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Franco Montagna’s work on provability logic and many-valued logic. (English) Zbl 1380.03004

This paper celebrates the work of Franco Motagna on provability logic, many-valued logic and probability theory on many-valued events. Note that scanning the zbMATH database for “au: Montagna, Franco” results in a list of 128 documents.
The paper opens with a overview of Montagna’s work on provability logic, many-valued logic and generalized probability theory. The main part of the paper (Section 2), concentrates on Montagna’s contribution to provability logic. Section 3 covers “algebraic properties of BL-algebras” investigated by Montagna in a series of papers starting with [P. Agliano and F. Montagna, J. Pure Appl. Algebra 181, No. 2–3, 105–129 (2003; Zbl 1034.06009)]. Finally, Section 4 describes Montagna’s research in many-valued logic to investigate the foundations of probability theory on many-valued events, in particular, de Fintetti’s betting game [B. de Finetti, Fundam. Math. 17, 298–329 (1931; Zbl 0003.16303); D. de Jongh et al., Stud. Log. 50, No. 1, 51–69 (1991; Zbl 0744.03057)].

MSC:

03-03 History of mathematical logic and foundations
03B45 Modal logic (including the logic of norms)
03F45 Provability logics and related algebras (e.g., diagonalizable algebras)
03B48 Probability and inductive logic
03B50 Many-valued logic
03B52 Fuzzy logic; logic of vagueness
03G25 Other algebras related to logic
01A70 Biographies, obituaries, personalia, bibliographies

Biographic References:

Montagna, Franco
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References:

[1] Aglianò, P., I. M. A. Ferreirim, and F. Montagna, Basic Hoops: an Algebraic Study of Continuous t-norms. Studia Logica 87(1): 73-98, 2007. · Zbl 1127.03049
[2] Aglianò P., Montagna F.: Varieties of BL-Algebras I: General Properties. Journal of Pure and Applied Algebra 181, 105-129 (2003) · Zbl 1034.06009 · doi:10.1016/S0022-4049(02)00329-8
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[9] Baaz M., Hájek P., Montagna F., Veith H.: Complexity of t-tautologies. Annals of Pure and Applied Logic 113(1-3), 3-11 (2001) · Zbl 1006.03022 · doi:10.1016/S0168-0072(01)00048-3
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[17] Bova S., Montagna F.: The consequence relation in the logic of commutative GBL-algebras is PSPACE-complete. Theoretical Computer Science 410(12-13), 1143-1158 (2009) · Zbl 1159.03045 · doi:10.1016/j.tcs.2008.10.024
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[21] Ciabattoni A., Metcalfe G., Montagna F.: Algebraic and proof-theoretic characterizations of truth stressers for MTL and its extensions. Fuzzy Sets and Systems 161(3), 369-389 (2010) · Zbl 1190.03026 · doi:10.1016/j.fss.2009.09.001
[22] Ciabattoni A., Montagna F.: Proof theory for locally finite many-valued logics: Semi-projective logics. Theoretical Computer Science 480, 26-42 (2013) · Zbl 1322.03019 · doi:10.1016/j.tcs.2013.02.003
[23] Cignoli, R., I. M. L. D’Ottaviano, and D. Mundici, Algebraic Foundations of Many-valued Reasoning. Kluwer, Dordrecht, 2000. · Zbl 0937.06009
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[27] Corsi, E. A., and F. Montagna, The Rényi-Ulam games and many-valued logics. Fuzzy Sets and Systems, in print. DOI:10.1016/j.fss.2015.09.006. · Zbl 1396.03042
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[31] de Jongh D., Visser A.: Explicit fixed points in interpretability logic. Studia Logica 50(1), 39-49 (1991) · Zbl 0744.03020 · doi:10.1007/BF00370386
[32] Di Nola A.: Representation and reticulation by quotients of MV-algebras. Ricerche di Matematica (Naples) 40, 291-297 (1991) · Zbl 0767.06013
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[36] Esteva F., Gispert J., Godo L., Montagna F.: On the standard and rational completeness of some axiomatic extensions of the monoidal t-norm logic. Studia Logica 71(2), 199-226 (2002) · Zbl 1011.03015 · doi:10.1023/A:1016548805869
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[39] Esteva, F., L. Godo, and F. Montagna, The Ł \[{\prod}\]∏ and Ł \[{\prod}\]∏\[{\frac{1}{2}}12\] logics: two complete fuzzy systems joining Łukasiewicz and Product Logics. Archive for Mathematical Logic 40(1): 39-67, 2001. · Zbl 0966.03022
[40] Fedel M., Hosni H., Montagna F.: A logical characterization of coherence for imprecise probabilities. International Journal of Approximate Reasoning 52(8), 1147-1170 (2011) · Zbl 1244.03082 · doi:10.1016/j.ijar.2011.06.004
[41] Fedel, M., K. Keimel, F. Montagna, and W. Roth, Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic. Forum Mathematicum 25: 405-441, 2013. · Zbl 1266.06011
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[45] Flaminio, T., H. Hosni, and F. Montagna, A characterization of strict coherence for infinite-valued events. Manuscript. · Zbl 1447.03001
[46] Flaminio T., Montagna F.: A logical and algebraic treatment of conditional probability. Archive for Mathematical Logic 44, 245-262 (2005) · Zbl 1064.03016 · doi:10.1007/s00153-004-0253-z
[47] Flaminio T., Montagna F.: MV-algebras with internal states and probabilistic fuzzy logics. International Journal of Approximate Reasoning 50(1), 138-152 (2009) · Zbl 1185.06007 · doi:10.1016/j.ijar.2008.07.006
[48] Flaminio T., Montagna F.: Models for many-valued probabilistic reasoning. Journal of Logic and Computation 21(3), 447-464 (2011) · Zbl 1237.06005 · doi:10.1093/logcom/exp013
[49] Friedman S.-D., Rathjen M., Weiermann A.: Slow consistency. Annals of Pure and Applied Logic 164(3), 382-393 (2013) · Zbl 1263.03055 · doi:10.1016/j.apal.2012.11.009
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[56] Hájek P., Montagna F.: A note on the first-order logic of complete BL-chains. Mathematical Logic Quarterly 54(4), 435-446 (2008) · Zbl 1152.03019 · doi:10.1002/malq.200710058
[57] Hčrcík R.: Standard completeness theorem for \[{\prod}\]∏MTL. Archive for Mathematical Logic 44(4), 413-424 (2005) · Zbl 1071.03013 · doi:10.1007/s00153-004-0254-y
[58] Hosni, H., and F. Montagna, Stable Non-standard Imprecise Probabilities. Proceedings of IPMU2014 (A. Laurent, et al. eds.), Communication in Computer and Information Science 444: 436-445, 2014. · Zbl 1461.03022
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[61] Jipsen P., Montagna F.: On the structure of generalized BL-algebras. Algebra Universalis 55, 226-237 (2006) · Zbl 1109.06011 · doi:10.1007/s00012-006-1960-6
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