THREE QUESTIONS TO DECIO KRAUSE

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

The first time I heard about PL was in 1975, when Prof. Newton da Costa came to Curitiba for a short course of one week on logic. I was an undergraduate student of mathematics. He mentioned PL only en passant, but it was a surprise to hear that there were logics where contradictions could be dealt with in an adequate way (without trivialization). I read about PL again in 1980, when his book Ensaio sobre os Fundamentos da Lógica (São Paulo, Hucitec) was published. But I was in Curitiba and there no one was specialized in logic, so I started learning by myself because I was interested in the subject. Fortunately, Newton's brother Haroldo was a professor at the Department of Mathematics of the Federal University of Paraná, teaching (mainly) Foundations of Mathematics. I had contact with Haroldo and liked very much his patience and cordiality in teaching basic and interesting things on foundations and related subjects. Then I attended a course with some disciplines in a post-graduate level (there were no post-graduate courses on maths in Curitiba) and one of them was about logic, but given by a philosopher, Alvino Moser, who had made his PhD in Louvain under Jean Ladrière and attended courses in logic with Ladrière and with Joseph Dopp. But although he has emphasized Aristotelian logic, the subject was terrific anyway and helped me in much (he used Dopp's book Formal Logic as the main reference). Later, in 1984, I started to participate of seminars at the Pontifical Catholic University of São Paulo, to where I traveled every week, with courses in logic delivered by Elias Humberto Alves and Lafayette de Moraes. Then I started studying the subject seriously. Newton, who was the mentor of the seminars, was in the US. When he came back, I was already inserted in the group and he invited me to accompany him at the University of São Paulo, were I began to attend his courses and talks. My contact with Newton never ended till today.

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

I am not someone you can call a paraconsistent logician properly speaking. I have done some few works on the subject, despite my interest on the theme. The first work was on Niels Bohr notion of complementarity. Quantum mechanics and its logic and metaphysics is the primary subject of my interest. But reading Bohr and commentators, I have found a way to define complementary propositions in such a way that a paraconsistent logic could be used. It is well known that no one can say that has understood Bohr's concept in a clear way, for he was never clear about it. Anyway, roughly speaking my definition goes as follows: we say that two propositions or sentences A and B are complementary if there exists a sentence C such that A ⊦ C and B ⊦ ¬C but without leaving to a contradiction (of course this cannot be done in a classical setting). For instance (just to illustrate, without careful details), take C to be "the electron is a particle". Let A, again, to be "the electron is a particle", hence (in most logics) A ⊦ C, but let B be "the electron is a wave", hence it is not a particle (in this case, B is equivalent to ¬A in the standard interpretations, but not in general). Consequently, B ⊦ ¬C. The problem is that the conjunction A ∧ B entails a contradiction. But not if the logic is a paraclassical logic, previously defined by da Costa to cope with some details in law in a work he has with the Argentinian philosopher of law Roberto Vernengo (Sobre algunas lógicas paraclásicas y el análisis del razonamiento jurídico), and adapted to the quantum case. Paraclassical logic is non-adjunctive, so not always we can perform the conjunction of two sentences, in the case, of complementary sentences, A ∧ B above. It results that two "complementary" sentences may not trivialize the whole system. So, using this definition and logic, the propositions "it is a particle" and "it is a wave" can be seen as implying a certain negation of one another and the system doesn't become trivial, so (at least in my readings) fulfilling Bohr's claim that we need to consider "The existence of different aspects of the description of a physical system, seemingly incompatible but both needed for a complete de-scription of the system. In particular, the wave-particle duality." In special, I think that this definition copes quite well with Roland Omnès interpretation of quantum mechanics (The Interpretation of Quantum Mechanics), but I never pursued this idea in deep. I have shown my ideas to Newton and we published a joint chapter in 2006 in (The Age of Alternative Logics).
The second work (or the first, in the temporal order) started from an idea of a computer scientist, Celso Kaestner. He came asking for some help in formalizing Wolfgang Bibel's matrix connection method of theorem proving (which was developed for classical logic) using another PL called annotated logic. We developed such an idea in a paper published in the Proceedings of a conference held in Bahia Blanca (1996.) Later our method was implemented by Martin Musicante and Emerson Nobre ("Bibel's Matrix Connection Method in Paraconsistent Logic: General Concepts and Implementation") and, as theoretically expected, we realized that it converges much more quickly than standard resolution methods. A resume of our approach (without the implementation details) is in the Handbook's paper I have with Newton and Otávio Bueno "Paraconsistent logics and paraconsistency", where we revise da Costa's systems and uniformize the notation. Complementarity is also mentioned there: Philosophy of Logic (Handbook of the Philosophy of Science).

