An abstract algebraic logic approach to tetravalent modal logics

2000 ◽  
Vol 65 (2) ◽  
pp. 481-518 ◽  
Author(s):  
Josep Maria Font ◽  
Miquel Rius
Keyword(s):  
Algebraic Logic ◽  
Modal Logics ◽  
The Other ◽  
Abstract Algebraic Logic ◽  
Algebraic Semantics ◽  
Modal Algebras ◽  
Logic Approach ◽  
Abstract Logics ◽  
Joint Study

AbstractThis paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their “A General Algebraic Semantics for Sentential Logics”. The logics studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo. Landini and Ziliani, at the suggestion of the late Antonio Monteiro.

scholarly journals Categorical Abstract Algebraic Logic: Equivalential π-Institutions

2008 ◽  
Vol 6 ◽  
Author(s):  
George Voutsadakis
Keyword(s):  
Matrix Theory ◽  
Algebraic Logic ◽  
Characterization Theorem ◽  
Abstract Algebraic Logic ◽  
Transfer Theorem ◽  
Deductive Systems

The theory of equivalential deductive systems, as introduced by Prucnal and Wrónski and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as π-institutions. More precisely, the notion of an N-equivalence system for a given π-institution is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equivalence systems, is revisited and a “transfer theorem” for N-equivalence systems is proven. For a π-institution I having an N-equivalence system, the maximum such system is singled out and, then, an analog of Herrmann’s Test, characterizing those N-protoalgebraic π-institutions having an N-equivalence system, is formulated. Finally, some of the rudiments of matrix theory are revisited in the context of π-institutions, as they relate to the existence of N-equivalence systems.

Paraconsistency, self-extensionality, modality

2018 ◽  
Vol 28 (5) ◽  
pp. 851-880
Author(s):  
Arnon Avron ◽  
Anna Zamansky
Keyword(s):  
Intuitionistic Logic ◽  
Modal Logics ◽  
The Other ◽  
Paraconsistent Logics ◽  
Classical Negation ◽  
Inconsistent Theories ◽  
Replacement Property ◽  
General Method

Abstract Paraconsistent logics are logics that, in contrast to classical and intuitionistic logic, do not trivialize inconsistent theories. In this paper we take a paraconsistent view on two famous modal logics: B and S5. We use for this a well-known general method for turning modal logics to paraconsistent logics by defining a new (paraconsistent) negation as $\neg \varphi =_{Def} \sim \Box \varphi$ (where $\sim$ is the classical negation). We show that while that makes both B and S5 members of the well-studied family of paraconsistent C-systems, they differ from most other C-systems in having the important replacement property (which means that equivalence of formulas implies their congruence). We further show that B is a very robust C-system in the sense that almost any axiom which has been considered in the context of C-systems is either already a theorem of B or its addition to B leads to a logic that is no longer paraconsistent. There is exactly one notable exception, and the result of adding this exception to B leads to the other logic studied here, S5.

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