On Łukasiewicz Logic with Truth Constants

Cignoli, Roberto; Esteva, Francesc; Godo, Lluı́s

doi:10.1007/978-3-540-72434-6_88

Cited by 4 publications

(5 citation statements)

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“…Similar expansions for a big class of other propositional t-norm based fuzzy logics (and even for a few distinguished first-order t-norm based fuzzy logics) have been analogously defined in Esteva et al (2006aEsteva et al ( , 2007aEsteva et al ( , b, 2009, Savický et al (2006), Cignoli et al (2007), but Pavelka-style completeness could not be obtained, as Łukasiewicz Logic is the only t-norm based logic whose truth-functions are continuous. Thus, in these papers rather than Pavelka-style completeness the authors have focused on the usual notion of completeness of a logic with respect to a significant class of linearly ordered algebras.…”

Section: Introductionmentioning

confidence: 79%

Expanding the propositional logic of a t-norm with truth-constants: completeness results for rational semantics

2009

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In this paper we consider the expansions of logics of a left-continuous t-norm with truth-constants from a subalgebra of the rational unit interval. From known results on standard semantics, we study completeness for these propositional logics with respect to chains defined over the rational unit interval with a special attention to the completeness with respect to the canonical chain, i.e. the algebra over ½0; 1 \ Q where each truth-constant is interpreted in its corresponding rational truth-value. Finally, we study rational completeness results when we restrict ourselves to deductions between the so-called evaluated formulae.

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Section: Introductionmentioning

confidence: 79%

Expanding the propositional logic of a t-norm with truth-constants: completeness results for rational semantics

2009

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“…As an easy consequence of this fact one obtains that L * (C) is a conservative expansion of L * [17]. The issue of which kinds of real completeness properties hold for the logics L * (C), both with respect to all standard chains or only the canonical one, has been addressed in the literature [22,5,16,36,13,17]. The results obtained in these papers are collected in Table 5, where we use the notations CanRC, CanFSRC and CanSRC to denote respectively the three kinds of completeness properties with respect to the canonical standard algebra.…”

Section: Expansions With Truth-constantsmentioning

confidence: 99%

First-order t-norm based fuzzy logics with truth-constants: Distinguished semantics and completeness properties

Esteva¹,

Godo²,

Noguera

2009

Annals of Pure and Applied Logic

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Dedicated to Franco Montagna in the occasion of his 60th birthday. MSC: 03B50 03B52 06B99Keywords: Algebraic logic Mathematical fuzzy logic First-order predicate non-classical logics Residuated lattices T-norm based fuzzy logics Truth-constants a b s t r a c t This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous tnorms (mainly continuous and weak nilpotent minimum t-norms). We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical real and rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind r → ϕ, where ϕ has no additional truth-constants.

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“…Goguen's considerations thus lead to the notion of a residuated lattice, first considered by Ward and Dilworth [61]. Thus he significantly contributed to the development of algebraic fuzzy logic, which was later on pursued by other researchers 4 : see the three volumes of the Handbook of Mathematical Fuzzy Logic [14,12] and the references therein for general overview, and in particular [51,25,45,24,26,27,10,47,4,29,30] for expansions of Lukasiewicz logic with constants; more generally [18,11] for expansions of wider classes of logics with propositional constants. Furthemore the survey [48] and the references therein complete the picture for logics with graded formulas.…”

Section: Introductionmentioning

confidence: 99%

Rational Pavelka logic: the best among three worlds?

Haniková¹

2021

Preprint

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This comparative survey explores three formal approaches to reasoning with partly true statements and degrees of truth, within the family of Lukasiewicz logic. These approaches are represented by infinite-valued Lukasiewicz logic ( L), Rational Pavelka logic (RPL) and a logic with graded formulas that we refer to as Graded Rational Pavelka logic (GRPL). Truth constants for all rationals between 0 and 1 are used as a technical means to calibrate degrees of truth. Lukasiewicz logic ostensibly features no truth constants except 0 and 1; Rational Pavelka logic includes constants in the basic language, with suitable axioms; Graded Rational Pavelka logic works with graded formulas and proofs, following the original intent of Pavelka, inspired by Goguen's work. Historically, Pavelka's papers precede the definition of GRPL, which in turn precedes RPL; retrieving these steps, we discuss how these formal systems naturally evolve from each other, and we also recall how this process has been a somewhat contentious issue in the realm of Lukasiewicz logic. This work can also be read as a case study in logics, their fragments, and the relationship of the fragments to a logic.

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On Łukasiewicz Logic with Truth Constants

Abstract: Abstract. Canonical completeness results for Ł(C), the expansion of Łukasiewicz logic Ł with a countable set of truth-constants C, have been recently proved in [5] for the case when the algebra of truth constants C is a subalgebra of the rational interval

Cited by 4 publications

References 11 publications

Expanding the propositional logic of a t-norm with truth-constants: completeness results for rational semantics

Expanding the propositional logic of a t-norm with truth-constants: completeness results for rational semantics

First-order t-norm based fuzzy logics with truth-constants: Distinguished semantics and completeness properties

Rational Pavelka logic: the best among three worlds?

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