Intensional extensionality: The extension of a sentence is an intension.

Combine this with a plausible Fregean thesis about composition.

Extensional compositionality: The extension of a sentence is determined by the extensions of its sub-sentential parts.

And we can derive the following.

Thesis. Sub-sentential expressions designate intensions.

Proof. Assume (to reach a contradiction) that sub-sentential expressions do not designate intensions. Consider a pair of non-cointensive sentences Ψa and Ψb that only differ on the sub-sentential parts 'a' and 'b', where 'a' and 'b' are coextensive but not cointensive. Since every sub-sentential part of Ψa is coextensive with a sub-sentential part of Ψb (and vice versa), it follows by Extensional Compositionally that Ψa and Ψb are coextensive. By Intensional Extensionality it follows that Ψa and Ψb are cointensive. Contradiction.

Extensional compositionality: The extension of a sentence is determined by the extensions of its sub-sentential parts.

And we can derive the following.

Thesis. Sub-sentential expressions designate intensions.

Proof. Assume (to reach a contradiction) that sub-sentential expressions do not designate intensions. Consider a pair of non-cointensive sentences Ψa and Ψb that only differ on the sub-sentential parts 'a' and 'b', where 'a' and 'b' are coextensive but not cointensive. Since every sub-sentential part of Ψa is coextensive with a sub-sentential part of Ψb (and vice versa), it follows by Extensional Compositionally that Ψa and Ψb are coextensive. By Intensional Extensionality it follows that Ψa and Ψb are cointensive. Contradiction.

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This is some kind of "modus tollens-ing" of the Frege-Church-Goedel slingshot argument that sentences refer to truth-values. The argument above starts with the assumption (which I tried to give some [insincere] motivation) that sentences designate intensions; functions from indicies to truth-values. The slingshot, on the other hand, starts with the assumption that it is not the case that all sub-sentential expressions designate intensions. Its not clear, to me, that this is a substantial issue (see reference in last post).

The semantic value of a sentence is determined by the semantic values of its parts. Whatever these "semantic values" of sentences are, we should ensure them to be the same in kind as the semantic values of the sub-sentential parts.

## 1 comment:

I agree, but I think it is misleading to use 'designation' for 'semantic value'. It seems to me that 'designation' is analytically tied to a disquotation principle:

'x' designates an F iff x is F.

Since London is not a function, 'London' therefore does not designate a function.

Cresswell usually takes the semantic value of sentences to be propositions (intensions), that of names and first-order variables to be individuals, and that of predicates to be properties (functions from indices to sets of individuals). Then it would seem as if names do designate their semantic value, in the ordinary sense of 'designate'. Your refutation assumes that there are constructions that are sensitive to more than the designatum. But depending on the language, there may be no such constructions (even if the language is modal).

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