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On the standard and rational completeness of some axiomatic extensions of the monoidal t-norm logic. (English) Zbl 1011.03015
Hájek’s Basic fuzzy logic BL [P. Hájek, Mathematics of fuzzy logic. Dordrecht: Kluwer (1998; Zbl 0937.03030)] is the logic axiomatizing the family of calculi in which the (strong) conjunction & and implication \( \rightarrow \) are interpreted by a continuous t-norm and its residuum respectively. Any continuous t-norm can be generated, using ordinal sum constructions, by only three distinguished t-norms: Łukasiewicz, Gödel and Product t-norms. Each one gives rise to an important extension of BL, namely Łukasiewicz logic Ł (extension of BL with a double negation axiom), Gödel logic G (idempotency axiom) and Product logic P with two axioms, one of which capturing the Strict Basic fuzzy logic SBL of strict continuous t-norms, i.e., those having Gödel negation as associated negation [F. Esteva, L. Godo, P. Hájek and M. Navara, Arch. Math. Log. 39, 103-124 (2000; Zbl 0965.03035)]. Each such logic is complete with respect to the calculus defined by their respective defining t-norms. The monoidal t-norm logic MTL was introduced by F. Esteva and L. Godo [Fuzzy Sets Syst. 124, 271-288 (2001; Zbl 0994.03017)] as a weakening of Hájek’s Basic fuzzy logic BL by weakening the divisibility condition. MTL is equivalently obtained as the extension of Höhle’s Monoidal logic with the pre-linearity axiom. Since BL is in fact the logic of continuous t-norms [Hájek, loc. cit.], and since the divisibility condition is in the real interval \([0,1]\), the direct counterpart of the continuity of the truth function for &, the intuition behind MTL is to capture the logic of left-continuous t-norms and their residua [S. Jenei and F. Montagna, Stud. Log. 70, 183-192 (2002; Zbl 0997.03027)]. Since left-continuity is the necessary and sufficient condition for a t-norm to have a residuum, one cannot weaken BL any more if one wants to stick in the framework of residuated logics with semantics on \([0,1]\) given by t-norms and their residua. The paper addresses the issue of standard completeness of some axiomatic extensions of MTL, corresponding with well-known parallel extensions of BL. In particular, the following logics are considered: (i) the logic IMTL, the extension of MTL by the double negation axiom, shown not to collapse with Łukasiewicz logic, (ii) the logic \(\Pi \)MTL, an extension of MTL with Product logic axioms [P. Hájek, Fuzzy Sets Syst. 132, 107-112 (2002; Zbl 1012.03035)] and shown not to collapse with Product logic, and (iii) the logic SMTL, the extension of MTL with a pseudo-complementation axiom, shown not to collapse with SBL logic. The logic GMTL, the extension of MTL with the idempotency axiom, is just Gödel logic G, so it is not considered further in the paper. All these logics enjoy completeness with respect to their respective class of linearly ordered algebras. The problem of standard completeness (i.e., completeness with respect only to their corresponding algebra in the (real) unit interval \([0,1]\)) is solved positively in this paper for IMTL and SMTL. The standard completeness of SMTL is proved by extending the proof of Jenei and Montagna [loc. cit.], and the construction of the new proof is further refined to show standard completeness for IMTL. For \(\Pi\)MTL, the authors show the possibility of embedding any finite or countable \(\Pi\)MTL-chain into an \(\Pi\)MTL-chain on the rational unit interval (and \(\Pi\)MTL is called rational complete), but not on the real unit interval. An attempt to extend the left-continuous t-norm on the rational unit interval to the whole real unit interval by means of suprema similar to the proof of Jenei and Montagna [loc. cit.] is not possible, since this would not preserve in general the cancellative property of the t-norm in the proof, which is demonstrated by a clear example. Therefore, whether \(\Pi\)MTL is standard complete (i.e., with respect to the usual semantics on the real unit interval) remains an open question. In the last section the authors proceed to investigate in much detail sub-varieties \(\mathbb{V}\) of MTL algebras whose linearly ordered countable algebras embed into algebras of \(\mathbb{V}\) whose lattice reduct is the rational interval \([0,1]\). The embedding property, the embedding into ultraproducts of algebras of \(\mathbb{V}\) whose lattice reduct is the rational interval \([0,1]\), is shown to be directly related to the finite strong rational completeness of the corresponding logic.

MSC:
03B52 Fuzzy logic; logic of vagueness
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