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## Publications

Publications (154)

We discuss the history of the revival of the theory of opposition, with its emerging paradigms of research, and the related events that are organized in this perspective, including the latest one in Leuven in 2022.

In this chapter, we explain why Ex contradictione sequitur quodlibet is a confusing expression to denote the statement \(p, \neg p \vdash q\), and we also explain why this statement is ambiguous. We start by setting out a framework about consequence relation and truth. We proceed by presenting the basic concepts of the theory of opposition and the...

In this paper we explain the different meanings of the word “logic” and the circumstances in which it makes sense to use its singular or plural form. We discuss the multiplicity of logical systems and the possibility of developing a unifying theory about them, not itself a logical system. We undertake some comparisons with other sciences, such as b...

We examine in which sense logic can be considered as exceptional. We start by emphasizing the difference between Logic as reasoning and logic as the science of reasoning, an essential distinction to launch the discussion. We then investigate if reasoning itself can be seen as exceptional, in particular an exceptional feature of human beings, and ne...

The theory of the square of opposition has been studied for over 2,000 years and has seen a resurgence in new theories and research since the second half of the twentieth century. This volume collects papers presented at the Sixth World Congress on the Square of Opposition, held in Crete in 2018, developing an interdisciplinary exploration of the t...

We present many figures of opposition (triangles and hexagons) for simple and double turnstiles. We start with one-sided turnstiles, corresponding to sets of tautologies, and then we go to double-sided turnstiles corresponding to consequence relations. In both cases, we consider proof-theoretic (with the simple turnstile) and model-theoretic (with...

We first explain the origin and development of the theory of opposition, its generalization to many concepts, and figures of opposition, particularly the hexagon of opposition. We also survey the organization of a series of events on the topic since 2007 in Montreux. We then talk in details about the sixth edition of the world congress on the squar...

We discuss the evolution of the World Logic Prizes Contest. In a first section, we describe how this contest developed, on the on hand by the creation of the Universal Logic Prize starting in 2005 at the 1st World Congress and School on Universal Logic in Montreux, Switzerland, on the other hand by the creation of the Newton da Costa Logic Prize in...

We first start by clarifying what axiomatizing everything can mean. We then study a famous case of axiomatization, the axiomatization of natural numbers, where two different aspects of axiomatization show up, the model-theoretical one and the proof-theoretical one. After that we discuss a case of axiomatization in a sense opposed to the one of arit...

In this paper we discuss the nature of artificial intelligence (AI) and present a hexagon of opposition (generalization of the square of opposition) to characterize what intelligence is, its relation with computability, creativity, understanding and undecidability. In a first part, we make some general comments about the history, development and ob...

The importance of silence has been emphasized in both ancient and modern traditions (Teschner, 1981; Davies and Turner, 2002; Stratton, 2015). In Eastern traditions, silence has been linked to the inner stillness of the mind, a sense of equanimity and unity (Feuerstein, 1996; Lin et al., 2008). At the same time, Western scholars such as Kierkegaard...

We discuss a theory presented in a posthumous paper by Alfred Tarski entitled “What are logical notions?”. Although the theory of these logical notions is something outside of the main stream of logic, not presented in logic textbooks, it is a very interesting theory and can easily be understood by anybody, especially studying the simplest case of...

In this paper we examine what is the relation between God and Paraconsistency. We first start by some general considerations about paraconsistent negations and paraconsistent things. We then examine in which sense the “thing” called “God” is paraconsistent or not. Finally, we look the other way round and discuss the divinity of paraconsistency.

We start by presenting various ways to define and to talk about many-valued logic(s). We make the distinction between on the one hand the class of many-valued logics and on the other hand what we call “many-valuedness”: the meta-theory of many-valued logics and the related meta-theoretical framework that is useful for the study of any logical syste...

