Bulletin of the Section of Logic 47/4 (2018)http://hdl.handle.net/11089/279332020-01-23T11:57:52Z2020-01-23T11:57:52ZOn Injective MV-ModulesBorzooei, Rajabali A.Goraghani, S. Saidihttp://hdl.handle.net/11089/284832019-05-25T01:17:23Z2018-01-01T00:00:00ZOn Injective MV-Modules
Borzooei, Rajabali A.; Goraghani, S. Saidi
In this paper, by considering the notion of MV-module, which is the structure that naturally correspond to lu-modules over lu-rings, we study injective MV-modules and we investigate some conditions for constructing injective MV-modules. Then we define the notions of essential A-homomorphisms and essential extension of A-homomorphisms, where A is a product MV-algebra, and we get some of there properties. Finally, we prove that a maximal essential extension of any A-ideal of an injective MV-module is an injective A-module, too.
2018-01-01T00:00:00ZRule-Generation Theorem and its ApplicationsIndrzejczak, Andrzejhttp://hdl.handle.net/11089/284822019-05-25T01:17:24Z2018-01-01T00:00:00ZRule-Generation Theorem and its Applications
Indrzejczak, Andrzej
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 successfully 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.
2018-01-01T00:00:00ZOn the Definability of Leśniewski’s Copula ‘is’ in Some Ontology-Like TheoriesŁyczak, MarcinPietruszczak, Andrzejhttp://hdl.handle.net/11089/284812019-05-25T01:17:22Z2018-01-01T00:00:00ZOn the Definability of Leśniewski’s Copula ‘is’ in Some Ontology-Like Theories
Łyczak, Marcin; Pietruszczak, Andrzej
We formulate a certain subtheory of Ishimoto’s [1] quantifier-free 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 set-theoretic interpretation. Furthermore, we introduce the name constant 1 (which corresponds to the universal name ‘object’) and we prove its adequacy with respect to the set-theoretic 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.
2018-01-01T00:00:00ZLabeled Sequent Calculus for OrthologicKawano, Tomoakihttp://hdl.handle.net/11089/284802019-05-25T01:17:25Z2018-01-01T00:00:00ZLabeled Sequent Calculus for Orthologic
Kawano, Tomoaki
Orthologic (OL) is non-classical logic and has been studied as a part of quantumlogic. 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 examinedfor several decades. Although there are many studies on sequent calculus forOL, these sequent calculi have some problems. In particular, they do not includeimplication connective and they are mostly incompatible with the cut-eliminationtheorem. In this paper, we introduce new labeled sequent calculus called LGOI,and show that this sequent calculus solve the above problems. It is alreadyknown that OL is decidable. We prove that decidability is preserved when theimplication connective is added to OL.
2018-01-01T00:00:00Z