Abstract
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.
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Bianchi, M., Montagna, F. Supersound many-valued logics and Dedekind-MacNeille completions. Arch. Math. Logic 48, 719–736 (2009). https://doi.org/10.1007/s00153-009-0145-3
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DOI: https://doi.org/10.1007/s00153-009-0145-3
Keywords
- Many-valued logics
- Basic properties of first-order languages and structures
- Lattices and related structures
- Complete lattices
- Completions