The gentzenization and decidability of RW

Studia Logica, 1986

The logic oi the weak law of excluded middle KOm is obtMned by adding the formula-tAv-1-TA as an axiom scheme to Heyting's in~uitionistie logic Hr. A cut-free sequent cMeulus for this logic is given. As the consequences of the cut-eNmination theorem, we get the decidabihty of the propositional part of this calculus, its separability, equality of the negationless fragments of KG~ and Hp, interpolation theorems and so on. From the proof-theoretical point of view, the formulation presented in this paper makes clearer the relations between KCcp, /~p and the classical logic. In the end, an interpretation of cI~ssical propositional logic in the proposi~ionM part of KC D is given.

Journal of Applied Non-classical Logics, 2005

Some investigations into the algebraicconstructive aspects of a decision procedure for various fragments of Relevant Logics are presented. Decidability of these fragments relies on S. Kripke's gentzenizations and on his combinatorial lemma known as Kripke's lemma that B. Meyer has shown equivalent to Dickson's lemma in number theory and to his own infinite divisor lemma, henceforth, Meyer's lemma orIDP. These

Annals of Pure and Applied Logic, 2010

We develop a general algebraic and proof-theoretic study of sub- structural logics that may lack associativity, along with other structural rules. Our study extends existing work on (associative) substructural logics over the full Lambek Calculus FL (see e.g. (36, 19, 18)). We present a Gentzen-style sequent system GL that lacks the structural rules of contraction, weakening, exchange and associativity, and

Studia Logica - An International Journal for Symbolic Logic, 2001

We introduce two Gentzen-style sequent calculus axiomatizations for conservative extensions of basic propositional logic. Our first axiomatization is an ipmrovement of, in the sense that it has a kind of the subformula property and is a slight modification of. In this system the cut rule is eliminated. The second axiomatization is a classical conservative extension of basic propositional logic. Using

The Journal of Symbolic Logic, Vol. 49, 1059-1073, 1984

Introduction. In this paper we show that the propositional logics E of entailment, R of relevant implication and T of ticket entailment are undecidable. The decision problem is also shown to be unsolvable in an extensive class of related logics. The main tool used in establishing these results is an adaptation of the von Neumann coordinatization theorem for modular lattices.

2017

We show how Leibnitz’s indiscernibility principle and Gentzen’s original work lead to extensions of the sequent calculus to first order logic with equality and investigate the cut elimination property. Furthermore we discuss and improve the nonlengthening property of Lifschitz and Orevkov in [5] and [8].

arXiv: Logic, 2009

We introduce a sequent calculus FL' for non-commutative substructural logic. It has at most one formula on the right side of sequent, and excludes three structural inference rules, i.e. contraction, weakening and exchange. (FL' is based on our investigations of the Gentzen-style natural deduction for non-commutative substructural logics.) FL' has the same proof strength as the standard sequent calculus FL (Full Lambek), which is the basic sequent calculus for all other substructural logics. For the standard FL, we use Ono's formulation. Although FL' and the standard FL are equivalent, there is a subtle difference in the left rule of implication. In the standard FL, two parameters $\Gamma_1$ and $\Gamma_2$(resp.), each of which is just an finite sequence of arbitrary formulas, appear on the left and right side (resp.) of a formula which is placed on the left side of the sequent on the upper left side of the left rule $\imply$ (which corresponds to $\imply'$ in...

Lecture Notes in Computer Science, 2008

Efficient, automated elimination of cuts is a prerequisite for proof analysis. The method CERES, based on Skolemization and resolution has been successfully developed for classical logic for this purpose. We generalize this method to Gödel logic, an important intermediate logic, which is also one of the main formalizations of fuzzy logic. RESolution. logic . We show that essential features of CERES can be adapted to the calculus HG [1, for G that uses hypersequents, a generalization of Gentzen's sequents to multisets of sequents. This adaption is far from trivial and, among other novel features, entails a new concept of 'resolution': hyperclause resolution, which combines most general unification and cuts on atomic hypersequents. It also provides clues to a better understanding of resolution based cut elimination for sequent and hypersequent calculi, in general.

Theoretical Computer Science, 2002

We present a natural generalization of Girard's (ÿrst order) phase semantics of linear logic (Theoret. Comput. Sci. 50 (1987)) to intuitionistic and higher-order phase semantics. Then we show that this semantic framework allows us to derive a uniform semantic proof of the (ÿrst order and) higher order cut-elimination theorem (as well as a (ÿrst order and) higher order phase-semantic completeness theorem) for various di erent logical systems at the same time. Our semantic proof works for various di erent logical systems uniformly in a strong sense (without any change of the argument of proof): it works for both ÿrst order and higher order versions and for linear, substructural, and standard logics uniformly, and for both their intuitionistic and classical versions uniformly.

Slogica, 1985