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Arnon Avron on Semantics and Proof Theory of Non-Classical Logics pp 107–139Cite as

Degree-Preserving Gödel Logics with an Involution: Intermediate Logics and (Ideal) Paraconsistency

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Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 21))

Abstract

In this paper, we study intermediate logics between the logic \({\mathrm {G}^{\le }_{\sim }}\), the degree-preserving companion of Gödel fuzzy logic with involution \(\mathrm {G}_\sim \), and classical propositional logic CPL, as well as the intermediate logics of their finite-valued counterparts \(\mathrm {G}^{\le }_{n\sim }\). Although \({\mathrm {G}^{\le }_{\sim }}\) and \(\mathrm {G}^{\le }_{n\sim }\) are explosive w.r.t. Gödel negation \(\lnot \), they are paraconsistent w.r.t. the involutive negation \(\sim \). We introduce the notion of saturated paraconsistency, a weaker notion than ideal paraconsistency, and we fully characterize the ideal and the saturated paraconsistent logics between \(\mathrm {G}^{\le }_{n\sim }\) and CPL. We also identify a large family of saturated paraconsistent logics in the family of intermediate logics for degree-preserving finite-valued Łukasiewicz logics.

This paper is a humble tribute to our friend and colleague Arnon Avron and his outstanding contributions on nonclassical logics, proof theory, and foundations of mathematics.

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Notes

  1. 1.

    Wansing and Odintsov (2016), p. 179.

  2. 2.

    Ibid., p. 180.

  3. 3.

    Ibid., p. 181.

  4. 4.

    Ibid., p. 181.

  5. 5.

    Ibid., p. 183.

  6. 6.

    It is worth noting that, more recently, the authors have changed the terminology “ideal paraconsistent logic” in Arieli et al. (2011b) to “fully maximal and normal paraconsistent logic,” e.g., in Avron et al. (2018). According to them, they choose the latter “to use a more neutral terminology” (see Avron et al. (2018, Footnote 9, p. 57)).

  7. 7.

    In fact, \({\mathrm {G}^{\le }_{\sim }}\) is then a paradefinite logic (w.r.t. \(\sim \)) in the sense of Arieli and Avron (2017), as it is both paraconsistent and paracomplete, since the law of excluded middle \(\varphi \vee {\sim }\varphi \) fails, as in all fuzzy logics. Logics with a negation which is both paraconsistent and paracomplete were already considered in the literature under different names: non-alethic logics (da Costa) and paranormal logics (Beziau).

  8. 8.

    This axiomatization comes from adding axiom (A7) to the axioms of Hájek’s BL logic (Hájek 1998). Later it was shown that axioms (A2) and (A3) were, in fact, redundant, see Běhounek (2011) for a detailed exposition and the references therein.

  9. 9.

    Called standard Gödel algebra.

  10. 10.

    These are the original axioms from Esteva et al. (2000), see again Běhounek (2011) and the references therein for a shorter axiomatization.

  11. 11.

    Equivalently, as the expansion of \(\mathrm{G}_n\) with \(\sim \) along with the axioms \((\sim 1)\)-\((\sim 3)\), \((\Delta 1)\)-\((\Delta 3)\), and the necessitation rule for \(\Delta \).

  12. 12.

    For practical purposes, we can assume in this paper that \(\mathrm {L}\) is an axiomatic extension of Hájek’s BL logic.

  13. 13.

    Given a class of Kripke models, a formula \(\varphi \) follows locally from a set \(\Gamma \) of formulas if, for any Kripke model M in the class and every world w in M, \(\varphi \) is true in \(\langle M,w\rangle \) whenever every formula in \(\Gamma \) is true in \(\langle M,w\rangle \) as well. On the other hand, \(\varphi \) follows globally from \(\Gamma \) in the class if, for every Kripke model M, \(\varphi \) is true in \(\langle M,w\rangle \) for every w whenever every formula in \(\Gamma \) is true in \(\langle M,w\rangle \) for every w.

  14. 14.

    For \(\mathrm {L} = \mathrm{G}_\sim \).

  15. 15.

    Recall that the finitary companion of a logic \((L, \vdash )\) is given by \((L, \vdash ^f)\) where, for every \(\Gamma \cup \{\varphi \} \subseteq L\), \(\Gamma \vdash ^f \varphi \) iff there exists a finite \(\Gamma _0 \subseteq \Gamma \) such that \(\Gamma _0 \vdash \varphi \).

  16. 16.

    In the following we use p and n to denote positive and negative values in [0, 1] with respect to the negation \(\sim x = 1-x\); in other words, \(p > 1/2\) and \(n < 1/2\).

  17. 17.

    In Avron (2016), the symbols \(\lnot \) and \(\rightarrow \) were used instead of \({\sim }\) and \(\rightarrow _\mathrm {FT}\). We adopt this notation in order to keep the notation of the present paper uniform.

  18. 18.

    Indeed, every formula \(\varphi \in Fm_\mathrm{FT}\) gets the value 1/2 in any evaluation e over \(\mathbf{M}_{[0,1]}\) such that e assigns the value 1/2 to any propositional variable occurring in \(\varphi \).

