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First-order Gödel logics are a family of infinite-valued logics where the sets of truth values V are closed subsets of [0, 1] containing both 0 and 1. Different such sets V in general determine different Gödel logics G V (sets of those formulas which evaluate to 1 in every interpretation into V). It is shown that G V is axiomatizable iff V is finite, V is(More)
Preface Many-valued logic is not much younger than the whole field of symbolic logic. It was introduced in the early twenties of this century by Lukasiewicz [1920] and Post [1921] and has since developed into a very large area of research. Most of the early work done has concentrated on problems of axiomatizability on the one hand, and algebraical/model(More)
Entailment in propositional Gödel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinite-valued Gödel logics, only one of which is compact. It is also shown that the compact infinite-valued Gödel logic is the(More)
A construction principle for natural deduction systems for arbitrary finitely-many-valued first order logics is exhibited. These systems are systematically obtained from sequent calculi, which in turn can be automatically extracted from the truth tables of the logics under consideration. Soundness and cut-free completeness of these sequent calculi translate(More)
A uniform construction for sequent calculi for nite-valued rst-order logics with distribution quantiiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand's theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this(More)
A general class of labeled sequent calculi is investigated, and necessary and suucient conditions are given for when such a calculus is sound and complete for a nite-valued logic if the labels are interpreted as sets of truth values (sets-assigns). Furthermore, it is shown that any nite-valued logic can be given an axiomatization by such a labeled calculus(More)