# Workshop da Sociedade Brasileira de Lógica

## Informações importantes

- Local: Auditório do Instituto de Matemática e Estatística, UFBA (Salvador-BA)
- Data: 24 de Outubro de 2017, 14:00h

## Sobre o evento

O Workshop da Sociedade Brasileira de Lógica é uma iniciativa da Sociedade Brasileira de Lógica (SBL), consistindo de palestras e comunicações convidadas, com o objetivo de apresentar um painel da pesquisa em Lógica no Brasil (e em particular no Nordeste), em todas as suas vertentes: Filosofia, Ciência da Computação, Matemática.

Os keynote speakers do Workshop são: Cezar Mortari (UFSC, atual Presidente da SBL), Walter Carnielli (UNICAMP) e Maximo Dickmann (Paris 7).

O Workshop realizar-se-á no Auditório do Instituto de Matemática e Estatística da UFBA, na mesma semana em que ocorrem em Salvador outros dois eventos relacionados à Lógica (e organizados pelo Prof. Abel Lassale Casanave, do Departamento de Filosofia da UFBA):

- o Colóquio Conesul (do Grupo Conesul de Filosofia das Ciências Formais), de 20 a 23 de Outubro, com programação já disponibilizada em http://gcfcf.com.br/pt/coloquios/
- e também o Fourth Meeting APMP (Association for the Philosophy of Mathematical Practice), de 23 a 26 de Outubro, com programação já disponibilizada em http://www.philmathpractice.org/

O Workshop é isento de inscrição. Todos os interessados em participar do Fourth Meeting APMP como ouvintes são solicitados a confirmar sua participação preenchendo a Registration Form e pagando a taxa de inscrição (R$ 150,00) até 30 de setembro. A inscrição no Meeting também permite participação no Colóquio Conesul.

## Programação

Horário | Palestrante | Título |

14:00h-14:40h | Cezar Mortari (UFSC, atual presidente da SBL) | Teaching logic |

14:50h-15:30h | Walter Carnielli (CLE/UNICAMP) | The principle of the Ariadne and the Axiom of choice |

15:35h-16:00h | Marco Cerami (UFBA) | Towards a Pavelka style modal logic |

16:05h-16:30h | Francicleber M. Ferreira (UFC) | Complexity of bounded-degree fixed-points |

16:30h-17:00h | Coffee break | |

17:00h-17:25h | Darllan C. Pinto (UFBA) | Congruential filter pairs and their relation with Leibniz operator |

17:30h-17:55h | Abílio A. Rodrigues Filho (UFMG) | An epistemic approach to paraconsistency: a logic of evidence and truth |

18:00h-18:25h | Samir Gorsky (UFRN) | Logic, epistemology and information |

18:30h-19:10h | Maximo Dickmann (Paris 7) | Model theory and the algebraic theory of quadratic forms: joint research and progress since the 1990's |

## Resumos das palestras

### Teaching logic (Cezar Mortari)

Based on my experience of teaching logic, mostly to undergraduate philosophy students, in this talk I would like to discuss some issues about logic education. My main concerns are what to teach, and when, using the ASL guidelines for logic education as a starting point.

### The principle of the Ariadne and the Axiom of choice (Walter Carnielli)

I intend to survey some results on combinatorial aspects of infinite Ramsey-type problems inspired by finite properties, discussing the relevance of an alternative set-theoretical principle, the so-called “Principle of Ariadne”. This principle, a rival of the Axiom of Choice and connected to the polarized partition relations, can be consistently added to the usual axiomatic stock of ZF set theory under certain conditions. The new axiom, which preserves all the finite contents of mathematics but deviates from the standard in the infinite contents, may help us to understand the finite-infinite divide in mathematics, making clear that there is more than one way to generalize from finite principles of order (or choice) to the infinite.

This is joint work with Carlos di Prisco.

### Towards a Pavelka style modal logic (Marco Cerami)

The existing literature on many-valued modal logic generalizes the classical Tarski style semantic consequence relation to the many-valued case. In this sense, even though a formula can be not only true or false, it can only be or not a consequence of a set of formulas. Another possibility, started in the '70s by J.A. Goguen and J. Pavelka is to consider fuzzy notions of sets of formulas, axioms, theorems and of consequence relation. The idea is that a set of formulas can be seen as its characteristic function both in the crisp and in the fuzzy cases. The same idea can be applied to the set of consequences of a given set of formulas. The study of logic based on degrees of truth has been already addressed for propositional and predicate logics. The present contribution is a proposal to define and systematically study of Modal Logics with degrees of truth. The idea is to extend Pavelka's framework from to the case of many-valued Kripke frame semantics. The aim of our presentation is showing how the notion of many-valued consequence relation, developed for propositional and predicate logics, can be adapted to define a many-valued consequence relation between modal formulas with a semantics based on many-valued Kripke frames.

