It is
dialectic. infallible, while, whatever the case with perception of the
special sensibles,65 the process which combines
particulars is not. On the side of induction we find that experience is said to
give the specific principles,66 “the phenomena being apprehended in
sufficiency.” On the side of intuition, self-evidence of scientific principles is
spoken of.67 Yet dialectic is auxiliary and of methodological importance in
their establishment.68 Mutually limiting statements occur almost or quite
side by side. We cannot take first principles “as the bare precipitate of a
progressively refined analysis”69 nor on the other as constitutive a priori
forms. The solution seems to lie in the conception of a process that has a
double aspect. On the one hand we have confrontation with fact, in which,
in virtue of the rational principle which is the final cause of the phenomenal
order, intelligence will find satisfaction. On the other we have a stage at
which the rational but as yet not reasoned concepts developed in the
medium of the psychological mechanism are subjected to processes of
reflective comparison and analysis, and, with some modification,
maintained against challenge, till at length the ultimate universals emerge,
which rational insight can posit as certain, and the whole hierarchy of
concepts from the “first” universals to τὰ ἀμερῆ are intuited in a coherent
system. Aristotle’s terminology is highly technical, but, as has often been
observed, not therefore clear. Here two words at least are ambiguous,

“principle” and “induction.” By the first he means any starting-point, “that
from which the matter in question is primarily to be known,”70 particular
facts therefore, premises, and what not. What then is meant by principles
when we ask in the closing chapter of his logic how they become known?
The data of sense are clearly not the principles in question here. The
premises of scientific syllogisms may equally be dismissed. Where they are
not derivative they clearly are definitions or immediate transcripts from
definitions. There remain, then, primary definitions and the postulates of
their realization, and the axioms or common principles, “which he must
needs have who is to reach any knowledge.”71 In the case of the former,
special each to its own science, Aristotle may be thought to hold that they
are the product of the psychological mechanism, but are ascertained only
when they have faced the fire of a critical dialectic and have been accepted
from the point of view of the integral rationality of the system of concepts.
Axioms, on the other hand, in which the sciences interconnect72 through the
employment of them in a parity of relation, seem to be implicit indeed in
the psychological mechanism, but to come to a kind of explicitness in the
first reflective reaction upon it, and without reference to any particular
content of it. They are not to be used as premises but as immanent laws of
thought, save only when an inference from true or admitted premises and
correct in form is challenged. The challenge must be countered in a reductio
ad impossibile in which the dilemma is put. Either this conclusion or the
denial of rationality. Even these principles, however, may get a greater
explicitness by dialectical treatment.73 The relation, then, of the two orders
of principle to the psychological mechanism is different. The kind of
warrant that intelligence can give to specific principles falls short of
infallibility. Celestial physics, with its pure forms and void of all matter
save extension, is not such an exemplary science after all. Rationality is
continuous throughout. A λόγος emerges with some beings in direct
sequence upon the persistence of impressions.74 Sense is of the “first”

universal, the form, though not of the ultimate universal. The rally from the
rout in Aristotle’s famous metaphor is of units that already belong together,
that are of the same regiment or order. On the other hand, rationality has
two stages. In the one it is relatively immersed in sense, in the other
relatively free. The same break is to be found in the conception of the
relation of receptive to active mind in the treatise Of the Soul.75 The one is
impressed by things and receives their form without their matter. The other
is free from impression. It thinks its system of concepts freely on the
occasion of the affections of the receptivity. Aristotle is fond of declaring
that knowledge is of the universal, while existence or reality is individual. It
seems to follow that the cleavage between knowledge and reality is not
bridged by the function of νοῦς in relation to “induction.” What is known is
not real, and what is real is not known. The nodus76 has
Knowledge and its cause in the double sense of the word “universal” and
reality. a possible solution in the doctrine of εἶδος. The “form”
of a thing constitutes it what it is, and at the same time,
therefore, is constitutive of the group to which it belongs. It has both
individual and universal reference. The individual is known in the εἶδος,
which is also the first universal in which by analysis higher universals are
discoverable. These are predicates of the object known, ways of knowing it,
rather than the object itself. The suggested solution removes certain
difficulties, but scarcely all. On seeing Callias my perception is of man, not
Callias, or even man-Callias. The recognition of the individual is a matter of
his accidents, to which even sex belongs, and the gap from lowest universal
to individual may still be conceived as unbridged. It is in induction, which
claims to start from particulars and end in universals,77 that we must, if
anywhere within the confines of logical inquiry, expect to find the required
bridge. The Aristotelian conception of induction, however, is somewhat
ambiguous. He had abandoned for the most part the Platonic sense of the
corresponding verb, viz. to lead forward to the as yet unknown, and his

substitute is not quite clear. It is scarcely the military
Conclusions as to metaphor. The adducing of a witness for which he uses
induction. the verb78 is not an idea that covers all the uses.79
Perhaps confrontation with facts is the general meaning.
But how does he conceive of its operation? There is in the first place the
action of the psychological mechanism in the process from discriminative
sense upwards wherein we realize “first” universals.80 This is clearly an
unreflective, pre-logical process, not altogether lighted up by our
retrojection upon it of our view of dialectical induction based thereon. The
immanent rationality of this first form, in virtue of which at the stage when
intelligence acts freely on the occasion of the datum supplied it recognizes
continuity with its own self-conscious process, is what gives the dialectical
type its meaning. Secondly we have this dialectical “induction as to
particulars by grouping of similars”81 whose liability to rebuttal by an
exception has been already noted in connexion with the limits of dialectic.
This is the incomplete induction by simple enumeration which has so often
been laughed to scorn. It is a heuristic process liable to failure, and its
application by a nation of talkers even to physics where non-expert opinion
is worthless somewhat discredited it. Yet it was the fundamental form of
induction as it was conceived throughout the scholastic period. Thirdly we
have the limiting cases of this in the inductive syllogism διὰ πάντων,82 a
syllogism in the third figure concluding universally, and yet valid because
the copula expresses equivalence, and in analogy83 in which, it has been
well said, instances are weighed and not counted. In the former it has been
noted84 that Aristotle’s illustration does not combine particular facts into a
lowest concept, but specific concepts into a generic concept, and85 that in
the construction of definite inductions the ruling thought with Aristotle is
already, though vaguely, that of causal relation. It appears safer,
notwithstanding, to take the less subtle interpretation86 that dialectical
induction struggling with instances is formally justified only at the limit,

and that this, where we have exhausted and know that we have exhausted
the cases, is in regard to individual subjects rarely and accidentally reached,
so that we perforce illustrate rather from the definite class-concepts falling
under a higher notion. After all, Aristotle must have had means by which he
reached the conclusions that horses are long-lived and lack gall. It is only
then in the rather mystical relation of νοῦς to the first type of induction as
the process of the psychological mechanism that an indication of the
direction in which the bridge from individual being to universal knowledge
is to be found can be held to lie.

Enough has been said to justify the great place assigned to Aristotle in the
history of logic. Without pressing metaphysical formulae in logic proper, he
analysed formal implication grounded implication as a mode of knowledge
in the rationality of the real, and developed a justificatory
Summary. metaphysic. He laid down the programme which the after
history of logic was to carry out. We have of course
abandoned particular logical positions. This is especially to be noted in the
theory of the proposition. The individualism with which he starts,
howsoever afterwards mitigated by his doctrine of τὸ τὶ ἦν εἰναι or εἶδος
constituting the individual in a system of intelligible relations, confined him
in an inadmissible way to the subject-attribute formula. He could not
recognize such vocables as the impersonals for what they were, and had
perforce to ignore the logical significance of purely reciprocal judgments,
such as those of equality. There was necessarily a “sense” or direction in
every proposition, with more than the purely psychological import that the
advance was from the already mastered and familiar taken as relatively
stable, to the new and strange. Many attributes, too, were predicable, even
to the end, in an external and accidental way, not being derivable from the
essence of the subject. The thought of contingency was too easily applied to
these attributes, and an unsatisfactory treatment of modality followed. It is
indeed the doctrine of the intractability of matter to form that lies at the

base of the paradox as to the disparateness of knowledge and the real
already noted. On the one hand Aristotle by his doctrine of matter admitted
a surd into his system. On the other, he assigned to νοῦς with its insight
into rationality too high a function with regard to the concrete in which the
surd was present, a power to certify the truth of scientific principles. The
example of Aristotle’s view of celestial physics as a science of pure forms
exhibits both points. On the Copernican change the heavenly bodies were
recognized as concrete and yet subject to calculable law. Intelligence had
warranted false principles. The moral is that of the story of the heel of
Achilles.

To return to logic proper. The Aristotelian theory of the universal of
science as secure from dependence on its instances and the theory of linking
in syllogism remain a heritage for all later logic, whether accepted in
precisely Aristotle’s formula or no. It is because the intervening centuries
had the Aristotelian basis to work on, sometimes in reduced quantity and
corrupt form, but always in some quantity and some form, that the rest of
our logical tradition is what it is. We stand upon his shoulders.

iii. Later Greek Logic.

After Aristotle we have, as regards logic, what the verdict of after
times has rightly characterized as an age of Epigoni. So far as the
Aristotelian framework is accepted we meet only minor corrections and
extensions of a formal kind. If there is conscious and purposed
divergence from Aristotle, inquiry moves, on the whole, within the
circle of ideas where Aristotelianism had fought its fight and won its
victory. Where new conceptions emerge, the imperfection of the
instruments, mechanical and methodological, of the sciences renders
them unfruitful, until their rediscovery in a later age. We have activity

without advance, diversity without development. Attempts at
comprehensiveness end in the compromises of eclecticism.

Illustrations are not far to seek. Theophrastus and in general the elder
Peripatetics, before the rise of new schools with new lines of cleavage
and new interests had led to new antagonisms and new alliances, do not
break away from the Aristotelian metaphysic. Their
The Peripatetics. interests, however, lie in the sublunary sciences in
which the substantive achievement of the school was
to be found. With Theophrastus, accordingly, in his botanical inquiries,
for example, the alternatives of classification, the normal sequence of
such and such a character upon such another, the conclusion of rational
probability, are what counts. It is perhaps not wholly fanciful to
connect with this attitude the fact that Aristotle’s pupils dealt with a
surer hand than the master with the conclusions from premises of
unlike modality, and that a formal advance of some significance
attributable to Theophrastus and Eudemus is the doctrine of the
hypothetical and disjunctive syllogisms.

The Stoics are of more importance. Despite the fact that their
philosophic interests lay rather in ethics and physics, their activity in
what they classified as the third department of speculation was
enormous and has at least left ineffaceable traces on
The Stoics. the terminology of philosophy. Logic is their word,
and consciousness, impression and other technical
words come to us, at least as technical words, from Roman Stoicism.
Even inference, though apparently not a classical word, throws back to
the Stoic name for a conclusion.87 In the second place, it is in the form
in which it was raised in connexion with the individualistic theory of
perception with which the Stoics started, that one question of
fundamental importance, viz. that of the criterion of truth, exercised its

influence on the individualists of the Renaissance. Perception, in the
view of the Stoics, at its highest both revealed and guaranteed the being
of its object. Its hold upon the object involved the discernment that it
could but be that which it purported to be. Such “psychological
certainty” was denied by their agnostic opponents, and in the history of
Stoicism we have apparently a modification of the doctrine of
φαντασία καταληπτικὴ with a view to meet the critics, an
approximation to a recognition that the primary conviction might meet
with a counter-conviction, and must then persist undissipated in face of
the challenge and in the last resort find verification in the haphazard
instance, under varying conditions, in actual working. The controversy
as to the self-evidence of perception in which the New Academy
effected some sort of conversion of the younger Stoics, and in which
the Sceptics opposed both, is one of the really vital issues of the
decadence.

Another doctrine of the Stoics which has interest in the light of
certain modern developments is their insistence on the place of the
λεκτόν in knowledge. Distinct alike from thing and mental happening,
it seems to correspond to “meaning” as it is used as a technical phrase
now-a-days. This anticipation was apparently sterile. Along the same
lines is their use of the hypothetical form for the universal judgment,
and their treatment of the hypothetical form as the typical form of
inference.

The Stoical categories, too, have an historical significance. They are
apparently offered in place of those of Aristotle, an acquaintance with
whose distinctions they clearly presume. Recognizing a linguistic side
to “logical” theory with a natural development in rhetoric, the Stoics
endeavour to exorcise considerations of language from the contrasted
side. They offer pure categories arising in series, each successive one

presupposing those that have gone before. Yet the substance, quality,
condition absolute (πῶς ἔχον) and condition relative of Stoicism have
no enduring influence outside the school, though they recur with
eclectics like Galen. The Stoics were too “scholastic” in their
speculations.

In Epicureanism logic is still less in honour. The practical end,
freedom from the bondage of things with the peace it brings, is all in
all, and even scientific inquiry is only in place as a means to this end.
Of the apparatus of method the less the better. We
Epicureans. are in the presence of a necessary evil. Yet, in falling
back, with a difference, upon the atomism of
Democritus, Epicurus had to face some questions of logic. In the
inference from phenomena to further phenomena positive verification
must be insisted on. In the inference from phenomena to their non-
phenomenal causes, the atoms with their inaccessibility to sense, a
different canon of validity obtains, that of non-contradiction.88 He
distinguishes too between the inference to combination of atoms as
universal cause, and inference to special causes beyond the range of
sense. In the latter case alternatives may be acquiesced in.89 The
practical aim of science is as well achieved if we set forth possible
causes as in showing the actual cause. This pococurantism might easily
be interpreted as an insight into the limitations of inverse method as
such or as a belief in the plurality of causes in Mill’s sense of the
phrase. More probably it reflects the fact that Epicurus was, according
to tradition through Nausiphanes, on the whole dominated by the
influences that produced Pyrrhonism. Democritean physics without a
calculus had necessarily proved sterile of determinate concrete results,
and this was more than enough to ripen the naturalism of the utilitarian
school into scepticism. Some reading between the lines of Lucretius

has led the “logic” of Epicurus to have an effect on the modern world,
but scarcely because of its deserts.

The school of Pyrrho has exercised a more legitimate influence.
Many of the arguments by which the Sceptics enforced their advocacy
of a suspense of judgment are antiquated in type, but many also are,
within the limits of the individualistic theory of
The Sceptics. knowledge, quite unanswerable. Hume had constant
recourse to this armoury. The major premise of
syllogism, says the Pyrrhonist, is established inductively from the
particular instances. If there be but one of these uncovered by the
generalization, this cannot be sound. If the crocodile moves its upper,
not its lower, jaw, we may not say that all animals move the lower jaw.
The conclusion then is really used to establish the major premise, and if
we still will infer it therefrom we fall into the circular proof.90 Could
Mill say more? But again. The inductive enumeration is either of all
cases or of some only. The former is in an indeterminate or infinite
subject-matter impossible. The latter is invalid.91 Less familiar to
modern ears is the contention that proof needs a standard or criterion,
while this standard or criterion in turn needs proof. Or still more the
dialectical device by which the sceptic claims to escape the riposte that
his very argument presumes the validity of this or that principle, viz.
the doctrine of the equipollence of counter-arguments. Of course the
counter-contention is no less valid! So too when the reflection is made
that scepticism is after all a medicine that purges out itself with the
disease, the disciple of Pyrrho and Aenesidemus bows and says,
Precisely! The sceptical suspension of judgment has its limits,
however. The Pyrrhonist will act upon a basis of probabilities. Nay, he
even treats the idea of cause92 as probable enough so long as nothing
more than action upon expectation is in question. He adds, however,
that any attempt to establish it is involved in some sort of dilemma.

That, for instance, cause as the correlate of effect only exists with it,
and accordingly, cause which is come while effect is still to come is
inconceivable.93 From the subjectivist point of view, which is
manifestly fundamental through most of this, such arguments suasory
of the Pyrrhonist suspense of judgment (ἐποχή) are indeed hard to
answer. It is natural, then, that the central contribution of the Sceptics
to the knowledge controversy lies in the modes (τρόποι) in which the
relativity of phenomena is made good, that these are elaborated with
extreme care, and that they have a modern ring and are full of
instruction even to-day. Scepticism, it must be confessed, was at the
least well equipped to expose the bankruptcy of the post-Aristotelian
dogmatism.

