We present two embeddings of infinite-valued ̷Lukasiewicz logic ̷L into Meyer and Slaney’s abelian logic A, the logic of lattice-ordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for ̷L. These include: hypersequent calculi for A and ̷L and terminating versions of these calculi; labelled single sequent calculi for A and ̷L of complexity co-NP; unlabelled single sequent calculi for A and ̷L. 1

Herbrand’s Theorem £¥ ¤ ¦ for, i.e., Gödel logic enriched by the projection § operator is proved. As a consequence we obtain a “chain normal form” and a translation of £ ¤ ¦ prenex into (order) clause logic, referring to the classical theory of dense total orders with endpoints. A chaining calculus provides a basis for efficient theorem proving.

Hypersequent calculi arise by generalizing standard sequent calculi to refer to whole contexts of sequents instead of single sequents. We present a number of results using hypersequents to obtain a Gentzen-style characterization for the family of Gödel logics. We first describe analytic calculi for propositional finite and infinite-valued Gödel logics. We then show that the framework of hypersequents allows one to move straightforwardly from the propositional level to first-order as well as propositional quantification. A certain type of modalities, enhancing the expressive power of Gödel logic, is also considered. 1

We present a terminating contraction-free calculus GLC for the propositional fragment of Godel-Dummett Logic LC. GLC uses hypersequents, and unlike other Gentzen-type calculi for LC, all its rules have at most two premises. These rules are all invertible. Hence it can be used as a basis for a deterministic tableaux system for LC. This tableaux system is presented in the last section. I A Review of LC and GLC In [Go33] Godel introduced a sequence fG n g of n-valued logics, as well as an infinite-valued matrix G ! in which all the G n s can be embedded. He used these matrices to show some important properties of intuitionistic logic. The logic of G ! was later axiomatized by Dummett in [Du59] and is known since then as Dummett's LC. It probably is the most important intermediate logic, one that turns up in several places, like the provability logic of Heyting's Arithmetics ([Vi82]) and relevance logic ([DM71]) and recently fuzzy logic([Ha98]). semantically LC corresponds to lin...

In [5] the analytic calculus RG1 for G odel logic has been introduced. RG1 operates on "sequents of relations ". We show constructively how to eliminate cuts from RG1 -derivations. The version of the cut rule we consider allows to derive other forms of cut as well as a rule corresponding to the "communication rule" of Avron's hypersequent calculus for G1 . Moreover, we give an explicit description of all the axioms of RG1 and prove their completeness. 1.

We present a general framework that allows to construct systematically analytic calculi for a large family of (propositional) many-valued logics --- called projective logics --- characterized by a special format of their semantics. All finite-valued logics as well as infinite-valued Godel logic are projective. As a case-study, sequent of relations calculi for Godel logics are derived. A comparison with some other analytic calculi is provided.

A classical Gentzen-type system is one which employs two-sided sequents, together with structural and logical rules of a certain characteristic form. A decent Gentzentype system should allow for direct proofs, which means that it should admit some useful forms of cut elimination and the subformula property. In this tutorial we explain the main difficulty in developing classical Gentzen-type systems with these properties for many-valued logics. We then illustrate with numerous examples the various possible ways of overcoming this difficulty. Our examples include practically all 3-valued logics, the most important class of 4-valued logics, as well as central infinite-valued logics (like GodelDummett logic, S5 and some substructural logics). 1

Abstract: A dialogue game for fuzzy logic, based on the comparison of truth degrees, is presented. It is shown that the game is adequate for G △ ∞, i.e., intuitionistic fuzzy logic enriched by the projection operator △. Any given countermodel to a formula can be used to construct a winning strategies for one of the players, called Opponent. Conversely, counter-models can be extracted from each winning strategy for Opponent. Winning strategies for the other player, Proponent, correspond to proofs of validity. The systematic construction of so-called complete dialogue trees can be viewed as tableau style proof search procedure.

Abstract. We present a graph-based decision procedure for Gödel-Dummett logics and an algorithm to compute counter-models. A formula is transformed into a conditional bi-colored graph in which we detect some specific cycles and alternating chains using matrix computations. From an instance graph containing no such cycle (resp. no (n + 1)-alternating chain) we extract a counter-model in LC (resp. LCn). Keywords: Gödel-Dummett logic, sequent calculus, decision procedures, graphs, counter-models. 1.

. A key property in the definition of logic programming languages is the completeness of goal-directed proofs. This concept originated in the study of logic programming languages for intuitionistic logic in the (single-conclusioned) sequent calculus LJ, but has subsequently been adapted to multiple-conclusioned systems such as those for linear logic. Given these developments, it seems interesting to investigate the notion of goal-directed proofs for a multiple-conclusioned sequent calculus for intuitionistic logic, in that this is a logic for which there are both single-conclusioned and multiple-conclusioned systems (although the latter are less well known). In this paper we show that the language obtained for the multiple-conclusioned system differs from that for the single-conclusioned case, show how hereditary Harrop formulae can be recovered, and investigate contraction-free fragments of the logic. 1 Introduction Logic programming is based upon the observation that if ...