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Soft Computing
Erschienen in: Soft Computing 3/2015

01.03.2015 | Foundations

Paraconsistency properties in degree-preserving fuzzy logics

verfasst von: Rodolfo Ertola, Francesc Esteva, Tommaso Flaminio, Lluís Godo, Carles Noguera

Erschienen in: Soft Computing | Ausgabe 3/2015

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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.
Vorheriger Artikel On a new construction of pseudo effect algebras
Nächster Artikel Dynamic optimization facilitated by the memory tree

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Fußnoten
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).
 
Literatur
Avron A, Zamansky A (2007) Non-deterministic multi-valued matrices for first-order logics of formal inconsistency. In: Proceedings of the 37th international symposium on multiple-valued logic, ISMVL. IEEE Press, Oslo, p 14
Batens D (1980) Paraconsistent extensional propositional logics. Logique et Analyse 23:195–234MATHMathSciNet
Besnard P, Hunter A (eds) (1998) Reasoning with actual and potential contradictions. Handbook of defeasible reasoning and uncertainty management systems, vol 2. Kluwer, Dordrecht
Blok WJ, Pigozzi DL (1989) Algebraizable logics, vol 77. Memoirs of the American Mathematical Society, Providence
Bou F, Esteva F, Font JM, Gil À, Godo L, Torrens A, Verdú V (2009) Logics preserving degrees of truth from varieties of residuated lattices. J Log Comput 19(6):1031–1069CrossRefMATH
Carnielli W, Coniglio ME, Marcos J (2007) Logics of formal inconsistency. In: Gabbay D, Guenthner F (eds) Handbook of philosophical logic, vol 14, 2nd edn. Springer, Berlin, pp 1–93CrossRef
Castiglioni JL, Ertola RC (2014) Strict paraconsistency of truth-degree preserving intuitionistic logic with dual negation. Log J IGPL 22(2):268–273
Cignoli R, D’Ottaviano IML, Mundici D (1999) Algebraic foundations of many-valued reasoning. Trends in logic, vol 7. Kluwer, Dordrecht
Cintula P, Noguera C (2011) A general framework for mathematical fuzzy logic. In: Cintula P, Hájek P, Noguera C (eds) Handbook of mathematical fuzzy logic-volume 1. Studies in logic, mathematical logic and foundations, vol 37. College Publications, London, pp 103–207
Cintula P, Klement E-P, Mesiar R, Navara M (2010) Fuzzy logics with an additional involutive negation. Fuzzy Sets Syst 161(3):390–411CrossRefMATHMathSciNet
Cintula P, Hájek P, Noguera C (eds) (2011) Handbook of mathematical fuzzy logic. Studies in logic, mathematical logic and foundations, vols 37, 38. College Publications, London
da Costa N (1974) On the theory of inconsistent formal systems. Notre Dame J Form Log 15:497–510CrossRefMATH
da Costa N, Alves E (1977) A semantical analysis of the calculi \(C_n\). Notre Dame J Form Log 18:621–630CrossRefMATH
Dunn M (1976) Intuitive semantics for first degree entailment and ‘coupled trees’. Philos Stud 29:149–168CrossRefMathSciNet
Dunn M, Restall G (2002) Relevance logic. In: Gabbay D, Guenthner F (eds) Handbook of philosophical logic, vol 6. Kluwer, Dordrecht, pp 1–136CrossRef
Ertola RC (2009) On some operations using the min operator. In: Carnielli W, Coniglio ME, D’Ottaviano IML (eds) The many sides of logic. Studies in logic, vol 21. College Publications, London, pp 353–368
Esteva F, Godo L (2001) Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst 124(3):271–288CrossRefMATHMathSciNet
Esteva F, Godo L, Hájek P, Navara M (2000) Residuated fuzzy logics with an involutive negation. Arch Math Log 39(2):103–124CrossRefMATH
Flaminio T, Marchioni E (2006) T-norm based logics with an independent involutive negation. Fuzzy Sets Syst 157(4):3125–3144CrossRefMATHMathSciNet
Font JM, Gil A, Torrens A, Verdú V (2006) On the infinite-valued Łukasiewicz logic that preserves degrees of truth. Arch Math Log 45(7):839–868
Goodman ND (1981) The logic of contradiction. Zeitschrift für Mathmatische Logik und Grundlagen der Mathematik 27:119–126
Haniková Z (2011) Computational complexity of propositional fuzzy logics. In: Cintula P, Hájek P, Noguera C (eds) Handbook of mathematical fuzzy logic-vol 2. Studies in logic, mathematical logic and foundations, vol 38. College Publications, London, pp 793–851
Jansana R (2013) On deductive systems associated with equationally orderable quasivarieties. In: Proceedings of the 19th international conference on logic for programming, artificial intelligence and reasoning LPAR, Stellenbosch, South Africa
Jaśkowski S (1969) Propositional calculus for contradictory deductive systems. Stud Log 24(1):143–157CrossRefMATH
Johansson I (1936) Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus. Compositio Mathematica 4(1):119–136
Moisil GC (1942) Logique modale. Disquisitiones math. et phys., Bucarest, II, vol 1, pp 3–98 [Reprinted. In: Moisil GC (ed) (1972) Essais sur les logiques non chrysippiennes. Éditions de l’Académie de la République Socialiste de Roumanie, Bucarest]
Noguera C (2007) Algebraic study of axiomatic extensions of triangular norm based fuzzy logics. Monografies de l’Institut d’Investigació en Intel\(\cdot \)ligència artificial, vol 27. CSIC, Barcelona
Noguera C, Esteva F, Gispert J (2005a) On some varieties of MTL-algebras. Log J IGPL 13(4):443–466
Noguera C, Esteva F, Gispert J (2005b) Perfect and bipartite IMTL-algebras and disconnected rotations of prelinear semihoops. Arch Math Log 44(7):869–886
Priest G (2002a) Paraconsistent logic. In: Gabbay D, Guenthner F (eds) Handbook of philosophical logic, vol 6, 2nd edn. Kluwer, Dordrecht, pp 287–393
Priest G (2002b) Fuzzy relevant logic. In: Carnielli WA et al (eds) Chapter 11 in paraconsistency: the logical way to the inconsistent. CRC Press, Boca Raton
Priest G (2009) Dualising intuitionistic negation. Principia 13(2):165–184CrossRef
Rasiowa H (1974) An algebraic approach to non-classical logics. North-Holland, AmsterdamMATH
Rauszer C (1974) Semi-boolean algebras and their application to intuitionistic logic with dual operators. Fundamenta Mathematicae 85:219–249MathSciNet
Skolem TA (1970) Untersuchungen über die Axiome des Klassenkalküls und über Produktations- und Summationsprobleme, welche gewisse Klassen von Aussagen betreffen. Skrifter utgit av Videnskabsselskapet i Kristiania 3, 1919 [Reprinted Skolem T. Selected works in logic. In: Fenstad JE (ed.) Universitetsforlaget]
Metadaten
Titel
Paraconsistency properties in degree-preserving fuzzy logics
verfasst von
Rodolfo Ertola
Francesc Esteva
Tommaso Flaminio
Lluís Godo
Carles Noguera
Publikationsdatum
01.03.2015
Verlag
Springer Berlin Heidelberg
Erschienen in
Soft Computing / Ausgabe 3/2015
Print ISSN: 1432-7643
Elektronische ISSN: 1433-7479
DOI
https://doi.org/10.1007/s00500-014-1489-0

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