The Antinomy of the Variable: A Tarskian Resolution

Abstract

The theory of quantification and variable binding developed by Tarski is a fixed point for many debates in metaphysics, formal semantics, and philosophy of logic. However, recent critics-most forcefully, Kit Fine (2003, 2007)-have posed an intriguing set of challenges to Tarskiis account, which re-expose long sublimated anxieties about the variable from the infancy of analytic philosophy. The problem is a version of a puzzle confronted by Russell, which Fine dubs the antimony of the variablee. This paradox arises from seemingly contradictory things that we wish to say about the variable. On the one hand, there are strong reasons to deny that x and y are synonymous, since they make different contributions when they jointly occur within a sentence. Consider, for instance, the sentence ∃∃ ≤. One cannot replace the second occurrence of x with y yielding ∃∃ ≤ without change of meaning. On the other hand, there is a strong temptation to say that distinct variables x and y are synonymous, since sentences differing by the total, proper substitution of x for y always agree in meaning. For instance, ∀ ≤ and ∀ ≤ are synonymous in the strongest possible sense. As Fine says, they are mere notational variantss. We suggest that it is best to construe this very strong synonymy as an identity in structured meanings: the sentences and their corresponding parts are synonymous all the way down. This suggests that the variables occurring in corresponding positions in these formulas are also synonymous. One of the innovations of Tarskiis (1935) semantics is that a variable refers to or designates an object only relative to a sequence. One might hope that this goes some way towards resolving the antinomy, since Tarski need not assign any sort of referent to the variable. But this is not enough, since the antinomy concerns whether two variables are synonymous. As we formulate the antinomy, it concerns the variablees contribution to the structured meaning of a sentence that 2 contains it. Even on Tarskiis sequence-relative semantics, x and y may designate different individuals even relative to the same sequence. This suggests that their meanings are different. But this leaves Tarski unable to account for the felt sameness of meaning between distinct but corresponding variables in alphabetic variants. These challenges would overturn seemingly settled doctrines about the relationship between language and the world. A dramatic reconceptualization of the role of variables in mathematical practice, in natural language semantics, and even in first-order logic would be called for. Fine suggests semantic relationism, a radical departure from standard compositional semantics. (owever, Tarskiis semantics for variables has the resources to resolve the antinomy without abandoning standard compositional semantics. In a neglected passage, Tarski worried about how to determine the value of a variable relative to a sequence. He suggests that, in a given sentence, the first variable should be associated with the first position, the second variable with ...

Relational approaches to Frege's puzzle

“…(8a) and (8b) are notational variants; they do not differ in meaning. It follows—given assumptions about synonymy and compositionality (Pickel & Rabern, )— that ‘ x ” and “ y ” have the same meaning. But (9a) and (9b) are not synonymous; (9b) is satisfiable on the intended interpretation, and (9a) is not.…”

“…That is, we must hold that there are semantic relations that are not reducible to semantic properties. The rules for the language must, in addition to assigning semantic properties to individual occurrences of variables, assign semantic relations to sequences of variables (for critical discussion, see (Pickel & Rabern, , )).…”

Does Semantic Relationism Solve Frege’s Puzzle?

“…Given this it follows that 'x' and 'y' must be synonymous. See [18] for a detailed discussion the antinomy framed in these terms and a resolution of the problem within a broadly Tarskian framework. semantic relationships that hold among the sequence of its constituents as a whole.…”

Understanding Semantic Coordination in Cognition

“…For detailed discussion of these issues, from Frege to Fine, seePickel and Rabern (2016).292Gurpreet Rattan © 2019 The Author dialectica © 2019 Editorial Board of dialectica…”