Theorems of Alternatives for Substructural Logics

Abstract

AbstractA 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 \(\mathrm {RM}\) 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 \(\mathrm {A}\). 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.