  19. 19.

    Recall that the \(\Delta \) connective is definable in any algebra \(\mathbf {\L } V_n\).

  20. 20.

    In fact, this is a discriminator term in the whole variety of \(\mathrm{G}_{\sim }\)-algebras. For a definition of discriminator term and discriminator variety, see Burris and Sankappanavar (1981).

  21. 21.

    A filter F of an algebra \(\mathbf{A}\) is compatible with a logic L if, whenever \(\Gamma \vdash _L \varphi \), the following holds: for every \(\mathbf{A}\)-evaluation e, if \(e(\gamma ) \in F\) for every \(\gamma \in \Gamma \) then \(e(\varphi ) \in F\).

  22. 22.

    Strictly speaking, this notation becomes ambiguous if n is not clear from the context and we identify rational numbers such as \(i/(n-1)\) and \(i.k/(n-1).k\), for instance, 1/2, 2/4, 3/6, and so on. In this case, the notation \(F_{\frac{1}{2}}\) is problematic, since it could denote any of an infinite sequence of different filters in \(GV_3\), \(GV_5\), \(GV_7\),...  respectively. The right notation for order filters in \(GV_n\) should be \(F^n_t\). However, the superscript n will be avoided when there is no risk of confusion.

  23. 23.

    The authors, as it was mentioned in Sect. 6.1, have changed the terminology “ideal paraconsistent logic” to “fully maximal and normal paraconsistent logic.” However, it should be noticed that being normal, according to Avron et al. (2018, Definition 1.32), means that the logic L has, besides a deductive implication, a conjunction and a disjunction satisfying the usual properties in terms of consequence relations. Here we decide to keep the original definition of ideal paraconsistency. It is worth noting that all the ideal (and saturated) logics considered in this paper and in Coniglio et al. (2019) are normal in the sense of Avron et al. (2018).

  24. 24.

    Observe that in Coniglio et al. (2019), \(\lnot \) denotes the Łukasiewicz negation, while the Gödel negation for \(\mathsf {J}_3\) and \(\mathsf {J}_4\) is, respectively, denoted by \(\sim ^1_2\) and \(\sim ^1_3\).

  25. 25.

    Using the terminology of Avron et al. (2018), a saturated paraconsistent logic is a logic such that: (i) it has an implication, (ii) it is F-contained in CPL, and (iii) it is strongly maximal.

  26. 26.

    This was denoted by \(L \in \mathbf{Triv}_\Theta \bot \{r\}\) in Ribeiro and Coniglio (2012), where \(\mathbf{Triv}_\Theta \) denotes the trivial logic over the signature \(\Theta \).

  27. 27.

    Indeed, by means of the notion of remainder set \(L \bot R\) of a logic L w.r.t. a set of rules R introduced in Ribeiro and Coniglio (2012, Definition 7), several concepts relative to maximality proposed in the literature can be easily represented, see Ribeiro and Coniglio (2012, p. 273).

  28. 28.

    An algebra is said to be critical if it is a finite algebra not belonging to the quasivariety generated by all its proper subalgebras.

  29. 29.

    This is the case of any pseudo-complemented logic L where \(\Delta \) is definable as \(\Delta \varphi := \lnot {\sim }\varphi \), in particular, the case of Gödel fuzzy logic \(\mathrm {G}\).

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Acknowledgements

The authors are indebted to two anonymous reviewers for their careful and insightful suggestions and remarks that have definitively helped to improve the paper. Coniglio acknowledges support from the National Council for Scientific and Technological Development (CNPq), Brazil, under research grant 306530/2019-8. Gispert acknowledges partial support by the Spanish FEDER/MINECO projects (PID2019-110843GA-100, MTM2016-74892 and MDM-2014-044) and grant 2017-SGR-95 of Generalitat de Catalunya. Esteva and Godo acknowledge partial support by the Spanish FEDER/MINECO project TIN2015-71799-C2-1-P and the project PID2019-111544GB-C21.

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Authors and Affiliations

  1. Dept. of Philosophy - IFCH and Centre for Logic, Epistemology and the History of Science, University of Campinas, Campinas, Brazil

    Marcelo E. Coniglio

  2. Artificial Intelligence Research Institute (IIIA) - CSIC, Barcelona, Spain

    Francesc Esteva & Lluis Godo

  3. Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona, Spain

    Joan Gispert

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  1. Marcelo E. Coniglio
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  2. Francesc Esteva
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  3. Joan Gispert
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  4. Lluis Godo
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Correspondence to Lluis Godo .

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Editors and Affiliations

  1. School of Computer Science, Academic College of Tel Aviv-Yafo, Tel Aviv, Israel

    Ofer Arieli

  2. Information Systems Department, University of Haifa, Haifa, Israel

    Anna Zamansky

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Coniglio, M.E., Esteva, F., Gispert, J., Godo, L. (2021). Degree-Preserving Gödel Logics with an Involution: Intermediate Logics and (Ideal) Paraconsistency. In: Arieli, O., Zamansky, A. (eds) Arnon Avron on Semantics and Proof Theory of Non-Classical Logics. Outstanding Contributions to Logic, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-71258-7_6

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