It was only gradually that the Sceptic’s art of fence was developed.
From the time of Pyrrho overlapping Aristotle himself, who seems to
have been well content to use the feints of more than one school among
his predecessors, while showing that none of them could claim to get
past his guard, down through a period in which the decadent academy
under Carneades, otherwise dogmatic in its negations, supplied new
thrusts and parries, to Aenesidemus in the late Ciceronian age, and
again to Sextus Empiricus, there seems to have been something of
plasticity and continuous progress. In this matter the dogmatic schools
offer a marked contrast. In especial it is an outstanding characteristic of
the younger rivals to Aristotelianism that as they sprang up suddenly
into being to contest the claims of the Aristotelian system in the
moment of its triumph, so they reached maturity very suddenly, and
thereafter persisted for the most part in a stereotyped tradition,
modified only when convicted of indefensible weakness. The 3rd
century b.c. saw in its first half the close of Epicurus’ activity, and the
life-work of Chrysippus, the refounder of Stoicism, is complete before
its close. And subsequent variations seem to have been of a negligible

where not of an eclectic character. In the case of Epicureanism we can
happily judge of the tyranny of the literal tradition by a comparison of
Lucretius with the recorded doctrine of the master. But the rule
apparently obtains throughout that stereotype and compromise offer
themselves as the exhaustive alternative. This is perhaps fortunate for
the history of doctrine, for it produces the commentator, your Aspasius
or Alexander of Aphrodisias, and the substitute for the critic, your
Cicero, or your Galen with his attempt at comprehension of the Stoic
categories and the like while starting from Aristotelianism. Cicero in
particular is important as showing the effect or philosophical
eclecticism upon Roman cultivation, and as the often author and
always popularizer of the Latin terminology of philosophy.

The cause of the stereotyping of the systems, apart from political
conditions, seems to have been the barrenness of science. Logic and
theory of knowledge go together, and without living science, theory of
knowledge loses touch with life, and logic becomes a perfunctory
thing. Under such circumstances speculative interest fritters itself and
sooner or later the sceptic has his way. Plato is full of the faith of
mathematical physics. Aristotle is optimistic of achievement over the
whole range of the sciences. But the divorce of science of nature from
mathematics, the failure of biological inquiry to reach so elementary a
conception as that of the nerves, the absence of chemistry from the
circle of the sciences, disappointed the promise of the dawn and the
relative achievement of the noon-day. There is no development.
Physical science remains dialectical, and a physical experiment is as
rare in the age of Lucretius as in that of Empedocles. The cause of
eclecticism is the unsatisfying character of the creeds of such science,
in conjunction with the familiar law that, in triangular or plusquam-
triangular controversies a common hatred will produce an alliance
based on compromise. A bastard Platonism through hostility to

Stoicism may become agnostic. Stoicism through hostility to its
sceptical critics may prefer to accept some of the positions of the
dogmatic nihilist.

Of the later schools the last to arise was Neoplatonism. The
mathematical sciences, at least, had not proved disappointing. For
those of the school of Plato who refused the apostasy of the new
academy, there was hope either in the mathematical
Neoplatonism. side of the Pythagoreo-Platonic tradition, or in its
ritual and theological side. Neoplatonism is
philosophy become theosophy, or it is the sermon on the text that God
geometrizes. It is of significance in the general history of thought as the
one great school that developed after the decadence had set in. In its
metaphysic it showed no failure in dialectical constructiveness. In the
history of logic it is of importance because of its production of a whole
series of commentators on the Aristotelian logic. Not only the
Introduction of Porphyry, which had lasting effects on the Scholastic
tradition, but the commentaries of Themistius, and Simplicius. It was
the acceptance of the Aristotelian logic by Neoplatonism that
determined the Aristotelian complexion of the logic of the next age. If
Alexander is responsible for such doctrines as that of the intellectus
acquisitus, it is to Porphyry, with his characteristically Platonist
preference for the doctrine of universals, and for classification, that we
owe the scholastic preoccupation with the realist controversy, and with
the quinque voces, i.e. the Aristotelian predicables as restated by
Porphyry.

B. Scholasticism

The living force in the spiritual life of the Roman empire was, after all,
not philosophy, but religion, and specifically Christianity. With the

extension of Christianity to the Gentile world it at length became necessary
for it to orientate itself towards what was best in Greek culture. There is a
Stoic element in the ethic of the Pauline epistles, but the theological affinity
that the Johannine gospel, with its background of philosophic ideas,
exhibits to Platonic and Neoplatonist teaching caused the effort at
absorption to be directed rather in that direction. Neoplatonism had
accepted the Aristotelian logic with its sharper definition than anything
handed down from Plato, and, except the logic of the Sceptics, there was no
longer any rival discipline of the like prestige. The logic of the Stoics had
been discredited by the sceptical onset, but in any case there was no
organon of a fitness even comparable to Aristotle’s for the task of drawing
out the implications of dogmatic premises. Aristotelian logic secured the
imprimatur of the revived Platonism, and it was primarily because of this
that it passed into the service of Christian theology. The contact of the
Church with Platonism was on the mystical side. Orthodoxy needed to
counter heretical logic not with mysticism, itself the fruitful mother of
heresies, but with argument. Aristotelianism approved itself as the
controversial instrument, and in due course held the field alone. The upshot
is what is called Scholasticism. Scholasticism is the Aristotelianism of
medieval orthodoxy as taught in the “schools” or universities of Western
Europe. It takes form as a body of doctrine drawing its premises from
authority, sometimes in secular matters from that of Aristotle, but normally
from that of the documents and traditions of systematic theology, while its
method it draws from Aristotle, as known in the Latin versions,94 mainly by
Boethius, of some few treatises of the Organon together with the Isagoge of
Porphyry. It dominates the centres of intellectual life in the West because,
despite its claim to finality in its principles or premises, and to universality
for its method, it represents the only culture of a philosophic kind available
to the adolescent peoples of the Western nations just becoming conscious of
their ignorance. Christianity was the one organizing principle that pulsed

with spiritual life. The vocation of the student could find fulfilment only in
the religious orders. Scholasticism embodied what the Christian community
had saved from the wreckage of Greek dialectic. Yet with all its effective
manipulation of the formal technique of its translated and mutilated
Aristotle, Scholasticism would have gone under long before it did through
the weakness intrinsic to its divorce of the form and the matter of
knowledge, but for two reasons. The first is the filtering through of some
science and some new Aristotelian learning from the Arabs. The second is
the spread of Greek scholarship and Greek manuscripts westward, which
was consequent on the Latin occupation of Constantinople in 1204. It was
respited by the opportunity which was afforded it of fresh draughts from the
Aristotle of a less partial and purer tradition, and we have, accordingly, a
golden age of revived Scholasticism beginning in the 13th century,
admitting now within itself more differences than before. It is to the
schoolmen of the two centuries preceding the Turkish capture of
Constantinople that the controversial refinements usually associated with
the name of Scholasticism are attributable. The Analytics of Aristotle now
entered quite definitely into the logical thought of Scholasticism and we
have the contrast of a logica vetus and logica nova. That other matters, the
parva logicalia and Mnemonics adapted from Psellus and possibly of Stoic
origin, entered too did not outweigh this advantage. Confrontation with the
historical Aristotle may have brought but little comfort to the orthodox
system, but it was a stimulus to dialectical activity within the schools. It
provoked the distinction of what was true secundum fidem and what was
true secundum rationem among even sincere champions of orthodoxy, and
their opponents accepted with a smile so admirable a mask for that thinking
for themselves to which the revival of hope of progress had spurred them.
The pioneers of the Renaissance owe something of their strength to their
training in the developments which the system that they overthrew
underwent during this period. The respite, however, was short. The flight of

Byzantine scholarship westward in the 15th century revealed, and finally,
that the philosophic content of the Scholastic teaching was as alien from
Aristotle as from the spirit of the contemporary revolt of science, with its
cry for a new medicine, a new nautical astronomy and the like. The doom of
the Scholastic Aristotle was nevertheless not the rehabilitation of the Greek
Aristotle. Between him and the tide of feeling at the Renaissance lay the
whole achievement of Arab science. That impatience of authority to which
we owe the Renaissance, the Reformation and the birth of Nationalism, is
not stilled by the downfall of Aristotle as the nomen appellativum of the
schools. The appeal is to experience, somewhat vaguely defined, as against
all authority, to the book of nature and no other. At last the world
undertakes to enlarge the circle of its ideas.

C. The Renaissance

Accordingly what is in one sense the revival of classical learning is in
another a recourse to what inspired that learning, and so is a new beginning.
There is no place for a reformed Aristotelian logic, though the genius of
Zabarella was there to attempt it. Nor for revivals of the competing systems,
though all have their advocates. Scientific discovery was in the air. The
tradition of the old world was too heavily weighted with the Ptolemaic
astronomy and the like to be regarded as other than a bar to progress. But
from the new point of view its method was inadequate too, its contentment
with an induction that merely leaves an opponent silent, when experiment
and the application of a calculus were within the possibilities. The
transformation of logic lay with the man of science, hindered though he
might be by the enthusiasm of some of the philosophers of nature.
Henceforth the Aristotelian logic, the genuine no less than the traditional,
was to lie on the other side of the Copernican change.

The demand is for a new organon, a scientific method which shall face
the facts of experience and justify itself by its achievement in the reduction
of them to control. It is a notable feature of the new movement, that except
verbally, in a certain licence of nominalist expression, due to the swing of
the pendulum away from the realist doctrine of universals, there is little that
we can characterize as Empiricism. Facts are opposed to abstract universals.
Yes. Particulars to controlling formulae. No. Experience is appealed to as
fruitful where the formal employment of syllogism is barren. But it is not
mere induction, with its “unanalysed concretes taken as ultimate” that is set
up as the substitute for deduction. Rather a scientific process, which as
experiential may be called inductive, but which is to other regards
deductive as syllogism, is set up in contrast to syllogism and enumeration
alike. This is to be seen in Zabarella,95 in Galilei,96 and in Bacon. The
reformed Aristotelian logic of the first-named with its inductio
demonstrativa, the mathematico-physical analysis followed by synthesis of
the second, the exclusiva, or method of exclusions of the last, agree at least
in this, that the method of science is one and indivisible, while containing
both an inductive and a deductive moment. That what, e.g., Bacon says of
his method may run counter to this is an accident of the tradition of the
quarrel with realism. So, too, with the scholastic universals. Aristotle’s
forms had been correlated, though inadequately, with the idea of function.
Divorced from this they are fairly stigmatized as mental figments or
branded as ghostly entities that can but block the path. But consider Bacon’s
own doctrine of forms. Or watch the mathematical physicist with his
formulae. The faith of science looks outward as in the dawn of Greek
philosophy, and subjectivism such as Hume’s has as yet no hold. Bacon
summing up the movement so far as he understood it, in a rather belated
way, has no theory of knowledge beyond the metaphor of the mirror held up
to nature. Yet he offers an ambitious logic of science, and the case is typical.

The science of the Renaissance differs from that of the false dawn in
Greek times in the fact of fruitfulness. It had the achievement of the old
world in the field of mathematics upon which to build. It was in reaction
against a dialectic and not immediately to be again
Galilei. entrapped. In scientific method, then, it could but
advance, provided physics and mathematics did not
again fail of accord. Kepler and Galilei secured it against that disaster. The
ubi materia ibi geometria of the one is the battle-cry of the mathematico-
physical advance. The scientific instrument of the other, with its moments
of analysis and construction, metodo risolutivo and metodo compositivo,
engineers the road for the advance. The new method of physics is verifiable
by its fruitfulness, and so free of any immediate danger from dialectic. Its
germinal thought may not have been new, but, if not new, it had at least
needed rediscovery from the beginning. For it was to be at once certain and
experiential. A mathematico-physical calculus that would work was in
question. The epistemological problem as such was out of the purview. The
relation of physical laws to the mind that thought them was for the time a
negligible constant. When Descartes, having faithfully and successfully
followed the mathematico-physical inquiry of his more strictly scientific
predecessors, found himself compelled to raise the question how it was
possible for him to know what in truth he seemed to know so certainly, the
problem entered on a new phase. The scientific movement had happily been
content for the time with a half which, then and there at least, was more
than the whole.

Bacon was no mathematician, and so was out of touch with the main
army of progress. By temperament he was rather with the Humanists. He
was content to voice the cry for the overthrow of the dominant system as
such, and to call for a new beginning, with no realist
Bacon. presuppositions. He is with the nominalists of the later
Scholasticism and the naturalists of the early

Renaissance. He echoes the cry for recourse to nature, for induction, for
experiment. He calls for a logic of discovery. But at first sight there is little
sign of any greater contribution to the reconstruction than is to be found in
Ramus or many another dead thinker. The syllogism is ineffective,
belonging to argumentation, and constraining assent where what we want is
control of things. It is a mechanical combination of propositions as these of
terms which are counters to express concepts often ill-defined. The flight
from a cursory survey of facts to wide so-called principles must give way to
a gradual progress upward from propositions of minimum to those of
medium generality, and in these consists the fruitfulness of science. Yet the
induction of the Aristotelians, the dialectical induction of the Topics,
content with imperfect enumeration and with showing the burden of
disproof upon the critic, is puerile, and at the mercy of a single instance to
the contrary. In all this there is but little promise for a new organon. It is
neither novel nor instrumental. On a sudden Bacon’s conception of a new
method begins to unfold itself. It is inductive only in the sense that it is
identical in purpose with the ascent from particulars. It were better called
exclusiva or elimination of the alternative, which Bacon proposes to
achieve, and thereby guarantee his conclusion against the possibility of
instance to the contrary.

Bacon’s method begins with a digest into three tables of the facts
relevant to any inquiry. The first contains cases of the occurrence of the
quality under investigation, colour, e.g., or heat, in varying
combinations. The second notes its absence in
His three Methods. combinations so allied to certain of these that its
presence might fairly have been looked for. The
third registers its quantitative variation according to quantitative
changes in its concomitants. The method now proceeds on the basis of
the first table to set forth the possible suggestions as to a general
explanatory formula for the quality in question. In virtue of the

remaining tables it rejects any suggestion qualitatively or quantitatively
inadequate. If one suggestion, and one alone, survives the process of
attempted rejection it is the explanatory formula required. If none, we
must begin afresh. If more than one, recourse is to be had to certain
devices of method, in the enumeration of which the methods of
agreement, difference and concomitant variations97 find a place, beside
the crucial experiment, the glaring instance and the like. An appeal,
however, to such devices, though a permissible “first vintage” is
relatively an imperfection of method, and a proof that the tables need
revision. The positive procedure by hypothesis and verification is
rejected by Bacon, who thinks of hypothesis as the will o’ the wisp of
science, and prefers the cumbrous machinery of negative reasoning.

Historically he appears to have been under the dominance of the
Platonic metaphor of an alphabet of nature, with a consequent belief in
the relatively small number of ultimate principles to be determined, and
of Plato’s conception of Division, cleared of its dialectical associations
and used experientially in application to his own molecular physics.
True it is that the rejection of all the cospecies is a long process, but
what if therein their simultaneous or subsequent determination is
helped forward? They, too, must fall to be determined sometime, and
the ideal of science is fully to determine all the species of the genus.
This will need co-operative effort as described in the account of
Solomon’s House in the New Atlantis.98 But once introduce the
conception of division of labour as between the collector of data on the
one hand and the expert of method, the interpreter of nature at
headquarters, on the other, and Bacon’s attitude to hypothesis and to
negative reasoning is at least in part explained. The hypothesis of the
collector, the man who keeps a rain-gauge, or the missionary among
savages, is to be discounted from as a source of error. The expert on the
other hand may be supposed, in the case of facts over which he has not

himself brooded in the course of their acquisition, to approach them
without any presumption this way or that. He will, too, have no interest
in the isolation of any one of several co-ordinate inquiries. That Bacon
underestimates the importance of selective and of provisional
explanatory hypotheses even in such fields as that of chemistry, and
that technically he is open to some criticism from the point of view that
negative judgment is derivate as necessarily resting on positive
presuppositions, may be true enough. It seems, however, no less true
that the greatness of his conception of organized common effort in
science has but rarely met with due appreciation.

In his doctrine of forms, too, the “universals” of his logic, Bacon
must at least be held to have been on a path which led forward and not
back. His forms are principles whose function falls entirely within
knowledge. They are formulae for the control of the
Forms. activities and the production of the qualities of
bodies. Forms are qualities and activities expressed
in terms of the ultimates of nature, i.e. normally in terms of
collocations of matter or modes of motion. (The human soul is still an
exception.) Form is bound up with the molecular structure and change
of structure of a body, one of whose qualities or activities it expresses
in wider relations. A mode of motion, for instance, of a certain definite
kind, is the form of heat. It is the recipe for, and at the same time is,
heat, much as H2O is the formula for and is water. Had Bacon analysed
bodies into their elements, instead of their qualities and ways of
behaviour, he would have been the logician of the chemical formula.
Here, too, he has scarcely received his meed of appreciation.

