19/11 - Esko Turunen, PhD*

A Paraconsistent Version of Pavelka's Fuzzy Logic

In 1979 Jan Pavelka introduced a very general framework to deal with many valued logics. Pavelka's  idea was to process Zadeh's Fuzzy Sets such that theories, rules of inference, proofs as well as  tautologies may be only partial, i.e. fuzzy. Pavelka defined all his concepts in complete residuated lattices. The main issue was to study the circumstances under which the fuzzy semantic consequence operation and fuzzy syntactic operation coincide; such a property is called Pavelka style completeness.  Pavelka solved the problem in the special case that the set of truth values is the Lukasiewicz sturucure, i.e. the real unit interval equipped with standard MV-structure. The present author has recently proved that Pavelka style completeness holds if, and only if the set of truth values is a complete MV-algebra. Thus, if in particular ,the set M of truth values is a certain collection of 2x2-matrices equipped with suitable operations, then M is a complete MV-algebra. In fact, the set M extends Belnap's four valued para consistent logic. Such an approach results a complete many-valued logic that behaves consistently when looking from outside: the structure in related to Lukasiewicz logic which is a consistent logic. However, looking the logic from inside, i.e. a single 2x2-matrix, para consistency steps in. Truth and falsehood are not opposites of each other, and also contradictions and lack of knowledge is involved.

*Tampere University of Technology (Tampere, Finland)

12/11 - Francesc Esteva* & Lluis Godo*

On paraconsistent fuzzy logics

Paraconsistent logics are specially tailored to deal with inconsistency, while fuzzy logics primarily deal with graded truth and vagueness. Aiming at studying logics that can handle inconsistency and graded truth at once, this talk will report about recent investigations on how a notion of paraconsistent fuzzy logic can be cast within the framework of the so-called logics of formal inconsistency (LFIs).

As in classical logic, it is clear that the notion of truth-preserving deduction commonly used in systems of mathematical fuzzy logic is incompatible with any form of paraconsistency. However, in the first part of the seminar we will show that, instead, some degree-preserving fuzzy logics exhibit interesting paraconsistency features. We will also consider expansions of these logics with additional negation connectives and study their paraconsistency properties as LFIs.

In the second part of the seminar, we will address a kind of converse problem, namely how to extend a given fuzzy logic with a new “consistency” operator in the style of the LFIs. We will introduce a set of postulates for this type of operators over the corresponding algebras, leading to the definition and axiomatization of a family of logics, expansions of MTL, whose degree-preserving counterpart are paraconsistent and moreover LFIs.

 

In the third and final part of the seminar, we will talk about some remarks on ongoing work on the study on intermediate paraconsistent fuzzy logics between the truth-preserving and degree-preserving logics.

 

*IIIA - CSIC, Barcelona, Spain

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.