COHERENCE AND TRUTH In memoriam

We report on research by Montagna and collaborators on the combinatorial and computational aspects of generalized basic logic. In the first part of the talk, we focus on the PSPACEcompleteness of the tautology and entailment problems. In the second part of the talk, we discuss a syntactic relaxation of the disjunction property leading to an uncountable family of substructural logics with a PSPACE-hard tautology problem (it is known that substructural logics enjoying the full disjunction property have a PSPACE-hard tautology problem). Preining, Norbert: A (quite) general method to prove non-re for Kripke frames and real-valued based logics ABSTRACT: We present a general method to show that large classes of logics of Kripke frames (linear or not, constant or increasing domains) as well as large classes of logics with truth values over the reals can be shown to be not recursively enumerable. We present a general method to show that large classes of logics of Kripke frames (linear or not, constant or increasing domains) as well as large classes of logics with truth values over the reals can be shown to be not recursively enumerable. ======================================================================= December 17, Thursday Ono, Hiroakira: Analytic cut and interpolation for bi-intuitionistic logic Esteva, Francesc: Paraconsistency and Fuzzy Logic: The case of Łuksiewicz Logic ABSTRACT: Among the plethora of fuzzy logics defined in [3] as many-valued logics withsemantics over the structure defined on the real unit interval by a continuous t-norm and its residuumwe take the special case of the well known propositional Łukasiewicz logic. This logic is finitelyaxiomatizable and finitely strong complete (complete for deductions from a finite set of premises). Inthe preliminaries of the talk we introduce this logic together with its degree preserving companion (see[1]) and their relationships. In particular we show that truth preserving Łukasiewicz logic Ł isexplosive while its degree preserving companion is paraconsistent.In the main part of the talk we will present the results in [2]. In that work we have started the study ofintermediate logics between Ł, and . We show that there are infinitely-many explosive andparaconsistent logics in between and we provide some general results about these logics. A moredetailed description of the family of intermediate logics is presented in the particular case of finitely-valued Łukasiewicz logics .(Joint work with M. Coniglio (CLE, Campinas) and L. Godo (IIIA CSIC, Barcelona))References[1] F. Bou, F. Esteva, J.M. Font, A. Gil, L. Godo, A. Torrens, and V. Verdú. Logics preservingdegrees of truth from varieties of residuated lattices. Journal of Logic and Computation, 19(6):1031-1069, 2009.[2] M. E. Coniglio, F. Esteva, and L. Godo. On the set of intermediate logics between the truth anddegree preserving Lukasiewicz logics. Logical Journal of the IGPL. In Press.[3] P. Hájek. Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer, Dordrecht, 1998 Among the plethora of fuzzy logics defined in [3] as many-valued logics withsemantics over the structure defined on the real unit interval by a continuous t-norm and its residuumwe take the special case of the well known propositional Łukasiewicz logic. This logic is finitelyaxiomatizable and finitely strong complete (complete for deductions from a finite set of premises). Inthe preliminaries of the talk we introduce this logic together with its degree preserving companion (see[1]) and their relationships. In particular we show that truth preserving Łukasiewicz logic Ł isexplosive while its degree preserving companion is paraconsistent.In the main part of the talk we will present the results in [2]. In that work we have started the study ofintermediate logics between Ł, and . We show that there are infinitely-many explosive andparaconsistent logics in between and we provide some general results about these logics. A moredetailed description of the family of intermediate logics is presented in the particular case of finitely-valued Łukasiewicz logics .(Joint work with M. Coniglio (CLE, Campinas) and L. Godo (IIIA CSIC, Barcelona))References[1] F. Bou, F. Esteva, J.M. Font, A. Gil, L. Godo, A. Torrens, and V. Verdú. Logics preservingdegrees of truth from varieties of residuated lattices. Journal of Logic and Computation, 19(6):1031-1069, 2009.[2] M. E. Coniglio, F. Esteva, and L. Godo. On the set of intermediate logics between the truth anddegree preserving Lukasiewicz logics. Logical Journal of the IGPL. In Press.[3] P. Hájek. Metamathematics of Fuzzy Logic, volume 4 of Trends in Logic. Kluwer, Dordrecht, 1998 Godo, Lluís: On a class of modal expansions of left-continuous t-norm based logicsJipsen, Peter: Duality for partial algebras, bunched implication algebras and GBL-algebras Cicalese, Ferdinando: Aguzzoli, Stefano: Equivalences of varieties built using prelinear semihoops ABSTRACT: (Joint work in progress with Brunella Gerla, Tommaso Flaminio and Sara Ugolini.) In2015 Franco Montagna and Sara Ugolini proved a categorical equivalence between the variety ofproduct algebras and a category whose objects are triples ( ) where is a Boolean algebra, isa cancellative hoop andsatisfies suitable properties. In this work we show how tobuild several categorical equivalences between varieties of MTL-algebras, using varieties of prelinearsemihoops as building blocks. Further, we generalise Montagna-Ugolini triples to show that each oneof the considered varieties of MTL-algebras is equivalent to a category of triples ( ) for Hpicked in a variety of prelinear semihoops. (Joint work in progress with Brunella Gerla, Tommaso Flaminio and Sara Ugolini.) In2015 Franco Montagna and Sara Ugolini proved a categorical equivalence between the variety ofproduct algebras and a category whose objects are triples ( ) where is a Boolean algebra, isa cancellative hoop andsatisfies suitable properties. In this work we show how tobuild several categorical equivalences between varieties of MTL-algebras, using varieties of prelinearsemihoops as building blocks. Further, we generalise Montagna-Ugolini triples to show that each oneof the considered varieties of MTL-algebras is equivalent to a category of triples ( ) for Hpicked in a variety of prelinear semihoops. Marra, Vincenzo: Tarski’s theorem on intuitionistic logic, for polyhedraABSTRACT: In 1938, Tarski proved his landmark result that intuitionistic logic is complete withrespect to interpretations into the (complete) Heyting algebras of open sets of topological spaces. Infact, as Tarski showed, one can restrict attention to all metrisable spaces, or even just the real line orthe Cantor space, without impairing completeness. I prove a version of Tarski’s theorem where thespaces are restricted to compact polyhedra, and the accompanying (not necessarily complete) Heytingalgebras are restricted to those given by open subpolyhedra. The key property turns out to betopological dimension, which I show is captured by the bounded-depth axioms. Theorem: Theintermediate logic of the class of all polyhedra of dimension at most is intuitionistic logic extendedby the bounded-depth axiom schema of order . Proofs are self-contained to within standard PL-topology. I discuss the research directions these results point to. (Partly based on joint work with NickBezhanishvili, Dan McNeill, and Andrea Pedrini.)