Supersound many-valued logics and Dedekind-MacNeille completions

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Matteo Bianchi
Franco Montagna
Title: Supersound many-valued logics and Dedekind-MacNeille completions
Journal: Archive for Mathematical Logic
Volume 48
Number 8
Pages: 719-736
Year: 2009
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In Hájek et al. (J Symb Logic 65(2):669–682, 2000) the authors 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 Hájek and Shepherdson (Ann Pure Appl Logic 109(1–2):65–69, 2001) 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 in Labuschagne and van Alten (Proceedings of the ninth international conference on intelligent technologies, 2008) and van Alten (2009). 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 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 ΠMTL, WCMTL and of each non-n-potent axiomatic extension of BL are not supersound.