For example, e.g. "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 selfcontradictory.
However there is no single, precise definition for analytic proposition, but instead related, but not precisely identical definitions exists.
The term was first defined by Emmanuel Kant:
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 subjectpredicate form. As a result, the definition of analytic proposition was expanded to include statements that are not in subjectpredicate form.
Two principle definitions for 'analytic proposition' have since been advanced:
Analytic propositions needed 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 case of mathematical and geometrical 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 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).
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