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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 Gordans 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.
This paper is a survey of two kinds of compressed proof schemes, the emph{matrix method} and emph{proof nets}, as applied to a variety of logics ranging along the substructural hierarchy from classical all the way down to the nonassociative Lambek sy
We give a sufficient condition for Kripke completeness of modal logics enriched with the transitive closure modality. More precisely, we show that if a logic admits what we call definable filtration (ADF), then such an expansion of the logic is compl
In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form $forall x_0 exists x_1 dots exists x_n bigwedge x_i R_lambda x_j$. We prove that many properties of these logics, such as finite axiom
This paper shows how to transform explosive many-valued systems into paraconsistent logics. We investigate especially the case of three-valued systems showing how paraconsistent three-valued logics can be obtained from them.
Given a class $mathcal C$ of models, a binary relation ${mathcal R}$ between models, and a model-theoretic language $L$, we consider the modal logic and the modal algebra of the theory of $mathcal C$ in $L$ where the modal operator is interpreted via