Algebraic proof theory for LE-logics

Studia Logica, 2006

Substructural logics have received a lot of attention in recent years from the communities of both logic and algebra. We discuss the algebraization of substructural logics over the full Lambek calculus and their connections to residuated lattices, and establish a weak form of the deduction theorem that is known as parametrized local deduction theorem. Finally, we study certain interpolation properties and explain how they imply the amalgamation property for certain varieties of residuated lattices.

arXiv (Cornell University), 2020

We introduce proper display calculi for basic monotonic modal logic, the conditional logic CK and a number of their axiomatic extensions. These calculi are sound, complete, conservative and enjoy cut elimination and subformula property. Our proposal applies the multi-type methodology in the design of proper display calculi, starting from a semantic analysis which motivates syntactic translations from single-type non-normal modal logics to multi-type normal poly-modal logics.

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.

Logica Universalis, 2007

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.

Annals of Pure and Applied Logic, 2010

We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart AlgS of any finitary and congruential logic S. This definition is logic-based rather than purely order-theoretic and is in general different from the one given e.g. in , but it agrees with it whenever the algebras in AlgS are based on lattices. As a case study on logics purely based on implication, we prove that the varieties of Hilbert and Tarski algebras are canonical in this new sense.

Logic Journal of IGPL, 1997

In this paper we give relational semantics and an accompanying relational proof system for a variety of intuitionistic substructural logics, including (intuitionistic) linear logic with exponentials. Starting with the (Kripke-style) semantics for F L as discussed in [13], we developed, in [11], a relational semantics and a relational proof system for full Lambek calculus. Here, we take this as a base and extend the results to deal with the various structural rules of exchange, contraction, weakening and expansion, and also to deal with an involution operator and with the operators ! and ? of linear logic. To accomplish this, for each extension X of F L we develop a Kripke-style semantics, RelKripke X semantics, as a bridge to relational semantics. The RelKripke X semantics consists of a set with distinguished elements, ternary relations and a list of conditions on the relations. For each extension X, RelKripke X semantics is accompanied by a Kripke-style valuation system analogous to that in [13]. Soundness and completeness theorems with respect to F L X hold for RelKripke X-models. Then, in the spirit of the work of Orlowska [16], [17], and Buszkowski & Orlowska [4], we develop relational logic RF L X for each extension X. The adjective relational is used to emphasize the fact that RF L X has a semantics wherein formulas are interpreted as relations. We prove that a sequent Γ → α in F L X is provable iff, a translation, t(γ 1 • ... • γn ⊃ α)ǫvu, has a cut-complete proof tree which is fundamental. This result is constructive: that is, if a cut-complete proof tree for t(γ 1 • ... • γn ⊃ α)ǫvu is not fundamental, we can use the failed proof search to build a relational countermodel for t(γ 1 • ... • γn ⊃ α) and from this, build a RelKripke X countermodel for γ 1 • ... • γn ⊃ α. 1

2000

Theoretical Computer Science - TCS, 1989

Linear Logic, we concisely write LL, has been introduced recently by Jean Yves Girard in Theoretical Computer Science ~0 (1987). Born from the semantics of second order lambda calculus, LL is more expressive than traditional logic (both classical and intuitionistic). Due to the absence of structural rules and to a partict:!ar treatment of negation, which is denoted by ~, proofs in LL do not have a "directional character". The constructive meaning of a proof of A-, B is a function mapping all proofs of A into proofs of B; in LL A-~B has a similar meaning, but B±-oA ± represents the same formula and has the same proofs: so one of such proofs can map a proof of A into one of B as well as a proof of B x into one of.4±~ In this paper we are interested in the multiplicative second order subsystem L,:~* of linear logic; we introduce a calculus (called z-calculus) whose terms are canonical represent: ~ions of proofs. The aim of the calculus is to give a be~ter comprehension of the computational aspects of the process of cut-elimination. We prove that the z-calculus obeys strong normalization and the Church-Rosser properties.