The Strong Version of a Sentential Logic

Albuquerque, Hugo; Font, Josep Maria; Jansana, Ramón

doi:10.1007/s11225-017-9709-0

Cited by 4 publications

(13 citation statements)

References 26 publications

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“…The results now published arose during the preparation of the author's PhD dissertation, under the supervision of Jansana and Font, whose main topics were published in [2, 3]. As such, the present note forms part of a broader joint work with Jansana and Font, and should be credited to them as well.…”

Section: Introductionmentioning

confidence: 97%

The Suszko operator relative to truth‐equational logics

Albuquerque

2021

Mathematical Logic Qtrly

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This note presents some new results from [1] about the Suszko operator and truth‐equational logics, following the works of Czelakowski [11] and Raftery [17]. It is proved that the Suszko operator relative to a truth‐equational logic scriptS preserves suprema and commutes with endomorphisms. Together with injectivity, proved by Raftery in [17], the Suszko operator relative to a truth‐equational logic is a structural representation, as defined in [15]. Furthermore, if Alg(S) is a quasivariety, then the Suszko operator relative to a truth‐equational logic is continuous. Finally, it is proved that truth is equationally definable in the class sans-serifLModSufalse(scriptSfalse) if and only if Alg(S) is a τ-0.16em-0.16em∞‐algebraic semantics for scriptS and the Suszko operator boldΩ∼scriptSbold-italicFm:ThscriptS→CoAlg(S)Fm preserves suprema and commutes with substitutions.

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Section: Introductionmentioning

confidence: 97%

The Suszko operator relative to truth‐equational logics

Albuquerque

2021

Mathematical Logic Qtrly

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“…finite matrices. 1 Semantical investigations of da Costa's C -systems were also present since the beginning. Da Costa himself proposed the notion of C n -algebras as early as 1966 in [21], defined the notions of filters and ideals for C n -algebras in [22], and later in a collaboration with Sette also defined the notion of C ω -algebras in [29].…”

Section: Introductionmentioning

confidence: 99%

“…The study of C 1 eventually led us to consider the logic Cilo [26], a paraconsistent extension of C 1 introduced by da Costa, Béziau and Bueno in [26], and coined as Cilo in [17], as well as other paraconsistent axiomatic extensions of C 1 covered in the literature, including the algebraizable logics P 1 , P 2 and P 3 . The choice of which extensions of C 1 to consider was mainly guided by [17,Section 3.10] and is depicted in Figure 1, along with the inclusion relations among them.…”

Section: Introductionmentioning

confidence: 99%

An Abstract Algebraic Logic Study of Da Costa’s Logic and Some of Its Paraconsistent Extensions

Albuquerque¹,

Caleiro²

2022

Bull. symb. log

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Two famous negative results about da Costa's paraconsistent logic C1 (the failure of the Lindenbaum-Tarski process [44] and its non-algebraizability [39]) have placed C1 seemingly as an exception to the scope of Abstract Algebraic Logic (AAL). In this paper we undertake a thorough AAL study of da Costa's logic C1. On the one hand, we strengthen the negative results about C1 by proving that it does not admit any algebraic semantics whatsoever in the sense of Blok and Pigozzi (a weaker notion than algebraizability also introduced in the monograph [6]). On the other hand, C1 is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of C1 covered in the literature. We prove that for extensions S such as Cilo [26], every algebra in Alg * (S) contains a Boolean subalgebra; and for extensions S such as P 1 , P 2 or P 3 [17,53], every subdirectly irreducible algebra in Alg * (S) has cardinality at most 3. We also characterize the quasivariety Alg * (S) and the intrinsic variety V(S), with

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“…One general situation where the above results apply is that of the pairs of an arbitrary logic S and its strong version S + , a notion introduced and studied in [1,3]. It may be defined, among several ways, as the logic determined by the class of all matrices whose filter is the smallest S-filter on the corresponding algebra.…”

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confidence: 99%

“…The quoted publications show that the logic S + is an extension of S with a privileged status among all its extensions, and Theorem 3 tells us how to find its full g-models from those of S. Moreover, it often happens, although this is not a general fact, that AlgS = AlgS + ; in these cases, the stronger Theorem 4 applies and the correspondence between the full g-models of S and those of S + is optimal. Several notable, large classes of examples of this situation are reviewed in [3], for instance those where S is the logic that preserves degrees of truth with respect to an arbitrary variety of commutative integral residuated lattices, and S + is the logic that preserves truth with respect to the same variety. Notice that in those cases the logic S is not truth-equational while the logic S + is so.…”

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confidence: 99%

A note of the full generalized models of the extensions of a logic

Albuquerque

Font

Jansana

2017

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In this short note we show that the full generalized models of any extension of a logic can be determined from the full generalized models of the base logic in a simple way. The result is a consequence of two central theorems of the theory of full generalized models of sentential logics. As applications we investigate when the full generalized models of an extension can also be full generalized models of the base logic, and we prove that each Suszko filter of a logic determines a Suszko filter of each of its extensions, also in a simple way. We use the terminology and notations that are standard in abstract algebraic logic, as given for instance in [7]. We identify a sentential logic S with a structural (i.e., substitution invariant) consequence relation S on the set of formulas of some algebraic

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The Strong Version of a Sentential Logic

Cited by 4 publications

References 26 publications

The Suszko operator relative to truth‐equational logics

The Suszko operator relative to truth‐equational logics

An Abstract Algebraic Logic Study of Da Costa’s Logic and Some of Its Paraconsistent Extensions

A note of the full generalized models of the extensions of a logic

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