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A Basic Logic of Formal Inconsistency: mbC

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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 begins a formal study of Logics of Formal Inconsistency (LFIs) by offering a careful survey of the basic logic of formal inconsistency, mbC. The chapter also lays out the main notation, ongoing definitions and main ideas that will be used throughout the book.

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Notes

  1. 1.

    In this book the following standard notation will be adopted: given a function f and a subset A of its domain, f[A] will denote the set \(\{f(a) \ : \ a \in A\}\).

  2. 2.

    It should be observed that the weak LFIs investigated in the three references mentioned above are also strong LFIs in the sense of Definition 2.1.9.

  3. 3.

    Since it is well-known, it is equivalent to the Axiom of Choice.

  4. 4.

    However, such tables do not correspond to non-deterministic matrices in the sense of Avron and Lev (see [10, 11]). The relationship between valuations for LFIs and non-deterministic matrices will be analyzed in Chap. 6.

  5. 5.

    There are logics which are simultaneously paraconsistent and paracomplete, that is, that allow the fourth scenario in which \(\alpha \) and \(\lnot \alpha \) are both false. Logics of this kind are frequently called paranormal by the literature. One example of paranormality is a tetravalent modal logic that can be associated with Monteiro’s tetravalent modal algebras, see [12]. This example will be analyzed in Chap. 5.

  6. 6.

    It should be observed that Ivlev (see [14]) and other authors use the term ‘quasi-matrices’ to refer to non-deterministic matrices in the sense of of A. Avron and I. Lev .

  7. 7.

    In formal terms, t is recursively defined as follows: \(t(p)=p\) if \(p \in Var\); \(t(\lnot \alpha )={\sim }t(\alpha )\); \(t({\circ }\alpha )={\circ }t(\alpha )\); and \(t(\alpha \, \#\, \beta )= t(\alpha )\, \# \,t(\beta )\) if \(\# \in \{\vee , \wedge , \rightarrow \}\).

  8. 8.

    In formal terms, \(t_0\) is recursively defined as follows: \(t_0(p)=p\) if \(p \in Var\); \(t_0(\lnot \alpha )={\sim }t_0(\alpha )\); and \(t_0(\alpha \, \#\, \beta )= t_0(\alpha )\, \# \,t_0(\beta )\) if \(\# \in \{\vee , \wedge , \rightarrow \}\).

  9. 9.

    In formal terms, \(t'\) is defined recursively as follows: \(t'(p)=p\) if \(p \in Var\); \(t'({\sim }\alpha )=\lnot t'(\alpha )\); and \(t'(\alpha \, \#\,\beta )= t'(\alpha )\, \#\, t'(\beta )\) if \(\# \in \{\vee , \wedge , \rightarrow \}\).

  10. 10.

    Notice that the mapping \(t'\) is necessary only because we consider different signatures for mbC and CPL.

  11. 11.

    Not to be confused with Sette’s P1, see Sect. 4.4.4. To be more precise \(P_1\) contains, besides MP, the inference rule of Uniform Substitution, since the axioms are presented by using propositional variables instead of schema formulas.

References

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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). A Basic Logic of Formal Inconsistency: mbC . 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_2

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