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Paraconsistency and the need for infinite semantics

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Abstract

We show that most of the paraconsistent logics which have been investigated in the literature have no finite characteristic matrices, and in the most important cases not even finite characteristic non-deterministic matrices (Nmatrices).

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Notes

  1. It should be emphasized that these results do not imply that the logics investigated in this paper are undecidable. In fact, with one exception, all of them are decidable. More information on this topic is given below in Notes 3.3, 4.4, 4.10, 4.13, and 4.18.

  2. Since \(\vdash _\mathbf{L}\) is structural, this implies that \(p,\lnot p \not \vdash _\mathbf{L} q\) whenever p and q are distinct propositional variables.

  3. Nmatrices were introduced in Avron and Lev (2001), Avron and Lev (2005). Independently, non-deterministic truth tables were used in Crawford and Etherington (1998) and Ivlev (2000). Special cases of the idea were practically anticipated already by Schütte (1960), Tait (1966), Quine (1974), Girard (1987), and Batens (1998). For a comprehensive survey on Nmatrices and their applications, see Avron and Zamansky (2011).

  4. It is easy to see (Avron et al. 2013) that [k] is derivable in HBl, so HBl is also an extension of HBk.

  5. Theorem 11 in Avron (2007) does not refer to most of the axioms in \(\mathsf{A}_{PAC}\). However, its proof remains valid in their presence, as long the [o]-axioms from Carnielli et al. (2007), Carnielli and Marcos (2002) are excluded. (Unlike here, the [o]-axioms are included in Theorem 11 of Avron (2007).) The situation when S may contain both [o]-axioms and axioms from \(\mathsf{A}_{PAC}\) is complicated, and we shall not dwell on this case here.

  6. Other methods for showing decidability of LFIs, including those treated here, can be found in Carnielli and Coniglio (2016). They include: bivaluations, possible-translations semantics, and Fidel structures.

  7. The decidability of \(\mathbf{RM}\) was established by Meyer using Sugihara’s matrix. (See Anderson and Belnap (1975) or Avron (2016).) The decidability of \(\mathbf{RMI}\) was shown in Avron (1990a).

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Acknowledgements

This research was supported by the Israel Science Foundation under Grant Agreement No. 817/15.

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  1. School of Computer Science, Tel Aviv University, Tel Aviv, Israel

    Arnon Avron

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  1. Arnon Avron
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Avron, A. Paraconsistency and the need for infinite semantics. Soft Comput 23, 2167–2175 (2019). https://doi.org/10.1007/s00500-018-3272-0

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