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Matrices and Algebraizability

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Paraconsistent Logic: Consistency, Contradiction and Negation

Part of the book series: Logic, Epistemology, and the Unity of Science ((LEUS,volume 40))

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Abstract

This chapter deals with matrices and algebraizability and their consequences, investigating in particular, the question of characterizability by finite matrices, as well as the algebraizability of (extensions of) mbC. Some negative results, in the style of the well-known Dugundji’s theorem for modal logics, are proved for several extensions of mbC.

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Notes

  1. 1.

    Despite this, and as in the case of Halldén, it should be clear that paraconsistency was not the main motivation for Segerberg’s nonsense logic.

  2. 2.

    It is worth noting that the logic \(\Phi _v\) was exclusively presented in [37] by means of a Hilbert style calculus, and not as a 3-valued matrix logic.

  3. 3.

    The definition of quantifiers in LPT1 differs from [39] to [40], the latter coinciding with that of [29, 34] (see [40]).

  4. 4.

    Our clarification.

  5. 5.

    Recall that, if \(\sigma \) is a substitution for variables, then \(\hat{\sigma }\) denote its unique extension to an endomorphism over the algebra of formulas.

  6. 6.

    This somewhat unclear aspect of Asenjo’s logic was already observed by Priest in [52], p. 228.

  7. 7.

    The author refers to [52].

  8. 8.

    It should be clear that, in the axioms involving disjunction \(\vee \) and consistency \({\circ }\), these operators are not the primitive ones from \(\Sigma \), but the corresponding abbreviations in \(\Sigma _{PT}\). Moreover, the implication symbol \(\rightarrow \) of \(\Sigma \) must be replaced by the corresponding symbol \(\Rightarrow \) of \(\Sigma _{PT}\).

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Authors and Affiliations

  1. Department of Philosophy and Centre for Logic, Epistemology and the History of Science (CLE), University of Campinas (UNICAMP), Campinas, São Paulo, Brazil

    Walter Carnielli & Marcelo Esteban Coniglio

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  1. Walter Carnielli
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  2. Marcelo Esteban Coniglio
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Correspondence to Walter Carnielli .

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Carnielli, W., Coniglio, M.E. (2016). Matrices and Algebraizability. In: Paraconsistent Logic: Consistency, Contradiction and Negation. Logic, Epistemology, and the Unity of Science, vol 40. Springer, Cham. https://doi.org/10.1007/978-3-319-33205-5_4

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