His influence on his successors has rather lain in the general
stimulus of his enthusiasm for experience, or in the success with which
he represents the cause of nominalism and in certain special devices of

method handed down till, through Hume or Herschel, they affected the
thought of Mill. For the rest he was too Aristotelian, if we take the
word broadly enough, or, as the result of his Cambridge studies, too
Ramist,99 when the interest in scholastic issues was fading, to bring his
original ideas to a successful market.

Bacon’s Logic, then, like Galilei’s, intended as a contribution to
scientific method, a systematization of discovery by which, given the
fact of knowledge, new items of knowledge may be acquired, failed to
convince contemporaries and successors alike of its efficiency as an
instrument. It was an ideal that failed to embody itself and justify itself
by its fruits. It was otherwise with the mathematical instrument of
Galilei.

Descartes stands in the following of Galilei. It is concurrently with signal
success in the work of a pioneer in the mathematical advance that he comes
to reflect on method, generalizes the method of mathematics to embrace
knowledge as a whole, and raises the ultimate issues of
Descartes. its presuppositions. In the mathematics we determine
complex problems by a construction link by link from
axioms and simple data clearly and distinctly conceived. Three moments
are involved. The first is an induction, i.e. an exhaustive enumeration of the
simple elements in the complex phenomenon under investigation. This
resolution or analysis into simple, because clear and distinct, elements may
be brought to a standstill again and again by obscurity and indistinctness,
but patient and repeated revision of all that is included in the problem
should bring the analytic process to fruition. It is impatience, a perversity of
will, that is the cause of error. Upon the analysis there results intuition of
the simple data. With Descartes intuition does not connote givenness, but its
objects are evident at a glance when induction has brought them to light.
Lastly we have deduction the determination of the most complex

phenomena by a continuous synthesis or combination of the simple
elements. Synthesis is demonstrative and complete. It is in virtue of this
view of derived or mediate knowledge that Descartes speaks of the
(subsumptive) syllogism as “of avail rather in the communication of what
we already know.” Syllogism is not the synthesis which together with
analysis goes to constitute the new instrument of science. The celebrated
Regulae of Descartes are precepts directed to the achievement of the new
methodological ideal in any and every subject matter, however reluctant.

It is the paradox involved in the function of intuition, the acceptance of
the psychological characters of clearness and distinctness as warranty of a
truth presumed to be trans-subjective, that leads to Descartes’s distinctive
contribution to the theory of knowledge. In order to lay bare the ground of
certainty he raises the universal doubt, and, although, following
Augustine,100 he finds its limit in the thought of the doubter, this of itself is
not enough. Cogito, ergo sum. That I think may be admitted. What I think
may still need validation. Descartes’s guarantee of the validity of my clear
and distinct perceptions is the veracity of God.101 Does the existence of
God in turn call for proof? An effect cannot contain more than its cause, nor
the idea of a perfect Being find adequate source save in the actuality of such
a Being. Thus the intuition of the casual axiom is used to prove the
existence of that which alone gives validity to intuitions. Though the logical
method of Descartes has a great and enduring influence, it is the dualism
and the need of God to bridge it, the doctrine of “innate” ideas, i.e. of ideas
not due to external causes nor to volition but only to our capacity to think,
our disposition to develop them, and finally the ontological proof, that
affect the thought of the next age most deeply. That essence in the supreme
case involves existence is a thought which comes to Spinoza more easily,
together with the tradition of the ordo geometricus.

D. Modern Logic

i. The Logic of Empiricism

The path followed by English thought was a different one. Hobbes
developed the nominalism which had been the hallmark of revolt against
scholastic orthodoxy, and, when he brings this into relation with the
analysis and synthesis of scientific method, it is at the expense of the
latter.102 Locke, when Cartesianism had raised the problem of the contents
of consciousness, and the spirit of Baconian positivism could not accept of
anything that bore the ill-omened name of innate ideas, elaborated a theory
of knowledge which is psychological in the sense that its problem is how
the simple data with which the individual is in contact in sensation are
worked up into a system. Though he makes his bow to mathematical
method, he, even more than Hobbes, misses its constructive character. The
clue of mathematical certainty is discarded in substance in the English form
of “the new way of ideas.”

With Hobbes logic is a calculus of marks and signs in the form of names.
Naming is what distinguishes man from the brutes. It enables him to fix
fleeting memories and to communicate with his fellows. He alone is
capable of truth in the due conjunction or disjunction of
Hobbes. names in propositions. Syllogism is simply summation of
propositions, its function being communication merely.
Analysis is the sole way of invention or discovery. There is more, however,
in Hobbes, than the paradox of nominalism. Spinoza could draw upon him
for the notion of genetic definition.103 Leibnitz probably owes to him the
thought of a calculus of symbols, and the conception of demonstration as
essentially a chain of definitions.104 His psychological account of
syllogism105 is taken over by Locke. Hume derived from him the
explanatory formula of the association of ideas,106 which is, however, still

with Hobbes a fact to be accounted for, not a theory to account for facts,
being grounded physically in “coherence of the matter moved.” Finally Mill
took from him his definition of cause as sum of conditions,107 which played
no small part in the applied logic of the 19th century.

Locke is of more importance, if not for his logical doctrine, at least for
the theory of knowledge from which it flows. With Locke the mind is
comparable to white paper on which the world of things records itself in
ideas of sensation. Simple ideas of sensation are the only
Locke. points of contact we have with things. They are the
atomic elements which “the workmanship of the
understanding” can thereafter do no more than systematically compound
and the like. It is Locke’s initial attribution of the primary rôle in mental
process to the simple ideas of sensation that precludes him from the
development of the conception of another sort of ideas, or mental contents
that he notes, which are produced by reflection on “the operations of our
own mind within us.” It is in the latter group that we have the explanation
of all that marks Locke as a forerunner of the critical philosophy. It contains
in germ a doctrine of categories discovered but not generated in the
psychological processes of the individual. Locke, however, fails to
“deduce” his categories. He has read Plato’s Theaetetus in the light of
Baconian and individualist preconceptions. Reflection remains a sort of
“internal sense,” whose ideas are of later origin than those of the external
sense. His successors emphasize the sensationist elements, not the
workmanship of the mind. When Berkeley has eliminated the literal
materialism of Locke’s metaphors of sense-perception, Hume finds no
difficulty in accepting the sensations as present virtually in their own right,
any nonsensible ground being altogether unknown. From a point of view
purely subjectivist he is prepared to explain all that is to be left standing of
what Locke ascribes to the workmanship of the mind by the principle of
association or customary conjunction of ideas, which Locke had added a

chapter to a later edition of his Essay explicitly to reject as an explanatory
formula. Condillac goes a step farther, and sees no necessity for the
superstructure at all, with its need of explanation valid or invalid. Drawing
upon Gassendi for his psychological atomism and upon Hobbes for a
thoroughgoing nominalism, he reproduces, as the logical conclusion from
Locke’s premises, the position of Antisthenes. The last word is that “une
science bien traitée n’est qu’une langue bien faite.”108

Locke’s logic comprises, amid much else, a theory of general terms109
and of definition, a view of syllogism110 and a declaration as to the
possibility of inference from particular to particular,111 a distinction
between propositions which are certain but trifling, and those which add to
our knowledge though uncertain, and a doctrine of mathematical
certainty.112 As to the first, “words become general by being made the signs
of general ideas, and ideas become general by separating from them” all
“that may determine them to this or that particular existence. By this way of
abstraction they are made capable of representing more individuals than
one.” This doctrine has found no acceptance. Not from the point of view for
which idea means image. Berkeley, though at length the notions of spirits,
acts and relations113 give him pause, prefers the formula which Hume
expresses in the phrase that “some ideas are particular in their nature but
general in their representation,”114 and the after-history of “abstraction” is a
discussion of the conditions under which one idea “stands for” a group. Not
from those for whom general ideas mean schematic concepts, not
imageable. The critic from this side has little difficulty in showing that
abstraction of the kind alleged still leave the residuum particular this
redness, e.g. not redness. It is, however, of the sorts constituted by the
representation which his abstraction makes possible that definition is given,
either by enumeration of the simple ideas combined in the significance of
the sortal name, or “to save the labour of enumerating,” and “for quickness

and despatch sake,” by giving the next wider general name and the
proximate difference. We define essences of course in a sense, but the
essences of which men talk are abstractions, “creatures of the
understanding.” Man determines the sorts or nominal essences, nature the
similitudes. The fundamentally enumerative character of the process is
clearly not cancelled by the recognition that it is possible to abbreviate it by
means of technique. So long as the relation of the nominal to the real
essence has no other background than Locke’s doctrine of perception, the
conclusion that what Kant afterwards calls analytical judgments a priori and
synthetic judgments a posteriori exhaust the field follows inevitably, with
its corollary, which Locke himself has the courage to draw, that the natural
sciences are in strictness impossible. Mathematical knowledge is not
involved in the same condemnation, solely because of the “archetypal”
character, which, not without indebtedness to Cumberland, Locke attributes
to its ideas. The reality of mathematics, equally with that of the ideals of
morals drawn from within, does not extend to the “ectypes” of the outer
world. The view of reasoning which Locke enunciates coheres with these
views. Reasoning from particular to particular, i.e. without the necessity of
a general premise, must be possible, and the possibility finds warranty in a
consideration of the psychological order of the terms in syllogism. As to
syllogism specifically, Locke in a passage,115 which has an obviously
Cartesian ring, lays down four stages or degrees of reasoning, and points
out that syllogism serves us in but one of these, and that not the all-
important one of finding the intermediate ideas. He is prepared readily to
“own that all right reasoning may be reduced to Aristotle’s forms of
syllogism,” yet holds that “a man knows first, and then he is able to prove
syllogistically.” The distance from Locke to Stuart Mill along this line of
thought is obviously but small.

Apart from the adoption by Hume of the association of ideas as the
explanatory formula of the school—it had been allowed by Malebranche

within the framework of his mysticism and employed by Berkeley in his
theory of vision—there are few fresh notes struck in the
Hume. logic of sensationalism. The most notable of these are
Berkeley’s treatment of “abstract” ideas and Hume’s
change of front as to mathematical certainty. What, however, Hume
describes as “all the logic I think proper to employ in my reasoning,” viz.
his “rules by which to judge cause and effects,”116 had, perhaps, farther-
reaching historical effects than either. In these the single method of Bacon
is already split up into separate modes. We have Mill’s inductive methods in
the germ, though with an emphasis quite older than Mill’s. Bacon’s form
has already in transmission through Hobbes been transmuted into cause as
antecedent in the time series. It may, perhaps, be accounted to Hume for
righteousness that he declares—whether consistently or not is another
matter—that “the same effect never arises but from the same cause,” and
that he still follows Bacon in the conception of absentia in proximo. It is
“when in any instance we find our expectation disappointed” that the effect
of one of “two resembling objects” will be like that of the other that Hume
proposes to apply his method of difference.

No scientific discipline, however, with the doubtful exception of
descriptive psychology, stands to gain anything from a temper like that of
Hume. The whittling away of its formal or organizing rubrics, as e.g.,
sameness into likeness, is disconcerting to science wherever the
significance of the process is realized. It was because the aftermath of
Newtonian science was so rich that the scientific faith of naturalism was
able to retain a place besides its epistemological creed that a logician of the
school could arise whose spirit was in some sort Baconian, but who, unlike
Bacon, had entered the modern world, and faced the problems stated for it
by Hume and by Newton.

Stuart Mill’s System of Logic marked a fresh stage in the history of
empiricism, for the reason that it made the effort to hold an even balance
between the two moments in the thought of the school. Agreement in the
use of a common watchword had masked as it seems a
J. S. Mill. real divergence of meaning and purpose. The apostles of
inductive method had preached recourse to experience,
but had meant thereby nature as a constituted order. They had devised
canons for the investigation of the concrete problems of this, but had either
ignored altogether the need to give an account of the mirroring mind, or, in
the alternative had been, with some naïveté, content to assume that their
nominalist friends, consistently their allies in the long struggle with
traditionalism, had adequately supplied or could adequately supply the
need. The exponents of psychological atomism, on the other hand, with the
association of ideas for their one principle of agglutination had come to
mean by experience the mental phantasmagoria of the individual. They had
undermined the foundations of scientific certainty, and so far as the
fecundity of contemporary science did not give them pause, were ready,
notwithstanding the difference of their starting-point, to acquiesce in the
formula as well as the temper of Pyrrhonism. They could concede the
triumphant achievement of science only with the proviso that it must be
assumed to fall within the framework of their nominalism. Mill aspired
after a doctrine of method such as should satisfy the needs of the natural
sciences, notably experimental physics and chemistry as understood in the
first half of the 19th century and, mutatis mutandis, of the moral sciences
naturalistically construed. In uniting with this the Associationism which he
inherited, through his father, from Hume, he revealed at once the strength
and weakness of the dual conception of naturalism. His rare thoroughness
and rarer candour made it at once unnecessary and impossible that the work
should be done again.

If judged by what he denies, viz. the formal logic of Hamilton and
Mansel, whose Aristotelian and scholastic learning did but accentuate their
traditionalism, and whose acquiescence in consistency constituted in Mill’s
view a discouragement of research, such as men now incline to attribute at
the least equally to Hume’s idealism, Mill is only negatively justified. If
judged by his positive contribution to the theory of method he may claim to
find a more than negative justification for his teaching in its success. In the
field covered by scholastic logic Mill is frankly associationist. He aims at
describing what he finds given, without reference to insensible implications
of doubtful validity and value. The upshot is a psychological account of
what from one aspect is evidence, from the other, belief. So he explains
“concepts or general notions”117 by an abstraction which he represents as a
sort of alt-relief operated by attention and fixed by naming, association with
the name giving to a set of attributes a unity they otherwise lack. This is
manifestly, when all is said, a particular psychological event, a collective
fact of the associative consciousness. It can exercise no organizing or
controlling function in knowledge. So again in determining the “import” of
propositions, it is no accident that in all save existential propositions it is to
the familiar rubrics of associationism—co-existence, sequence, causation
and resemblance—that he refers for classification, while his general
formula as to the conjunctions of connotations is associationist through and
through. It follows consistently enough that inference is from particular to
particular. Mill holds even the ideas of mathematics to be hypothetical, and
in theory knows nothing of a non-enumerative or non-associative universal.
A premise that has the utmost universality consistent with this view can
clearly be of no service for the establishment of a proposition that has gone
to the making of it. Nor again of one that has not. Its use, then, can only be
as a memorandum. It is a shorthand formula of registration. Mill’s view of
ratiocinative process clearly stands and falls with the presumed
impossibility of establishing the necessity for universals of another type

than his, for what may be called principles of construction. His critics
incline to press the point that association itself is only intelligible so far as it
is seen to depend on universals of the kind that he denies.

In Mill’s inductive logic, the nominalistic convention has, through his
tendency to think in relatively watertight compartments,118 faded somewhat
into the background. Normally he thinks of what he calls phenomena no
longer as psychological groupings of sensations, as “states of mind,” but as
things and events in a physical world howsoever constituted and
apprehended. His free use of relating concepts, that of sameness, for
instance, bears no impress of his theory of the general notion, and it is
possible to put out of sight the fact that, taken in conjunction with his
nominalism, it raises the whole issue of the possibility of the equivocal
generation of formative principles from the given contents of the individual
consciousness, in any manipulation of which they are already implied.
Equally, too, the deductive character, apparently in intention as well as in
actual fact, of Mill’s experimental methods fails to recall the point of theory
that the process is essentially one from particular to particular. The nerve of
proof in the processes by which he establishes causal conjunctions of
unlimited application is naturally thought to lie in the special canons of the
several processes and the axioms of universal and uniform causation which
form their background. The conclusions seem not merely to fall within, but
to depend on these organic and controlling formulae. They follow not
merely according to them but from them. The reference to the rule is not
one which may be made and normally is made as a safeguard, but one
which must be made, if thought is engaged in a forward and constructive
movement at all. Yet Mill’s view of the function of “universal” propositions
had been historically suggested by a theory—Dugald Stewart’s—of the use
of axioms!119 Once more, it would be possible to forget that Mill’s ultimate
laws or axioms are not in his view intuitions, nor forms constitutive of the
rational order, nor postulates of all rational construction, were it not that he

has made the endeavour to establish them on associationist lines. It is
because of the failure of this endeavour to bring the technique of induction
within the setting of his Humian psychology of belief that the separation of
his contribution to the applied logic of science from his sensationism
became necessary, as it happily was easy. Mill’s device rested special
inductions of causation upon the laws that every event has a cause, and
every cause has always the same effect. It rested these in turn upon a
general induction enumerative in character of enormous and practically
infinite range and always uncontradicted. Though obviously not exhaustive,
the unique extent of this induction was held to render it competent to give
practical certainty or psychological necessity. A vicious circle is obviously
involved. It is true, of course, that ultimate laws need discovery, that they
are discovered in some sense in the medium of the psychological
mechanism, and that they are nevertheless the grounds of all specific
inferences. But that truth is not what Mill expounds, nor is it capable of
development within the limits imposed by the associationist formula.

