THREE QUESTIONS TO JEAN-YVES BEZIAU

1) When and how did you first hear about paraconsistent logic and start your work?

In the winter 1989 I was relaxing in the house of the father of my girlfriend near Angers in France. He was a psychoanalyst, having done analysis with Jacques Lacan. I saw in the Lacanian Magazine L'âne an interview of Newton da Costa about the inconscious and paracaconsitent logic.
Back to Paris where I was studying mathematical logic and philosophy I went to the library and started to investigate the topic. Since the principle of non-contradiction had been considered as one of the basic laws of thinking and/or reality since antiquity I was curious to see how it was possible to have a logical system disobeying this pirnciple. I then did a master thesis on this, in particular building a sequent system for the paraconsistent logic C1 and proving cut-elimination for it.
Few months later da Costa was visiting Paris, he was enthusiastic about these results and he also asked me why I was interested in paraconsistent logic. Happy with my reply he invited me to work with him in Brazil for one year (1991-1992).

2. How did you further develop your work on paraconsistent logic?

I wanted to investigate the very nature of paraconsistent negation both from a mathematical and philosophical point of views. At the 1st WCP (World Congress on Parconsistent Logic) in 1997 I presented the paper "What is paraconsistent logic?" and at the 2nd WCP the paper "Are paraconsistent negations negations?".
On the other hand I was looking for an intuitive basis for a paraconsistent negation, that's how I discovered that the famous modal logic S5 is a paraconsistent logic. I found a good intuitive definition of a paraconsistent negation: ¬p is false iff p is true from all points of views, objectively (this can be applied to modern physics, I was working with David Bohm at some point) or subjectively (this includes Jaśkowski's motivation for discussive logic - my talk on this topic was presented at the 50th birthday of Jaśkowski's paper in Torun in 1998).
Replying to Slater's paper "Paraconsistent logics?" with a paper entitled "Paraconsistent logic!", I started to work on the connection between paraconsistent negation and the square of opposition. One of the offsprings is my paper "Round squares are no contradictions" that I presented at the 4th WCP that I organized with Mihir Chakraborty in Kolkata in 2014. During this event we organized the contest Picturing contradiction to see if anyone had enough imagination to figure what contradiction is or could be. The result was rather disappointing: bridge to nowhere.

With Newton da Costa and Arthur Buchsbaum at the
2nd World Congress on Paraconsistency in Juquehy, Brazil, 2000

More recently I have been working on some three-valued paraconsistent logics in which neither p,¬p ⊢ q nor ¬(p&¬p) is valid, I call them genuine paraconsistent logics.
On the philosophical side I have been developing a theory of concepts according to which most of the concepts can be considered as paraconsistent (and at the same time paracomplete, i.e. paranormal). This can be expressed by the equation "Cats are not cats". Big cats like Siberian Tigers are indeed cats in some sense but they are not like domestic cats. Facing pussy cats, you also have someone like Felix the cat, a small cat (size does not matter!), who is not a real cat, but a product of imagination.

3) How do you see the evolution of paraconsistent logic? What are the future challenges?

I think one major problem of paraconsistent logic, like in fact with other non-classical logics or logic itself as it is nowadays, is the lack of connection between the philosophical and the mathematical sides. Logic is very interesting exactly because it can link both sides, but the danger is to have something which ultimately is both philosophically and mathematically trivial. You end up with gaps rather than with guts ...
On the mathematical side we have many different logical systems, built using different techniques, used also for the development of other logics: logical matrices, sequent systems, possible worlds semantics, etc. There is not a proper technique that has been created and motivated for/by paraconsistency. And there is a lack of unification, no general theory of paraconsistent logical systems. It would be great to have such a theory.
On the philosophical side we still need good intuitive motivations supporting the idea of paraconsistent negation. It cannot only be a negationist approach, rejecting the ex-contradictione sequitur quodlibet. We need postive ideas and we have to come back to the fascinating question of what is the basic law of thought and/or reality, if any. All the rest is fashionable sophistry, neo-pleonasmatically speaking.