05/11 - Gabriele Pulcini

A Uniform Setting for Classical, Non-Monotonic and Paraconsistent Logic

In this talk, we propose a uniform syntactical framework encompassing classical, non monotonic and paraconsistent logic. Such a uniform framework is obtained by means of the control sets logical device. Control sets leave the underlying syntax unchanged, while affecting the very combinatorial structure of sequents and proofs. Moreover, we prove the cut-elimination theorem for a version of controlled propositional classical logic, i.e. the sequent calculus for classical propositional logic to which a control sets system is applied. Our goals are two-folds: i) to overcome the conceptual gap between classical and non-classical logics; ii) to give, in particular, a new (positive) account of paraconsistency (and non-monotonicity) in terms of concurrency.

22/10 - Rodrigo Freire

Funções de primeira ordem, Parte 2

Esta apresentação é dedicada às funções de primeira ordem, que são uma generalização das funções de verdade. Os conceitos de tabela de verdade e de sistema de funções de verdade, ambos introduzidos na lógica proposicional por Emil Post, são também generalizados e estudados no caso quantificacional. O tema central desta exposição é a relação de definição entre noções expressas por fórmulas da lógica de primeira ordem. Enfatizamos que a lógica não se ocupa apenas da relação de consequência entre noções expressas por fórmulas, em que uma noção é consequência de outras. A lógica também se ocupa da relação de definição entre noções, em que uma noção é definida a partir de outras. Em uma segunda parte, vamos analisar a relação de definição entre noções expressas por fórmulas da lógica de primeira ordem. Nós vemos a lógica de primeira ordem como uma estrutura matemática cujo domínio é o sistema de todas as funções e primeira ordem, munida das operações básicas e da relação de consequência entre funções de primeira ordem. Em particular, os domínios de subestruturas da lógica de primeira ordem são os sistemas de funções de primeira ordem.

01/10 - Rodrigo Freire

Funções de primeira ordem, Parte 1

Esta apresentação é dedicada às funções de primeira ordem, que são uma generalização das funções de verdade. Os conceitos de tabela de verdade e de sistema de funções de verdade, ambos introduzidos na lógica proposicional por Emil Post, são também generalizados e estudados no caso quantificacional. O tema central desta exposição é a relação de definição entre noções expressas por fórmulas da lógica de primeira ordem. Enfatizamos que a lógica não se ocupa apenas da relação de consequência entre noções expressas por fórmulas, em que uma noção é consequência de outras. A lógica também se ocupa da relação de definição entre noções, em que uma noção é definida a partir de outras. Em uma segunda parte, vamos analisar a relação de definição entre noções expressas por fórmulas da lógica de primeira ordem. Nós vemos a lógica de primeira ordem como uma estrutura matemática cujo domínio é o sistema de todas as funções e primeira ordem, munida das operações básicas e da relação de consequência entre funções de primeira ordem. Em particular, os domínios de subestruturas da lógica de primeira ordem são os sistemas de funções de primeira ordem.

24/09 - Mathieu Beirlaen*

Inconsistency-adaptive dialogical logic, or how to dialogue sensibly in the presence of inconsistencies

Even when inconsistencies are present, we can sensibly distinguish between good and bad arguments relying on these premises. Not anything goes: the mere presence of inconsistencies does not warrant the inference to any conclusion whatsoever. In order to separate good and bad inferences in the possible presence of inconsistency, we nowadays have a wide range of paraconsistent logics to our disposal.

Many of these logics, however, lack the inferential power and the dynamics to model how we actually treat information tainted by inconsistency. An exception in this respect is Batens’ inconsistency-adaptive approach, in which all rules of classical logic are applicable to those parts of our premise set which we can safely consider untainted by inconsistency, without having to specify beforehand which parts of our premises behave consistently.

In order to bring this dynamic approach to paraconsistency closer to our actual argumentative practice, we use its machinery to extend the paraconsistent approach to dialogical logic as developed by Rahman and Carnielli. This way, we obtain a very powerful formalism for the systematic study of dialogues in which two parties exchange arguments over a central claim, in the possible presence of inconsistent information.

