Sequent Calculus in the Topos of Trees

Nakano's "later" modality, inspired by G\"{o}del-L\"{o}b provability logic, has been applied in type systems and program logics to capture guarded recursion. Birkedal et al modelled this modality via the internal logic of the topos of trees. We show that the semantics of the propositional fragment of this logic can be given by linear converse-well-founded intuitionistic Kripke frames, so this logic is a marriage of the intuitionistic modal logic KM and the intermediate logic LC. We therefore call this logic $\mathrm{KM}_{\mathrm{lin}}$. We give a sound and cut-free complete sequent calculus for $\mathrm{KM}_{\mathrm{lin}}$ via a strategy that decomposes implication into its static and irreflexive components. Our calculus provides deterministic and terminating backward proof-search, yields decidability of the logic and the coNP-completeness of its validity problem. Our calculus and decision procedure can be restricted to drop linearity and hence capture KM.Comment: Extended version, with full proof details, of a paper accepted to FoSSaCS 2015 (this version edited to fix some minor typos

On the injectivity of the Leibniz operator

The class of weakly algebrizable logics is defined as the class of logics having monotonic and injective Leibniz operator. We show that \monotonicity" can- not be discarded on this definition, by presenting an example of a system with injective and non monotonic Leibniz operator. We also show that the non injectivity of the non protoalgebraic inf-sup fragment of the Classic Propositional Calculus, CPC_{inf,sup}, holds only from the fact that the empty set is a CPC_{inf,sup}-filter.FCT via UIM

Semantic Incompleteness of del Cerro and Herzig's Hilbert System for a Combination of Classical and Intuitionistic Propositional Logic

This paper shows Hilbert system (C+J)-, given by del Cerro and Herzig (1996) is semantically incomplete. This system is proposed as a proof theory for Kripke semantics for a combination of intuitionistic and classical propositional logic, which is obtained by adding the natural semantic clause of classical implication into intuitionistic Kripke semantics. Although Hilbert system (C+J)- contains intuitionistic modus ponens as a rule, it does not contain classical modus ponens. This paper gives an argument ensuring that the system (C+J)- is semantically incomplete because of the absence of classical modus ponens. Our method is based on the logic of paradox, which is a paraconsistent logic proposed by Priest (1979)

On subreducts of subresiduated lattices and logic

Subresiduated lattices were introduced during the decade of 1970 by Epstein and Horn as an algebraic counterpart of some logics with strong implication previously studied by Lewy and Hacking. These logics are examples of subuintuitionistic logics, i.e., logics in the language of intuitionistic logic that are defined semantically by using Kripke models, in the same way as intuitionistic logic is defined, but without requiring of the models some of the properties required in the intuitionistic case. Also in relation with the study of subintuitionistic logics, Celani and Jansana get these algebras as the elements of a subvariety of that of weak Heyting algebras. Here, we study both the implicative and the implicative-infimum subreducts of subresiduated lattices. Besides, we propose a calculus whose algebraic semantics is given by these classes of algebras. Several expansions of this calculi are also studied together to some interesting properties of them