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Logical deducibility and substitution in Bolzano and beyond (William Nava)

The Logic and Metaphysics Workshop will meet on April 12th from 4:15-6:15 (NY time) via Zoom for a talk by William Nava (NYU).

Title: Logical deducibility and substitution in Bolzano (and beyond)

Abstract: Bolzano is famously responsible for an influential substitutional account of logical consequence (or, as he calls it, logical deducibility): a proposition, 𝜑, is logically deducible from a set of propositions, Γ, iff every uniform substitution of non-logical ideas in Γ∪{𝜑} that makes every proposition in Γ true also makes 𝜑 true. There are two problems with making sense of Bolzano’s proposal, however. One is that Bolzano argues that every proposition is of the form a has B—in other words, is a monadic atomic predication. So, for Bolzano, logically complex propositions like ‘𝜑 and 𝜓’ cannot have the semantic structure they appear to. This can be addressed, roughly, by taking complex propositions to predicate logical ideas of collections of propositions. But this introduces the second problem: for Bolzano, familiar logical ideas like ‘and’, ‘or’, and ‘not’ are complex ideas with compositional structure. I’ll show that, as a result of this structure, we cannot use the simple and familiar notion of uniform substitution in order to understand logical deducibility. We must instead use what I’ll call form-sensitive substitution. I will end by drawing some general lessons about substitutional definitions of logical consequence in languages with the resources to generate complex predicates of propositions.

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