It is deservedly, nevertheless, that Mill’s applied logic has retained its
pride of place amid what has been handed on, if in modified shape, by
writers, e.g., Sigwart, and Professor Bosanquet, whose theory of knowledge
is quite alien from his. He prescribed regulative or limiting formulae for
research as it was actually conducted in his world. His grasp of the
procedure by which the man of science manipulated his particular concrete
problems was admirable. In especial he showed clear understanding of the
functions of hypothesis and verification in the investigations of the solitary
worker, with his facts still in course of accumulation and needing to be
lighted up by the scientific imagination. He was therefore enabled to
formulate the method of what Bacon had tended to despise as merely the
“first vintage.” Bacon spent his strength upon a dream of organization for
all future discovery. Mill was content to codify. The difference between
Bacon and Mill lies chiefly in this, and it is because of this difference that

Mill’s contribution, spite of its debt to the Baconian tradition, remains both
characteristic and valuable. It is of course possible to criticise even the
experimental canons with some severity. The caveats, however, which are
relevant within the circle of ideas within which Mill’s lesson can be learned
and improved on,120 seem to admit of being satisfied by relatively slight
modifications in detail, or by explanations often supplied or easily to be
supplied from points brought out amid the wealth of illustration with which
Mill accompanied his formal or systematic exposition of method. The critic
has the right of it when he points out, for example, that the practical
difficulty in the Method of Agreement is not due to plurality of causes, as
Mill states, but rather to intermixture of effects, while, if the canon could be
satisfied exactly, the result would not be rendered uncertain in the manner
or to the extent which he supposes. Again the formula of the Joint-Method,
which contemplates the enumeration of cases “which have nothing in
common but the absence of one circumstance,” is ridiculously unsound as it
stands. Or, on rather a different line of criticism, the use of corresponding
letters in the two series of antecedents and consequents raises, it is said, a
false presumption of correlation. Nay, even the use of letters at all suggests
that the sort of analysis that actually breaks up its subject-matter is
universally or all but universally applicable in nature, and this is not the
case. Finally, the conditions of the methods are either realized or not. If they
are realized, the work of the scientist falls entirely within the field of the
processes preliminary to the satisfaction of the canon. The latter becomes a
mere memorandum or formula of registration. So is it possible “to have the
enginer hoist with his own petar.” But the conditions are not realized, and in
an experiential subject-matter are not realizable. Not one circumstance only
in common but “apparently one relevant circumstance only in common” is
what we are able to assert. If we add the qualification of relevance we
destroy the cogency of the method. If we fail to add it, we destroy the
applicability.

The objections turn on two main issues. One is the exaggeration of the
possibilities of resolution into separate elements that is due to the
acceptance of the postulate of an alphabet of nature. This so soon as noted
can be allowed for. It is to the combination of this doctrine with a tendency
to think chiefly of experiment, of the controlled addition or subtraction of
these elements one at a time, that we owe the theoretically premature
linking of a as effect to A as cause. This too can be met by a modification of
form. The other issue is perhaps of more significance. It is the oscillation
which Mill manifests between the conception of his formula as it is actually
applicable to concrete problems in practice, and the conception of it as an
expression of a theoretical limit to practical procedure. Mill seems most
often to think of the former, while tending to formulate in terms of the
latter. At any rate, if relevance in proximo is interpolated in the peccant
clause of the canon of the Joint-Method, the practical utility of the method
is rehabilitated. So too, if the canon of the Method of Agreement is never
more than approximately satisfied, intermixture of effects will in practice
mean that we at least often do not know the cause or antecedent equivalent
of a given effect, without the possibility of an alternative. Finally, it is on
the whole in keeping with Mill’s presuppositions to admit even in the case
of the method of difference that in practice it is approximative and
instructive, while the theoretical formula, to which it aims at approaching
asymptotically as limit, if exact, is in some sense sterile. Mill may well
have himself conceived his methods as practically fruitful and normally
convincing with the limiting formula in each case more cogent in form but
therewith merely the skeleton of the process that but now pulsed with life.

Enough has been said to show why the advance beyond the letter of Mill
was inevitable while much in the spirit of Mill must necessarily affect
deeply all later experientialism. After Mill experientialism takes essentially
new forms. In part because of what Mill had done. In part also because of
what he had left undone. After Mill means after Kant and Hegel and

Herbart, and it means after the emergence of evolutionary naturalism. Mill,
then, marks the final stage in the achievement of a great school of thought.

ii. The Logic of Rationalism.

A fundamental contrast to the school of Bacon and of Locke is afforded
by the great systems of reason, owning Cartesian inspiration, which are
identified with the names of Spinoza and Leibnitz. In the history of logic
the latter thinker is of the more importance. Spinoza’s
Spinoza. philosophy is expounded ordine geometrico and with
Euclidean cogency from a relatively small number of
definitions, axioms and postulates. But how we reach our assurance of the
necessity of these principles is not made specifically clear. The invaluable
tractate De Intellectus emendatione, in which the agreement with and
divergence from Descartes on the question of method could have been fully
elucidated, is unhappily not finished. We know that we need to pass from
what Spinoza terms experientia vaga,121 where imagination with its
fragmentary apprehension is liable to error and neither necessity nor
impossibility can be predicated, right up to that which fictionem terminat—
namely, intellectio. And what Spinoza has to say of the requisites of
definition and the marks of intellection makes it clear that insight comes
with coherence, and that the work of method on the “inductive” side is by
means of the unravelling of all that makes for artificial limitation to lay bare
what can then be seen to exhibit nexus in the one great system. When all is
said, however, the geometric method as universalized in philosophy is
rather used by Spinoza than expounded.

With Leibnitz, on the other hand, the logical problem holds the foremost
place in philosophical inquiry.122 From the purely logical thesis, developed
at quite an early stage of his thinking,123 that in any true proposition the
predicate is contained in the subject, the main principles of his doctrine of

Monads are derivable with the minimum of help from his
Leibnitz. philosophy of dynamics. Praedicatum inest subjecto. All
valid propositions express in the last resort the relation of
predicate or predicates to a subject, and this Leibnitz holds after considering
the case of relational propositions where either term may hold the position
of grammatical subject, A = B and the like. There is a subject then, or there
are subjects which must be recognized as not possible to be predicated, but
as absolute. For reasons not purely logical Leibnitz declares for the plurality
of such subjects. Each contains all its predicates: and this is true not only in
the case of truths of reason, which are necessary, and ultimately to be
exhibited as coming under the law of contradiction, “or, what comes to the
same thing, that of identity,” but also in the case of truths of fact which are
contingent, though a sufficient reason can be given for them which
“inclines” without importing necessity. The extreme case of course is the
human subject. “The individual notion of each person includes once for all
what is to befall it, world without end,” and “it would not have been our
Adam but another, if he had had other events.” Existent subjects, containing
eternally all their successive predicates in the time-series, are substances,
which when the problems connected with their activity, or dynamically
speaking their force, have been resolved, demand—and supply—the
metaphysic of the Monadology.

Complex truths of reason or essence raise the problem of definition,
which consists in their analysis into simpler truths and ultimately into
simple—i.e. indefinable ideas, with primary principles of another kind—
axioms, and postulates that neither need nor admit of proof. These are
identical in the sense that the opposite contains an express contradiction.124
In the case of non-identical truths, too, there is a priori proof drawn from
the notion of the terms, “though it is not always in our power to arrive at
this analysis,”125 so that the question arises, specially in connexion with the
possibility of a calculus, whether the contingent is reducible to the

necessary or identical at the ideal limit. With much that suggests an
affirmative answer, Leibnitz gives the negative. Even in the case of the
Divine will, though it be always for the best possible, the sufficient reason
will “incline without necessitating.” The propositions which deal with
actual existence are still of a unique type, with whatever limitation to the
calculus.

Leibnitz’s treatment of the primary principles among truths of reason as
identities, and his examples drawn inter alia from the “first principles” of
mathematics, influenced Kant by antagonism. Identities some of them
manifestly were not. The formula of identity passed in another form to
Herbart and therefore to Lotze. In recognizing, further, that the relation of
an actual individual fact to its sufficient ground was not reducible to
identity, he set a problem diversely treated by Kant and Herbart. He brought
existential propositions, indeed, within a rational system through the
principle that it must be feasible to assign a sufficient reason for them, but
he refused to bring them under the conception of identity or necessity, i.e. to
treat their opposites as formally self-contradictory. This bore interest in the
Kantian age in the treatment alike of cause and effect, and of the ontological
proof of existence from essence. Not that the Law of Sufficient Reason is
quite free from equivoque. Propositions concerning the possible existence
of individuals put Leibnitz to some shifts, and the difficulty accounts for the
close connexion established in regard to our actual world between the law
of sufficient reason and the doctrine of the final cause. This connexion is
something of an afterthought to distinguish from the potential contingency
of the objectively possible the real contingency of the actual, for which the
“cause or reason” of Spinoza126 could not account. The law, however, is not
invalidated by these considerations, and with the degree of emphasis and
the special setting that Leibnitz gives the law, it is definitely his own.

If we may pass by the doctrine of the Identity of Indiscernibles, which
played a part of some importance in subsequent philosophy, and the Law of
Continuity, which as Leibnitz represents it is, if not sheer dogma, reached
by something very like a fallacy, we have as Leibnitz’s remaining legacy to
later logicians the conception of Characteristica Universalis and Ars
Combinatoria, a universal denoting by symbols and a calculus working by
substitutions and the like. The two positions that a subject contains all its
predicates and that all non-contingent propositions—i.e. all propositions not
concerned with the existence of individual facts ultimately analyse out into
identities—obviously lend themselves to the design of this algebra of
thought, though the mathematician in Leibnitz should have been aware that
a significant equation is never an identity. Leibnitz, fresh from the battle of
the calculus in the mathematical field, and with his conception of logic, at
least in some of its aspects, as a generalized mathematic,127 found a fruitful
inspiration, harmonizing well with his own metaphysic, in Bacon’s alphabet
of nature. He, too, was prepared to offer a new instrument. That the most
important section, the list of forms of combination, was never achieved—
this too was after the Baconian example while the mode of symbolization
was crude with a = ab and the like—matters little. A new technique of
manipulation—it is, of course, no more—had been evolved.

It may be said that among Leibnitz’s successors there is no Leibnitzian.
The system as a whole is something too artificial to secure whole-hearted
allegiance. Wolff’s formalism is the bastard outcome of the speculation of
Leibnitz, and is related to it as remotely as Scholasticism is to Aristotle.
Wolff found a sufficient reason for everything and embodied the results of
his inquiries in systematic treatises, sometimes in the vernacular. He also,
by a transparent petitio principii, brought the law of the sufficient reason
under that of non-contradiction. Wolff and his numerous followers account
for the charge of dogmatism against “the Leibnitzio-Wolffian school.” They
are of importance in the history of logic for two reasons only: they affected

strongly the German vocabulary of philosophy and they constituted the
intellectual environment in which Kant grew to manhood.

A truer continuator of Leibnitz in the spirit was Herbart.

iii. Kant’s Logic.

Herbart’s admitted allegiance, however, was Kantian with the
qualification, at a relatively advanced stage of his thinking, that it was “of
the year 1828”—that is, after controversy had brought out implications of
Kant’s teaching not wholly contemplated by Kant himself. The critical
philosophy had indeed made it impossible to hark back to Leibnitz or any
other master otherwise than with a difference.

Yet it is not a single and unambiguous logical movement that derives
from Kant. Kant’s lesson was variously understood. Different moments in it
were emphasized, with a large diversity of result. As interpreted it was
acquiesced in or revolted from and revolt ranged from a desire for some
modifications of detail or expression to the call for a radical transformation.
Grounds for a variety of developments are to be found in the imperfect
harmonization of the rationalistic heritage from the Wolffian tradition which
still dominates Kant’s pure general logic with the manifest epistemological
intention of his transcendental theory. Or again, within the latter in his
admission of a duality of thought and “the given” in knowledge, which
within knowledge was apparently irreducible, concurrently with hints as to
the possibility, upon a wider view, of the sublation of their disparateness at
least hypothetically and speculatively. The sense in which there must be a
ground of the unity of the supersensible128 while yet the transcendent use of
Reason—i.e. its use beyond the limits of experience was denied theoretical
validity—was not unnaturally regarded as obscure.

Kant’s treatment of technical logic was wholly traditional, and in itself is
almost negligible. It is comprised129 in an early essay on the mistaken
subtlety of the syllogistic figures, and a late compilation by a pupil from the
introductory matter and running annotations with which
Formal Logic. the master had enriched his interleaved lecture-room
copy of Meyer’s Compendium of 1752. Wolff’s general
logic, “the best,” said Kant, “that we possess,” had been abridged by
Baumgarten and the abridgment then subjected to commentation by Meyer.
With this traditional body of doctrine Kant was, save for matters of minor
detail, quite content. Logic was of necessity formal, dealing as it must with
those rules without which no exercise of the understanding would be
possible at all. Upon abstraction from all particular methods of thought
these rules were to be discerned a priori or without dependence on
experience by reflection solely upon the use of the understanding in
general. The science of the form of thought abstracted in this way from its
matter or content was regarded as of value both as propaedeutic and as
canon. It was manifestly one of the disciplines in which a position of
finality was attainable. Aristotle might be allowed, indeed, to have omitted
no essential point of the understanding, what the moderns had achieved
consisted in an advance in accuracy and methodical completeness. “Indeed,
we do not require any new discoverers in logic,”130 said the discoverer of a
priori synthesis, “since it contains merely the form of thought.” Applied
logic is merely psychology, and not properly to be called logic at all. The
technical logic of Kant, then, justifies literally a movement among his
successors in favour of a formal conception of logic with the law of
contradiction and the doctrine of formal implication for its equipment.
Unless the doctrine of Kant’s “transcendental logic” must be held to supply
a point of view from which a logical development of quite another kind is
inevitable, Kant’s mantle, so far as logic is concerned, must be regarded as
having fallen upon the formal logicians.

Kant’s transcendental teaching is summarily as follows:
“Transcendental” is his epithet for what is neither empirical—i.e. to be
derived from experience—nor yet transcendent—i.e. applicable beyond the
limits of experience, the mark of experience being the
Definition of implication of sense or of something which thought
“Transcendental.” contra-distinguishes from its own spontaneous activity as
in some sense “the given.” Those features in our
organized experience are to be regarded as transcendentally established
which are the presuppositions of our having that experience at all. Since
they are not empirical they must be structural and belong to “the mind”—
i.e. the normal human intelligence, and to like intelligence so far as like. If
we set aside such transcendental conditions as belong to sensibility or to the
receptive phase of mind and are the presuppositions of juxtaposition of
parts, the remainder are ascribable to spontaneity or understanding, to
thought with its unifying, organizing or focussing function, and their
elucidation is the problem of transcendental analytic. It is still logic, indeed,
when we are occupied with the transcendent objects of the discursive
faculty as it is employed beyond the limits of experience where it cannot
validate its ideas. Such a logic, however, is a dialectic of illusion, perplexed
by paralogisms and helpless in the face of antinomies. In transcendental
analytic on the other hand we concern ourselves only with the
transcendental “deduction” or vindication of the conditions of experience,
and we have a logic of cognition in which we may establish our
epistemological categories with complete validity. Categories are the forms
according to which the combining unity of self-consciousness (synthetic
unity of apperception) pluralizes itself through the various functions
involved in the constitution of objectivity in different types of the one act of
thought, viz. judgment. The clue to the discovery of transcendental
conditions Kant finds in the existence of judgments, most manifest in
mathematics and in the pure science of nature, which are certain, yet not

trifling, necessary and yet not reducible to identities, synthetic therefore and
a priori, and so accounted for neither by Locke nor by Leibnitz. “There lies
a transcendental condition at the basis of every necessity.”

