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Supersound many-valued logics and Dedekind-MacNeille completions. (English) Zbl 1185.03040
Authors’ abstract: “In [J. Symb. Log. 65, No. 2, 669–682 (2000; Zbl 0971.03025)], P. Hájek, J. Paris and J. Shepherdson introduce the concept of supersound logic, proving that first-order Gödel logic enjoys this property, whilst first-order Łukasiewicz and product logics do not; in [P. Hájek and J. Shepherdson, Ann. Pure Appl. Logic 109, No. 1–2, 65–69 (2001; Zbl 1004.03020)] this result is improved showing that, among the logics given by continuous t-norms, Gödel logic is the only one that is supersound. In this paper we will generalize the previous results. Two conditions will be presented: the first one implies the supersoundness and the second one non-supersoundness. To develop these results we will use, between the other machineries, the techniques of completions of MTL-chains developed by C. C. A. Labuschagne and C. J. van Alten. We list some of the main results. The first-order versions of MTL, SMTL, IMTL, WNM, NM, RDP are supersound; the first-order version of an axiomatic extension of BL is supersound if and only if it is \(n\)-potent (i.e. it proves the formula \({\varphi^{n}\,\to\,\varphi^{n+1}}\) for some \({n\,\in\,\mathbb{N}^+}\)). Concerning the negative results, we have that the first-order versions of \(\Pi\)MTL, WCMTL and of each non-\(n\)-potent axiomatic extension of BL are not supersound.”

MSC:
03B50 Many-valued logic
06B23 Complete lattices, completions
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