Stabilizing Quantum Disjunction

Abstract

Since the appearance of Prior's tonk, inferentialists tried to formulate conditions that a collection of inference rules for a logical constant has to satisfy in order to succeed in conferring an acceptable meaning to it. Dummett proposed a pair of conditions, dubbed 'harmony' and 'stability' that have been cashed out in terms of the existence of certain transformations on natural deduction derivations called reductions and expansions. A long standing open problem for this proposal is posed by quantum disjunction: although its rules are intuitively unstable, they pass the test of existence of expansions. Although most authors view instabilities of this kind as too subtle to be detected by the requirement of existence of expansions, we first discuss a case showing that this requirement can indeed detect instabilities of this kind, and then show how the expansions for disjunctionlike connectives have to be reformulated to rule out quantum disjunction. We show how the alternative pattern for expansions can be formulated for connectives and quantifiers whose rules satisfy a scheme originally developed by Prawitz and Schroeder-Heister. Finally we compare our proposal with a recent one due to Jacinto and Read.

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“…Generalized permuations were first introduced by Seely[18] and have been recently investigated in the context of proof-theoretic semantics by Tranchini[20,21].…”