Kant’s mode of conceiving the activity of thought in the constitution of
objects and of their connexion in experience was thought to lie open to an
interpretation in conformity with the spirit of his logic, in the sense that the
form and the content in knowledge are not merely
Form of Matter of distinguishable functions within an organic whole, but
Thought. either separable, or at least indifferent one to the other in
such a way as to be clearly independent. Thought as form
would thus be a factor or an element in a composite unit. It would clearly
have its own laws. It would be the whole concern of logic, which, since in it
thought has itself for object, would have no reference to the other term of
the antithesis, nor properly and immediately to the knowledge which is
compact of thought in conjunction with something which, whatever it may
be, is prima facie other than thought. There is too much textual warrant for
this interpretation of Kant’s meaning. Doubtless there are passages which
make against an extreme dualistic interpretation. Even in his “logic” Kant
speaks of abstraction from all particular objects of thought rather than of a
resolution of concrete thinking into thought and its “other” as separable co-
operating factors in a joint product. He spoke throughout, however, as if
form and content were mutually indifferent, so that the abstraction of form
from content implied nothing of falsification or mutilation. The reserve,
therefore, that it was abstraction and not a decomposing that was in
question remained to the admirers of his logic quite nugatory. They failed to
realize that permissible abstraction from specific contents or methods of
knowledge does not obliterate reference to matter or content. They passed
easily from the acceptance of a priori forms of thinking to that of forms of a
priori thinking, and could plead the example of Kant’s logic.

Kant’s theory of knowledge, then, needed to be pressed to other
consequences for logic which were more consonant with the spirit of the
Critique. The forms of thought and what gives thought its particular content
in concrete acts of thinking could not be regarded as subsisting in a purely
external and indifferent relation one to the other. “Laws according to which
the subject thinks” and “laws according to which the object is known”
cannot be the concern of separate departments of inquiry. As soon divorce
the investigation of the shape and material of a mirror from the laws of the
incidence of the rays that form images in it, and call it a science of
reflection! An important group of writers developed the conception of an
adaptation between the two sides of Kant’s antithesis, and made the
endeavour to establish some kind of correlation between logical forms and
the process of “the given.” There was a tendency to fall back upon the
conception of some kind of parallelism, whether it was taken to be
interpretative or rather corrective of Kant’s meaning. This device was never
remote from the constructions of writers for whom the teaching of Spinoza
and Leibnitz was an integral part of their intellectual equipment. Other
modes of correlation, however, find favour also, and in some variety. Kant
is seldom the sole source of inspiration. His unresolved antithesis131 is
interpreted either diversely or with a difference of emphasis. And the light
that later writers bring to bear on Kant’s logic and epistemology from other
sides of his speculation varies in kind and in degree.

Another logical movement springs from those whom a correlation of fact
within the unity of a system altogether failed to satisfy. There must also be
development of the correlated terms from a single principle. Form and
content must not only correspond one to the other. They must be exhibited
as distinguishable moments within a unity which can at one and the same
time be seen to be the ground from which the distinction springs and the
ground in virtue of which it is over-ruled. Along this line of speculation we
have a logic which claims that whatsoever is in one plane or at one stage in

the development of thought a residuum that apparently defies analysis must
at another stage and on a higher plane be shown so to be absorbed as to fall
altogether within thought. This is the view of Hegel upon which logic
comes to coincide with the progressive self-unfolding of thought in that
type of metaphysic which is known as absolute, i.e. all-inclusive idealism.
The exponent of logic as metaphysic, for whom the rational is the real is
necessarily in revolt against all that is characteristically Kantian in the
theory of knowledge, against the transcendental method itself and against
the doctrine of limits which constitutes the nerve of “criticism.” Stress was
to be laid upon the constructive character of the act of thought which Kant
had recognized, and without Kant’s qualifications of it. In all else the claim
is made to have left the Kantian teaching behind as a cancelled level of
speculation.

Transcendental method is indeed not invulnerable. A principle is
transcendentally “deduced” when it and only it can explain the validity of
some phase of experience, some order of truths. The order of truths, the
phase of experience and its certainty had to be taken for
Limitation of granted. The sense, for example, in which the
Transcendental irreversibility of sequence which is the more known in
Method. ordine ad hominem in the case of the causal principle
differs from merely psychological conviction is not made
fully clear. Even so the inference to the a priori ground of its necessity is, it
has been often pointed out, subject to the limitation inherent in any process
of reduction, in any regress, that is, from conditionate to condition, viz. that
in theory an alternative is still possible. The inferred principle may hold the
field as explanation without obvious competitor potential or actual.
Nevertheless its claim to be the sole possible explanation can in nowise be
validated. It has been established after all by dialectic in the Aristotelian
sense of the word. But if transcendental method has no special pride of
place, Kant’s conclusion as to the limits of the competence of intellectual

faculty falls with it. Cognition manifestly needs the help of Reason even in
its theoretical use. Its speculation can no longer be stigmatized as
vaticination in vacuo, nor its results as illusory.

Finally, to logic as metaphysic the polar antithesis is psychology as logic.
The turn of this also was to come again. If logic were treated as merely
formal, the stress of the problem of knowledge fell upon the determination
of the processes of the psychological mechanism. If
Logic and alleged a priori constituents of knowledge—such rubrics
Psychology. as substance, property, relation—come to be explained
psychologically, the formal logic that has perforce to
ignore all that belongs to psychology is confined within too narrow a range
to be able to maintain its place as an independent discipline, and tends to be
merged in psychology. This tendency is to be seen in the activity of Fries
and Herbart and Beneke, and was actualized as the aftermath of their
speculation. It is no accident that it was the psychology of apperception and
the voluntaryist theory or practice of Herbart, whose logical theory was so
closely allied to that of the formal logicians proper, that contributed most to
the development of the post-Kantian psychological logic. Another
movement helped also; the exponents of naturalistic evolution were
prepared with Spencer to explain the so-called a priori in knowledge as in
truth a posteriori, if not to the individual at any rate to the race. It is of
course a newer type of psychological logic that is in question, one that is
aware of Kant’s “answer to Hume.” Stuart Mill, despite of his relation of
antagonism to Hamilton and Mansel, who held themselves to be Kantian in
spirit, is still wholly pre-Kantian in his outlook.

Kant’s influence, then, upon subsequent logic is least of all to be
measured by his achievement in his professed contribution to technical
logic. It may be attributed in some slight degree, perhaps, to incidental
flashes of logical insight where his thought is least of what he himself calls

logic, e.g. his exposition of the significance of synthetic
Summary. judgments a priori, or his explanation of the function of
imagery in relation to thought, whereby he offers a
solution of the problem of the conditions under which one member of a
group unified through a concept can be taken to stand for the rest, or again
the way in which he puts his finger on the vital issue in regard to the alleged
proof from essence to existence, and illustrations could be multiplied. But
much more it belongs to his transformation of the epistemological problem,
and to the suggestiveness of his philosophy as a whole for an advance in the
direction of a speculative construction which should be able to cancel all
Kant’s surds, and in particular vindicate a “ground of the unity of the
supersensible which lies back of nature with that which the concept of
freedom implies in the sphere of practice,”132 which is what Kant finally
asserts.

iv. After Kant.

Starting from the obvious antithesis of thought and that of which it is the
thought, it is possible to view the ultimate relation of its term as that of
mutual indifference or, secondly, as that of a correspondence such that
while they retain their distinct character modification of the one implies
modification of the other, or thirdly and lastly, as that of a mergence of one
in the other of such a nature that the merged term, whichever it be, is fully
accounted for in a complete theory of that in which it is merged.

The first way is that of the purely formal logicians, of whom Twesten133
and in England H. L. Mansel may be regarded as typical. They take thought
and “the given” as self-contained units which, if not in fact separable, are at
any rate susceptible of an abstraction the one from the other so decisive as
to constitute an ideal separation. The laws of the pure activity of thought
must be independently determined, and since the contribution of thought to

knowledge is form they must be formal only. They
The Formal cannot go beyond the limits of formal consistency or
Logicians. analytic correctness. They are confined to the
determination of what the truth of any matter of thought,
taken for granted upon grounds psychological or other, which are
extraneous to logic, includes or excludes. The unit for logic is the concept
taken for granted. The function of logic is to exhibit its formal implications
and repulsions. It is questionable whether even this modest task could be
really achieved without other reference to the content abstracted from than
Mansel, for example, allows. The analogy of the resolution of a chemical
compound with its elements which is often on the lips of those who would
justify the independence of thought and the real world, with an agnostic
conclusion as to non-phenomenal or trans-subjective reality, is not really
applicable. The oxygen and hydrogen, for example, into which water may
be resolved are not in strictness indifferent one to the other, since both are
members of an order regulated according to laws of combination in definite
ratios. Or, if applicable, it is double-edged. Suppose oxygen to be found
only in water. Were it to become conscious, would it therefore follow that it
could infer the laws of a separate or independent activity of its own?
Similarly forms of thinking, the law of contradiction not excepted, have
their meaning only in reference to determinate content, even though
distributively all determinate contents are dispensable. The extreme
formalist is guilty of a fallacy of composition in regard to abstraction.

It does not follow, however, that the laws asserted by the formal logicians
are invalid or unimportant. There is a permissible abstraction, and in
general they practise this, and although they narrow its range unduly, it is
legitimately to be applied to certain characters of thinking. As the living
organism includes something of mechanism—the skeleton, for example—
so an organic logic doubtless includes determinations of formal consistency.
The skeleton is meaningless apart from reference to its function in the life

of an organism, yet there are laws of skeleton structure which can be
studied with most advantage if other characters of the organism are
relegated to the background. To allow, however, that abstraction admits of
degrees, and that it never obliterates all reference to that from which it is
abstracted, is to take a step forward in the direction of the correlation of
logical forms with the concrete processes of actual thinking. What was true
in formal logic tended to be absorbed in the correlationist theories.

Those formal logicians of the Kantian school, then, may be summarily
dismissed, though their undertaking was a necessary one, who failed to
raise the epistemological issue at all, or who, raising it, acquiesced in a
naïve dualism agnostic of the real world as Kant’s essential lesson. They
failed to develop any view which could serve either in fact or in theory as a
corrective to the effect of their formalism. What they said with justice was
said as well or better elsewhere.

Among them it is on the whole impossible not to include the names of
Hamilton and Mansel. The former, while his erudition in respect to the
history of philosophical opinion has rarely been equalled, was not a clear
thinker. His general theory of knowledge deriving from Kant and Reid, and
including among other things a contaminatio of their theories of
perception,134 in no way sustains or mitigates his narrow view of logic. He
makes no effective use of his general formula that to think is to condition.
He appeals to the direct testimony of consciousness in the sense in which
the appeal involves a fallacy. He accepts an ultimate antinomy as to the
finiteness or infinity of “the unconditioned,” yet applies the law of the
excluded middle to insist that one of the two alternatives must be true,
wherefore we must make the choice. And what is to be said of the judgment
of a writer who considers the relativity of thought demonstrated by the fact
that every judgment unites two members? Hamilton’s significance for the
history of logic lies in the stimulus that he gave to the development of

symbolic logic in England by his new analytic based upon his discovery or
adoption of the principle of the quantification of the predicate. Mansel, too,
was learned, specially in matters of Aristotelian exegesis, and much that is
of value lies buried in his commentation of the dry bones of the Artis
Logicae Rudimenta of Locke’s contemporary Aldrich. And he was a clearer
thinker than Hamilton. Formal logic of the extremest rigour is nowhere to
be found more adequately expressed in all its strength, and it must be added
in all its weakness, than in the writings of Mansel. But if the view
maintained above that formal logic must compromise or mitigate its rigour
and so fail to maintain its independence, be correct, the logical consistency
of Mansel’s logic of consistency does but emphasize its barrenness. It
contains no germ for further development. It is the end of a movement.

The brief logic of Herbart135 is altogether formal too. Logical forms have
for him neither psychological nor metaphysical reference, we are concerned
in logic solely with the systematic clarification of concepts which are
wholly abstract, so that they are not merely not ultimate
Herbart. realities, but also in no sense actual moments of our
concrete thinking. The first task of logic is to distinguish
and group such concepts according to their marks, and from their
classification there naturally follows their connexion in judgment. It is in
the logic of judgment that Herbart inaugurates a new era. He is not, of
course, the first to note that even categorical judgments do not assert the
realization of their subject. That is a thought which lies very near the
surface for formal logic. He had been preceded too by Maimon in the
attempt at a reduction of the traditional types of judgment. He was,
however, the first whose analysis was sufficiently convincing to exorcise
the tyranny of grammatical forms. The categorical and disjunctive judgment
reduce to the hypothetical. By means of the doctrine of the quantification of
the predicate, in which with his Leibnitzian conception of identity he
anticipated Beneke and Hamilton alike, universal and particular judgments

are made to pull together. Modal, impersonal, existential judgments are all
accounted for. Only the distinction of affirmative and negative judgments
remains unresolved, and the exception is a natural one from the point of
view of a philosophy of pluralism. There was little left to be done here save
in the way of an inevitable mutatis mutandis, even by Lotze and F. H.
Bradley. From the judgment viewed as hypothetical we pass by affirmation
of the antecedent or denial of the consequent to inference. This point of
departure is noteworthy, as also is the treatment of the inductive syllogism
as one in which the middle term is resoluble into a group or series (Reihe).
In indicating specifically, too, the case of conclusion from a copulative
major premise with a disjunctive minor, Herbart seems to have suggested
the cue for Sigwart’s exposition of Bacon’s method of exclusions.

That it was the formal character of Herbart’s logic which was ultimately
fatal to its acceptance outside the school as an independent discipline is not
to be doubted. It stands, however, on a different footing from that of the
formal logic hitherto discussed, and is not to be condemned upon quite the
same grounds. In the first place, Herbart is quite aware of the nature of
abstraction. In the second, there is no claim that thought at one and the
same time imposes form on “the given” and is susceptible of treatment in
isolation by logic. With Herbart the forms of common experience, and
indeed all that we can regard as his categories, are products of the
psychological mechanism and destitute of logical import. And lastly,
Herbart’s logic conforms to the exigencies of his system as a whole and the
principle of the bare or absolute self-identity of the ultimate “reals” in
particular. It is for this reason that it finally lacks real affinity to the “pure
logic” of Fries. For at the basis of Herbart’s speculation there lies a
conception of identity foreign to the thought of Kant with his stress on
synthesis, in his thoroughgoing metaphysical use of which Herbart goes
back not merely to Wolff but to Leibnitz. It is no mere coincidence that his
treatment of all forms of continuance and even his positive metaphysic of

“reals” show affinity to Leibnitz. It was in the pressing to its extreme
consequences of the conception of uncompromising identity which is to be
found in Leibnitz, that the contradictions took their rise which Herbart
aimed at solving, by the method of relations and his doctrine of the ultimate
plurality of “reals.” The logic of relations between conceptual units,
themselves unaltered by the relation, seems a kind of reflection of his
metaphysical method. To those, of course, for whom the only real identity is
identity in difference, while identity without difference, like difference
without identity, is simply a limit or a vanishing point, Herbart’s logic and
metaphysic will alike lack plausibility.

The setting of Herbart’s logic in his thought as a whole might of itself
perhaps justify separate treatment. His far-reaching influence in the
development of later logic must certainly do so. Directly he affected a
school of thought which contained one logician of first-rate importance in
Moritz Wilhelm Drobisch (1802-1896), professor at Leipzig. In less direct
relation stands Lotze, who, although under other influences he developed a
different view even in logic, certainly let no point in the doctrine of his
great predecessor at Göttingen escape him. A Herbartian strain is to be met
with also in the thought of writers much further afield, for example F. H.
Bradley, far though his metaphysic is removed from Herbart’s. Herbart’s
influence is surely to be found too in the evolution of what is called
Gegenstandstheorie. Nor did he affect the logic of his successors through
his logic alone. Reference has been made above to the effect upon the rise
of the later psychological logic produced by Herbart’s psychology of
apperception, when disengaged from the background of his metaphysic
taken in conjunction with his treatment in his practical philosophy of the
judgment of value or what he calls the aesthetic judgment. Emerson’s
verdict upon a greater thinker—that his was “not a mind to nestle in”—may
be true of Herbart, but there can be no doubt as to the stimulating force of
this master.

