Non-deterministic algebraization of logics by swap structures1

Abstract

Multialgebras (or hyperalgebras, or non-deterministic algebras) have been very 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) which 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 logic J3 the usual class of algebraic models is recovered, and the swap structures semantics became twist-structures semantics (as introduced by Fidel-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 considered as non-deterministic twist structures, opening so interesting possibilities for dealing with non-algebraizable logics by means of multialgebraic semantics.

Modal logic with non-deterministic semantics: Part I—Propositional case

“…16 According to Noonan, Kripke semantics "brought back into the philosophical mainstream the Leibnizian language according to which necessity is truth in all possible worlds and possibility is truth in some. ", see [29], p. 8. 17 We could use as the only modal operator, defining ♦ ≡ def ¬ ¬.…”

“…Now, a simple but relevant technical result concerning Nmatrices will be stated. Recall from [8] the notion of submultialgebra. It states that a multialgebra A is a submultialgebra of another B, denoted by A ⊆ sm B provided that: both are defined over the same signature; the domain A of A is contained in the domain B of B; and c A ( a) ⊆ c B ( a) for every n-ary connective c and any a ∈ A n .…”

“…In another line of research, the generalization of modal Nmatrices to swap structures semantics proposed in [9] opens interesting possibilities for Ivlev-like modal systems with respect to algebraizability in a wide sense (see [8]). Thus, it would be interesting the study of the modal systems proposed here and the ones analyzed in [10] from the broader perspective of swap structures.…”

Recovery operators, paraconsistency and duality

“…Given a swap structure B for a given logic L, it originates a non-deterministic matrix (in the sense of Avron and Lev, see for instance [1]) such that the class of such Nmatrices semantically characterizes L. In this section, this technique (which was additionally developed from the algebraic point of view in [13]) will be used in order to semantically characterize the logics mbD, mbCD and mbCDE (in the latter, snapshots will be quadruples instead of triples). Moreover, a decision procedure will be obtained for such logics from this semantics.…”

“…As it happens with several LFIs and other logics characterized by swap structures defined over Boolean algebras (see [7,13,14,24]), the swap structure B L A 2 with domain B L A 2 over the 2-element Boolean algebra A 2 (with domain 0, 1 ) is enough to characterize the logics L ∈ mbD, mbCD, mbCDE . This produces a decision procedure for each L by means of a finite Nmatrix, thanks to the semantical characterization of these logics through valuations (recall Section 4.1).…”