José R. Arrazola Ramírez

Logics with Common Weak Completions

We introduce the notion of X-stable models parametrized by a given logic X. Such notion is based on a construction that we call weak completions: a set of atoms M is an X-stable model of a theory T if M is a model of T, in the sense of classical logic, and the weak completion of T (namely T[:e M) can prove, in the sense given by logic X, every atom in the set M. We prove that, for normal logic programs, the result obtained by these weak completions is invariant with respect to a large family of logics. Two kinds of logics are mainly considered: paraconsistent logics and normal modal logics. For modal logics we use a translation proposed by Gelfond that identifies:œa with:a. As a consequence we prove that several semantics (recently introduced) for non-monotonic reasoning (NMR) are equivalent for normal programs. In addition, we show that such semantics can be characterized by a fixed-point operator. Also, as a side effect, we provide new results for the stable model semantics.

Logical Weak Completions of Paraconsistent Logics

Let P be an arbitrary theory and let X be any given logic. Let M be a set of atoms. We say that M is a X-stable model of P if M is a classical model of P and P∪¬M~ proves in logic X all atoms in M, this is denoted by P∪¬M~ ⊩xM. We prove that being an X-stable model is an invariant property for disjunctive programmes under a large class of logics. Two kinds of logics are mainly considered: paraconsistent logics and normal modal logics. For modal logics we use a translation proposed by Gelfond that replaces ¬a with ¬□a. As a consequence we prove that several semantics (recently introduced) for non-monotonic reasoning are equivalent for disjunctive programmes. In addition, we show that such semantics can be characterized by a fixed-point operator in terms of classical logic. We also present a simple translation of a disjunctive programme D into a normal programme N, such that the PStable model semantics of N corresponds to the stable semantics of D over the common language. We present the formal proof of this statement.

Ground Nonmonotonic Modal Logic S5: New Results

We study logic programs under Gelfonds translation in the context of modal logic S5. We show that for arbitrary logic programs (propositional theories where logic negation is associated with default negation) ground nonmonotonic modal logics between T and S5 are equivalent. Furthermore, we also show that these logics are equivalent to a nonmonotonic logic that we construct using the well known F O U R bilattice. We will call this semantic GNM-S5 as a reminder of its origin in the logic S5. Finally we show that, for normal programs, our approach is closely related to theWell-Founded-by-Cases Semantics introduced by Schlipf and the WFS+ proposed by Dix. We prove that GNM-S5 has the properties of classicality and extended cut. While WFS+ also supports classicality it fails to satisfy the extended cut principle, an important property available in other semantics such as stable models. Hence, we claim that GNM-S5 is a good candidate for defining a nonmonotonic semantics closer to the direction of classical logic.

Revisiting da Costa logic

Abstract In [25] Priest developed the da Costa logic (daC); this is a paraconsistent logic which is also a co-intuitionistic logic that contains the logic C ω . Due to its interesting properties it has been studied by Castiglioni, Ertola and Ferguson, and some remarkable results about it and its extensions are shown in [8] , [11] . In the present article we continue the study of daC, we prove that a restricted Hilbert system for daC, named DC, satisfies certain properties that help us show that this logic is not a maximal paraconsistent system. We also study an extension of daC called P H 1 and we give different characterizations of it. Finally we compare daC and P H 1 with several paraconsistent logics.

The Pursuit of an Implication for the Logics L3A and L3B

The authors of Beziau and Franceschetto (New directions in paraconsistent logic, vol 152, Springer, New Delhi, 2015) work with logics that have the property of not satisfying any of the formulations of the principle of non contradiction, Béziau and Franceschetto also analyze, among the three-valued logics, which of these logics satisfy this property. They prove that there exist only four of such logics, but only two of them are worthwhile to study. The language of these logics does not consider implication as a connective. However, the enrichment of a language with an implication connective leads us to more interesting systems, therefore we look for one implication for these logics and we study further properties that the logics obtain when this connective is added to these systems.

On Automatic Theorem Proving with ML

In this paper, we describe the development of a series of automatic theorem provers for a variety of logics. Provers are developed from a functional approach. The first prover is for Classical Propositional Calculus (CPC), which is based on a constructive proof of Kalmars Theorem. We also provide the implementation of a cut and contraction free sequent calculus for Intuitionistic Propositional Logic (IPC). Next, it is introduced a prover for ALCS4, which is the description logic ALC with transitive and reflexive roles only. This prover is also based on a cut and contraction free sequent calculus. We also provide a complexity analysis for each prover.

Possibilistic Minimal Models for Possibilistic Normal Programs

In this paper we present possibilistic minimal models for possibilistic normal programs, we relate them to the possibilistic C ω logic, PC ω L, and to minimal models of normal logic programs. Possibilistic stable models for possibilistic normal programs have been presented previously, but we present a more general type. We also characterize the provability of possibilistic atoms from possibilistic normal programs in terms of PC ω L.