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NONASSOCIATIVE SUBSTRUCTURAL LOGICS AND THEIR SEMILINEAR EXTENSIONS: AXIOMATIZATION AND COMPLETENESS PROPERTIES

Published online by Cambridge University Press:  15 May 2013

PETR CINTULA ,
ROSTISLAV HORČÍK  and
CARLES NOGUERA
PETR CINTULA*
Affiliation:
Institute of Computer Science, Academy of Sciences of the Czech Republic
ROSTISLAV HORČÍK*
Affiliation:
Institute of Computer Science, Academy of Sciences of the Czech Republic
CARLES NOGUERA*
Affiliation:
Artificial Intelligence Research Institute (IIIA - CSIC) and Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic
*
*INSTITUTE OF COMPUTER SCIENCE, ACADEMY OF SCIENCES OF THE CZECH REPUBLIC POD VODÁRENSKOU VEŽÍ 2 182 07 PRAGUE, CZECH REPUBLIC E-mail: {cintula,horcik}@cs.cas.cz
*INSTITUTE OF COMPUTER SCIENCE, ACADEMY OF SCIENCES OF THE CZECH REPUBLIC POD VODÁRENSKOU VEŽÍ 2 182 07 PRAGUE, CZECH REPUBLIC E-mail: {cintula,horcik}@cs.cas.cz
†INSTITUTE OF INFORMATION THEORY AND AUTOMATION ACADEMY OF SCIENCES OF THE CZECH REPUBLIC POD VODÁRENSKOU VEŽÍ 4 182 08 PRAGUE, CZECH REPUBLIC ARTIFICIAL INTELLIGENCE RESEARCH INSTITUTE SPANISH NATIONAL RESEARCH COUNCIL CAMPUS DE LA UNIVERSITAT AUTÒNOMA DE BARCELONA S/N,08193 BELLATERRA, CATALONIA, SPAIN E-mail: noguera@utia.cas.cz
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Abstract

Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL (which we call SL) has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP-based. This presentation is then used to obtain, in a uniform way applicable to most (both associative and nonassociative) substructural logics, a form of local deduction theorem, description of filter generation, and proper forms of generalized disjunctions. A special stress is put on semilinear substructural logics (i.e., logics complete with respect to linearly ordered algebras). Axiomatizations of the weakest semilinear logic over SL and other prominent substructural logics are provided and their completeness with respect to chains defined over the real unit interval is proved.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 6 , Issue 3 , September 2013 , pp. 394 - 423
Copyright
Copyright © Association for Symbolic Logic 2013 

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