Décio with friends in Florianópolis in 2015

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

Here is not the forum to make a history of PL, since by definition I am sure that the readers of these Letters know already about the subject and its history (and stories as well). Evandro Gomes and Itala D'Ottaviano's book (Para além das Colunas de Hércules, uma história da paraconsistência - De Heráclito a Newton da Costa) shows us the history with details and interpretations. The book, published in Portuguese, gained an English version which will appear soon.
But the evolution of PL is something impressive both in variety and in profundity of results. Computation is perhaps the field where relevant applications have being recently found (Towards Paraconsistent Engineering), but I still regard the role of PL in foundational issues as quite important, both in mathematics and in the sciences. In particular in the empirical sciences, we can look for the meaning and of course the existence of contradictions in scientific theories, but firstly we need to understand what a contradiction means in this field. An example may help. Take for instance quantum mechanics. It is usually said that Schrödinger's cat is both alive and dead (before measurement), that a particle can be in two places at once, that the particle passed trough both slits simultaneously (unfortunately I need to suppose certain knowledge of these experiments here), so that the use of some PL would be in need. I don't agree, at least in what concerns the mentioned cases; for instance, a particle has not position, but what we can call "state-positions".
To emphasize, the problem, as said before, is with the concept of negation, something I report quite important to make clear as one of the PL challenges. Firstly, it is agreed by quantum logicians that `quantum negation' is not `classical negation' (look at (Contradiction, Quantum Mechanics, and the Square of Opposition). The best explanation I know about these negations, came from Priest and Routley and was explored by Beziau (“Round squares are no contradictions”) ; it is the reading of paraconsistent negation as subcontrariety in the Square of Opposition. Although these mentioned authors didn't mention quantum negation, we can consider that quantum negation is mapped by contrariety. In fact (in a very simple way), before measurement the cat is in a superposition of states "alive" and "dead", but these are not actual states of the cat, so it definitively is not both alive and dead. Both "to be dead" and "to be alive" are not states of the cat; they are both false, so to say, for the cat is in another state, described by a superposition. In the same vein, subcontraries can be both true in a sense, giving a more intuitive idea (although still vague) of what a paraconsistent negation means. It could be guessed that it would be interesting to define a dual of a superposition involving sub-contrary propositions and to explore this idea. Recently, Kherian Gracher, a former student of mine, has analysed these negations in his PhD thesis at UFSC.
PL, and paraconsistency in general, is a huge subject with potentially infinitely many possibilities. Its link with fuzzyness is something relevant, and (in my readings) can be achieved by means of annotated logics. Perhaps the most interesting application would be to enable intelligent systems to deal with huge data bases where contradictions may be found due to both the quantity of stocked information and the differences of opinion, such as in medicine and law. These logics probably will of course not save someone from jail, but at least will avoid that the system collapses in the presence of a contradiction. The pursue of the logical foundations of scientific theories, so going in the direction of Hilbert's 6th mathematical problem is something still in line, in particular in the axiomatization of theories where (apparently) some form of contradiction might occur, such as Bohr's theory for the atom (although this is also disputable). We cannot anticipate, but the future of these logics is to be great, so that maybe in the future some form of PL turns to be our `classical logic'.