Disimplicature

Speranza

In §93 of "The Principles of Mathematics," Baron B. A. W. Russell (born in England -- now Wales) observes that 

"the variable is a very complicated logical entity, by no means easy to analyze correctly."

He is using 'analyse' alla Grice.

(Russell was a Griceian at heart -- and vice versa -- vide "Definite descriptions in Russell and in the vernacular," by Grice).

This 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 -- Grice's "System Q" in honour of Quine, later labelled System G, by Myro, and System G-hp (hopefully plausible/highly powerful version of system G).

In a standard Tarskian semantics, variables are treated as meaning-bearing entities.

But Grice read no Polish!

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. 

We may develop an alternative, Fregeian 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.

And that would be loved by Grice in that it would be expressible within System G-hp.

Or not, of course!