Critique of the Pure Reason (II. "Method of Transcendentalism,"
Chap. I., Sec. 1), where the distinction between these two
employments of the reason is sufficiently explained. So far
concerning the sources of metaphysical cognition.

            Sect. 2. Concerning the Kind of Cognition which can
                       alone be called Metaphysical

     a. Of the Distinction between Analytical and Synthetical
judgments in general. -- The peculiarity of its sources demands
that metaphysical cognition must consist of nothing but a priori
judgments. But whatever be their origin, or their logical form,
there is a distinction in judgments, as to their content,
according to which they are either merely explicative, adding
nothing to the content of the cognition, or expansive, increasing
the given cognition: the former may be called analytical, the
latter synthetical, judgments.

     Analytical judgments express nothing in the predicate but
what has been already actually thought in the concept of the
subject, though not so distinctly or with the same (full)
consciousness. When I say: All bodies are extended, I have not
amplified in the least my concept of body, but have only analyzed
it, as extension was really thought to belong to that concept
before the judgment was made, though it was not expressed, this
judgment is therefore analytical. On the contrary, this judgment,
All bodies have weight, contains in its predicate something not
actually thought in the general concept of the body; it amplifies
my knowledge by adding something to my concept, and must
therefore be called synthetical.

     b. The Common Principle of all Analytical Judgments is the
Law of Contradiction. -- All analytical judgments depend wholly
on the law of Contradiction, and are in their nature a priori
cognitions, whether the concepts that supply them with matter be
empirical or not. For the predicate of an affirmative analytical
judgment is already contained in the concept of the subject, of
which it cannot be denied without contradiction. In the same way
its opposite is necessarily denied of the subject in an
analytical, but negative, judgment, by the same law of
contradiction. Such is the nature of the judgments: all bodies
are extended, and no bodies are unextended (i. e., simple).

     For this very reason all analytical judgments are a .priori
even when the concepts are empirical, as, for example, Gold is a
yellow metal; for to know this I require no experience beyond my
concept of gold as a yellow metal: it is, in fact, the very
concept, and I need only analyze it, without looking beyond it
elsewhere.

     c. Synthetical judgments require a different Principle from
the Law of Contradiction.-There are synthetical a posteriori
judgments of empirical origin; but there are also others which
are proved to be certain a priori, and which spring from pure
Understanding and Reason. Yet they both agree in this, that they
cannot possibly spring from the principle of analysis, viz., the
law of contradiction, alone; they require a quite different
principle, though, from whatever they may be deduced, they must
be subject to the law of contradiction, which must never be
violated, even though everything cannot be deduced from it. I
shall first classify synthetical judgments.

     1. Empirical judgments are always synthetical. For it would
be absurd to base an analytical judgment on experience, as our
concept suffices for the purpose without requiring any testimony
from experience. That body is extended, is a judgment established
a priori, and not an empirical judgment. For before appealing to
experience, we already have all the conditions of the judgment in
the concept, from which we have but to elicit the predicate
according to the law of contradiction, and thereby to become
conscious of the necessity of the judgment, which experience
could not even teach us.

     2. Mathematical judgments are all synthetical. This fact
seems hitherto to have altogether escaped the observation of
those who have analyzed human reason; it even seems directly
opposed to all their conjectures, though incontestably certain,
and most important in its consequences. For as it was found that
the conclusions of mathematicians all proceed according to the
law of contradiction (as is demanded by all apodictic certainty),
men persuaded themselves that the fundamental principles were
known from the same law. This was a great mistake, for a
synthetical proposition can indeed be comprehended according to
the law of contradiction, but only by presupposing another
synthetical proposition from which it follows, but never in
itself.

     First of all, we must observe that all proper mathematical
judgments are a priori, and not empirical, because they carry
with them necessity, which cannot be obtained from experience.
But if this be not conceded to me, very good; I shall confine my
assertion pure Mathematics, the very notion of which implies that
it contains pure a priori and not empirical cognitions.

     It might at first be thought that the proposition 7 + 5 = 12
is a mere analytical judgment, following from the concept of the
sum of seven and five, according to the law of contradiction. But
on closer examination it appears that the concept of the sum Of
7+5 contains merely their union in a single number, without its
being at all thought what the particular number is that unites