Non-deterministic algebraization of logics by swap structures

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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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Archival date: 2019-01-08
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References found in this work BETA
An Alternative Approach for Quasi-Truth.Coniglio, Marcelo E. & Da Cruz Silvestrini, Luiz H.
Modal Logics with Belnapian Truth Values.Odintsov, Serge P. & Wansing, Heinrich

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Citations of this work BETA
Swap Structures Semantics for Ivlev-Like Modal Logics.Coniglio, Marcelo E. & Golzio, Ana Claudia
Recovery Operators, Paraconsistency and Duality.Carnielli, Walter; Coniglio, Marcelo E. & Rodrigues, Abilio
Recovery Operators, Paraconsistency and Duality.Carnielli, Walter A.; Coniglio, Marcelo E. & Rodrigues Filho, Abilio
Modal Logic with Non-Deterministic Semantics: Part I—Propositional Case.Coniglio, Marcelo E.; Luis, Fariñas Del Cerro & Newton, Marques Peron

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