Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science

Abstract Multialgebras (or hyperalgebras or non-deterministic algebras) have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency (or LFIs) that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. Specifically, a formal study of swap structures for LFIs is developed, by adapting concepts of universal algebra to multialgebras in a suitable way. A decomposition theorem similar to Birkhoff’s representation theorem is obtained for each class of swap structures. Moreover, when applied to the 3-valued algebraizable logics J3 and Ciore, their classes of algebraic models are retrieved, and the swap structures semantics become twist structures semantics (as independently introduced by M. Fidel and D. Vakarelov). This fact, together with the existence of a functor from the category of Boolean algebras to the category of swap structures for each LFI (which is closely connected with Kalman’s functor), suggests that swap structures can be seen as non-deterministic twist structures. This opens new avenues for dealing with non-algebraizable logics by the more general methodology of multialgebraic semantics.

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Categorical Abstract Algebraic Logic: Meet-Combination of Logical Systems

Journal of Mathematics ◽

10.1155/2013/126347 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8

Author(s):

George Voutsadakis

Keyword(s):

Computer Science ◽

Algebraic Logic ◽

Abstract Algebraic Logic ◽

Combined Logics ◽

Component Systems ◽

Combining Logics

The widespread and rapid proliferation of logical systems in several areas of computer science has led to a resurgence of interest in various methods for combining logical systems and in investigations into the properties inherited by the resulting combinations. One of the oldest such methods isfibring. In fibring the shared connectives of the combined logics inherit properties frombothcomponent logical systems, and this leads often to inconsistencies. To deal with such undesired effects, Sernadas et al. (2011, 2012) have recently introduced a novel way of combining logics, calledmeet-combination, in which the combined connectives share only thecommonlogical properties they enjoy in the component systems. In their investigations they provide a sound and concretely complete calculus for the meet-combination based on available sound and complete calculi for the component systems. In this work, an effort is made to abstract those results to a categorical level amenable tocategorical abstract algebraic logictechniques.

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Applications of Algebraic Logic and Universal Algebra to Computer Science

10.21236/ada210556 ◽

1989 ◽

Author(s):

IOWA STATE UNIV AMES DEPT OF MATHEMATICS

Keyword(s):

Computer Science ◽

Universal Algebra ◽

Algebraic Logic

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An abstract algebraic logic approach to tetravalent modal logics

Journal of Symbolic Logic ◽

10.2307/2586552 ◽

2000 ◽

Vol 65 (2) ◽

pp. 481-518 ◽

Cited By ~ 15

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.

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Categorical Abstract Algebraic Logic: Equivalential π-Institutions

The Australasian Journal of Logic ◽

10.26686/ajl.v6i0.1790 ◽

2008 ◽

Vol 6 ◽

Cited By ~ 2

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.

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