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Some (non)tautologies of Łukasiewicz and product logic. (English) Zbl 1209.03017

The paper deals with first-order infinite-valued logics based on continuous t-norms and their residua, as defined by P. Hájek in his monograph [Metamathematics of fuzzy logic. Dordrecht: Kluwer Academic Publishers (1998; Zbl 0937.03030)]. Recall that a formula in a first-order language \(L\) is a general tautology of one of these logics if it evaluates to the top value in each interpretation \((M,A)\), where \(A\) is a chain in the variety constituting the algebraic semantics of the logic, and \(M\) is an \(A\)-valued first-order structure for the language \(L\); analogously, this formula is a standard tautology if it evaluates to 1 in each interpretation \((M,A)\), for \(A\) a standard algebra in the algebraic semantics, that is, an algebra \(([0,1],*,\to,0)\), where \(*\) is a continuous t-norm and \(\to\) is its residuum. In particular, for what concerns the three main logics of continuous t-norms, only first-order Gödel logic is complete with respect to both standard and general semantics, while first-order Łukasiewicz and product logics are both complete with respect to the general semantics, but are incomplete with respect to standard semantics. So there exist formulas that are standard tautologies but not general tautologies of the latter two logics.
The paper finds concrete examples of such formulas, and moreover explicitly builds general models where these formulas fail. The paper first deals with Łukasiewicz logic with crisp equality (and function symbols). The formula which is shown to be a standard but not general Łukasiewicz tautology is obtained using Robinson’s arithmetic (which is finitely axiomatized) and a bounded version of the statement asserting consistency of arithmetic. An analogous result for Łukasiewicz logic with fuzzy equality (and without function symbols) is then proved. For what concerns product logic, the author uses a finitely axiomatised theory, introduced by F. Montagna in the paper [“Three complexity problems in quantified fuzzy logic”, Stud. Log. 68, No. 1, 143–152 (2001; Zbl 0985.03014)], to build, from each formula true in the standard model of arithmetic but unprovable from Peano’s axioms, a new formula with one unary fuzzy predicate which is a standard tautology of product logic, but not one of its general tautologies.

MSC:

03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness
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References:

[1] Łukasiewicz, Ruch filozoficzny 5 pp 170– (1920)
[2] Hájek, Metamathematics of First-Order Arithmetic (1993) · Zbl 0781.03047 · doi:10.1007/978-3-662-22156-3
[3] Hájek, Archive for Mathematical Logic 35 pp 191– (1996) · Zbl 0848.03005 · doi:10.1007/BF01268618
[4] DOI: 10.1002/malq.200610027 · Zbl 1110.03013 · doi:10.1002/malq.200610027
[5] Hájek, Logic versus Approximation. Essays Dedicated to Michael M. Richter on the Occasion of His 65th Birthday pp 1– (2004) · Zbl 1052.68005
[6] Łukasiewicz, C.R. de la Societé des Sciences et des Letters de Varsovie 23 pp 51– (1930)
[7] Hájek, Metamathematics of Fuzzy Logic (1998) · Zbl 0937.03030 · doi:10.1007/978-94-011-5300-3
[8] Cignoli, Algebraic Foundations of Many-Valued Reasoning (2000) · Zbl 0937.06009 · doi:10.1007/978-94-015-9480-6
[9] DOI: 10.2307/2964111 · Zbl 0112.24503 · doi:10.2307/2964111
[10] DOI: 10.1023/A:1011958407631 · Zbl 0985.03014 · doi:10.1023/A:1011958407631
[11] Hájek, Proceedings East West Fuzzy Colloquium pp 2– (2000)
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