### Complexity of bounded-degree fixed-points (Francicleber M. Ferreira)

The characterization of the expressive power of logics over finite structures sheds light on the connections between finite model theory and complexity theory, making it possible to transfer results and insights from one side to the other. In particular, questions about the separation of complexity classes reduce to questions regarding the separation of logics with respect to expressive power. Also, the logical characterization of complexity classes, in the spirit of descriptive complexity, provides a machine-independent representation of such classes and provides decidable languages where all problems in a complexity class can be expressed. In this work we investigate the complexity of problems expressible by fixed-point logics where both the fixed-points computed and the structures over which problems are defined have bounded Gaifman degree, which are related to linear time complexity.

### Congruential filter pairs and their relation with Leibniz operator (Darllan C. Pinto)

We consider the special case of filter pairs (G,i) where the functor G = Co_K is given by congruences relative to some quasivariety K , and give criteria when the associated logic is protoalgebraic, equivalential, algebraizable, truth- equational, self-extentional or Lindenbaum algebraizable, by just analyzing the relation between Leibniz operator and the adjoint of i , improving our previous results. Also, we give a way of producing a logic from a quasivariety and a given set of equations, yielding many interesting new logics. We introduce a notion of morphism of filter pairs and show that it encodes translations between their associated logics. Moreover, we show that the category of abstract logics is isomorphic to a full and reflective subcategory of the category of filter pairs.

This is joint work with Peter Arndt, Ramon Jansana and Hugo Luiz Mariano.

### An epistemic approach to paraconsistency: a logic of evidence and truth (Abílio A. Rodrigues Filho)

The aim of this paper is to offer an approach to paraconsistency different from both metaphysical neutrality and dialetheism. If we assume that there are no 'real contradictions', hence, no pair of contradictory true propositions, and yet pairs of propositions A and ¬A are accepted in some contexts of reasoning, we have to explain what it means, if it is not the case that they are both true. Although the topic of epistemic contradictions is not new, and its origins can be traced back at least to Kant, the issue has not yet been properly developed in terms of a paraconsistent non-dialetheist formal system designed to represent contexts of reasoning in which contradictions have a strictly epistemic character. We start from two ideas. First: contradictions are epistemic rather than ontological. Second: logic is not restricted to truth preservation. The acceptance of a pair of contradictory propositions A and ¬A does not commit us to their truth. Rather, we understand it as some kind of 'conflicting information' about A, namely, that there is conflicting evidence for A. 'Evidence for A' is understood in broad terms as reasons for believing that A is true. Evidence is an epistemic notion, weaker than truth in the sense that there may be evidence for a proposition A even in the case A is not true. This paper introduces a natural deduction system designed to express preservation of evidence rather than preservation of truth. The system is paraconsistent and paracomplete, since neither explosion nor excluded middle hold. The inference rules for conjunction, disjunction and implication are obtained in two steps. First, we ask about the sufficient conditions for having evidence that a given proposition is true. Then, we ask what would be sufficient conditions for having evidence that a given proposition is false. The falsity of A is represented by ¬A. We thus obtain rules whose conclusions are disjunctions, conjunctions, conditionals and negations of these formulas. Once we have the introduction rules, the elimination rules are obtained, as suggested by Gentzen, as 'consequences', so to speak, of the introduction rules. Although the system so obtained is able to express the notion of preservation of evidence, and not preservation of truth, by applying the resources of the logics of formal inconsistency, classical logic is recovered within the domain of propositions whose truth value has been conclusively established. Once classical logic is recovered, the system turns out to be able to give also an account of preservation of truth.

This is joint work with Walter Carnielli.

### Logic, epistemology and information (Samir Gorsky)

The objective of this communication is to extend the concept of semantics for the context of modal logic. This work is situated in the field of research on logic and information which, in turn, is a branch of philosophy of information. The references that will be used for this purpose include works of Bar-Hillel, Carnap, D’Agostino, Floridi, Hintikka and the speaker himself.

### Model theory and the algebraic theory of quadratic forms: joint research and progress since the 1990's (Maximo Dickmann)

I will briefly outline the most significant results of joint research carried out since the middle of the 1990's by F. Miraglia, A. Petrovich and myself on the algebraic theory of quadratic forms and its relationship with the model theory of semi-real rings and of preordered fields.