A strong completeness theorem for the gentzen systems associated with finite algebras

Preprint enviat per a la seva publicació en una revista científica: Journal of Applied Non-Classical Logics. 1999, vol. 9, issue 1, pp 9-36. [http://doi.org/10.1080/11663081.1999.10510956] In this paper we study consequence relations on the set of many sided sequents over a propositional language. We deal with the consequence relations axiomatized by the sequent calculi defined in [2] and associated with arbitrary finite algebras. These consequence relations are examples of what we call Gentzen systems. We define a semantics for these systems and prove a Strong Completeness Theorem, which is an extension of the Completeness Theorem for provable sequents stated in [2]. For the special case of the finite linear MV-algebras, the Strong Completeness Theorem was proved in [10], as a consequence of McNaughton's Theorem. The main tool to prove this result for arbitrary algebras is the deduction-detachment theorem for Gentzen systems.