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

My first encounter with paraconsistent logic took place during the Memorial Symposium Parainconsistent Logic, Logical Philosophy, Informatics and Mathematics on the occasion of the 50th anniversary of Stanisław Jaśkowski's seminal paper. The symposium was held at Nicolaus Copernicus University in Torun (NCU), starting from July 15th till July 18th 1998. The organizer of this symposium was the Department of Logic NCU. Next, as one of two editors of Logic and Logical Philosophy, I prepared three volumes 7-9, "Parainconsistency. Part I, II and III" (1999-2001), containing materials from this conference. The majority of papers in these three volumes concerned paraconsistent logic. At that time, I had an opportunity to get acquainted with some interesting ideas due to a paper by Jean-Yves Beziau, later on published in our journal Logic and Logical Philosophy.

Born July 21, 1954 in Tomaszów Mazowiecki, Poland

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

All my published works on paraconsistent logic were written together with Marek Nasieniewski. Basically, these papers are devoted to the question of characterising some modal logics weaker than S5 that are still suitable for the definition of Jaśkowski's discussive logic or Jaśkowski's discussive consequence. Jaśkowski's aim was to find a logic which could be treated as a basis of an inconsistent system but which would not entail overcompleteness, in general. To obtain such a deductive system that would be able to (i) cope with inconsistencies but (ii) prevent the entailment of an overfilled set of conclusions, Jaśkowski introduced the idea of a model of discussion.
The idea is that during a discussion some inconsistent voices can appear, but participants of the discussion as well as external observers are not inclined to deduce every sentence from statements expressed in the discussion. So, in this model two types of point of view are considered: the internal ones of particular participants and the external ones of observers. To express his logic, Jaśkowski used the normal logic S5. However, in already 70's there were given other normal modal logics (T. Furmanowski and J.Perzanowski) weaker than S5, but still reach enough to express the very same discussive logic.
Continuing this work we found even weaker respective logics and finally, we developed a general method that for a given class of modal logics fulfilling some initial conditions rise the smallest modal logic defining D2. In Jaśkowski paper in fact we have also a formulation of consequence relation based on S5. And also in this case we can consider a similar question concerning other modal logics expressing the very same relation. While formulating his logic, Jaśkowski used a discussive implication that was meant to express an "internal" interactions taking place between participants of the discussion. Intuitively, it is meant as saying: if anyone states that p, then q, that is read in terms of modal language: if it is possible that p, then q. In his short second paper on issue of discussive logic, Jaśkowski added to the language discussive conjunction (in the first paper he used classical conjunction) that can be meant as a summary made be a given participant and after translating into modal language can be read: p and it is possible that q. But one could consider other translations. So, we also worked on similar tasks with reference to some variants of D2. Recently, together with Krystyna Mruczek-Nasieniewska and Marek Nasieniewski, we started to work on modal extensions of discussive logic and this work is in progress.
Let me add that this year we will have the 70th birthday of the first Jaśkowski's paper on the discussive logic. We organise at this occasion a conference that will be held in Torun, September 24-27, 2018.

Jaśkowski's 50 Memorial Symposium, Torun 1998

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

I think that paraconsistent logic is a very important tool for the formal examination of various aspects of people's statements made in "everyday life". Certainly, it can also be useful in the analysis of some philosophical issues. However, what seems to be the main challenge is the widespread use of paraconsistent logic. There are known applications of specific paraconsistent logics similar to those of fuzzy logic, but it seems that the future of the paraconsistent logic lies in even more widespread applications in decision systems, interface software, control systems (for example quality control system, machine control system), artificial intelligence, etc.

1. "New axiomatizations of the weakest regular modal logic defining Jaśkowski's logic D2", Bulletin of the Section of Logic, vol. 38, no. 1-2 (2009), pp. 45-50.
2. "Semantics for regular logics connected with Jaśkowski's D2", Bulletin of the Section of Logic, vol. 38, no. 3/4 (2009), pp. 173-188.
3. "A method of generating modal logics defining Jaśkowski's discussive logic D2", Studia Logica, vol. 97, no. 1 (2011), pp. 161-182 (special issue "The Legacy of Newton da Costa").
4. "On the weakest modal logics defining Jaśkowski's logic D2 and the D2-consequence", Bulletin of the Section of Logic, vol. 41, no. 3/4 (2012), pp. 215-232.
5. "On modal logics defining Jaśkowski's D2-consequence", pages 141-160 in K. Tanaka, F. Berto, E. Mares and F. Paoli (eds.), Paraconsistency: Logic and Applications, vol. 26 of series "Logic, Epistemology and the Unity of Science", Springer 2013 (2012).
6. "A method of generating modal logics defining Jaskowski's discussive D2-consequence", pages 95-123 in E. Weber, D. Wouters and J. Meheus (eds.), Logic, Reasoning, and Rationality, vol. 5 of series "Logic, Argumentation, and Reasoning", Springer 2014.
7. "Modal logics connected to Jaśkowski's logic D2", in J-Y. Béziau, A. Buchsbaum and A. Altair (eds.), Handbook of the 5th World Congress on Paraconsistency, Indian Statistical Institute, Kolkata 2014.
8. "On modal logics defining a Jaśkowski-like discussive logic", in J-Y. Béziau, A. Buchsbaum and A. Altair (eds.), Handbook of the 5th World Congress on Paraconsistency, Indian Statistical Institute, Kolkata 2014.
9. "Axiomatisations of minimal modal logics defining Jaśkowski-like discussive logics", pages 149-163 in A. Indrzejczak, J. Kaczmarek and M. Zawidzki (eds.), "Trends in Logic XIII", Wydawnictwo Uniwersytetu Lódzkiego, Lódz 2014.
10. "On modal logics defining Jaśkowski-like discussive logics", pages 213-228 in J.-Y. Beziau, M. Chakraborty and S. Dutta (eds.), New Directions in Paraconsistent Logic, vol. 152 of series "Springer Proceedings in Mathematics & Statistics", Springer Indie, 2015.
11. "Modal logics defining Jaśkowski's and Jaśkowski-like discussive logics", pages 100-104 in V.I. Markin (ed.), Desatye Smirnovskie ctenia po logike: Materialy mezdunarodnoj naucnoj konferencii, Moskva 2017.