
Authors:
Tomoaki KAWANO
Title:Labelled Sequent Calculus for Orthologic
Pages:217232
File:bibtex
Abstract ( + )
Orthologic (OL) is nonclassical logic and has been studied as a part of quantum logic. OL is based on an ortholattice and is also called minimal quantum logic. Sequent calculus is used as a tool for proof in logic and has been examined for several decades. Although there are many studies on sequent calculus for OL, these sequent calculi have some problems. In particular, they do not include implication connective and they are mostly incompatible with the cutelimination theorem. In this paper, we introduce new labeled sequent calculus called LGOI, and show that this sequent calculus solve the above problems. It is already known that OL is decidable. We prove that decidability is preserved when the implication connective is added to OL.
Keywords: Quantum logic, Sequent calculus, cutelimination theorem, Decidability, Kripke Model

Authors:
Marcin ŁYCZAK and Andrzej PIETRUSZCZAK
Title:On the Definability of Leśniewski's Copula 'is' in Some Ontologylike Theories
Pages:233263
File:bibtex
Abstract ( + )
We formulate a certain subtheory of Ishimoto's [1] quantifierfree fragment of Leśniewski's ontology, and show that Ishimoto's theory can be reconstructed in it. Using an epimorphism theorem we prove that our theory is complete with respect to a suitable settheoretic interpretation. Furthermore, we introduce the name constant 1 (which corresponds to the universal name `object') and we prove its adequacy with respect to the settheoretic interpretation (again using an epimorphism theorem). Ishimoto's theory enriched by the constant 1 is also reconstructed in our formalism with into which 1 has been introduced. Finally we examine for both our theories their quantifier extensions and their connections with Leśniewski's classical quantified ontology.
Keywords: Leśniewski's ontology, elementary ontology, quantifierfree fragment of ontology, copula `is', calculus of names, ontologylike theories, subtheories of Leśniewski's ontology

Authors:
Andrzej INDRZEJCZAK
Title:RuleGeneration Theorem and Its Applications
Pages:265281
File:bibtex
Abstract ( + )
In several applications of sequent calculi going beyond pure logic, an introduction of suitably defined rules seems to be more profitable than addition of extra axiomatic sequents. A program of formalization of mathematical theories via rules of special sort was developed sucessfully by Negri and von Plato. In this paper a general theorem on possible ways of transforming axiomatic sequents into rules in sequent calculi is proved. We discuss its possible applications and provide some case studies for illustration.
Keywords: sequent calculus, cut elimination, proof theory, extralogical rules

Authors:
Rajabali A. BORZOOEI and S. Saidi GORAGHANI
Title:On Injective MVmodules
Pages:283298
File:bibtex
Abstract ( + )
In this paper, by considering the notion of MVmodule, which is the structure that naturally correspond to lumodules over lurings, we study injective MVmodules and we investigate some conditions for constructing injective MVmodules. Then we define the notions of essential Ahomomorphisms and essential extension of Ahomomorphisms, where A is a product MValgebra, and we get some of there properties. Finally, we prove that a maximal essential extension of any Aideal of an injective MVmodule is an injective Amodule, too.
Mathematical Subject Classification(2010): 06D35, 06F99, 16D80
Keywords: (MV, PMV)algebra, MVmodule, Injective MVmodule, Essential extension