COHERENCE AND TRUTH

In memoriam Franco Montagna

16-18 December 2015, Certosa di Pontignano (Siena, Italy)

Invited speakers Paolo Aglianò, Stefano Aguzzoli, Matthias Baaz, Lev Beklemishev, Claudio Bernardi, Simone Bova, Agata Ciabattoni, Ferdinando Cicalese, Dick de Jongh, Francesc Esteva, Tommaso Flaminio, Lluís Godo, Zuzana Haniková, Hykel Hosni, Peter Jipsen, Enrico Marchioni, Vincenzo Marra, George Metcalfe, Carles Noguera, Hiroakira Ono, Norbert Preining, Giovanni Sambin, Luca Spada, Albert Visser.

Program committee Antonio Di Nola (Salerno), Graziano Gentili (Firenze), Daniele Mundici (Firenze), Santina Rocchi (Siena), Andrea Sorbi (Siena), Carlo Toffalori (Camerino), Aldo Ursini (Siena).

Main sponsors University of Siena, PRIN 2010-2011 (Metodi logici per il trattamento dell’informazione), AILA (Associazione Italiana di Logica e sue Applicazioni), KGS (Kurt Gödel Society), GNSAGA (Gruppo Nazionale Strutture Algebriche, Geometriche e Applicazioni).

WEDNESDAY THURSDAY FRIDAY

9,30 9,00 9,00 DGREETINGS ONO SPADA

10,00 9,50 9,50 EBEKLEMISHEV ESTEVA METCALFE

10,50 10,40 10,40 P

11,10 11,00 11,00 ABAAZ GODO HOSNI12,00 11,50 11,50 R

DE JONGH JIPSEN FLAMINIO12,50 12,40 12,40 T

14,30 14,30 14,30 UVISSER CICALESE AGLIANO'

15,20 15,20 15,20 RA NOGUERA AGUZZOLI BERNARDI

16,10 16,10 16,10 ER HANIKOVA MARRA SAMBIN

17,00 17,00 17,00R

17,20 17,20I BOVA MARCHIONI

18,10 18,10

V PREINING CIABATTONI

A21,00 21,00 21,00

L KGS Meeting Jazz & Alessandra Manganelli

Coherence & Truth-In memoriam Franco MontagnaProgram (titles to be announced)

15 December 16 December 17 December 18 December 19 December

coffee break coffee break coffee break

lunch lunch lunch

coffee break coffee break coffee break

dinner dinner dinner

Concert Paolo Aglianò Concert Stefano Bellissima

honouring Franco Montagna

Brahms and Dvořák, music

for piano four-hands

Coherence and Truth – In memoriam Franco Montagna

Titles of Talks

December 16, Wednesday

Beklemishev, Lev: On Franco's contributions to provability logic

Baaz, Matthias: 10 problems in Gödel logics

De Jongh, Dick: Subminimal Negation

Visser, Albert: Big Models

ABSTRACT: In this talk, I discuss an idea that has been around for some time, but never was taken

up seriously. It is the idea to study the modal logic of the models of a theory with a natural

accessibility relation such as *model extension* and *internal model*. This project is similar in spirit

to the study by Joel Hamkins and Benedikt Löwe of models of ZF with *forcing extension* as

accessibility relation.

First, I will consider the question whether one can prove Solovay's Theorem using a Big Model. This

turns out to be the case. There is a reasonable accessibility relation for which necessity and provability

coincide. Regrettably, the scope of proofs in this style seems to be much smaller than the scope of the

usual more syntactic proof. (A detailed verification of a version of the result can be found in a paper

by Paula Henk.)

Secondly, I discuss cases in which the notions of necessity and possibility are not arithmetically

definable. I will illustrate the power of the modal language to define familiar notions like

interpretability. Then, I will present some of the results known so far. In particular, I will zoom in on

the questions of what arithmetical sentences are necessarily possible and what arithmetical sentences

are possibly necessary.

Noguera, Carles: Dense semantics for fuzzy logics

Hanikova, Zuzana: Syntactic fragments in FLew-algebras

Bova, Simone: Generalized Basic Logic, Polynomial Space, and Disjunction Property

ABSTRACT: 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 PSPACE-

completeness 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.

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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 with

semantics over the structure defined on the real unit interval by a continuous t-norm and its residuum

we take the special case of the well known propositional Łukasiewicz logic. This logic is finitely

axiomatizable and finitely strong complete (complete for deductions from a finite set of premises). In

the 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 is

explosive 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 of

intermediate logics between Ł, and . We show that there are infinitely-many explosive and

paraconsistent logics in between and we provide some general results about these logics. A more

detailed 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 preserving

degrees 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 and

degree 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 logics

Jipsen, 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.) In

2015 Franco Montagna and Sara Ugolini proved a categorical equivalence between the variety of

product algebras and a category whose objects are triples ( ) where is a Boolean algebra, is

a cancellative hoop and satisfies suitable properties. In this work we show how to

build several categorical equivalences between varieties of MTL-algebras, using varieties of prelinear

semihoops as building blocks. Further, we generalise Montagna-Ugolini triples to show that each one

of the considered varieties of MTL-algebras is equivalent to a category of triples ( ) for H

picked in a variety of prelinear semihoops.

Marra, Vincenzo: Tarski’s theorem on intuitionistic logic, for polyhedra

ABSTRACT: In 1938, Tarski proved his landmark result that intuitionistic logic is complete with

respect to interpretations into the (complete) Heyting algebras of open sets of topological spaces. In

fact, as Tarski showed, one can restrict attention to all metrisable spaces, or even just the real line or

the Cantor space, without impairing completeness. I prove a version of Tarski’s theorem where the

spaces are restricted to compact polyhedra, and the accompanying (not necessarily complete) Heyting

algebras are restricted to those given by open subpolyhedra. The key property turns out to be

topological dimension, which I show is captured by the bounded-depth axioms. Theorem: The

intermediate logic of the class of all polyhedra of dimension at most is intuitionistic logic extended

by 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 Nick

Bezhanishvili, Dan McNeill, and Andrea Pedrini.)

Marchioni, Enrico:

Ciabattoni, Agata: Proof Search and Co-NP Completeness for Many-Valued Logics

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December 18, Friday

Spada, Luca: An extension of Basic Logic with fixed points

Metcalfe, George: Density Revisited

Hosni, Hykel: Strictly coherent bets on real-valued events

Flaminio, Tommaso: Strictly coherent bets on real-valued events

Aglianò, Paolo: Varieties of BL-algebras

Bernardi, Claudio: Graphs of real functions with pathological behaviors

Sambin, Giovanni: Dynamic foundations of mathematics as space for communication

ABSTRACT: A short mail exchange with Franco Montagna, on communication between different

views in the foundations of mathematics, is taken as a starting point for some considerations. I will

argue that a dynamic view is the best way to foster awareness and hence communication between

different foundational choices. I will illustrate this in practice with some principles and some

examples.