ON SOME AXIOMATIC EXTENSIONS OF THE MONOIDAL T-NORM BASED LOGIC MTL: AN ANALYSIS IN THE PROPOSITIONAL AND IN THE FIRST-ORDER CASE
Creators: name: BIANCHI, MATTEO
Publisher: name: Universit� degli Studi di Milano
Year: 2010

description: The scientific area this thesis belongs to are many-valued logics: in particular, the logic MTL and some of its extensions, in the propositional and in the first-order case (see [8],[9],[6],[7]). The thesis is divided in two parts: in the first one the necessary background about these logics, with some minor new results, are presented. The second part is devoted to more specific topics: there are five chapters, each one about a different problem. In chapter 6 a temporal semantics for Basic Logic BL is presented. In chapter 7 we move to first-order logics, by studying the supersoundness property: we have improved some previous works about this theme, by expanding the analysis to many extensions of the first-order version of MTL. Chapter 8 is dedicated to four different families of n-contractive axiomatic extensions of BL, analyzed in the propositional and in the first-order case: completeness, computational and arithmetical complexity, amalgamation and interpolation properties are studied. Finally, chapters 9 and 10 are about Nilpotent Minimum logic (NM, see [8]): in chapter 9 the sets of tautologies of some NM-chains (subalgebras of [0,1]_NM) are studied, compared and the problems of axiomatization and undecidability are tackled. Chapter 10, instead, concerns some logical and algebraic properties of (propositional) Nilpotent Minimum logic. The results (or an extended version of them) of these last chapters have been also presented in papers [1, 4, 5, 2, 3]. ---------------------------------References--------------------------------------------- [1] S. Aguzzoli, M. Bianchi, and V. Marra. A temporal semantics for Basic Logic. Studia Logica, 92(2), 147-162, 2009. doi:10.1007/s11225-009-9192-3. [2] M. Bianchi. First-order Nilpotent Minimum Logics: first steps. Submitted for publication,2010. [3] M. Bianchi. On some logical and algebraic properties of Nilpotent Minimum logic and its relation with G�del logic. Submitted for publication, 2010. [4] M. Bianchi and F. Montagna. Supersound many-valued logics and Dedekind-MacNeille completions. Arch. Math. Log., 48(8), 719-736, 2009. doi:10.1007/s00153-009-0145-3. [5] M. Bianchi and F. Montagna. n-contractive BL-logics. Arch. Math. Log., 2010. doi:10.1007/s00153-010-0213-8. [6] P. Cintula, F. Esteva, J. Gispert, L. Godo, F. Montagna, and C. Noguera. Distinguished algebraic semantics for t-norm based fuzzy logics: methods and algebraic equivalencies. Ann. Pure Appl. Log., 160(1), 53-81, 2009. doi:10.1016/j.apal.2009.01.012. [7] P. Cintula and P. H�jek. Triangular norm predicate fuzzy logics. Fuzzy Sets Syst., 161(3), 311-346, 2010. doi:10.1016/j.fss.2009.09.006. [8] F. Esteva and L. Godo. Monoidal t-norm based logic: Towards a logic for left-continuous t-norms. Fuzzy sets Syst., 124(3), 271-288, 2001. doi:10.1016/S0165-0114(01)00098-7. [9] P. H�jek. Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer Academic Publishers, paperback edition, 1998. ISBN:9781402003707.