# Analytic proposition

Loosely defined, an analytic proposition is a proposition the negation of which is self-contradictory, or a proposition that is true in every conceivable world, or a proposition that is true by definition. (See also synthetic proposition.)

For example, "All white cats are white" is not only true, but also necessarily true – since a negation of it – "Not all white cats are white" is self-contradictory.

There is no single, generally accepted, precise definition for analytic proposition, but philosophers have proposed a small number of closely related definitions, some of which are presented in the remainder of this article.

The term was first defined by Immanuel Kant (17241804):

Either the predicate B belongs to subject A, as something which is contained (though covertly) in the conception of A; or the predicate B lies entirely out of the conception of A, although it stands in connection with it. In the first instance, I term the judgement analytical, in the second, synthetical. Analytical judgements (affirmative) are therefore those in which the connection of the predicate with the subject is cogitated through identity; those in which this connection is cogitated without identity, are called synthetical judgments. --(From the Introduction to The Critique of Pure Reason.)

This definition is narrower than definitions currently in use.

Later philosophers pointed out that if Kant’s definition is accepted, some propositions that are true by definition are not analytic.

For example, 'A is A' is analytic by Kant’s definition.

But an equally obvious 'If A, then A' is not analytic since it is not framed in the subject-predicate form. As a result, the definition of analytic proposition was expanded to include statements that are not in subject-predicate form.

Two principle definitions for 'analytic proposition' have since been advanced:

1. An analytic proposition is one the negation of which is self-contradictory. If you deny a true analytic proposition, you always get a self-contradictory proposition.

2. An analytic proposition is a proposition the truth of which can be determined solely through the analysis of the meaning of its words. Nothing in the world apart from language needs to be examined.

For example, "All bachelors are unmarried" is true if we take "bachelor" to mean "unmarried man" – and its negation is self-contradictory – so it is an analytic proposition. Its truth is apparent through the definition of its words.

But if by "bachelors" we mean "individuals who have received a certain kind of academic degree" then we have a statement that may or may not be true, but certainly one that can be negated with no contradiction. In other words, in this case we no longer have an analytic proposition, but a synthetic one.

Analytic propositions need not be trivial tautologies like "All white cats are cats". Complex mathematical and geometrical theorems are analytic propositions, since a denial of such theorems leads to a contradiction. However, in the case of mathematical and geometric theorems, the statement that analytic propositions are true in any conceivable world breaks down.

For example, the theorems of Euclidean geometry are analytic – but only if the axioms of Euclidean geometry are assumed. In other words, these theorems are analytic within a specific deductive system rather than "any conceivable world".

### Analytic propositions and a priori knowledge

Analytic propositions and a priori knowledge are related, though not the same.

Analytic propositions are propositions of a certain kind.

A priori knowledge is knowledge that can be acquired without experience of the world.

So knowledge of analytic propositions is commonly held to be a priori knowledge. Whether other kinds of a priori knowledge can exist is a matter of considerable debate within philosophy (see synthetic proposition).