A General Algebraic Semantics for Sentential Logics

Font, Josep Maria; Jansana, Ramón

doi:10.1017/9781316716915

Cited by 75 publications

(111 citation statements)

References 48 publications

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“…Given arbitrary formulas α and β, there are exactly six relevant scenarios in mbCciw to be analyzed, in which either •α or •β are true and, at the same time, α ∧ β is also true: Through analogous reasoning, it is easy to characterize the strong consistency propagation for disjunction and implication: It was claimed that in [18] it is possible to define a non-trivial congruence for C + 1 . The existence of such congruence would give us a chance to obtain some kind of algebraizability for this logic (see [19]), contrary to what happens with the weaker logic C 1 . Unfortunately, this is not the case since the congruence relation proposed in [18] fails to be reflexive, as we shall see in Sect.…”

Section: A Stronger Consistency Propagationmentioning

confidence: 96%

Some Extensions of mbC

Carnielli

Coniglio

2016

Paraconsistent Logic: Consistency, Contradiction and Negation

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Clearly, the logic mbC is the minimal extension of CPL + having a consistency operator • and a paraconsistent negation ¬, and such that it is an LFI. In this chapter, several extensions of mbC will be proposed, taking into account different features of mbC that can be strengthened or expanded. A Wider Form of Truth-Functionality for ConsistencyBy observing the quasi-matrices presented in Sect. 2.3 of Chap. 2, it becomes clear that not every connective of mbC is truth-functional (w.r.t. valuations): the tables for ¬ and • show that non-determinism exists, to a certain extent, in the evaluation process. This aspect is not a defect of the valuations for mbC: it can be proven that mbC (and several extensions of it) are not characterizable by finite matrices, adapting J. Dugundji's argument for modal logics. This topic will be briefly analyzed in Sect. 4.2 of Chap. 4.Returning to the valuation semantics for mbC, it is clear that the truth-value of a formula α partially determines the truth-value of ¬α: if α is false then ¬α must be true, but when α is true the truth-value of ¬α cannot be determined: it can be either true or false. Similarly, the truth-values of α and ¬α partially determine the truthvalue of •α in mbC: if they are both true then •α must be false, but if one of {α, ¬α} is false and the other is true (that is, if the truth-values of α and ¬α are different), the truth-value of •α is undetermined: it can be either true or false. In [1] an extension of mbC called mCi was researched, which guarantees that the truth-values of α and ¬α are sufficient to determine the truth-value of •α. However, this extension also 1 In general, names of logic systems are acronyms for the names of the axioms involved.

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Section: A Stronger Consistency Propagationmentioning

confidence: 96%

Some Extensions of mbC

Carnielli

Coniglio

2016

Paraconsistent Logic: Consistency, Contradiction and Negation

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“…The last result shows that Ciore, unlike C 1 , admits a non-trivial congruence, suggesting the possibility of defining some kind of algebraization within the setting of Blok and Pigozzi (see Definition 4.3.1) or in the more general framework of Font and Jansana (see [19]). In fact, this is the case:…”

Section: Then Is a Congruence In The Algebra Of Formulas L That Ismentioning

confidence: 98%

Matrices and Algebraizability

Carnielli

Coniglio

2016

Paraconsistent Logic: Consistency, Contradiction and Negation

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“…Here we present a brief survey of the main definitions and results of abstract algebraic logic we will make use of along the article; a systematic exposition can be found for example in [6,7,11,14,15,16].…”

Section: Preliminariesmentioning

confidence: 99%

“…Abstract algebraic logic (AAL for short) is a field that studies uniformly propositional logics [14,11,15,16]. One of its main achievements is the development of the so-called Leibniz and Frege hierarchies in which propositional logics are classified according to two different criteria.…”

Section: Introductionmentioning

confidence: 99%

A computational glimpse at the Leibniz and Frege hierarchies

Moraschini

2018

Annals of Pure and Applied Logic

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In this paper we consider, from a computational point of view, the problem of classifying logics within the Leibniz and Frege hierarchies typical of abstract algebraic logic. The main result states that, for logics presented syntactically, this problem is in general undecidable. More precisely, we show that there is no algorithm that classifies the logic of a finite consistent Hilbert calculus in the Leibniz and in the Frege hierarchies.

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A General Algebraic Semantics for Sentential Logics

Cited by 75 publications

References 48 publications

Some Extensions of mbC

Some Extensions of mbC

Matrices and Algebraizability

A computational glimpse at the Leibniz and Frege hierarchies

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