The analytic/synthetic distinction as conceived by Kant.
"All definitions are either analytic or synthetic. The former are definitions of a given, the latter of a made concept" (Kant, Logic, Section 100).
The common philosophic understanding of these admittedly technical terms sufficiently matches what is said by Kant excepting two pretty big modifications. The first of these modifications is that, for Kant, the analyticity of any judgment does not depend upon the principle of non-contradiction (PNC). According to Kant, the PNC is a criterion of analyticity, but only in as much as it is first a criterion of apriority. Kant believes only that the PNC functions to determine the apriority of any analytic statement. That is, all analytic statements are a priori because they are true by virtue of the PNC. This modification is made explicit by Kant, and it will rescue him from claims that he offers two characterizations of analyticity that are sometimes mutually incompatible and aporetic.
The second, and more important, of these modifications is that the analytic requirement of one concept being "contained in" another concept is best understood through Kant's discussions of intensional clarification and the identity of concepts (to really understand Kant's distinction between intension and extension, read his Logic). Kant's link between analyticity and one concept being contained in the concept of another is a key to understanding his epistemology. This second modification will not only clarify Kant's intended criterion for analyticity, but it will also dispel any myths that Kant's notion of analyticity is stranded at the level of metaphor, as some critics argue.
Kant says about the meaning of the analytic/synthetic distinction in terms of the subject being "contained in" the predicate:
In all judgments in which the relation of a subject to the predicate is thought . . . this relation is possible in two different ways. Either the predicate B belongs to the subject A, as something which is (covertly) contained in this concept A; or B lies outside the concept A, although it does indeed stand in connection with it. In the one case I entitle the judgment analytic, in the other synthetic (Critique of Pure Reason, B10).
Kant continues to equate this analytic/synthetic distinction with the distinction between statements that express connections that are thought through identity and those that are not thought through identity. He also equates it with the distinction explicative/amplitative (see his Prolegomena to Any Future Metaphysics, Section 2a). All of these synonyms for "contained in" will be important in defining this term, particularly, as noted above, Kant's synonym of a predicate-subject connection that is "thought through identity."
Kant also conceived of analytic and synthetic as two contrasting forms of definition. A concept is synthetically defined if and only if the concept is defined either through "exposition" or through "construction" (see Logic, Section 102). A "constructed" concept is one that is arbitrarily made or contingent on some human practice. According to Kant, all mathematical concepts are constructed because they are all entirely contingent and entirely made, or constructed, that is, not found, or given, or forced upon us. An "exposed" concept is one that is empirically made, or made from the appearances presented to the senses. Kant offers as examples the concepts of water, fire, and air. They are synthetical, because "I shall not analyze what lies in them but learn through experience what belongs to them" (ibid). According to Kant's definition of syntheticity as it is shown through his discussion of intensionality, a statement, S, is synthetic if and only if the predicate, X, is attached to the subject, A, by means of exposition (empirical observation) or construction (invention).
What, then, of analytical definitions? Kant says that "All given concepts . . . can only be defined through analysis. For given concepts can only be made distinct by making their characteristics successively clear" (Logic, Section 104). What, though, is a given concept? A given concept is, presumably, one that is not made. Thus, a concept is given if and only if it can be defined by analyzing the concept itself and not by referring to experience (exposition) or invention (construction). An analytic concept, for Kant, is one that can be known by observing (to misappropriate a term normally associated with empiricism) the concept itself and not referring to anything outside of the concept. According to this, Kant is saying that a statement, S, is analytic if and only if the subject, A, already discloses the predicate, X. Likewise, Kant says in the Prolegomena that a statement is analytic if it is "merely explicative, adding nothing to the content of the cognition" (Section 2a). A statement is analytic if and only if the subject already expresses the predicate, that is if the predicate is "contained in" the defintion subject. This can be rephrased as: A statement, S, is analytic if and only if the predicate, P, is not known to be a predicate of the subject, A, unless A is A (that is, unless the subject is identical with itself). This is because A is A only if P is contained in it, and thought along with it. If A could be A if P were not attached to it, then A must be either an invented concept or one that is empirically observable.
(I used to think that this implied that all true statements are analytic, but I no longer hold this opinion (which I thought destructive for Kant's critical philosophy), rather, I think Kant is wrapped up in a series of questions we would be better to forget: epistemology.)