Theorems of Alternatives for Substructural Logics

Colacito, Almudena; Galatos, Nikolaos; Metcalfe, George (2021). Theorems of Alternatives for Substructural Logics. In: Arieli, Ofer; Zamansky, Anna (eds.) Arnon Avron on Semantics and Proof Theory of Non-Classical Logics. Outstanding Contributions to Logic: Vol. 21 (pp. 91-105). Switzerland: Springer 10.1007/978-3-030-71258-7_5

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A theorem of alternatives provides a reduction of validity in a substructural logic to validity in its multiplicative fragment. Notable examples include a theorem of Arnon Avron that reduces the validity of a disjunction of multiplicative formulas in the "R-mingle" logic RM to the validity of a linear combination of these formulas, and Gordan’s theorem for solutions of linear systems over the real numbers that yields an analogous reduction for validity in Abelian logic A. In this paper, general conditions are provided for axiomatic extensions of involutive uninorm logic without additive constants to admit a theorem of alternatives. It is also shown that a theorem of alternatives for a logic can be used to establish (uniform) deductive interpolation and completeness with respect to a class of dense totally ordered residuated lattices.

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