Disimplicature

Speranza

In §93 of The Principles of Mathematics, Bertrand Russell observes that “the variable is a very complicated logical entity, by no means easy to analyze correctly”. 

Grice found that funny!

Russell's assessment is borne out by the fact that even now we have no fully satisfactory understanding of the role of variables in a compositional semantics for first-order logic. In standard Tarskian semantics, variables are treated as meaning-bearing entities; moreover, they serve as the basic building blocks of all meanings, which are constructed out of variable assignments. But this has disquieting consequences, including Fine’s antinomy of the variable and an undue dependence of meanings on language. Here I develop an alternative, Fregean version of predicate logic that uses the traditional quantifier–variable apparatus for the expression of generality, possesses a fully compositional, non-representational semantics, and is not subject to the antinomy of the variable. 

The advantages of Fregean over Tarskian predicate logic are due to the former’s treating variables not as meaningful lexical items, but as mere marks of punctuation, similar to parentheses. I submit that this is indeed how the variables of predicate logic should be construed.

Balashov Sums Up Neale for us

The Gricean-Russellian Analysis of the Referential Use of DDs

(From Stephen Neale, Descriptions (MIT Press, 1990), Ch. 3

Consider the utterance of

The Φ is Ψ

by speaker S, where ‘the Φ’ (e.g., ‘Smith’s murderer’) is used referentially to refer to a (e.g., Jones).

1. S has expressed the proposition that [the x: Φx] Ψx
(i.e., the proposition expressed by the purely general “Russellian” sentence ‘(∃x) (Φx & ((∀y) (Φy ⊃ x=y) & Ψx))’ )

2. There is no reason to suppose that S is not observing the CP and maxims.

3. S could not be doing this unless he thought that Ψa (where ‘a’ is a name).
Gloss: on the assumption that S is observing the Maxim of Relation, he must be attempting to convey something beyond the general proposition that whoever is uniquely Φ is Ψ. On the assumption that S is adhering to the Maxim of Quality, he must have adequate evidence for thinking that the Φ is Ψ. I know S knows that a is the Φ, therefore S thinks that Ψa.

4. S knows (and knows that I know that he knows) that I know that a is the Φ, that I know that S knows that a is the Φ, and that I can see that S thinks the supposition that he thinks that Ψa is required.

5. S has done nothing to stop me thinking that Ψa.

6. S intends me to think, or at least willing to allow me to think, that Ψa.

7. And so, S has implicated that Ψa.

Balashov Sums Up Neale for us

The Gricean-Russellian Analysis of the Referential Use of DDs

(From Stephen Neale, Descriptions (MIT Press, 1990), Ch. 3

Consider the utterance of

The Φ is Ψ

by speaker S, where ‘the Φ’ (e.g., ‘Smith’s murderer’) is used referentially to refer to a (e.g., Jones).

1. S has expressed the proposition that [the x: Φx] Ψx
(i.e., the proposition expressed by the purely general “Russellian” sentence ‘(∃x) (Φx & ((∀y) (Φy ⊃ x=y) & Ψx))’ )

2. There is no reason to suppose that S is not observing the CP and maxims.

3. S could not be doing this unless he thought that Ψa (where ‘a’ is a name).
Gloss: on the assumption that S is observing the Maxim of Relation, he must be attempting to convey something beyond the general proposition that whoever is uniquely Φ is Ψ. On the assumption that S is adhering to the Maxim of Quality, he must have adequate evidence for thinking that the Φ is Ψ. I know S knows that a is the Φ, therefore S thinks that Ψa.

4. S knows (and knows that I know that he knows) that I know that a is the Φ, that I know that S knows that a is the Φ, and that I can see that S thinks the supposition that he thinks that Ψa is required.

5. S has done nothing to stop me thinking that Ψa.

6. S intends me to think, or at least willing to allow me to think, that Ψa.

7. And so, S has implicated that Ψa.