* Instituto de Investigaciones Filosóficas (IIF) - Universidad National Autonoma de México (UNAM)

Joint work with Matthieu Fontaine (IIF-UNAM)

Raymundo Morado's Talks at GTAL-CLE Seminars

In 1800, Kant famously declared exhausted our research into logic. According to him, “we do not require any new discoveries in Logic” (“wir brauchen auch zur Logik keine neuen Erfindungen”). Mankind had found practically all there was to find about inference and validity. Turned out the news of the end of logic were greatly exaggerated. So, are there any limits to logic? Certainly, we do very little syllogistic logic anymore, and we expect no big surprises from classical propositional calculus. Maybe logic ended in 1879, with the publication of the first complete system of first-order quantificational logic; maybe in 1932 with the normal systems of strict implication. Yet, logic keeps expanding both its depth and its breadth. We have discovered truths about the logical connectives that intrigued the stoics, and we have expanded the power of classical logic with amazing conservative extensions. Still, some developments seem to challenge our very notion of logicality. Kant was talking from a certain perspective of what logic is that excluded from the set go many recent developments. If logic is the science of necessary inference, non-deductive forms of reasoning must fall outside its realm; mathematical induction is in, induction in zoology is out. If logic is the science of abstract concepts, there can be a logical theory of classical quantifiers, but not of all fallacies. Each idea of logic sets its limits. Limits can be good, as Kant’s dove attests. But there can be also good reasons to evolve our concepts out of the old limits and to allow them to encompass new or unsuspected facets of reality.

September 10

I propose to see if we can find a principled extension of our ideas of logic when confronted with non-classical systems, especially with rival logics. I believe there are important lessons for the philosophy of logic to be gleaned from the examination of logics such as the intuitionistic, free, or quantum ones.

September 17

I will illustrate this with the case of the paraconsistent logics of relevance which are of great importance both theoretical and practical.

October 8

Then we shall examine some of the general problems of constraining excessively our notion of logicality and illustrate this with a discussion of the family of non-monotonic formalisms. This will lead us to consider some formal questions that can help us hone pertinent concepts.

October 15

And this in turn will be useful to tackle the ultimate limit: the general issue of what justifies logic itself. We shall finally mention some open problems in this area of the philosophy of logic(s). These are mostly fundamental topics, and we shall only require the minimal symbolic apparatus of a first semester in logic.

28/05 - Tony Marmo

Comparison of Logics: Some Issues and Perspectives

Throughout the recent history of logic, many logic systems have been proposed in accordance with their proponents’ philosophical standpoints. Additionally, the comparative endeavours require that one firstly defines the sense in which one system contains the other (specially when logics of different valences are at stake). In this talk we shall present some comparative methods available in the literature and, inasmuch as possible, some pertinent issues. We shall briefly try to show how philosophical arguments/objections reflect in different results, perhaps yielding unexpected results.

One of such issues will be Suszko’s claim against many-valuedness and his reduction method. Time permitting; we shall try to present Gehrke and Walker theorem, a proven result that goes in the opposite direction of Suszko’s arguments.

14/05 - Peter Verdée

(Paraconsistent) adaptive logics: a logico-philosophical introduction

In this talk I will introduce adaptive logics as models for rational defeasible reasoning. First I will explain what defeasible reasoning is and why is useful to distinguish rational from irrational defeasible reasoning patterns. I will illustrate that there exist very different forms of defeasible reasoning (induction, abduction, vagueness, inconsistency handling, belief merging, etc.) but that they nevertheless have some formal aspects in common.

Next, I will introduce the Standard Format of Adaptive Logic (SF). I will give a short introduction to the semantics and proof theory of (SF) and will give some examples of adaptive logics within the format of the SF, with special attention for paraconsistent adaptive logics. I will argue why adaptive logics defined within SF are good unifying formalisations of many aspects of defeasible reasoning.

Finally, I will discuss some issues concerning the (computational) complexity of adaptive logics.