Thursday, February 13, 2014

Intensions

The intension of a referring expression e in a language is a partial function Ie that assigns to a world w the referent Ie(w) of e there, when there is a referent of e in w. Thus, the intension of "the tallest woman" is a partial function that assigns to w the tallest woman in w.

The intension of a unary predicate P is a partial function IP that assigns to a world w the extension IP(w) of P at w, i.e., the set of all satisfiers of P there.

Intensions are meant to capture the semantic features of terms, with respect to intensional semantics. Now let e be the referring expression:

  • The set of even integers.
Let E be the predicate
  • is an even integer.
Then the intension of e assigns to w the set of all even integers, for each w. And the intension of E assigns to w the set of all even integers, too. So Ie=IE. But e and E are plainly not semantically equivalent, even within intensional semantics. So intensions are insufficient for characterizing the semantic features of expressions, even with respect to intensional semantics.

A longshot: Perhaps something like this led Frege to his weird "The concept horse is not a concept" claim.

2 comments:

David Balcarras said...

Literature on the concept horse paradox suggests the following line of thought:

(1) The referent of 'the concept horse' is not a concept.
(2) The referent of 'the concept horse' = the concept horse.
(3) So, the concept horse is not a concept.

(1) follows from Frege's type-theory and syntax-semantics isomorphism: noun phrases must refer to objects; no object is a concept.

One might interpret (2) just as an identity claim. Alternatively, it might be, for Frege, the claim that 'The referent of 'the concept horse'' and 'the concept horse' co-refer, and so, via Frege's so-called "Reference Principle", they must be intersubstitutable salva veritate.

Alexander R Pruss said...

Yeah, but this raises the question why not just take this to provide a counterexample to the thesis that concepts are not objects, or to the thesis that only objects are referred to.