Although it is very hard to establish a new and correct view of a subject
which for more than two thousand years has been handled by innumerable
writers, and which, moreover, does not receive additions through the
growth of experience, yet this must not deter me from presenting to the
thinker for examination the following attempt of this kind.
An inference is that operation of our reason by virtue of which, through
the comparison of two judgments a third judgment arises, without the
assistance of any knowledge otherwise obtained. The condition of this is
that these two judgments have _one_ conception in common, for otherwise
they are foreign to each other and have no community. But under this
condition they become the father and mother of a child that contains in
itself something of both. Moreover, this operation is no arbitrary act,
but an act of the reason, which, when it has considered such judgments,
performs it of itself according to its own laws. So far it is objective,
not subjective, and therefore subject to the strictest rules.
We may ask in passing whether he who draws an inference really learns
something new from the new proposition, something previously unknown to
him? Not absolutely; but yet to a certain extent he does. What he learns
lay in what he knew: thus he knew it also, but he did not know that he
knew it; which is as if he had something, but did not know that he had it,
and this is just the same as if he had it not. He knew it only
_implicite_, now he knows it _explicite_; but this distinction may be so
great that the conclusion appears to him a new truth. For example:
All diamonds are stones;
All diamonds are combustible:
Therefore some stones are combustible.
The nature of inference consequently consists in this, that we bring it to
distinct consciousness that we have already thought in the premisses what
is asserted in the conclusion. It is therefore a means of becoming more
distinctly conscious of one’s own knowledge, of learning more fully, or
becoming aware of what one knows. The knowledge which is afforded by the
conclusion was _latent_, and therefore had just as little effect as latent
heat has on the thermometer. Whoever has salt has also chlorine; but it is
as if he had it not, for it can only act as chlorine if it is chemically
evolved; thus only, then, does he really possess it. It is the same with
the gain which a mere conclusion from already known premisses affords: a
previously _bound or latent knowledge_ is thereby set _free_. These
comparisons may indeed seem to be somewhat strained, but yet they really
are not. For because we draw many of the possible inferences from our
knowledge very soon, very rapidly, and without formality, and therefore
have no distinct recollection of them, it seems to us as if no premisses
for possible conclusions remained long stored up unused, but as if we
already had also conclusions prepared for all the premisses within reach
of our knowledge. But this is not always the case; on the contrary, two
premisses may have for a long time an isolated existence in the same mind,
till at last some occasion brings them together, and then the conclusion
suddenly appears, as the spark comes from the steel and the stone only
when they are struck together. In reality the premisses assumed from
without, both for theoretical insight and for motives, which bring about
resolves, often lie for a long time in us, and become, partly through
half-conscious, and even inarticulate, processes of thought, compared with
the rest of our stock of knowledge, reflected upon, and, as it were,
shaken up together, till at last the right major finds the right minor,
and these immediately take up their proper places, and at once the
conclusion exists as a light that has suddenly arisen for us, without any
action on our part, as if it were an inspiration; for we cannot comprehend
how we and others have so long been in ignorance of it. It is true that in
a happily organised mind this process goes on more quickly and easily than
in ordinary minds; and just because it is carried on spontaneously and
without distinct consciousness it cannot be learned. Therefore Goethe
says: “How easy anything is he knows who has discovered it, he knows who
has attained to it.” As an illustration of the process of thought here
described we may compare it to those padlocks which consist of rings with
letters; hanging on the box of a travelling carriage, they are shaken so
long that at last the letters of the word come together in their order and
the lock opens. For the rest, we must also remember that the syllogism
consists in the process of thought itself, and the words and propositions
through which it is expressed only indicate the traces it has left behind
it—they are related to it as the sound-figures of sand are related to the
notes whose vibrations they express. When we reflect upon something, we
collect our data, reduce them to judgments, which are all quickly brought
together and compared, and thereby the conclusions which it is possible to
draw from them are instantly arrived at by means of the use of all the
three syllogistic figures. Yet on account of the great rapidity of this
operation only a few words are used, and sometimes none at all, and only
the conclusion is formally expressed. Thus it sometimes happens that
because in this way, or even merely intuitively, _i.e._, by a happy
_apperçu_, we have brought some new truth to consciousness, we now treat
it as a conclusion and seek premisses for it, that is, we desire to prove
it, for as a rule knowledge exists earlier than its proofs. We then go
through our stock of knowledge in order to see whether we can find some
truth in it in which the newly discovered truth was already implicitly
contained, or two propositions which would give this as a result if they
were brought together according to rule. On the other hand, every judicial
proceeding affords a most complete and imposing syllogism, a syllogism in
the first figure. The civil or criminal transgression complained of is the
minor; it is established by the prosecutor. The law applicable to the case
is the major. The judgment is the conclusion, which therefore, as
something necessary, is “merely recognised” by the judge.
