On real and rational completeness of some predicate fuzzy logics with truth-constants F. Esteva IIIA - CSIC Bellaterra, Spain
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L. Godo IIIA - CSIC Bellaterra, Spain
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C. Noguera University of Siena Siena, Italy
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Extended Abstract Throughout this paper, by a (propositional) fuzzy logic L we understand what Cintula and H` ajek call a core fuzzy logic [1, 9, 10], that is, an expansion of the Monoidal t-norm based logic MTL [3] satisfying the congruence condition: ϕ ↔ ψ `L χ(ϕ) ↔ χ(ψ), and the following local deduction theorem: Γ, ϕ `L ψ iff there a is natural number n such that Γ `L ϕ& . n. . &ϕ → ψ . Moreover the term predicate fuzzy logic L∀ will denote the predicate calculus obtained from the propositional core fuzzy logic L by adding the “classical” quantifiers (∀ and ∃) together with the same axioms and rule of generalization used by H` ajek in his book [7] when defining BL∀. On the other hand, given a left-continuous t-norm ∗, if L∗ denotes the (propositional) fuzzy logic which is standard complete with respect to the MTL-algebra [0, 1]∗ defined by ∗ and its residuum, L∗ (C) (resp. L∗ ∀(C)) will denote the core (predicate) fuzzy logic which is the expansion of L∗ (resp. of L∗ ∀) with a truth-constant for each element of a countable subalgebra C of [0, 1]∗ and the corresponding book-keeping axioms (see also H´ ajek’s book). The aim of this paper is to investigate at large conservativeness results for the expanded logics L∗ (C) and L∗ ∀(C), as well as real and rational completeness results in the sense defined in [2]. To this end, we take advantage of a number of already available results in the literature, namely • results about real and rational completeness of propositional fuzzy logics with truth constants: for real completeness we mainly refer to the results of the papers [5, 4] and for rational completeness we will use results in the manuscript [2]; 1
• results about real completeness of predicate fuzzy logics, mainly from [10]; • and results on conservativeness when adding truth-constants to a predicate fuzzy logic: H`ajek, Paris and Shepherdson’s result [11] showing that RPL∀ (rational predicate Pavelka logic) is a conservative expansion of L∀ (predicate Lukasiewicz logic) , and those in [6] showing that L∗ ∀(C) is a conservative expansion of L∗ ∀ when ∗ is either a pseudocomplemented or a nilpotent minimum t-norm. In this paper, besides of summarizing known results, we will show the following: (i) for any left-continuous t-norm ∗, the logic L∗ (C) is a conservative expansion of L∗ ; (ii) new results about rational completeness of the logics L∗ (C); (iii) new results about conservativeness for L∗ ∀(C) with respect to L∗ ∀; (iv) new results about real and rational completeness for most prominent L∗ ∀(C) logics, some of them are a direct consequence of the conservativeness (knowing that the corresponding predicate logic without constants does not have some completeness) and some of them require ad hoc proofs.
References [1] P. Cintula. From Fuzzy Logic to Fuzzy Mathematics PhD disertation , Prague 2004. [2] P. Cintula, F. Esteva, J. Gispert, L. Godo, F. Montagna and C. Noguera, Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies, Submitted, 2007. [3] F. Esteva and L. Godo. Monoidal t-norm based logic: Towards a logic for left-continuous t-norms. Fuzzy Sets and Systems, 124: 271–288, 2001. [4] F. Esteva, J. Gispert, L. Godo and C. Noguera. Adding truth-constants to logics of a continuous t-norm: axiomatization and completeness results. Fuzzy Sets and Systems 158: 597–618, 2007. [5] F. Esteva, L. Godo and C. Noguera. On rational weak nilpotent minimum logics, Journal of Multiple-Valued Logic and Soft Computing 12, Number 1-2, pp. 9-32, 2006. [6] F. Esteva, L. Godo and C. Noguera. On completeness results for the expansions with truth-constants of some predicate fuzzy logics Mathware and Soft Computing, to appear. [7] P. H´ ajek. Metamathematics of Fuzzy Logic, volume 4 of Trends in LogicStudia Logica Library. Dordrecht/Boston/London, 1998. 2
[8] P. H´ ajek. Computational complexity of t-norm based propositional fuzzy logics with rational truth constants. Fuzzy Sets and Systems 157: 677–682, 2006. [9] P. H´ ajek and P. Cintula. On theories and models in fuzzy predicate logics. The Journal of Symbolic Logic 71: 863–880, 2006. [10] P. H´ ajek and P. Cintula. Triangular Norm Predicate Fuzzy Logics. In Fuzzy Logics and Related Structures (S. Gottwald, P. H´ajek, U. H¨ohle and E.P. Klement eds.), to appear. [11] P. H´ ajek, J. Paris and J. Shepherdson. Rational Pavelka predicate logic is a conservative extension of Lukasiewicz predicate logic. Journal of Symbolic Logic, 65(2):669–682, 2000. [12] F. Montagna. On the predicate logics of continuous t-norm BL-algebras. Archive for Mathematical logic 44, 97–114 (2005).
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