Strong non-standard completeness for fuzzy logics. (English) Zbl 1132.03332

Summary: In this paper we introduce the notion of strong nonstandard completeness (SNSC) for fuzzy logics. This notion naturally arises from the well-known construction of ultraproducts. Roughly speaking, to say that a logic \(\mathcal C\) is strong nonstandard complete means that, for any countable theory \(\Gamma\) over \(\mathcal C\) and any formula \(\varphi\) such that \(\Gamma\not\vdash_{\mathcal C} \varphi\), there exists an evaluation \(e\) of \(\mathcal C\)-formulas into a \(\mathcal C\)-algebra \(\mathcal A\) such that the universe of \(\mathcal{A}\) is a non-Archimedean extension \([0,1]^*\) of the real unit interval \([0,1], e\) is a model for \(\Gamma\), but \(e (\varphi) < 1\). Then we apply SNSC to prove that various modal fuzzy logics allowing to deal with simple and conditional probability of infinite-valued events are complete with respect to classes of models defined starting from nonstandard measures, that is, measures taking values in \([0,1]^*\).


03B52 Fuzzy logic; logic of vagueness
03H05 Nonstandard models in mathematics
Full Text: DOI


[1] Belluce LP (1986) Semisimple MV-algebras of infinite-valued logic and bold fuzzy set theory. Can J Math 38(6):1356–1379 · Zbl 0625.03009
[2] Blok WJ, Pigozzi D (1989) Algebraizable logics. Memb Am Math Soc 396(77) · Zbl 0664.03042
[3] Burrus S, Sankappanavar HP (1980) A course in Universal Algebra. Springer, Heidelberg
[4] Chang CC, Keisler HJ (1973) Model theory. North-Holland, Amsterdam
[5] Cintula P (2003) Advances in the Ł{\(\Pi\)} and Ł{\(\Pi\)}1/2 logics. Arch Math Logic 42:449–468 · Zbl 1026.03017
[6] Cignoli R, D’Ottaviano IML, Mundici D (2000a) Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht
[7] Cignoli R, Esteva F, Godo L, Torrens A (2000b) Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Comput 4:106–112 · Zbl 02181428
[8] Di Nola A (1993) MV-algebras in the treatment of uncertainty. In: Lowen R, Rubens M (eds) Fuzzy Logic. Kluwer, Dordrecht
[9] Di Nola A, Georgescu G, Lettieri A (1999) Conditional states in finite-valued logic. In: Klement EP, Dubois D, Prade H (eds) Fuzzy sets, logics, and reasoning about knowledge. Kluwer, Dordrecht, pp 161–174 · Zbl 0942.06004
[10] Esteva F, Godo L (2001) Monoidal t-norm based logic: towards a logic for left-continuous t-norms. Fuzzy Sets Syst 124:271–288 · Zbl 0994.03017
[11] Esteva F, Godo L, Hájek P (2000) Reasoning about probability using fuzzy logic. Neural Netw World 10(5):811–824
[12] Esteva F, Godo L, Montagna F (2001) Ł{\(\Pi\)} and Ł{\(\Pi\)}1/2: two complete fuzzy systems joining Łukasiewicz and Product logics. Arch Math Logic 40:39–67 · Zbl 0966.03022
[13] Esteva F, Gispert J, Godo L, Montagna F (2002) On the standard and rational completeness for some axiomatic extension of the monoidal t-norm logic. Studia Logica 71:199–226 · Zbl 1011.03015
[14] Flaminio T (2005) A zero-layer based fuzzy probabilistic logic for conditional probability. In: Lluís Godo (ed) Lecture Notes in Artificial Intelligence, vol 3571, 8th European Conference on Symbolic and Quantitaive Approaches on Reasoning under Uncertainty ECSQARU’05, Barcelona, Spain, July 2005, pp 714–725 · Zbl 1109.03018
[15] Flaminio T (2007) NP-containment for the coherence tests of assessment of conditional probability: a fuzzy-logical approach. Arch Math Logic (in press) · Zbl 1110.03012
[16] Flaminio T, Godo L (2006) A logic for reasoning about the probability of fuzzy events. Fuzzy Sets Syst (in press) · Zbl 1116.03018
[17] Flaminio T, Montagna F (2005) A logical and algebraic treatment of conditional probability. Arch Math Logic 44:245–262 · Zbl 1064.03016
[18] Gerla B (2001) Many-valed Logics of Continuous t-norms and their functional representation. Ph.D. Thesis, University of Milan
[19] Grigolia R (1977) Algebraic analysis of Łukasiewicz-Tarski n-valued logical systems. In: Wójcicki R, Malinowski G (Eds) Selected Papers on Łukasiewicz Sentencial Calculi. Polish Academy of Science, Ossolineum, Wrocław, pp 81–91
[20] Hájek P (1998) Metamathematics fo fuzzy logic, Kluwer, Dordrecht
[21] Hájek P, Godo L, Esteva F (1995) Probability and fuzzy logic. In: Besnard, Hanks (eds) Proceedings of uncertainty in artificial intelligence UAI’95. Morgan Kaufmann, San Francisco, pp 237–244
[22] Horčík R (2005) Standard completeness theorem for {\(\Pi\)}MTL logic. Arch Math Logic 44:413–424 · Zbl 1071.03013
[23] Horčík R (2007) On the failure of standard completeness in {\(\Pi\)}MTL for infinite theories. Fuzzy Sets Syst (in press) · Zbl 1117.03032
[24] Horčík R, Cintula P (2004) Product Łukasiewicz logic. Arch Math Logic 43:477–503 · Zbl 1059.03011
[25] Hurd AE, Loeb PA (1985) An introduction to nonstandard real analysis. Academic, Orlando · Zbl 0583.26006
[26] Jenei S, Montagna F (2002) A proof of standard completeness for Esteva and Godo’s logic MTL. Studia Logica 70:183–192 · Zbl 0997.03027
[27] Kroupa T (2005) Conditional probability on MV-algebras. Fuzzy Sets Syst 149(2):369–381 · Zbl 1061.60004
[28] Kroupa T (2006) Every state on semisimple MV-algebra is integral. Fuzzy Sets Syst 157:2771–2782 · Zbl 1107.06007
[29] Marchioni E, Godo L (2004) A logic for reasoning about coherent conditional probability: a fuzzy modal logic approach. In: Alferes JJ, Leite J (eds) Lecture Notes in Artificial Intelligence, vol 3229. 9th European Conference on Logic in Artificial Intelligence JELIA’04. Lisbon, Portugal, September 2004, pp 213–225 · Zbl 1111.68683
[30] Montagna F, Panti G (2001) Adding structures to MV-algebras. J Pure Appl Algebra 164(3):365–387 · Zbl 0992.06012
[31] Montagna F, Noguera C, Horčík R (2006) On weakly cancellative fuzzy logics. J Logic Comput 16:423–450 · Zbl 1113.03021
[32] Mundici D (1995) Averaging the truth-value in Łukasiewicz logic. Studia Logica 55(1):113–127 · Zbl 0836.03016
[33] Noguera C (2006) Algebraic study of axiomatic extensions of triangular norm based fuzzy logics. Ph.D. Thesis, Barcelona
[34] Pavelka J (1979) On fuzzy logic I, II, III. Zeitschr f Math Logik Grundl Math 25:45–52, 119–134, 447–464 · Zbl 0435.03020
[35] Robinson A (1996) Non-standard analysis. North-Holland, Amsterdam · Zbl 0843.26012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.