Abstract
In this paper we prove the equivalence between the Gentzen system G LJ*\c , obtained by deleting the contraction rule from the sequent calculus LJ* (which is a redundant version of LJ), the deductive system IPC*\c and the equational system associated with the variety RL of residuated lattices. This means that the variety RL is the equivalent algebraic semantics for both systems G LJ*\c in the sense of [18] and [4], respectively. The equivalence between G LJ*\c and IPC*\c is a strengthening of a result obtained by H. Ono and Y. Komori [14, Corollary 2.8.1] and the equivalence between G LJ*\c and the equational system associated with the variety RL of residuated lattices is a strengthening of a result obtained by P.M. Idziak [13, Theorem 1].
An axiomatization of the restriction of IPC*\c to the formulas whose main connective is the implication connective is obtained by using an interpretation of G LJ*\c in IPC*\c.
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Adillon, R.J., Verdú, V. On a Contraction-Less Intuitionistic Propositional Logic with Conjunction and Fusion. Studia Logica 65, 11–30 (2000). https://doi.org/10.1023/A:1005238908087
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DOI: https://doi.org/10.1023/A:1005238908087