Schopenhauer used the word “metalogical” since his first work, On the Fourfold Root of the Principle of Sufficient Reason (1813), being the first to give it a precise meaning and a proper place within a philosophical system. One century later the word “Metalogic” started to be used and promoted in modern logic by the Russian logician Nicolai Vasili...

This special issue is related to the 6th World Congress on the Square of Opposition which took place at the Orthodox Academy of Crete in November 2018. In this introductory paper we explain the context of the event and the topics discussed.

We study the relation between classical propositional logic (CPL) as it is nowadays and how it appears in the Tractatus focusing on a specific feature expressed in the paragraph 5.141. In a first part we make some general considerations about CPL, pointing out that CPL is difficult to characterize and define, that there is no definite final version...

La nature du rire et ses différents aspects sont examinés. On commence par noter que le rire est un phénomène bien connu, cela ne veut pas dire grand-chose, mais c’est un point de départ. Ensuite on essaye de voir comment le classifier. On explique que le rire se présente comme un phénomène caractéristique de l’être humain en rapport notamment avec...

We assess the celebration of the 1st World Logic Day which recently took place all over the world. We then answer the question Why a World Logic Day? in two steps. First we explain why promoting logic, emphasizing its fundamental importance and its relations with many other fields. Secondly we examine the sense of a one-day celebration: how this ca...

I discuss the origin and development of logic prizes around the world. In a first section I describe how I started this project by creating the Newton da Costa Logic Prize in Brazil in 2014. In a second section I explain how this idea was extended into the world through the manifesto A Logic Prize in Every Country! and how was organized the Logic P...

This paper presents the special issue on Formal Approaches to the Ontological Argument and briefly introduces the ontological argument from the standpoint of logic and philosophy of religion (more specifically the debate on the rationality of theistic belief).

We discuss the origin and development of the universal logic project. We describe in particular the structure of UNILOG, a series of events created for promoting the universal logic project, with a school, a congress, a secret speaker and a contest. We explain how the contest has evolved into a session of logic prizes.

According to Boole it is possible to deduce the principle of contradiction from what he calls the fundamental law of thought and expresses as \(x^{2}=x\). We examine in which framework this makes sense and up to which point it depends on notation. This leads us to make various comments on the history and philosophy of modern logic.

I discuss various aspects of the Lvov-Warsaw School: its past, present and future; its location, evolution, mathematics; the variety of its members. I develop this analysis on the basis of my 25-year experience with Poland.

In a first part we discuss the different ways to go beyond dichotomies, using trichotomies and hexagons of opposition. In a second part we show how to produce a hexagon with analogy. In a third part we investigate the meaning given to analogy and related notions with this hexagon presenting some examples.

Colors can be understood in a logical way through the theory of opposition. This approach was recently developed by Dany Jaspers, giving a new and fresh approach to the theory of colors, in particular with a hexagon of colors close to Goethe’s intuitions. On the other hand colors can also be used at a metalogical level to understand and characteriz...

In this paper we examine up to which point Modern logic can be qualified as non-Aristotelian. After clarifying the difference between logic as reasoning and logic as a theory of reasoning, we compare syllogistic with propositional and first-order logic. We touch the question of formal validity, variable and mathematization and we point out that Gen...

Modern science has qualified human beings as homo sapiens. Is there a serious scientific theory backing this nomenclature? And can we proclaim ourselves as wise (sapiens)? The classical rational animals characterization has apparently the same syntactic form (a qualificative applied to a substantive) but it is not working exactly in the same way. M...

A proof is presented showing that there is no paraconsistent logics with a standard implication (or even semi-implication) which have a three-valued characteristic matrix, and in which the replacement principle holds.

We first describe how after having started in Montreux, Switzerland in 2007, the congress on the square of opposition moved to the American University of Beirut in Lebanon in 2012 after a stop at the University Pasquale Paoli in Corsica in 2010. We then describe the square congress at the Pontifical Lateran University in the Vatican in 2014 and the...

