Paraconsistency

Carnielli, Walter; Coniglio, Marcelo E.; D'ottaviano, Itala Maria Lof

doi:10.1201/9780203910139

Cited by 89 publications

(28 citation statements)

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“…In Table 8, we summarize the properties of the several logics we have explored in this section. Many of them can be found in the literature (see [7], [11], [12], [23] [28], [30], [33]), while others are verified in later sections.…”

Section: Summarizing the Properties Of Some Paraconsistent Logicsmentioning

confidence: 90%

“…It is always useful to know that some strong inference rules hold in a given logic. In this sense, a simple but useful and well-known result is the following (see Carnielli and Marcos [11]).…”

Section: Definition 21mentioning

confidence: 97%

“…We start with the 3-valued logics, among them logics that are maximal paraconsistent with respect to classical logic, P 1 , P 2 , P 3 , LFI2, and LFI1. We remark that the matrices of P 1 are axiomatized by Civw, the matrices of P 2 by Cive, the matrices of P 3 by Ciorw, the matrices of LFI2 by Ciore, and the matrices of LFI1 by Cije (see [11]). We continue with the review of 4-valued logics, and then we summarize some important facts and results about Z.…”

Section: Some Paraconsistent Logicsmentioning

confidence: 99%

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Revisiting Z

Osorio¹,

Carballido²,

Zepeda³

2014

Notre Dame J. Formal Logic

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Béziau developed the paraconsistent logic Z, which is definitionally equivalent to the modal logic S5, and gave an axiomatization of the logic Z: the system HZ. Omori and Waragai proved that some axioms of HZ are not independent and then proposed another axiomatization for Z that includes two inference rules and helps to understand the relation between S5 and classical propositional logic. In the present paper, we analyze logic Z in detail; in the process we also construct a family of paraconsistent logics that are characterized by different properties that are relevant in the study of logics. MotivationThe convenience of accepting local inconsistencies is supported by Minsky's comment 1 in [22, p. 76]:But I do not believe that consistency is necessary or even desirable in a developing intelligent system. No one is ever completely consistent. What is important

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Section: Summarizing the Properties Of Some Paraconsistent Logicsmentioning

confidence: 90%

Section: Definition 21mentioning

confidence: 97%

Section: Some Paraconsistent Logicsmentioning

confidence: 99%

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Revisiting Z

Osorio¹,

Carballido²,

Zepeda³

2014

Notre Dame J. Formal Logic

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“…In this category, the epics are precisely the epimorphisms, while any monomorphism is a monic. In turn, isomorphisms, as defined above, are exactly the isomorphisms in the categorical sense (see, for instance, [5], Section 2). Notice, however, that is not known whether all monics are monomorphisms.…”

Section: Multialgebras Of Non-deterministic Termsmentioning

confidence: 97%

“…Because of this, certain classes of paraconsistent logics, as the ones known as logics of formal inconsistency, 1 are characterized by means of semantics of non-deterministic character such as non-deterministic matrices, Fidel structures or swap structures (see for instance [3]). Besides giving a semantical characterization, as well as a decision procedure, for these logics, such non-deterministic structures constitute an interesting object of study by themselves (see, for instance, [3,Chapter 6], [5] and [7]).…”

Section: Introductionmentioning

confidence: 99%

Weakly Free Multialgebras

Coniglio

Toledo²

2021

B Sect Log

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In abstract algebraic logic, many systems, such as those paraconsistent logics taking inspiration from da Costa's hierarchy, are not algebraizable by even the broadest standard methodologies, as that of Blok and Pigozzi. However, these logics can be semantically characterized by means of non-deterministic algebraic structures such as Nmatrices, RNmatrices and swap structures. These structures are based on multialgebras, which generalize algebras by allowing the result of an operation to assume a non-empty set of values. This leads to an interest in exploring the foundations of multialgebras applied to the study of logic systems. It is well known from universal algebra that, for every signature \(\Sigma\), there exist algebras over \(\Sigma\) which are absolutely free, meaning that they do not satisfy any identities or, alternatively, satisfy the universal mapping property for the class of \(\Sigma\)-algebras. Furthermore, once we fix a cardinality of the generating set, they are, up to isomorphisms, unique, and equal to algebras of terms (or propositional formulas, in the context of logic). Equivalently, the forgetful functor, from the category of \(\Sigma\)-algebras to Set, has a left adjoint. This result does not extend to multialgebras. Not only multialgebras satisfying the universal mapping property do not exist, but the forgetful functor \(\mathcal{U}\), from the category of \(\Sigma\)-multialgebras to Set, does not have a left adjoint. In this paper we generalize, in a natural way, algebras of terms to multialgebras of terms, whose family of submultialgebras enjoys many properties of the former. One example is that, to every pair consisting of a function, from a submultialgebra of a multialgebra of terms to another multialgebra, and a collection of choices (which selects how a homomorphism approaches indeterminacies), there corresponds a unique homomorphism, what resembles the universal mapping property. Another example is that the multialgebras of terms are generated by a set that may be viewed as a strong basis, which we call the ground of the multialgebra. Submultialgebras of multialgebras of terms are what we call weakly free multialgebras. Finally, with these definitions at hand, we offer a simple proof that multialgebras with the universal mapping property for the class of all multialgebras do not exist and that \(\mathcal{U}\) does not have a left adjoint.

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Paraconsistent Logic

Costa

Bueno

2009

A Companion to Latin American Philosophy

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Paraconsistency

Cited by 89 publications

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Revisiting Z

Revisiting Z

Weakly Free Multialgebras

Paraconsistent Logic

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