A General Schema for Bilateral Proof Rules

Journal of Philosophical Logic (3):1-34 (2024)
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Abstract

Bilateral proof systems, which provide rules for both affirming and denying sentences, have been prominent in the development of proof-theoretic semantics for classical logic in recent years. However, such systems provide a substantial amount of freedom in the formulation of the rules, and, as a result, a number of different sets of rules have been put forward as definitive of the meanings of the classical connectives. In this paper, I argue that a single general schema for bilateral proof rules has a reasonable claim to inferentially articulating the core meaning of all of the classical connectives. I propose this schema in the context of a bilateral sequent calculus in which each connective is given exactly two rules: a rule for affirmation and a rule for denial. Positive and negative rules for all of the classical connectives are given by a single rule schema, harmony between these positive and negative rules is established at the schematic level by a pair of elimination theorems, and the truth-conditions for all of the classical connectives are read off at once from the schema itself.

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Ryan Simonelli
University of Chicago

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References found in this work

The logical basis of metaphysics.Michael Dummett - 1991 - Cambridge: Harvard University Press.
Making it Explicit.Isaac Levi & Robert B. Brandom - 1996 - Journal of Philosophy 93 (3):145.
First-order logic.Raymond Merrill Smullyan - 1968 - New York [etc.]: Springer Verlag.
Paradoxes and Failures of Cut.David Ripley - 2013 - Australasian Journal of Philosophy 91 (1):139 - 164.
Epistemic Multilateral Logic.Luca Incurvati & Julian J. Schlöder - 2022 - Review of Symbolic Logic 15 (2):505-536.

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