The theory of opposition has been famously crystallized in a square. One of the most common generalizations of the square is a cube of opposition. We show here that there is no cube such that each of its faces is a square of opposition. We discuss the question of generalization and present two other generalizations of the theory of opposition to th...

Open image in new windowOpen image in new window We discuss the many aspects and qualities of the number one: the different ways it can be represented, the different things it may represent. We discuss the ordinal and cardinal natures of the one, its algebraic behaviour as a neutral element and finally its role as a truth-value in logic.

We first start by describing the happening of the 1st World Congress on Logic and Religion. We then explain the motivation for developing the interaction between logic and religion. In a third part we discuss some papers presented at this event published in the present special issue.

This is a collection of new investigations and discoveries on the theory of opposition (square, hexagon, octagon, polyhedra of opposition) by the best specialists from all over the world.
The papers range from historical considerations to new mathematical developments of the theory of opposition including applications to theology, theory of argumen...

In this paper we present two genuine three-valued paraconsistent logics, i.e. logics obeying neither \(p, \lnot p \vdash q\) nor \(\vdash \lnot (p \wedge \lnot p)\). We study their basic properties and their relations with other paraconsistent logics, in particular da Costa’s paraconsistent logics C1 and its extension \(C1+\).

Contradiction is often confused with contrariety. We propose to disentangle contrariety from contradiction using the hexagon of opposition, providing a clear and distinct characterization of three notions: contrariety, contradiction, incompatibility. At the same time, this hexagonal structure describes and explains the relations between them.

After explaining the interdisciplinary aspect of the series of events organized around the square of opposition since 2007, we discuss papers related to the 4th World Congress on the Square of Opposition which was organized in the Vatican at the Pontifical Lateran University in 2014. We distinguish three categories of work: those dealing with the e...

http://dx.doi.org/10.5007/1808-1711.2016v20n1p99
In this paper we criticize the way possibility is characterized in contemporary modal logic through the diamond operator. We explain that it does not match with the usual notion of possibility and that this notion is better described by the vertex Y of the hexagon of opposition usually called contin...

In this paper we explain that the paraconsistent logic LP (Logic of Paradox) promoted by Graham Priest can only be supported by trivial dialetheists, i.e., those who believe that all sentences are dialetheias.

After describing my family background and interest in mathematics and philosophy at school, I explain how I became interested in logic when studying at university in Paris. I describe how I discovered the work of Newton da Costa on paraconsistent logic, how I met him in Paris, and then go to do research in Brazil, Poland and USA before defending my...

We investigate the notion of contradiction taking as a central point the idea of a round square. After discussing the question of images of contradiction, related to the contest Picturing Contradiction, we explain why from the point of view of the theory of opposition, a round square is not a contradiction. We then draw a parallel between different...

After describing the two formulations of the principle of non contradiction in modern logic \(T \vdash \lnot (p \wedge \lnot p)\) (NC) and \(T, p, \lnot p \vdash q\) (EC) and explaining that three-valued matrices can be used to easily prove their independence, we investigate the possibilities to construct strong paraconsistent negations, i.e., for...

The present book discusses all aspects of paraconsistent logic, including the latest findings, and its various systems. It includes papers by leading international researchers, which address the subject in many different ways: development of abstract paraconsistent systems and new theorems about them; studies of the connections between these system...

After recalling the distinction between logic as reasoning and logic as theory of reasoning, we first examine the question of relativity of logic arguing that the theory of reasoning as any other science is relative. In a second part we discuss the emergence of universal logic as a general theory of logical systems, making comparison with universal...

The compound word " truth-value", sometimes written " truth value", is a bit monstrous and ambiguous. It is the name of a central concept of modern logic, but has not yet invaded everyday language. An ordinary man will say: it is true that Paris is the capital of France, rather than: the truth-value of " Paris is the capital of France" is true. And...