But now I shall attempt to give the simplest and most correct exposition
of the peculiar mechanism of inference.
_Judging_, this elementary and most important process of thought, consists
in the comparison of two _conceptions_; _inference_ in the comparison of
two _judgments_. Yet ordinarily in text-books inference is also referred
to the comparison of conceptions, though of _three_, because from the
relation which two of these conceptions have to a third their relation to
each other may be known. Truth cannot be denied to this view also; and
since it affords opportunity for the perceptible demonstration of
syllogistic relations by means of drawn concept-spheres, a method approved
of by me in the text, it has the advantage of making the matter easily
comprehensible. But it seems to me that here, as in so many cases,
comprehensibility is attained at the cost of thoroughness. The real
process of thought in inference, with which the three syllogistic figures
and their necessity precisely agree, is not thus recognised. In inference
we operate _not_ with mere _conceptions_ but with whole _judgments_, to
which quality, which lies only in the copula and not in the conceptions,
and also quantity are absolutely essential, and indeed we have further to
add modality. That exposition of inference as a relation of _three
conceptions_ fails in this, that it at once resolves the judgments into
their ultimate elements (the conceptions), and thus the means of combining
these is lost, and that which is peculiar to the judgments as such and in
their completeness, which is just what constitutes the necessity of the
conclusion which follows from them, is lost sight of. It thus falls into
an error analogous to that which organic chemistry would commit if, for
example, in the analysis of plants it were at once to reduce them to their
_ultimate_ elements, when it would find in all plants carbon, hydrogen,
and oxygen, but would lose the specific differences, to obtain which it is
necessary to stop at their more special elements, the so-called alkaloids,
and to take care to analyse these in their turn. From three given
conceptions no conclusion can as yet be drawn. It may certainly be said:
the relation of two of them to the third must be given with them. But it
is just the _judgments_ which combine these conceptions, that are the
expression of this relation; thus _judgments_, not mere _conceptions_, are
the material of the inference. Accordingly inference is essentially a
comparison of two _judgments_. The process of thought in our mind is
concerned with these and the thoughts expressed by them, not merely with
three conceptions. This is the case even when this process is imperfectly
or not at all expressed in words; and it is as such, as a bringing
together of the complete and unanalysed judgments, that we must consider
it in order properly to understand the technical procedure of inference.
From this there will then also follow the necessity for three really
rational syllogistic figures.
As in the exposition of syllogistic reasoning by means of
_concept-spheres_ these are presented to the mind under the form of
circles, so in the exposition by means of entire judgments we have to
think these under the form of rods, which, for the purpose of comparison,
are held together now by one end, now by the other. The different ways in
which this can take place give the three figures. Since now every premiss
contains its subject and its predicate, these two conceptions are to be
imagined as situated at the two ends of each rod. The two judgments are
now compared with reference to the two _different_ conceptions in them;
for, as has already been said, the third conception must be the same in
both, and is therefore subject to no comparison, but is that _with which_,
that is, in reference to which, the other two are compared; it is the
_middle_. The latter is accordingly always only the means and not the
chief concern. The two different conceptions, on the other hand, are the
subject of reflection, and to find out their relation to each other by
means of the judgments in which they are contained is the aim of the
syllogism. Therefore the conclusion speaks only of them, not of the
middle, which was only a means, a measuring rod, which we let fall as soon
as it has served its end. Now if this conception which is _identical_ in
both propositions, thus the middle, is the subject of _one_ premiss, the
conception to be compared with it must be the predicate, and conversely.
Here at once is established _a priori_ the possibility of three cases;
either the subject of one premiss is compared with the predicate of the
other, or the subject of the one with the subject of the other, or,
finally, the predicate of the one with the predicate of the other. Hence
arise the three syllogistic figures of Aristotle; the fourth, which was
added somewhat impertinently, is ungenuine and a spurious form. It is
attributed to Galenus, but this rests only on Arabian authority. Each of
the three figures exhibits a perfectly different, correct, and natural
thought-process of the reason in inference.
If in the two judgments to be compared the relation between the _predicate
of the one and the subject of the other_ is the object of the comparison,
the _first figure_ appears. This figure alone has the advantage that the
conceptions which in the conclusion are subject and predicate both appear
already in the same character in the premisses; while in the two other
figures one of them must always change its roll in the conclusion. But
thus in the first figure the result is always less novel and surprising
than in the other two. Now this advantage in the first figure is obtained
by the fact that the predicate of the major is compared with the subject
of the minor, but not conversely, which is therefore here essential, and
involves that the middle should assume both the positions, _i.e._, it is
the subject in the major and the predicate in the minor. And from this
again arises its subordinate significance, for it appears as a mere weight
which we lay at pleasure now in one scale and now in the other. The course
of thought in this figure is, that the predicate of the major is
attributed to the subject of the minor, because the subject of the major
is the predicate of the minor, or, in the negative case, the converse
holds for the same reason. Thus here a property is attributed to the
things thought through a conception, because it depends upon another
property which we already know they possess; or conversely. Therefore here
the guiding principle is: _Nota notæ est nota rei ipsius, et repugnans
notæ repugnat rei ipsi_.
