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Section Two: Of the Mode of Cognition that can Alone be Termed Metaphysical
Of the distinction between synthetic and analytic judgments
Metaphysical knowledge must contain simply judgments apriori, so much is demanded by the speciality of its sources. But judgments, let
them have what origin they may, or let them even as regards logical form be constituted as they may, possess a distinction according to their content, by virtue of which they are either
simply explanatoryand contribute nothing to the content of a cognition, or they
are extensive, and enlarge the given cognition; the first may be
termed analytic, and the second syntheticjudgments.
Analytic judgments say nothing in the predicate, but what was already
cogitated in the conception of the subject, though perhaps not so clearly, or with the same degree of consciousness. When I say, all bodies are extended, I do not thereby enlarge my
conception of a body in the least, but simply analyse it, inasmuch as extension, although not expressly stated, was already cogitated in that conception; the judgment is, in other words,
analytic. On the other hand, the proposition, some bodies are heavy, contains something in the predicate which was not already cogitated in the general conception of a body; it enlarges,
that is to say, my knowl- edge, in so far as it adds something to my conception; and must therefore be termed a synthetic judgment,
- The common principle of all analytic judgments is the principle of contradiction.
All analytic judgments are based entirely on the principle of contradiction, and are by their nature cognitions apriori, whether the conceptions serving as their matter be empirical or not. For
inasmuch as the predicate of an affirmative analytic judgment is previously cogitated in the conception of the subject, it cannot without contradiction be denied of it; in the same way, its
contrary, in a negative analytic judgment, must necessarily be denied of the subject, likewise in accordance with the principle of contradiction. It is thus with the propo- sitions—every
body is extended; no body is unextended (simple). For this reason all analytic propositions are judgments apriori, although their conceptions may be empirical. Let us take as an instance
the proposition, gold is a yellow metal. Now, to know this, I require no further experience beyond my conception of gold, which contains the propositions that this body is yellow and a
metal; for this constitutes precisely my concep- tion, and therefore I have only to dissect it, without needing to look around for anything elsewhere.
- Synthetic judgments demand a principle other than that of contradiction
There are synthetic judgments a posterioriwhose origin is empirical; but there are also others of
an apriori certainty, that spring from the Understanding and Reason. But both are
alike in this, that they can never have their source solely in the axiom of analysis, viz., the principle of contradiction; they require an altogether dif- ferent principle, notwithstanding
that whatever principle they may be deduced from, they must always con- formtotheprincipleofcontradiction, for nothing can be opposed to this principle, although not everything can
be deduced from it. I will first of all bring synthetic judgments under certain classes.
Judgments of experience are always synthetic. It would be absurd to found an analytic judgment
on experience, as it is unnecessary to go beyond my own conception in order to construct the judgment, and therefore the confirmation of experience is unnecessary to it. That a body is
extended is a proposition possessing apriori certainty, and no judgment of experience. For before I go to
experience I have all the conditions of my judgment already present in the conception, out of which I simply draw the predicate in accordance with the principle of contradiction, and
thereby at the same time the necessityof the judgment may be known, a point which experience could never teach
Mathematicaljudgmentsare in their entirety synthetic. This truth seems hitherto to
have altogether escaped the analysts of human Reason;
indeed, to be directly opposed to all their suppositions, although it is indisputably certain and very important in its consequences. For, because it was found that the conclusions of
mathematicians all proceed according to the principle of contradiction (which the nature of every apodictic certainty demands), it was concluded that the axioms were also known through
the principle of contradiction, which was a great error; for though a synthetic proposition can be viewed in the light of the above principle, it can only be so by presupposing another
synthetic proposition from which it is derived, but never by itself.
It must be first of all remarked that essentially
mathematical propositions are always a
priori, and never empirical, because they involve
necessity, which cannot be inferred from experience. Should any one be unwilling to admit this, I will limit my assertion to puremathematics, the very conception of which itself brings with it the fact that it
contains nothing empirical, but simply pure knowledge apriori.
At first sight, one might be disposed to think the
proposition 7+5=12 merely analytic, resulting from the conception of a sum of seven and five, according to the principle of contradiction. But more closely considered it will be found
that the conception of the sum of 7 and 5 comprises noth- ing beyond the union of two numbers in a single one, and that therein nothing whatever is cogitat- ed as to what this single
number is, that comprehends both the others. The conception of twelve is by no means already cogitated, when I think merely of the union of seven and five, and I may dissect my
conception of such a possible sum as long as I please, without discovering therein the number twelve. One must leave these conceptions, and call to one’s aid an intuition corresponding
to one or other of them, as for instance one’s five fingers (or, like Segner in his Arithmetic, five points), and so gradually add the units of the five given in intuition to the
conception of the seven. One’s con- ception is therefore really enlarged by the proposition 7+5=12; to the first a new one being added, that was in nowise cogitated in the former; in
other words, arithmetical propositions are always synthetic, a truth which is more apparent when we take rather larger numbers, for we must then be clearly convinced, that turn and
twist our conceptions as we may, without calling intuition to our aid, we shall never find the sum required, by the mere dissection of them.
Just as little is any axiom of pure geometry analytic. That a straightline is the shortest between two points, is a synthetic proposition. For
my conception of straight, has no reference to size, but only to quality. The conception of the
“shortest” therefore is quite additional, and cannot be drawn from any analysis of the conception of a straight line. Intuition must therefore again be taken to our aid, by means of
which alone the synthesis is possible.
Certain other axioms, postulated by geometricians, are indeed really analytic and rest on the principle of contradiction, but they only serve, like identical propositions, as
links in the chain of method, and not themselves as principles; as for instance a=a, the whole is equal to itself, or (a+b) > a, i.e., the whole is greater than its part. But even
these, although they are contained in mere concep- tions, are only admitted in mathematics because they can be presented in intuition. What produces the common belief that the predicate
of such apodictic judgments lies already in our conception, and that the judgment is therefore analytic, is merely the ambiguity of expression. We ought, namely, to cogitate a certain predicate to a given conception, and
this necessity adheres even to the conceptions themselves. But the question is not what we oughtto, but what we actually do, although obscurely, cogitate in them; this shows us that the predicate
of those conceptions is dependent indeed neces- sarily, though not immediately (but by means of an added intuition), upon its subject.