Ouvrage de l'équipe Academos, liée aux Archives Poincaré LPHS-Archives Poincaré, UMR 7117

Quasi-set theory is a ZFU-like axiomatic set theory, which deals with two kinds of ur-elements: M-atoms, objects like the atoms of ZFU, and m-atoms, items for which the usual identity relation is not defined. One of the motivations to advance such a ...

The hexagon of opposition is an improvement of the square of opposition due to Robert Blanché. After a short presentation of the square and its various interpretations, we discuss two important problems related with the square: the problem of the I-corner and the problem of the O-corner. The meaning of the notion described by the I-corner does not...

In this paper I relate the story about the new rising of the square of opposition: how I got in touch with it and started to develop new ideas and organizing world congresses on the topic with subsequent publications. My first contact with the square was in connection with Slater's criticisms of paraconsistent logic. Then by looking for an intuitiv...

In the paper [“Paraconsistency, paracompleteness and valuations”, Logique Anal., Nouv. Sér. 27, 119–131 (1984; Zbl 0549.03023)], A. Loparić and N. C. A. da Costa use a general theory of bivaluations to develop a paranormal logic, i.e. a logic which is both paraconsistent and paracomplete. We explain how this theory works, in which sense it allows t...

One of the manuscripts of Buridan’s Summulae contains three figures, each in the form of an octagon. At each node of each octagon there are nine propositions. Buridan uses the figures to illustrate his doctrine of the syllogism, revising Aristotle’s theory of the modal syllogism and adding theories of syllogisms with propositions containing oblique...

In this paper we address some central problems of combination of logics through the study of a very simple but highly informative
case, the combination of the logics of disjunction and conjunction. At first it seems that it would be very easy to combine
such logics, but the following problem arises: if we combine these logics in a straightforward w...

In the first part of this work, the concepts of model and theory were analysed and discussed from an abstract standpoint, clarifying relations between both. This second part addresses the use of the terms “model” and “theory” and the associated concepts in scientific disciplines. Particular attention is given to the establishment of a relation betw...

In this paper several systems of modal logic based on four-valued matrices are presented. We start with pure modal logics, i.e. modal logics with modal operators as the only operators, using the Polish framework of structural consequence relation. We show that with a four-valued matrix we can define modal operators which have the same behavior as i...

In this paper we discuss in which sense truth is considered as a mathematical object in propositional logic. After clarifying how this concept is used in classical logic, through the notions of truth-table, truth-function and bivaluation, we examine some gen-eralizations of it in non-classical logics: many-valued matrix semantics with three and fou...

In this paper the concepts of model and theory are analyzed and discussed. We start with an informal approach, trying to give general definitions of what is a model and what is a theory, making a clear distinction between these two notions. This relation between model and theory is then understood in the light of modern logic, in particular model t...

In this paper we discuss the distinction between sentence and proposition from the perspective of identity. After criticizing
Quine, we discuss how objects of logical languages are constructed, explaining what is Kleene’s congruence—used by Bourbaki
with his square—and Paul Halmos’s view about the difference between formulas and objects of the fact...

This is a survey of the main contributions of the author to the field of paraconsistent logic. After a brief introduction explaining how the author entered this field, Section 2 describes his work on C-systems: reformula- tion of the semantics of C1, creation of a sequent systems for C1, proof of cut-elimintaion for this system, extension of C1 int...

We present a paraconsistent logic, called Z, based on an intuitive possible worlds semantics, in which the replacement theorem holds. We show how to axiomatize this logic and prove the completeness theorem. 1. Jaśkowski's problem and paraconsistent logic It seems that nowadays the main open problem in the field of paraconsistent logic is still : Do...

Many-valued and Kripke semantics are generalizations of classical semantics in two different "opposite" ways. Many-valued semantics keep the idea of homomorphisms between the structure of the language and an algebra of truth-functions, but the domain of the algebra may have more than two values. Kripke semantics keep only two values but a relation...

