Negative Equivalence of Extensions of Minimal Logic

Two logics L₁ and L₂ are negatively equivalent if for any set of formulas X and any negated formula ¬, ¬ can be deduced from the set of hypotheses X in L₁ if and only if it can be done in L₂. This article is devoted to the investigation of negative equivalence relation in the class of extensions of minimal logic.The author acknowledges support by the Alexander von Humboldt-StiftungPresented by Jacek Malinowski     

Equational Bases for Joins of Residuated-lattice Varieties

Given a positive universal formula in the language of residuated lattices, we construct a recursive basis of equations for a variety, such that a subdirectly irreducible residuated lattice is in the variety exactly when it satisfies the positive universal formula. We use this correspondence to prove, among other things, that the join of two finitely based varieties of commutative residuated lattices is also finitely based. This implies that the intersection of two finitely axiomatized substructural logics over FL ⁺ is also finitely axiomatized. Finally, we give examples of cases where the join of two varieties is their Cartesian product.     

The Revival of Rejective Negation

Whether assent (acceptance) and dissent (rejection) are thought of as speech acts or as propositional attitudes, the leading idea of rejectivism is that a grasp of the distinction between them is prior to our understanding of negation as a sentence operator, this operator then being explicable as applying to A to yield something assent to which is tantamount to dissent from A. Widely thought to have been refuted by an argument of Frege"s, rejectivism has undergone something of a revival in recent years, especially in writings by Huw Price and Timothy Smiley. While agreeing that Frege"s argument does not refute the position, we shall air some philosophical qualms about it in Section 5, after a thorough examination of the formal issues in Sections 1–4. This discussion draws on – and seeks to draw attention to – some pertinent work of Kent Bendall in the 1970s.     

The elimination of de re formulas

It is shown that de re formulas are eliminable in the modal logic S5 extended with the axiom scheme x x.     

A sequent calculus for a logic of contingencies

We introduce a sequent calculus that is sound and complete with respect to propositional contingencies, i.e., formulas which are neither provable nor refutable. Like many other sequent and natural deduction proof systems, this calculus possesses cut elimination and the subformula property and has a simple proof search mechanism.     

Glivenko Type Theorems for Intuitionistic Modal Logics

In this article we deal with Glivenko type theorems for intuitionistic modal logics over Prior's MIPC. We examine the problems which appear in proving Glivenko type theorems when passing from the intuitionistic propositional logic Intto MIPC. As a result we obtain two different versions of Glivenko's theorem for logics over MIPC. Since MIPCcan be thought of as a one-variable fragment of the intuitionistic predicate logic Q-Int, one of the versions of Glivenko's theorem for logics over MIPCis closely related to that for intermediate predicate logics obtained by Umezawa [27] and Gabbay [15]. Another one is rather surprising.     

A QBF-based formalization of abstract argumentation semantics

We introduce a unified logical theory, based on signed theories and Quantified Boolean Formulas (QBFs) that can serve as the basis for representing and computing various argumentation-based decision problems. It is shown that within our framework we are able to model, in a simple and modular way, a wide range of semantics for abstract argumentation theory. This includes complete, grounded, preferred, stable, semi-stable, stage, ideal and eager semantics. Furthermore, our approach is purely logical, making for instance decision problems like skeptical and credulous acceptance of arguments simply a matter of entailment and satisfiability checking. The latter may be verified by off-the-shelf QBF-solvers.     

Kripke Incompleteness of Predicate Extensions of the Modal Logics Axiomatized by a Canonical Formula for a Frame with a Nontrivial Cluster

We generalize the incompleteness proof of the modal predicate logic Q-S4+ p p + BF described in Hughes-Cresswell [6]. As a corollary, we show that, for every subframe logic Lcontaining S4, Kripke completeness of Q-L+ BF implies the finite embedding property of L.     

First Order Extensions of Classical Systems of Modal Logic; The role of the Barcan schemas

The paper studies first order extensions of classical systems of modal logic (see (Chellas, 1980, part III)). We focus on the role of the Barcan formulas. It is shown that these formulas correspond to fundamental properties of neighborhood frames. The results have interesting applications in epistemic logic. In particular we suggest that the proposed models can be used in order to study monadic operators of probability (Kyburg, 1990) and likelihood (Halpern-Rabin, 1987).