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 | :. S may be P. | :. S may be P.
is 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 [Sigma], 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 [Sigma], and [Sigma] 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 ([Greek: ho tou dioti
syllogismos]), and the other regressing a posteriori from consequent
fact to real ground ([Greek: ho tou hoti syllogismos]). 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.)