2000 Mathematics Subject Classification. 03B22 Universal logic is not a new logic, but a general theory of logics, considered as mathematical structures, much in the same way that universal algebra is a general theory of algebraic structures. The idea to consider logics in such an abstract way goes back to the work of Tarski on consequence operator...

The expression “universal logic” prompts a number of misunderstandings pressing up against to the confusion prevailing nowadays around the very notion of logic. In order to clear up such equivocations, I prepared a series of questions to Jean-Yves Béziau, who has been working for many years on his project of universal logic, recently in the Univers...

In this paper we study paraconsistent negation as a modal operator, considering the fact that the classical negation of necessity has a paraconsistent behavior. We examine this operator on the one hand in the modal logic S5 and on the other hand in some new four-valued modal logics.

In this paper we address some central problems of combina- tion of logics through the study of a very simple but highly informative case, the combination of the logics of disjunction and conjunction. At first it seems that it would be very easy to combine such logics, but the following problem arises: if we combine these logics in a straightforward...

In this paper, we examine some intuitive motivations to develop a para-consistent logic. These motivations are formally developed using semantic ideas, and we employ, in particular, bivaluations and truth-tables to characterise this logic. After discussing these ideas, we examine some applications of paraconsistent logic to various domains. With th...

We present an overview of the different frameworks and structures that have been proposed during the last century in order
to develop a general theory of logics. This includes Tarski's consequence operator, logical matrices, Hertz's Satzsysteme,
Gentzen's sequent calculus, Suszko's abstract logic, algebraic logic, da Costa's theory of valuation and...

We show by way of example how one can provide in a lot of cases simple modular semantics of rules of inference, so that the semantics of a system is obtained by joining the semantics of its rules in the most straightforward way. Our main tool for this task is the use of finite Nmatrices, which are multi-valued structures in which the value assigned...

This paper builds on the theory of institutions, a version of abstract model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representin...

When can we say that two distinct logical systems are, neverthe- less, essentially the “same”? In this paper we discuss the notion of “sameness” between logical systems, bearing in mind the expressive power of their associated spaces of theories, but without neglecting their syntactical dimension. Departing from a categorial analysis of the questio...

This article sets forth a detailed theoretical proposal of how the truth of ordinary empirical statements, often atomic in form, is computed. The method of computation draws on psychological concepts such as those of associative networks and spreading activation, rather that the concepts of philosophical or logical theories of truth. Axioms for a r...

This paper is an attempt to clear some philosophical questions about the nature of logic by setting up a mathematical framework. The no- tion of congruence in logic is defined. A logical structure in which there is no non-trivial congruence relation, like some paraconsistent logics, is called simple. The relations between simplicity, the replacemen...

It has been pointed out that there is no primitive name in natural and formal languages for one corner of the famous square of oppositions. We have all, some and no, but no primitive name for not all. It is true also in the modal version of the square, we have necessary, possible and impossible, but no primitive name for not necessary. I shed here...

Many-valued logics are standardly defined by logical matrices. They are truth-functional. In this paper non truth-functional many-valued se- mantics are presented, in a philosophical and mathematical perspective.

To know if paraconsistent negations are negations is a fundamental issue: if they are not, paraconsistent logic does not properly exist. In a first part we present a philosophical discussion about the existence of paraconsistent logic and the surrounding confusion about the emergence of possible paraconsistent negations. In a second part we have a...

We present and discuss the fact that the well-known modal logic S5 and classical first-order logic are paraconsistent logics.

We present the logic K/2 which is a logic with classical implication and only the left part of classical negation. We show that it is possible to define a classical negation into K/2 and that the classical proposition logic K can be translated into this apparently weaker logic. We use concepts from model-theory in order to characterized rigorously...

A sequent calculus S3 for Lukasiewicz's logic L3 is presented. The completeness theorem is proved relatively to a bivalent semantics equivalent to the non truth- functional bivalent semantics for L3 proposed by Suszko in 1975. A distinguishing property of the approach proposed here is that we are keeping the format of the classical sequent calculus...