intensional logic that part of deductive logic which treats arguments whose validity or invalidity depends on strict difference, or identity, of meaning. The denotation of a singular term (i.e., a proper name or definite description), the class of things of which a predicate is true, and the truth or falsity (the truth-value) of a sentence may be called the extensions of these respective linguistic expressions. Their intensions are their meanings strictly so called: the (individual) concept conveyed by the singular term, the property expressed by the predicate, and the proposition asserted by the sentence. The most extensively studied part of formal logic deals largely with inferences turning only on extensions. One principle of extensional logic is that if two singular terms have identical denotations, the truth-values of corresponding sentences containing the terms are identical. Thus the inference from ‘Bern is the capital of Switzerland’ to ‘You are in Bern if and only if you are in the capital of Switzerland’ is valid. But this is invalid: ‘Bern is the capital of Switzerland. Therefore, you believe that you are in Bern if and only if you believe that you are in the capital of Switzerland.’ For one may lack the belief that Bern is the capital of Switzerland. It seems that we should distinguish between the intensional meanings of ‘Bern’ and of ‘the capital of Switzerland’. One supposes that only a strict identity of intension would license interchange in such a context, in which they are in the scope of a propositional attitude. It has been questioned whether the idea of an intension really applies to proper names, but parallel examples are easily constructed that make similar use of the differences in the meanings of predicates or of whole sentences. Quite generally, then, the principle that expressions with the same extension may be interchanged with preservation of extension of the containing expression, seems to fail for such ‘intensional contexts.’
The range of expressions producing such sensitive contexts includes psychological verbs like ‘know’, ‘believe’, ‘suppose’, ‘assert’, ‘desire’, ‘allege’, ‘wonders whether’; expressions conveying modal ideas such as necessity, possibility, and impossibility; some adverbs, e.g. ‘intentionally’; and a large number of other expressions – ‘prove’, ‘imply’, ‘make probable’, etc. Although reasoning involving some of these is well understood, there is not yet general agreement on the best methods for dealing with arguments involving many of these notions.
See also MODAL LOGIC. C.A.A.