If, on the other hand, we compare two judgments with the intention of
bringing out the relation which the _subjects of both_ may have to each
other, we must take as the common measure their predicate. This will
accordingly be here the middle, and must therefore be the same in both
judgments. Hence arises the _second figure_. In it the relation of two
subjects to each other is determined by that which they have as their
common predicate. But this relation can only have significance if the same
predicate is attributed to the one subject and denied of the other, for
thus it becomes an essential ground of distinction between the two. For if
it were attributed to both the subjects this could decide nothing as to
their relation to each other, for almost every predicate belongs to
innumerable subjects. Still less would it decide this relation if the
predicate were denied of both the subjects. From this follows the
fundamental characteristic of the second figure, that the premisses must
be of _opposite quality_; the one must affirm and the other deny.
Therefore here the principal rule is: _Sit altera negans_; the corollary
of which is: _E meris affirmativis nihil sequitur_; a rule which is
sometimes transgressed in a loose argument obscured by many parenthetical
propositions. The course of thought which this figure exhibits distinctly
appears from what has been said. It is the investigation of two kinds of
things with the view of distinguishing them, thus of establishing that
they are _not_ of the same species; which is here decided by showing that
a certain property is essential to the one kind, which the other lacks.
That this course of thought assumes the second figure of its own accord,
and expresses itself clearly only in it, will be shown by an example:
All fishes have cold blood;
No whale has cold blood:
Thus no whale is a fish.
In the first figure, on the other hand, this thought exhibits itself in a
weak, forced, and ultimately patched-up form:
Nothing that has cold blood is a whale;
All fishes have cold blood:
Thus no fish is a whale,
And consequently no whale is a fish.
Take also an example with an affirmative minor:
No Mohamedan is a Jew;
Some Turks are Jews:
Therefore some Turks are not Mohamedans.
As the guiding principle for this figure I therefore give, for the mood
with the negative minor: _Cui repugnat nota, etiam repugnat notatum_; and
for the mood with the affirmative minor: _Notato repugnat id cui nota
repugnat_. Translated these may be thus combined: Two subjects which stand
in opposite relations to one predicate have a negative relation to each
other.
The third case is that in which we place two judgments together in order
to investigate the relation of their _predicates_. Hence arises the _third
figure_, in which accordingly the middle appears in both premisses as the
subject. It is also here the _tertium comparationis_, the measure which is
applied to both the conceptions which are to be investigated, or, as it
were, a chemical reagent, with which we test them both in order to learn
from their relation to it what relation exists between themselves. Thus,
then, the conclusion declares whether a relation of subject and predicate
exists between the two, and to what extent this is the case. Accordingly,
what exhibits itself in this figure is reflection concerning two
properties which we are inclined to regard either as _incompatible_, or
else as _inseparable_, and in order to decide this we attempt to make them
the predicates of one subject in two judgments. From this it results
either that both properties belong to the same thing, consequently their
_compatibility_, or else that a thing has the one but not the other,
consequently their _separableness_. The former in all moods with two
affirmative premisses, the latter in all moods with one negative; for
example:
Some brutes can speak;
All brutes are irrational:
Therefore some irrational beings can speak.
According to Kant (_Die Falsche Spitzfinigkeit_, § 4) this inference would
only be conclusive if we added in thought: “Therefore some irrational
beings are brutes.” But this seems to be here quite superfluous and by no
means the natural process of thought. But in order to carry out the same
process of thought directly by means of the first figure I must say:
“All brutes are irrational;
Some beings that can speak are brutes,”
which is clearly not the natural course of thought; indeed the conclusion
which would then follow, “Some beings that can speak are irrational,”
would have to be converted in order to preserve the conclusion which the
third figure gives of itself, and at which the whole course of thought has
aimed. Let us take another example:
All alkalis float in water;
All alkalis are metals:
Therefore some metals float in water.
[Alkalis and Metals overlapping circles]
Figure 1
[Metals circle with Alkalis circle inside it]
Figure 2
When this is transposed into the first figure the minor must be converted,
and thus runs: “Some metals are alkalis.” It therefore merely asserts that
some metals lie in the sphere “alkalis,” thus [Figure 1], while our actual
knowledge is that all alkalis lie in the sphere “metals,” thus [Figure 2]:
It follows that if the first figure is to be regarded as the only normal
one, in order to think naturally we would have to think less than we know,
and to think indefinitely while we know definitely. This assumption has
too much against it. Thus in general it must be denied that when we draw
inferences in the second and third figures we tacitly convert a
proposition. On the contrary, the third, and also the second, figure
exhibits just as rational a process of thought as the first. Let us now
consider another example of the other class of the third figure, in which
the separableness of two predicates is the result; on account of which one
premiss must here be negative:
No Buddhist believes in a God;
Some Buddhists are rational:
Therefore some rational beings do not believe in a God.