The second way of interpreting the antithesis of thought to what is
thought of, was taken by a group of thinkers among whom a central and
inspiring figure was Schleiermacher. They in no sense constitute a school
and manifest radical differences among themselves. They
Logic as the are agreed, however, in the rejection, on the one hand, of
rationale of the subjectivist logic with its intrinsic implication that
knowledge. knowledge veils rather than reveals the real world, and,
on the other hand, of the logic of the speculative
construction with its pretension to “deduce,” to determine, and finally at
once to cancel and conserve any antithesis in its all-embracing dialectic.
They agree, then, in a maintenance of the critical point of view, while all
alike recognize the necessity of bringing the thought-function in knowledge
into more intimate relation with its “other” than Kant had done, by means
of some formula of correlation or parallelism. Such an advance might have
taken its cue directly from Kant himself. As an historical fact it tended
rather to formulate itself as a reaction towards Kant in view of the course
taken by the speculative movement. Thus Schleiermacher’s posthumously
published Dialektik (1839) may be characterized as an appeal from the
absolutist element in Schelling’s philosophy to the conception of that
correlation or parallelism which Schelling had exhibited as flowing from
and subsisting within his absolute, and therein as a return upon Kant’s
doctrine of limits. Schleiermacher’s conception of
Schleiermacher. dialectic is to the effect that it is concerned with the
principles of the art of philosophizing, as these are
susceptible of a relatively independent treatment by a permissible
abstraction. Pure thinking or philosophizing is with a view to philosophy or
knowledge as an interconnected system of all sciences or departmental
forms of knowledge, the mark of knowledge being its identity for all
thinking minds. Dialectic then investigates the nexus which must be held to
obtain between all thoughts, but also that agreement with the nexus in being

which is the condition of the validity of the thought-nexus. In knowing
there are two functions involved, the “organic” or animal function of
sensuous experience in virtue of which we are in touch with being, directly
in inner perception, mediately in outer experience, and the “intellectual”
function of construction. Either is indispensable, though in different
departments of knowledge the predominant rôle falls to one or other, e.g.
we are more dependent in physics, less so in ethics. The idea of a perfect
harmony of thinking and being is a presupposition that underlies all
knowing but cannot itself be realized in knowledge. In terms of the
agreement of thought and being, the logical forms of the part of dialectic
correspondent to knowledge statically considered have parallels and
analogies in being, the concept being correlated to substance, the judgment
to causal nexus. Inference, curiously enough, falls under the technical side
of dialectic concerned with knowledge in process or becoming, a line of
cleavage which Ueberweg has rightly characterized as constituting a rift
within Schleiermacher’s parallelism.

Schleiermacher’s formula obviously ascribes a function in knowledge to
thought as such, and describes in a suggestive manner a duality of the
intellectual and organic functions, resting on a parallelism of thought and
being whose collapse into identity it is beyond human capacity to grasp. It
is rather, however, a statement of a way in which the relations of the terms
of the problem may be conceived than a system of necessity. It may indeed
be permitted to doubt whether its influence upon subsequent theory would
have been a great one apart from the spiritual force of Schleiermacher’s
personality. Some sort of correlationist conception, however, was an
inevitable development, and the list136 of those who accepted it in
something of the spirit of Schleiermacher is a long one and contains many
distinguished names, notably those of Trendelenburg and Ueberweg. The
group is loosely constituted however. There was scope for diversity of view
and there was diversity of view, according as the vital issue of the formula

was held to lie in the relation of intellectual function to organic function or
in the not quite equivalent relation of thinking to being. Moreover, few of
the writers who, whatsoever it was that they baptized with the name of
logic, were at least earnestly engaged in an endeavour to solve the problem
of knowledge within a circle of ideas which was on the whole Kantian,
were under the dominance of a single inspiration. Beneke’s philosophy is a
striking instance of this, with application to Fries and affinity to Herbart
conjoined with obligations to Schelling both directly and through
Schleiermacher. Lotze again wove together many threads of earlier thought,
though the web was assuredly his own. Finally it must not be forgotten that
the host of writers who were in reaction against Hegelianism tended to take
refuge in some formula of correlation, as a half-way house between that and
formalism or psychologism or both, without reference to, and often perhaps
without consciousness of, the way in which historically it had taken shape
to meet the problem held to have been left unresolved by Kant.

Lotze on the one hand held the Hegelian “deduction” to be untenable,
and classed himself with those who in his own phrase “passed to the order
of the day,” while on the other hand he definitely raised the question, how
an “object” could be brought into forms to which it was
Lotze. not in some sense adapted. Accordingly, though he
regards logic as formal, its forms come into relation to
objectivity in some sort even within the logical field itself, while when
taken in the setting of his system as a whole, its formal character is not of a
kind that ultimately excludes psychological and metaphysical reference, at
least speculatively. As a logician Lotze stands among the masters. His flair
for the essentials in his problem, his subtlety of analysis, his patient
willingness to return upon a difficulty from a fresh and still a fresh point of
view, and finally his fineness of judgment, make his logic137 so essentially
logic of the present, and of its kind not soon to be superseded, that nothing

more than an indication of the historical significance of some of its
characteristic features need be attempted here.

In Lotze’s pure logic it is the Herbartian element that tends to be
disconcerting. Logic is formal. Its unit, the logical concept, is a manipulated
product and the process of manipulation may be called abstraction.
Processes of the psychological mechanism lie below it. The paradox of the
theory of judgment is due to the ideal of identity, and the way in which this
is evaded by supplementation to produce a non-judgmental identity,
followed by translation of the introduced accessories with conditions in the
hypothetical judgment, is thoroughly in Herbart’s manner. The reduction of
judgments is on lines already familiar. Syllogism is no instrumental method
by which we compose our knowledge, but an ideal to the form of which it
should be brought. It is, as it were, a schedule to be filled in, and is
connected with the disjunctive judgment as a schematic setting forth of
alternatives, not with the hypothetical, and ultimately the apodictic
judgment with their suggestion that it is the real movement of thought that
is subjected to analysis. Yet the resultant impression left by the whole
treatment is not Herbartian. The concept is accounted for in Kantian terms.
There is no discontinuity between the pre-logical or sub-logical conversion
of impressions into “first universals” and the formation of the logical
concept. Abstraction proves to be synthesis with compensatory universal
marks in the place of the particular marks abstracted from. Synthesis as the
work of thought always supplies, beside the mere conjunction or disjunction
of ideas, a ground of their coherence or non-coherence. It is evident that
thought, even as dealt with in pure logic, has an objectifying function. Its
universals have objective validity, though this does not involve direct real
reference. The formal conception of pure logic, then, is modified by Lotze
in such a way as not only to be compatible with a view of the structural and
functional adequacy of thought to that which at every point at which we
take thinking is still distinguishable from thought, but even inevitably to

suggest it. That the unit for logic is the concept and not the judgment has
proved a stumbling-block to those of Lotze’s critics who are accustomed to
think in terms of the act of thought as unit. Lotze’s procedure is, indeed,
analogous to the way in which, in his philosophy of nature, he starts from a
plurality of real beings, but by means of a reductive movement, an
application of Kant’s transcendental method, arrives at the postulate or fact
of a law of their reciprocal action which calls for a monistic and idealist
interpretation. He starts, that is in logic, with conceptual units apparently
self-contained and admitting of nothing but external relation, but proceeds
to justify the intrinsic relation between the matter of his units by an appeal
to the fact of the coherence of all contents of thought. Indeed, if thought
admits irreducible units, what can unite? Yet he is left committed to his
puzzle as to a reduction of judgment to identity, which partially vitiates his
treatment of the theory of judgment. The outstanding feature of this is,
nevertheless, not affected, viz. the attempt that he makes, inspired clearly
by Hegel, “to develop the various forms of judgment systematically as
members of a series of operations, each of which leaves a part of its
problem unmastered and thereby gives rise to the next.”138 As to inference,
finally, the ideal of the articulation of the universe of discourse, as it is for
complete knowledge, when its disjunctions have been thoroughly followed
out and it is exhaustively determined, carried the day with him against the
view that the organon for gaining knowledge is syllogism. The Aristotelian
formula is “merely the expression, formally expanded and complete, of the
truth already embodied in disjunctive judgment, namely, that every S which
is a specific form of M possesses as its predicate a particular modification
of each of the universal predicates of M to the exclusion of the rest.”
Schleiermacher’s separation of inference from judgment and his attribution
of the power to knowledge in process cannot find acceptance with Lotze.
The psychologist and the formal logician do indeed join hands in the denial
of a real movement of thought in syllogism. Lotze’s logic then, is formal in

a sense in which a logic which does not find the conception of synthetic
truth embarrassing is not so. It is canon and not organon. In the one case,
however, where it recognizes what is truly synthesis, i.e. in its account of
the concept, it brings the statics of knowledge, so to speak, into integral
relation with the dynamics. And throughout, wherever the survival from
1843, the identity bug-bear, is for the moment got rid of in what is really a
more liberal conception, the statical doctrine is developed in a brilliant and
informing manner. Yet it is in the detail of his logical investigations,
something too volatile to fix in summary, that Lotze’s greatness as a
logician more especially lies.

With Lotze the ideal that at last the forms of thought shall be realized to
be adequate to that which at any stage of actual knowledge always proves
relatively intractable is an illuminating projection of faith. He takes courage
from the reflection that to accept scepticism is to presume the competence
of the thought that accepts. He will, however, take no easy way of
parallelism. Our human thought pursues devious and circuitous methods. Its
forms are not unseldom scaffolding for the house of knowledge rather than
the framework of the house itself. Our task is not to realise correspondence
with something other than thought, but to make explicit those justificatory
notions which condition the form of our apprehension. “However much we
may presuppose an original reference of the forms of thought to that nature
of things which is the goal of knowledge, we must be prepared to find in
them many elements which do not directly reproduce the actual reality to
the knowledge of which they are to lead us.”139 The impulse of thought to
reduce coincidence to coherence reaches immediately only to objectivity or
validity. The sense in which the presupposition of a further reference is to
be interpreted and in which justificatory notions for it can be adduced is
only determinable in a philosophic system as a whole, where feeling has a
place as well as thought, value equally with validity.

Lotze’s logic then represents the statical aspect of the function of thought
in knowledge, while, so far as we go in knowledge thought is always
engaged in the unification of a manifold, which remains contradistinguished
from it, though not, of course, completely alien to and unadapted to it. The
further step to the determination of the ground of harmony is not to be taken
in logic, where limits are present and untranscended.

The position of the search for truth, for which knowledge is a growing
organism in which thought needs, so to speak, to feed on something other
than itself, is conditioned in the post-Kantian period by antagonism to the
speculative movement which culminated in the dialectic
Logic as of Hegel. The radical thought of this movement was
Metaphysic. voiced in the demand of Reinhold140 that philosophy
should “deduce” it all from a single principle and by a
single method. Kant’s limits that must needs be thought and yet cannot be
thought must be thought away. An earnest attempt to satisfy this demand
was made by Fichte whose single principle was the activity of the pure Ego,
while his single method was the assertion of a truth revealed by reflection
on the content of conscious experience, the characterization of this as a half
truth and the supplementation of it by its other, and finally the
harmonization of both. The pure ego is inferred from the fact that the non-
ego is realized only in the act of the ego in positing it. The ego posits itself,
but reflection on the given shows that we must add that it posits also the
non-ego. The two positions are to be conciliated in the thought of reciprocal
limitation of the posited ego and non-ego. And so forth. Fichte cannot be
said to have developed a logic, but this rhythm of thesis, antithesis and
synthesis, foreshadowed in part for Fichte in Spinoza’s formula, “omnis
determinatio est negatio” and significantly in Kant’s triadic grouping of his
categories, gave a cue to the thought of Hegel. Schelling, too, called for a
single principle and claimed to have found it in his Absolute, “the night”
said Hegel, “in which all cows are black,” but his historical influence lay, as

we have seen, in the direction of a parallelism within the unity, and he also
developed no logic. It is altogether otherwise with Hegel.

Hegel’s logic,141 though it involves inquiries which custom regards as
metaphysical, is not to be characterized as a metaphysic with a method. It is
logic or a rationale of thought by thought, with a full development among
other matters of all that the most separatist of logicians
Hegel. regards as thought forms. It offers a solution of what has
throughout appeared as the logical problem. That
solution lies doubtless in the evolution of the Idea, i.e. an all-inclusive in
which mere or pure thought is cancelled in its separateness by a
transfiguration, while logic is nothing but the science of the Idea viewed in
the medium of pure thought. But, whatever else it be, this Panlogismus, to
use the word of J. E. Erdmann, is at least a logic. Thought in its progressive
unfolding, of which the history of philosophy taken in its broad outline
offers a pageant, necessarily cannot find anything external to or alien from
itself, though that there is something external for it is another matter. As
Fichte’s Ego finds that its non-ego springs from and has its home within its
very self, so with Hegel thought finds itself in its “other,” both subsisting in
the Idea which is both and neither. Either of the two is the all, as, for
example, the law of the convexity of the curve is the law of the curve and
the law of its concavity. The process of the development of the Idea or
Absolute is in one regard the immanent process of the all. Logically
regarded, i.e. “in the medium of mere thought,” it is dialectical method. Any
abstract and limited point of view carries necessarily to its contradictory.
This can only be atoned with the original determination by fresh negation in
which a new thought-determination is born, which is yet in a sense the old,
though enriched, and valid on a higher plane. The limitations of this in turn
cause a contradiction to emerge, and the process needs repetition. At last,
however, no swing into the opposite, with its primarily conflicting, if
ultimately complementary function, is any longer possible. That in which

no further contradiction is possible is the absolute Idea. Bare or
indeterminate being, for instance, the first of the determinations of Hegel’s
logic, as the being of that which is not anything determinate, of Kant’s
thing-in-itself, for example, positively understood, implicated at once the
notion of not-being, which negates it, and is one with it, yet with a
difference, so that we have the transition to determinate being, the transition
being baptized as becoming. And so forth. It is easy to raise difficulties not
only in regard to the detail in Hegel’s development of his categories,
especially the higher ones, but also in regard to the essential rhythm of his
method. The consideration that mere double negation leaves us precisely
where we were and not upon a higher plane where the dominant concept is
richer, is, of course, fatal only to certain verbal expressions of Hegel’s
intent. There is a differentiation in type between the two negations. But if
we grant this it is no longer obviously the simple logical operation
indicated. It is inferred then that Hegel complements from the stuff of
experience, and fails to make good the pretension of his method to be by
itself and of itself the means of advance to higher and still higher concepts
till it can rest in the Absolute. He discards, as it were, and takes in from the
stock while professing to play from what he has originally in his hand. He
postulates his unity in senses and at stages in which it is inadmissible, and
so supplies only a schema of relations otherwise won, a view supported by
the way in which he injects certain determinations in the process, e.g. the
category of chemism. Has he not cooked the process in the light of the
result? In truth the Hegelian logic suffers from the fact that the good to be
reached is presupposed in the beginning. Nature, e.g., is not deduced as real
because rational, but being real its rationality is presumed and, very
imperfectly, exhibited in a way to make it possible to conceive it as in its
essence the reflex of Reason. It is a vision rather than a construction. It is a
“theosophical logic.” Consider the rational-real in the unity that must be,
and this is the way of it, or an approximation to the way of it! It was

inevitable that the epistemologists of the search for truth would have none
of it. The ideal in whatsoever sense real still needs to be realized. It is from
the human standpoint regulative and only hypothetically or formally
constitutive. We must not confuse οὐσία with εἶναι, nor εἶναι with
γίγνεσθαι.

Yet in a less ambitious form the fundamental contentions of Hegel’s
method tend to find a qualified acceptance. In any piece of presumed
knowledge its partial or abstract character involves the presence of loose
edges which force the conviction of inadequacy and the development of
contradictions. Contradictions must be annulled by complementation, with
resultant increasing coherence in ascending stages. At each successive stage
in our progress fresh contradictions break out, but the ideal of a station at
which the thought-process and its other, if not one, are at one, is permissible
as a limiting conception. Yet if Hegel meant only this he has indeed
succeeded in concealing his meaning.

Hegel’s treatment of the categories or thought determinations which arise
in the development of the immanent dialectic is rich in flashes of insight,
but most of them are in the ordinary view of logic wholly metaphysical. In
the stage, however, of his process in which he is concerned with the notion
are to be found concept, judgment, syllogism. Of the last he declares that it
“is the reasonable and everything reasonable” (Encyk. § 181), and has the
phantasy to speak of the definition of the Absolute as being “at this stage”
simply the syllogism. It is, of course, the rhythm of the syllogism that
attracts him. The concept goes out from or utters itself in judgment to return
to an enhanced unity in syllogism. Ueberweg (System § 101) is, on the
whole, justified in exclaiming that Hegel’s rehabilitation of syllogism “did
but slight service to the Aristotelian theory of syllogism,” yet his treatment
of syllogism must be regarded as an acute contribution to logical criticism
in the technical sense. He insists on its objectivity. The transition from

judgment is not brought about by our subjective action. The syllogism of
“all-ness” is convicted of a petitio principii (Encyk. § 190), with consequent
lapse into the inductive syllogism, and, finally, since inductive syllogism is
involved in the infinite process, into analogy. “The syllogism of necessity,”
on the contrary, does not presuppose its conclusion in its premises. The
detail, too, of the whole discussion is rich in suggestion, and subsequent
logicians—Ueberweg himself perhaps, Lotze certainly in his genetic scale
of types of judgment and inference, Professor Bosanquet notably in his
systematic development of “the morphology of knowledge,” and others—
have with reason exploited it.

