Distributive-lattice semantics of sequent calculi with structural rules

Logica Universalis 3 (1):59-94 (2009)
The goal of the paper is to develop a universal semantic approach to derivable rules of propositional multiple-conclusion sequent calculi with structural rules, which explicitly involve not only atomic formulas, treated as metavariables for formulas, but also formula set variables (viz., metavariables for finite sets of formulas), upon the basis of the conception of model introduced in (Fuzzy Sets Syst 121(3):27–37, 2001). One of the main results of the paper is that any regular sequent calculus with structural rules has such class of sequent models (called its semantics ) that a rule is derivable in the calculus iff it is sound with respect to each model of the semantics. We then show how semantics of admissible rules of such calculi can be found with using a method of free models. Next, our universal approach is applied to sequent calculi for many-valued logics with equality determinant . Finally, we exemplify this application by studying sequent calculi for some of such logics.
Keywords sequent calculus  structural rule  inference  derivable rule  admissible rule  distributive lattice  many-valued logic
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DOI 10.1007/s11787-009-0001-6
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References found in this work BETA
Raymond M. Smullyan (1968). First-Order Logic. New York [Etc.]Springer-Verlag.
Raymond Balbes & Philip Dwinger (1977). Distributive Lattices. Journal of Symbolic Logic 42 (4):587-588.

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