As in the examples given above the _compatibility_ of two properties is
the problem of reflection, now their _separableness_ is its problem, which
here also must be decided by comparing them with _one_ subject and showing
that _one_ of them is present in it without the _other_. Thus the end is
directly attained, while by means of the first figure it could only be
attained indirectly. For in order to reduce the syllogism to the first
figure we must convert the minor, and therefore say: “Some rational beings
are Buddhists,” which would be only a faulty expression of its meaning,
which really is: “Some Buddhists are yet certainly rational.”
As the guiding principle of this figure I therefore give: for the
affirmative moods: _Ejusdem rei notœ, modo sit altera universalis, sibi
invicem sunt notœ particulares_; and for the negative moods: _Nota rei
competens, notœ eidem repugnanti, particulariter repugnat, modo sit altera
universalis_. Translated: If two predicates are affirmed of one subject,
and at least one of them universally, they are also affirmed of each other
particularly; and, on the contrary, they are denied of each other
particularly whenever one of them contradicts the subject of which the
other is affirmed; provided always that either the contradiction or the
affirmation be universal.
In the _fourth figure_ the subject of the major has to be compared with
the predicate of the minor; but in the conclusion they must both exchange
their value and position, so that what was the subject of the major
appears as the predicate of the conclusion, and what was the predicate of
the minor appears as the subject of the conclusion. By this it becomes
apparent that this figure is merely the _first_, wilfully turned upside
down, and by no means the expression of a real process of thought natural
to the reason.
On the other hand, the first three figures are the ectypes of three real
and essentially different operations of thought. They have this in common,
that they consist in the comparison of two judgments; but such a
comparison only becomes fruitful when these judgments have _one_
conception in common. If we present the premisses to our imagination under
the sensible form of two rods, we can think of this conception as a clasp
that links them to each other; indeed in lecturing one might provide
oneself with such rods. On the other hand, the three figures are
distinguished by this, that those judgments are compared either with
reference to the subjects of both, or to the predicates of both, or
lastly, with reference to the subject of the one and the predicate of the
other. Since now every conception has the property of being subject or
predicate only because it is already part of a judgment, this confirms my
view that in the syllogism only judgments are primarily compared, and
conceptions only because they are parts of judgments. In the comparison of
two judgments, however, the essential question is, in _respect of what_
are they compared? not _by what means_ are they compared? The former
consists of the concepts which are different in the two judgments; the
latter consists of the middle, that is, the conception which is identical
in both. It is therefore not the right point of view which Lambert, and
indeed really Aristotle, and almost all the moderns have taken in starting
from the _middle_ in the analysis of syllogisms, and making it the
principal matter and its position the essential characteristic of the
syllogisms. On the contrary, its role is only secondary, and its position
a consequence of the logical value of the conceptions which are really to
be compared in the syllogism. These may be compared to two substances
which are to be chemically tested, and the middle to the reagent by which
they are tested. It therefore always takes the place which the conceptions
to be compared leave vacant, and does not appear again in the conclusion.
It is selected according to our knowledge of its relation to both the
conceptions and its suitableness for the place it has to take up.
Therefore in many cases we can change it at pleasure for another without
affecting the syllogism. For example, in the syllogism:
All men are mortal;
Caius is a man:
I can exchange the middle “man” for “animal existence.” In the syllogism:
All diamonds are stones;
All diamonds are combustible:
I can exchange the middle “diamond” for “anthracite.” As an external mark
by which we can recognise at once the figure of a syllogism the middle is
certainly very useful. But as the fundamental characteristic of a thing
which is to be explained, we must take what is essential to it; and what
is essential here is, whether we place two propositions together in order
to compare their predicates or their subjects, or the predicate of the one
and the subject of the other.
Therefore, in order as premisses to yield a conclusion, two judgments must
have a conception in common; further, they must not both be negative, nor
both particular; and lastly, in the case in which the conceptions to be
compared are the subjects of both, they must not both be affirmative.
The voltaic pile may be regarded as a sensible image of the syllogism. Its
point of indifference, at the centre, represents the middle, which holds
together the two premisses, and by virtue of which they have the power of
yielding a conclusion. The two different conceptions, on the other hand,
which are really what is to be compared, are represented by the two
opposite poles of the pile. Only because these are brought together by
means of their two conducting wires, which represent the copulas of the
two judgments, is the spark emitted upon their contact—the new light of
the conclusion.