Hegel’s logic as a whole, however, stands and falls not with his thoughts
on syllogism, but with the claim made for the dialectical method that it
exhibits logic in its integral unity with metaphysic, the thought-process as
the self-revelation of the Idea. The claim was disallowed. To the formalist
proper it was self-condemned in its pretension to develop the content of
thought and its rejection of the formula of bare-identity. To the
epistemologist it seemed to confuse foundation and keystone, and to
suppose itself to build upon the latter in a construction illegitimately
appropriative of materials otherwise accumulated. At most it was thought to
establish a schema of formal unity which might serve as a regulative ideal.
To the methodologist of science in genesis it appeared altogether to fail to
satisfy any practical interest. Finally, to the psychologist it spelt the failure
of intellectualism, and encouraged, therefore, some form of rehabilitated
experientialism.

In the Hegelian school in the narrower sense the logic of the master
receives some exegesis and defence upon single points of doctrine rather
than as a whole. Its effect upon logic is rather to be seen in the rethinking of
the traditional body of logical doctrine in the light of an absolute
presupposed as ideal, with the postulate that a regulative ideal must

ultimately exhibit itself as constitutive, the justification of the postulate
being held to lie in the coherence and all-inclusiveness of the result. In such
a logic, if and so far as coherence should be attained, would be found
something akin to the spirit of what Hegel achieves, though doubtless alien
to the letter of what it is his pretension to have achieved. There is perhaps
no serious misrepresentation involved in regarding a key-thought of this
type, though not necessarily expressed in those verbal forms, as pervading
such logic of the present as coheres with a philosophy of the absolute
conceived from a point of view that is intellectualist throughout. All other
contemporary movements may be said to be in revolt from Hegel.

v. Logic from 1880-1910

Logic in the present exhibits, though in characteristically modified
shapes, all the main types that have been found in its past history. There is
an intellectualist logic coalescent with an absolutist metaphysic as
aforesaid. There is an epistemological logic with sometimes formalist,
sometimes methodological leanings. There is a formal-symbolic logic
engaged with the elaboration of a relational calculus. Finally, there is what
may be termed psychological-voluntaryist logic. It is in the rapidity of
development of logical investigations of the third and fourth types and the
growing number of their exponents that the present shows most clearly the
history of logic in the making. All these movements are logic of the present,
and a very brief indication may be added of points of historical significance.

Of intellectualist logic Francis Herbert Bradley142 (b. 1846) and Bernard
Bosanquet143 (1848) may be taken as typical exponents. The philosophy of
the former concludes to an Absolute by the annulment of contradictions,
though the ladder of Hegel is conspicuous by its absence. His metaphysical
method, however, is like Herbart’s, not identifiable with his logic, and the
latter has for its central characteristic its thorough restatement of the logical

forms traditional in language and the text-books, in such a way as to
harmonize with the doctrine of a reality whose organic unity is all-inclusive.
The thorough recasting that this involves, even of the thought of the masters
when it occasionally echoes them, has resulted in a phrasing uncouth to the
ear of the plain man with his world of persons and things in which the
former simply think about the latter, but it is fundamentally necessary for
Bradley’s purpose. The negative judgment, for example, cannot be held in
one and the same undivided act to presuppose the unity of the real, project
an adjective as conceivably applicable to it and assert its rejection. We need,
therefore, a restatement of it. With Bradley reality is the one subject of all
judgment immediate or mediate. The act of judgment “which refers an ideal
content (recognized as such) to a reality beyond the act” is the unit for
logic. Grammatical subject and predicate necessarily both fall under the
rubric of the adjectival, that is, within the logical idea or ideal content
asserted. This is a meaning or universal, which can have no detached or
abstract self-subsistence. As found in judgment it may exhibit differences
within itself, but it is not two, but one, an articulation of unity, not a fusion,
which could only be a confusion, of differences. With a brilliant subtlety
Bradley analyses the various types of judgment in his own way, with results
that must be taken into account by all subsequent logicians of this type. The
view of inference with which he complements it is only less satisfactory
because of a failure to distinguish the principle of nexus in syllogism from
its traditional formulation and rules, and because he is hampered by the
intractability which he finds in certain forms of relational construction.

Bosanquet had the advantage that his logic was a work of a slightly later
date. He is, perhaps, more able than Bradley has shown himself, to use
material from alien sources and to penetrate to what is of value in the
thought of writers from whom, whether on the whole or on particular
issues, he disagrees. He treats the book-tradition, however, a debt to which,
nowadays inevitable, he is generous in acknowledging,144 with a judicious

exercise of freedom in adaptation, i.e. constructively as datum, never
eclectically. In his fundamental theory of judgment his obligation is to
Bradley. It is to Lotze, however, that he owes most in the characteristic
feature of his logic, viz., the systematic development of the types of
judgment, and inference from less adequate to more adequate forms. His
fundamental continuity with Bradley may be illustrated by his definition of
inference. “Inference is the indirect reference to reality of differences within
a universal, by means of the exhibition of this universal in differences
directly referred to reality.”145 Bosanquet’s Logic will long retain its place
as an authoritative exposition of logic of this type.

Of epistemological logic in one sense of the phrase Lotze is still to be
regarded as a typical exponent. Of another type Chr. Sigwart (q.v.) may be
named as representative. Sigwart’s aim was “to reconstruct logic from the
point of view of methodology.” His problem was the claim to arrive at
propositions universally valid, and so true of the object, whosoever the
individual thinker. His solution, within the Kantian circle of ideas, was that
such principles as the Kantian principle of causality were justified as
“postulates of the endeavour after complete knowledge.” “What Kant has
shown is not that irregular fleeting changes can never be the object of
consciousness, but only that the ideal consciousness of complete science
would be impossible without the knowledge of the necessity of all
events.”146 “The universal presuppositions which form the outline of our
ideal of knowledge are not so much laws which the understanding
prescribes to nature ... as laws which the understanding lays down for its
own regulation in its investigation and consideration of nature. They are a
priori because no experience is sufficient to reveal or confirm them in
unconditional universality; but they are a priori ... only in the sense of
presuppositions without which we should work with no hope of success and
merely at random and which therefore we must believe.” Finally they are
akin to our ethical principles. With this coheres his dictum, with its far-

reaching consequences for the philosophy of induction, that “the logical
justification of the inductive process rests upon the fact that it is an
inevitable postulate of our effort after knowledge, that the given is
necessary, and can be known as proceeding from its grounds according to
universal laws.”147 It is characteristic of Sigwart’s point of view that he
acknowledges obligation to Mill as well as to Ueberweg. The transmutation
of Mill’s induction of inductions into a postulate is an advance of which the
psychological school of logicians have not been slow to make use. The
comparison of Sigwart with Lotze is instructive, in regard both to their
agreement and their divergence as showing the range of the epistemological
formula.

Of the formal-symbolic logic all that falls to be said here is, that from the
point of view of logic as a whole, it is to be regarded as a legitimate praxis
as long as it shows itself aware of the sense in which alone form is
susceptible of abstraction, and is aware that in itself it offers no solution of
the logical problem. “It is not an algebra,” said Kant148 of his technical
logic, and the kind of support lent recently to symbolic logic by the
Gegenstandstheorie identified with the name of Alexius Meinong (b.
1853)149 is qualified by the warning that the real activity of thought tends to
fall outside the calculus of relations and to attach rather to the subsidiary
function of denoting. The future of symbolic logic as coherent with the rest
of logic, in the sense which the word has borne throughout its history seems
to be bound up with the question of the nature of the analysis that lies
behind the symbolism, and of the way in which this is justified in the setting
of a doctrine of validity. The “theory of the object,” itself, while affecting
logic alike in the formal and in the psychological conception of it very
deeply, does not claim to be regarded as logic or a logic, apart from a
setting supplied from elsewhere.

Finally we have a logic of a type fundamentally psychological, if it be
not more properly characterized as a psychology which claims to cover the
whole field of philosophy, including the logical field. The central and
organizing principle of this is that knowledge is in genesis, that the genesis
takes place in the medium of individual minds, and that this fact implies
that there is a necessary reference throughout to interests or purposes of the
subject which thinks because it wills and acts. Historically this doctrine was
formulated as the declaration of independence of the insurgents in revolt
against the pretensions of absolutist logic. It drew for support upon the
psychological movement that begins with Fries and Herbart. It has been
chiefly indebted to writers, who were not, or were not primarily, logicians,
to Avenarius, for example, for the law of the economy of thought, to Wundt,
whose system, and therewith his logic,150 is a pendant to his psychology, for
the volitional character of judgment, to Herbert Spencer and others. A
judgment is practical, and not to be divorced without improper abstraction
from the purpose and will that informs it. A concept is instrumental to an
end beyond itself, without any validity other than its value for action. A
situation involving a need of adaptation to environment arises and the
problem it sets must be solved that the will may control environment and be
justified by success. Truth is the improvised machinery that is interjected,
so far as this works. It is clear that we are in the presence of what is at least
an important half-truth, which intellectuallism with its statics of the rational
order viewed as a completely articulate system has tended to ignore. It
throws light on many phases of the search for truth, upon the plain man’s
claim to start with a subject which he knows whose predicate which he does
not know is still to be developed, or again upon his use of the negative form
of judgment, when the further determination of his purposive system is
served by a positive judgment from without, the positive content of which is
yet to be dropped as irrelevant to the matter in hand. The movement has,
however, scarcely developed its logic151 except as polemic. What seems

clear is that it cannot be the whole solution. While man must confront
nature from the human and largely the practical standpoint, yet his control
is achieved only by the increasing recognition of objective controls. He
conquers by obedience. So truth works and is economical because it is
truth. Working is proportioned to inner coherence. It is well that the view
should be developed into all its consequences. The result will be to limit it,
though perhaps also to justify it, save in its claim to reign alone.

There is, perhaps, an increasing tendency to recognize that the organism
of knowledge is a thing which from any single viewpoint must be seen in
perspective. It is of course a postulate that all truths harmonize, but to give
the harmonious whole in a projection in one plane is an undertaking whose
adequacy in one sense involves an inadequacy in another. No human
architect can hope to take up in succession all essential points of view in
regard to the form of knowledge or to logic. “The great campanile is still to
finish.”

Bibliography.—Historical: No complete history of logic in the sense
in which it is to be distinguished from theoretical philosophy in general
has as yet been written. The history of logic is indeed so little
intelligible apart from constant reference to tendencies in philosophical
development as a whole, that the historian, when he has made the
requisite preparatory studies, inclines to essay the more ambitious task.
Yet there are, of course, works devoted to the history of logic proper.

Of these Prantl’s Geschichte der Logik im Abendlande (4 vols.,
1855-1870), which traces the rise, development and fortunes of the
Aristotelian logic to the close of the middle ages, is monumental. Next
in importance are the works of L. Rabus, Logik und Metaphysik, i.
(1868) (pp. 123-242 historical, pp. 453-518 bibliographical, pp. 514
sqq. a section on apparatus for the study of the history of logic), Die
neuesten Bestrebungen auf dem Gebiete der Logik bei den Deutschen
(1880), Logik (1895), especially for later writers § 17. Ueberweg’s
System der Logik und Geschichte der logischen Lehren (4th ed. and last
revised by the author, 1874, though it has been reissued later, Eng.
trans., 1871) is alone to be named with these. Harms’ posthumously
published Geschichte der Logik (1881) (Die Philosophie in ihrer
Geschichte, ii.) was completed by the author only so far as Leibnitz.
Blakey’s Historical Sketch of Logic (1851), though, like all this writer’s
works, closing with a bibliography of some pretensions, is now
negligible. Franck, Esquisse d’une histoire de la logique (1838) is the
chief French contribution to the subject as a whole.

Of contributions towards the history of special periods or schools of
logical thought the list, from the opening chapters of Ramus’s Scholae
Dialecticae (1569) downwards (v. Rabus loc. cit.) would be endless.
What is of value in the earlier works has now been absorbed. The
System der Logik (1828) of Bachmann (a Kantian logician of

distinction) contains a historical survey (pp. 569-644), as does the
Denklehre (1822) of van Calker (allied in thought to Fries) pp. 12 sqq.;
Eberstein’s Geschichte der Logik und Metaphysik bei den Deutschen
von Leibniz bis auf gegenwärtige Zeit (latest edition, 1799) is still of
importance in regard to logicians of the school of Wolff and the
origines of Kant’s logical thought. Hoffmann, the editor and disciple of
von Baader, published Grundzüge einer Geschichte der Begriffe der
Logik in Deutschland von Kant bis Baader (1851). Wallace’s
prolegomena and notes to his Logic of Hegel (1874, revised and
augmented 1892-1894) are of use for the history and terminology, as
well as the theory. Riehl’s article entitled Logik in Die Kultur der
Gegenwart, vi. 1. Systematische Philosophie (1907), is excellent, and
touches on quite modern developments. Liard, Les Logiciens Anglais
Contemporains (5th ed., 1907), deals only with the 19th-century
inductive and formal-symbolic logicians down to Jevons, to whom the
book was originally dedicated. Venn’s Symbolic Logic (1881) gave a
careful history and bibliography of that development. The history of
the more recent changes is as yet to be found only in the form of
unshaped material in the pages of review and Jahresbericht.(H. W. B.*)

1 Cf. Heidel, “The Logic of the Pre-Socratic Philosophy,” in Dewey’s Studies in
Logical Theory (Chicago, 1903).

2 Heraclitus, Fragmm. 107 (Diels, Fragmente der Vorsokratiker) and 2, on which see
Burnet, Early Greek Philosophy, p. 153 note (ed. 2).

3 e.g. Diog. Laërt. ix. 25, from the lost Sophistes of Aristotle.

4 Plato and Platonism, p. 24.

5 Nothing is. If anything is, it cannot be known. If anything is known it cannot be
communicated.

6 Metaphys. μ. 1078b 28 sqq.

7 Cf. Arist. Top. θ. i. 1 ad fin.

8 For whom see Dümmler, Antisthenica (1882, reprinted in his Kleine Schriften, 1901).

9 Aristotle, Metaphys. 1024b 32 sqq.

10 Plato, Theaetetus, 201 E. sqq., where, however, Antisthenes is not named, and the
reference to him is sometimes doubted. But cf. Aristotle, Met. H 3. 1043b 24-28.

11 Diog. Laërt. ii. 107.

12 Aristotle, An. Pr. i. 31, 46a 32 sqq.; cf. 91b 12 sqq.

13 Athenaeus ii. 59c. See Usener, Organisation der wissenschaftl. Arbeit (1884;
reprinted in his Vorträge und Aufsätze, 1907).

14 Socrates’ reference of a discussion to its presuppositions (Xenophon, Mem. iv. 6, 13)
is not relevant for the history of the terminology of induction.

15 Theaetetus, 186c.

16 Timaeus, 37a, b (quoted in H. F. Carlill’s translation of the Theaetetus, p. 60).

17 Theaetetus, 186d.

18 Sophistes, 253d.

19 Ib. id.; cf. Theaetetus, 197d.

20 Aristotle, de An. 430b 5, and generally iii. 2, iii. 5.

21 For Plato’s Logic, the controversies as to the genuineness of the dialogues may be
treated summarily. The Theaetetus labours under no suspicion. The Sophistes is apparently
matter for animadversion by Aristotle in the Metaphysics and elsewhere, but derives
stronger support from the testimonies to the Politicus which presumes it. The Politicus and
Philebus are guaranteed by the use made of them in Aristotle’s Ethics. The rejection of the
Parmenides would involve the paradox of a nameless contemporary of Plato and Aristotle
who was inferior as a metaphysician to neither. No other dialogue adds anything to the
logical content of these.

Granted their genuineness, the relative dating of three of them is given, viz. Theaetetus,
Sophistes and Politicus in the order named. The Philebus seems to presuppose Politicus,
283-284, but if this be an error, it will affect the logical theory not at all. There remains the
Parmenides. It can scarcely be later than the Sophistes. The antinomies with which it
concludes are more naturally taken as a prelude to the discussion of the Sophistes than as
an unnecessary retreatment of the doctrine of the one and the many in a more negative
form. It may well be earlier than the Theaetetus in its present form. The stylistic argument
shows the Theaetetus relatively early. The maturity of its philosophic outlook tends to give
it a place relatively advanced in the Platonic canon. To meet the problem here raised, the

theory has been devised of an earlier and a later version. The first may have linked on to
the series of Plato’s dialogues of search, and to put the Parmenides before it is impossible.
The second, though it might still have preceded the Parmenides might equally well have
followed the negative criticism of that dialogue, as the beginning of reconstruction. For
Plato’s logic this question only has interest on account of the introduction of an
Ἀριστοτέλης in a non-speaking part in the Parmenides. If this be pressed as suggesting
that the philosopher Aristotle was already in full activity at the date of writing, it is of
importance to know what Platonic dialogues were later than the début of his critical pupil.

