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Paraconsistency properties in degree-preserving fuzzy logics

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Abstract

Paraconsistent logics are specially tailored to deal with inconsistency, while fuzzy logics primarily deal with graded truth and vagueness. Aiming to find logics that can handle inconsistency and graded truth at once, in this paper we explore the notion of paraconsistent fuzzy logic. We show that degree-preserving fuzzy logics have paraconsistency features and study them as logics of formal inconsistency. We also consider their expansions with additional negation connectives and first-order formalisms and study their paraconsistency properties. Finally, we compare our approach to other paraconsistent logics in the literature.

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Notes

  1. Notice here that in the frame of LFIs the term consistent refers to formulas that basically exhibit a classical logic behavior, so in particular an explosive behavior.

  2. In a very general setting, one could argue what properties should be required for a unary connective to be properly called a negation. However, in the context of the fuzzy logic systems considered later in this paper, all the negation connectives that we will deal with are indeed proper negations, in the sense that their truth tables always revert to the classical negation truth-table as soon as we restrict ourselves to the classical 0 and 1 truth values.

  3. \(c(\chi _1,\dots , \varphi ,\dots ,\chi _n)\) and \(c(\chi _1,\dots , \psi ,\dots ,\chi _n)\) denote two instances of the \(n\)-ary connective \(c\) where \(\varphi \) and \(\psi \) appear in a same (arbitrary) \(i\)-th place in \(c\) (for \(1 \le i \le n\)), while keeping the same formulas \(\chi _j\)’s (with \(j \ne i\)) in the other places.

  4. Moreover, for a number of core fuzzy logics, including \(\mathrm{MTL}\), it has been shown that their corresponding varieties are also generated by the subclass of \(\mathrm{MTL}\)-chains defined on the real unit interval, called standard algebras. For instance, \(\mathrm{MTL}\) is also complete wrt standard \(\mathrm{MTL}\)-chains, that are of the form \([0,1]_*=\langle [0,1], \min , \max , *,\rightarrow _*,1,0\rangle \) of type \(\langle 2,2,2,2,0,0\rangle \), where \(*\) denotes a left-continuous t-norm and \(\rightarrow _*\) is its residuum (Jenei and Montagna 2002).

  5. It is worth noticing that, even if we drop in the above definition the condition of the existence of a finite \(\varGamma _0 \subseteq \varGamma \), the logic \(\mathrm{L}^{\le }\) remains finitary (Jansana 2013).

  6. Note that applications of inference rules in \(\varPhi \) are only to theorems of \(\mathrm{L}\).

  7. In Priest (2002b) it was already noted that the degree-preserving Łukasiewicz logic \(\L ^{\le }\) was paraconsistent.

  8. This type of chains are studied in Noguera et al. (2005a).

  9. Given a natural number \(n\), \(\varphi ^n\) is an abbreviation for \( \varphi \mathbin { \& }\mathop {\ldots }\limits ^{n} \mathbin { \& } \varphi \), that is, the formula obtained as conjunction of \(n\) times \(\varphi \).

  10. Note that \(x\) and \(y\) correspond, respectively, to \(e_1(\varphi )\) and \(e_2(\varphi )\).

  11. An S\(_n\) \(\mathrm{MTL}\)-chain \(\mathbf A\) is a \(\mathrm{MTL}\)-chain satisfying the equation \(x \vee \lnot x^{n-1} = \overline{1}^\mathbf{A}\).

  12. Of course, the interesting case is when the negation \(\lnot \) of \(\mathrm{L}\) is not involutive.

  13. As it occurs either in any pseudo-complemented logic where \(\triangle \) is definable as \(\triangle \varphi := \lnot \mathord \sim \varphi \) or in a finitely valued Łukasiewicz logic Ł\(_n\)where \(\triangle \) is definable as \(\triangle \varphi := \varphi ^n\).

  14. For more details and proofs see e.g. Cintula et al. (2011).

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Acknowledgments

The authors are indebted to two anonymous referees for their critical and interesting remarks that have led to a significant improvement of the paper. All the authors have been partially supported by the FP7 PIRSES-GA-2009-247584 Project MaToMUVI. Besides, Ertola was supported by FAPESP LOGCONS Project and CONICET Project PIP 112201101006336, Esteva and Godo were supported by the Spanish Project TIN2012-39348-C02-01, Flaminio was supported by the Italian project FIRB 2010 (RBFR10DGUA\(_-\)002) and Noguera was supported by the Grant P202/10/1826 of the Czech Science Foundation.

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Authors and Affiliations

  1. State University of Campinas, Rua Sérgio Buarque de Holanda 251, 13083-859, Campinas, São Paulo, Brazil

    Rodolfo Ertola

  2. Artificial Intelligence Research Institute-CSIC, Campus de la UAB s/n, 08193, Bellaterra, Catalonia, Spain

    Francesc Esteva & Lluís Godo

  3. Dipartimento di Scienze Teoriche e Applicate, Università dell’Insubria, Via Mazzini 5, 21100, Varese, Italy

    Tommaso Flaminio

  4. Institute of Information Theory and Automation-ASČR, Pod vodárenskou věží 4, 18208, Prague, Czech Republic

    Carles Noguera

  5. Institute of Computer Science-ASČR, Pod vodárenskou věží 2, 18207, Prague, Czech Republic

    Carles Noguera

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  1. Rodolfo Ertola
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  2. Francesc Esteva
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  3. Tommaso Flaminio
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  4. Lluís Godo
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  5. Carles Noguera
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Correspondence to Carles Noguera.

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Communicated by L. Spada.

This is an extended and revised version of the conference communication “Exploring paraconsistency in degree-preserving fuzzy logics”, Proceedings of the 8th conference of the European Society for Fuzzy Logic and Technology EUSFLAT 2013, Gabriela Pasi, Javier Montero, Davide Ciucci (eds), Atlantis Press, pp. 117–124.

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Ertola, R., Esteva, F., Flaminio, T. et al. Paraconsistency properties in degree-preserving fuzzy logics. Soft Comput 19, 531–546 (2015). https://doi.org/10.1007/s00500-014-1489-0

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