The proper treatment of variables in predicate logic

Linguistics and Philosophy 41 (2):209-249 (2018)
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Abstract

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”. 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. 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.

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Kai Wehmeier
University of California, Irvine

Citations of this work

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A Step Towards Absolute Versions of Metamathematical Results.Balthasar Grabmayr - 2024 - Journal of Philosophical Logic 53 (1):247-291.

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Semantics in generative grammar.Irene Heim & Angelika Kratzer - 1998 - Malden, MA: Blackwell. Edited by Angelika Kratzer.
Tractatus logico-philosophicus.Ludwig Wittgenstein - 1922 - Filosoficky Casopis 52:336-341.
What numbers could not be.Paul Benacerraf - 1965 - Philosophical Review 74 (1):47-73.

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