an:05646248
Zbl 1185.03040
Bianchi, Matteo; Montagna, Franco
Supersound many-valued logics and Dedekind-MacNeille completions
EN
Arch. Math. Logic 48, No. 8, 719-736 (2009).
0933-5846 1432-0665
2009
j
03B50 06B23
Gödel logic; many-valued logic; completion of MTL-chains; axiomatic extension
Authors' abstract: ``In [J. Symb. Log. 65, No.~2, 669--682 (2000; Zbl 0971.03025)], \textit{P. Hájek, J. Paris} and \textit{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 [\textit{P. Hájek} and \textit{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.''
Daniele Mundici (Firenze)
0971.03025; 1004.03020