On the stylistic argument as applied to Platonic controversies Janell’s Quaestiones
Platonicae (1901) is important. On the whole question of genuineness and dates of the
dialogues, H. Raeder, Platons philosophische Entwickelung (1905), gives an excellent
conspectus of the views held and the grounds alleged. See also Plato.

22 E.g. that of essence and accident. Republic, 454.

23 E.g. the discussion of correlation, ib. 437 sqq.

24 Politicus, 285d.

25 Sophistes, 261c sqq.

26 E.g. in Nic. Eth. i. 6.

27 Philebus, 16d.

28 Principal edition still that of Waitz, with Latin commentary, (2 vols., 1844-1846).
Among the innumerable writers who have thrown light upon Aristotle’s logical doctrine, St
Hilaire, Trendelenburg, Ueberweg, Hamilton, Mansel, G. Grote may be named. There are,
however, others of equal distinction. Reference to Prantl, op. cit., is indispensable. Zeller,
Die philosophie der Griechen, ii. 2, “Aristoteles” (3rd ed., 1879), pp. 185-257 (there is an
Eng. trans.), and Maier, Die Syllogistik des Aristoteles (2 vols., 1896, 1900) (some 900
pp.), are also of first-rate importance.

29 Sophist. Elench. 184, espec. b 1-3, but see Maier, loc. cit. i. 1.

30 References such as 18b 12 are the result of subsequent editing and prove nothing.
See, however, Aristotle.

31 Adrastus is said to have called them πρὸ τῶν τοπικῶν.

32 Metaphys. E. 1.

33 De Part. Animal. A. 1, 639a 1 sqq.; cf. Metaphys. 1005b 2 sqq.

34 De Interpretatione 16a sqq.

35 De Interpretatione 16a 24-25.

36 Ib. 18a 28 sqq.

37 Ib. 19a 28-29.

38 As shown e.g. by the way in which the relativity of sense and the object of sense is
conceived, 7b 35-37.

39 Topics 101a 27 and 36-b 4.

40 Topics 100.

41 Politics 1282a 1 sqq.

42 103b 21.

43 Topics 160a 37-b 5.

44 This is the explanation of the formal definition of induction, Prior Analytics, ii. 23,
68b 15 sqq.

45 25b 36.

46 Prior Analytics, i. 1. 24a 18-20, Συλλογισμὸς δὲ ἑστὶ λόγος ἐν ᾦ τεθέντων τινῶν
ἕτερόν τι τῶν κειμένων ἐξ ἀνάγκης σνμβαίνει τῷ ταῦτα εἶναι. The equivalent
previously in Topics 100a 25 sqq.

47 Prior Analytics, ii. 21; Posterior Analytics, i. 1.

48 67a 33-37, μὴ συνθεωρῶν τὸ καθ᾽ ἑκάτερον.

49 67a 39-63.

50 79a 4-5.

51 24b 10-11.

52 Posterior Analytics, i. 4 καθ᾽ αὐτὸ means (1) contained in the definition of the
subject; (2) having the subject contained in its definition, as being an alternative
determination of the subject, crooked, e.g. is per se of line; (3) self-subsistent; (4)
connected with the subject as consequent to ground. Its needs stricter determination
therefore.

53 73b 26 sqq., 74a 37 sqq.

54 90b 16.

55 Metaphys. Z. 12, H. 6 ground this formula metaphysically.

56 94a 12, 75b 32.

57 90a 6. Cf. Ueberweg, System der Logik, § 101.

58 78a 30 sqq.

59 Topics, 101b 18, 19.

60 Posterior Analytics, ii. 13.

61 Posterior Analytics, ii. 16.

62 Posterior Analytics, i. 13 ad. fin., and i. 27. The form which a mathematical science
treats as relatively self-subsistent is certainly not the constitutive idea.

63 Posterior Analytics, i. 3.

64 Posterior Analytics, ii. 19.

65 De Anima, 428b 18, 19.

66 Prior Analytics, i. 30, 46a 18.

67 Topics, 100b 20, 21.

68 Topics, 101a 25, 36-37, b1-4, &c.

69 Zeller (loc. cit. p. 194), who puts this formula in order to reject it.

70 Metaphys. Δ 1, 1013a 14.

71 Posterior Analytics, 72a 16 seq.

72 Posterior Analytics, 77a 26, 76a 37 sqq.

73 Metaphys. Γ.

74 Posterior Analytics, ii. 19.

75 de Anima, iii. 4-6.

76 Metaphys. M. 1087a 10-12; Zeller loc. cit. 304 sqq.; McLeod Innes, The Universal
and Particular in Aristotle’s Theory of Knowledge (1886).

77 Topics, 105a 13.

78 Metaphys. 995a 8.

79 E.g., Topics, 108b 10, “to induce” the universal.

80 Posterior Analytics, ii. 19, 100b 3, 4.

81 Topics, i. 18, 108b 10.

82 Prior Analytics, ii. 23.

83 Παράδειγμα, Prior Analytics, ii. 24.

84 Sigwart, Logik, Eng. trans. vol. ii. p. 292 and elsewhere.

85 Ueberweg, System, § 127, with a ref. to de Partibus Animalium, 667a.

86 See 67a 17 ἐξ ἁπάντων τῶν ἀτόμων.

87 Ἐπιφορά. Ἐπι = “in” as in ἐπαγωγὴ, inductio, and -φορὰ = -ferentia, as in
διαφορὰ, differentia.

88 Diog. Laërt. x. 33 seq.; Sext. Emp. Adv. Math. vii. 211.

89 Diog. Laërt. x. 87; cf. Lucretius, vi. 703 sq., v. 526 sqq. (ed. Munro).

90 Sextus Empiricus, Pyrrhon. Hypotyp. ii. 195, 196.

91 Sextus, op. cit. ii. 204.

92 Op. cit. iii. 17 sqq., and especially 28.

93 The point is raised by Aristotle, 95A.

94 See Jourdain, Recherches critiques sur l’âge et l’origine des traductions latines
d’Aristote (1843).

95 See E. Cassirer, Das Erkenntnisproblem, i. 134 seq., and the justificatory excerpts,
pp. 539 sqq.

96 See Riehl in Vierteljahrschr. f. wiss. Philos. (1893).

97 Bacon, Novum Organum, ii. 22, 23; cf. also Aristotle, Topics i. 12. 13, ii. 10. 11
(Stewart, ad Nic. Eth. 1139b 27) and Sextus Empiricus, Pyrr. Hypot. iii. 15.

98 Bacon’s Works, ed. Ellis and Spedding, iii. 164-165.

99 A notable formula of Bacon’s Novum Organum ii. 4 § 3 turns out, Valerius Terminus,
cap. 11, to come from Aristotle, Post. An. i. 4 via Ramus. See Ellis in Bacon’s Works, iii.
203 sqq.

100 De Civitate Dei, xi. 26. “Certum est me esse, si fallor.”

101 Cf. Plato, Republic, 381E seq.

102 Elementa Philosophiæ, i. 3. 20, i. 6. 17 seq.

103 Hobbes, Elementa Philosophiæ, i. 1. 5.

104 Id. ib. i. 6. 16.

105 Id. ib. i. 4. 8; cf. Locke’s Essay of Human Understanding, iv. 17.

106 Id. Leviathan, i. 3.

107 Id. Elem. Philos. i. 6. 10.

108 Condillac, Langue des Calculs, p. 7.

109 Locke, Essay, iii. 3.

110 Id. ib. iv. 17.

111 Loc. cit. § 8.

112 Id. ib. iv. 4, §§ 6 sqq.

113 Berkeley, Of the Principles of Human Knowledge, § 142.

114 Hume, Treatise of Human Nature, i. 1. 7 (from Berkeley, op. cit., introd., §§ 15-16).

115 Essay, iv. 17, § 3.

116 Hume, Treatise of Human Nature, i. 3. 15.

117 Mill, Examination of Sir William Hamilton’s Philosophy, cap. 17.

118 Cf. Mill, Autobiography, p. 159. “I grappled at once with the problem of Induction,
postponing that of Reasoning.” Ib. p. 182 (when he is preoccupied with syllogism), “I
could make nothing satisfactory of Induction at this time.”

119 Autobiography, p. 181.

120 The insight, for instance, of F. H. Bradley’s criticism, Principles of Logic, II. ii. 3,
is somewhat dimmed by a lack of sympathy due to extreme difference in the point of view
adopted.

121 Bacon, Novum organum, i. 100.

122 Russell’s Philosophy of Leibnitz, capp. 1-5.

123 See especially remarks on the letter of M. Arnauld (Gerhardt’s edition of the
philosophical works, ii. 37 sqq.).

124 Gerhardt, vi. 612, quoted by Russell, loc. cit., p. 19.

125 Ibid., ii. 62, Russell, p. 33.

126 Spinoza, ed. van Vloten and Land, i. 46 (Ethica, i. 11).

127 Nouveaux essais, iv. 2 § 9, 17 § 4 (Gerhardt v. 351, 460).

128 Critique of Judgment, Introd. § 2, ad. fin. (Werke, Berlin Academy edition, vol. v. p.
176, l. 10).

129 Kant’s Introduction to Logic and his Essay on the Mistaken Subtlety of the Four
Figures, trans. T. K. Abbott (1885).

130 Loc. cit., p. 11.

131 Or antitheses. Kant follows, for example, a different line of cleavage between form
and content from that developed between thought and the “given.” And these are not his
only unresolved dualities, even in the Critique of Pure Reason. For the logical inquiry,
however, it is permissible to ignore or reduce these differences.

The determination too of the sense in which Kant’s theory of knowledge involves an
unresolved antithesis is for the logical purpose necessary so far only as it throws light upon
his logic and his influence upon logical developments. Historically the question of the
extent to which writers adopted the dualistic interpretation or one that had the like
consequences is of greater importance.

It may be said summarily that Kant holds the antithesis between thought and “the given”
to be unresolved and within the limits of theory of knowledge irreducible. The dove of
thought falls lifeless if the resistant atmosphere of “the given” be withdrawn (Critique of
Pure Reason, ed. 2 Introd. Kant’s Werke, ed. of the Prussian Academy, vol. iii. p. 32, ll. 10
sqq.). Nevertheless the thing-in-itself is a problematic conception and of a limiting or
negative use merely. He “had woven,” according to an often quoted phrase of Goethe, “a
certain sly element of irony into his method; ... he pointed as it were with a side gesture
beyond the limits which he himself had drawn.” Thus (loc. cit. p. 46, ll. 8, 9) he declares
that “there are two lineages united in human knowledge, which perhaps spring from a
common stock, though to us unknown—namely sense and understanding.” Some
indication of the way in which he would hypothetically and speculatively mitigate the
antithesis is perhaps afforded by the reflection that the distinction of the mental and what
appears as material is an external distinction in which the one appears outside to the other.
“Yet what as thing-in-itself lies back of the phenomenon may perhaps not be so wholly
disparate after all” (ib. p. 278, ll. 26 sqq.).

132 Critique of Judgment, Introd. § 2 (Werke, v., 276 ll. 9 sqq.); cf. Bernard’s
“Prolegomena” to his translation of this, (pp. xxxviii. sqq.).

133 Die Logik, insbesondere die Analytik (Schleswig, 1825). August Detlev Christian
Twesten (1789-1876), a Protestant theologian, succeeded Schleiermacher as professor in
Berlin in 1835.

134 See Sir William Hamilton: The Philosophy of Perception, by J. Hutchison Stirling.

135 Hauptpunkte der Logik, 1808 (Werke, ed. Hartenstein, i. 465 sqq.), and specially
Lehrbuch der Einleitung in die Philosophie (1813), and subsequently §§ 34 sqq. (Werke, i.
77 sqq.).

136 See Ueberweg, System of Logic and History of Logical Doctrines, § 34.

137 Drei Bücher der Logik, 1874 (E.T., 1884). The Book on Pure Logic follows in
essentials the line of thought of an earlier work (1843).

138 Logic, Eng. trans. 35 ad. fin.

139 Logic, Introd. § ix.

140 For whom see Höffding, History of Modern Philosophy, Eng. trans., vol. ii. pp. 122
sqq.; invaluable for the logical methods of modern philosophers.

141 Wissenschaft der Logik (1812-1816), in course of revision at Hegel’s death in 1831
(Werke, vols. iii.-v.), and Encyklopädie der philosophischen Wissenschaften, i.; Die Logik
(1817; 3rd ed., 1830); Werke, vol. vi., Eng. trans., Wallace (2nd ed., 1892).

142 The Principles of Logic (1883).

143 Logic, or The Morphology of Thought (2 vols., 1888).

144 Logic, Pref. pp. 6 seq.

145 Id. vol. ii. p. 4.

146 Logik (1873, 1889), Eng. trans. ii. 17.

147 Op. cit. ii. 289.

148 Introd. to Logic., trans. Abbott, p. 10.

149 Ueber Annahmen (1902, &c.).

150 Logik (1880, and in later editions).

151 Yet see Studies in Logic, by John Dewey and others (1903).

LOGOCYCLIC CURVE,
STROPHOID or FOLIATE, a cubic curve
generated by increasing or diminishing the radius
vector of a variable point Q on a straight line AB by
the distance QC of the point from the foot of the
perpendicular drawn from the origin to the fixed line.
The polar equation is r cos θ = a(1 ± sinθ), the upper
sign referring to the case when the vector is
increased, the lower when it is diminished. Both
branches are included in the Cartesian equation (x2 +
y2)(2a − x) = a2x, where a is the distance of the line
from the origin. If we take for axes the fixed line and
the perpendicular through the initial point, the equation takes the form y √(a
− x) = x √(a + x). The curve resembles the folium of Descartes, and has a
node between x = 0, x = a, and two branches asymptotic to the line x = 2a.

LOGOGRAPHI (λόγος, γράφω, writers of prose histories or
tales), the name given by modern scholars to the Greek historiographers
before Herodotus.1 Thucydides, however, applies the term to all his own
predecessors, and it is therefore usual to make a distinction between the
older and the younger logographers. Their representatives, with one
exception, came from Ionia and its islands, which from their position were
most favourably situated for the acquisition of knowledge concerning the

distant countries of East and West. They wrote in the Ionic dialect, in what
was called the unperiodic style, and preserved the poetic character of their
epic model. Their criticism amounts to nothing more than a crude attempt to
rationalize the current legends and traditions connected with the founding
of cities, the genealogies of ruling families, and the manners and customs of
individual peoples. Of scientific criticism there is no trace whatever. The
first of these historians was probably Cadmus of Miletus (who lived, if at
all, in the early part of the 6th century), the earliest writer of prose, author
of a work on the founding of his native city and the colonization of Ionia
(so Suïdas); Pherecydes of Leros, who died about 400, is generally
considered the last. Mention may also be made of the following: Hecataeus
of Miletus (550-476); Acusilaus of Argos,2 who paraphrased in prose
(correcting the tradition where it seemed necessary) the genealogical works
of Hesiod in the Ionic dialect; he confined his attention to the prehistoric
period, and made no attempt at a real history; Charon of Lampsacus (c.
450), author of histories of Persia, Libya, and Ethiopia, of annals (ὦροι) of
his native town with lists of the prytaneis and archons, and of the chronicles
of Lacedaemonian kings; Xanthus of Sardis in Lydia (c. 450), author of a
history of Lydia, one of the chief authorities used by Nicolaus of Damascus
(fl. during the time of Augustus); Hellanicus of Mytilene; Stesimbrotus of
Thasos, opponent of Pericles and reputed author of a political pamphlet on
Themistocles, Thucydides and Pericles; Hippys and Glaucus, both of
Rhegium, the first the author of histories of Italy and Sicily, the second of a
treatise on ancient poets and musicians, used by Harpocration and Plutarch;
Damastes of Sigeum, pupil of Hellanicus, author of genealogies of the
combatants before Troy (an ethnographic and statistical list), of short
treatises on poets, sophists, and geographical subjects.

On the early Greek historians, see G. Busolt, Griechische Geschichte
(1893), i. 147-153; C. Wachsmuth, Einleitung in das Studium der alten
Geschichte (1895); A. Schäfer, Abriss der Quellenkunde der