THREE QUESTIONS TO
THREE QUESTIONS TO
- When and how did you hear about paraconsistent logic and start your work?
Well, I started my graduate studies in "Logic and Philosophy of Science" in Campinas, right? It follows that it would have been impossible NOT to have heard about paraconsistent logic! I must say, though, that the topic did not immediately attract my attention at first — at least no more than any other standard logic-related topic (I was happy to devour everything that fell in my hands), or any other topic related to the abstract theory of consequence, or concerning the non-classical jungle out there. After studying FOL, Axiomatic Set Theory, Computability, and after a first semester following a route from Frege to Kripke, however, I dived into Wittgenstein's philosophy of mathematics and was deeply impressed with the latter's shocking aphorisms on the topics of contradiction, (relative) consistency and incompleteness.
Well, I started my graduate studies in "Logic and Philosophy of Science" in Campinas, right? It follows that it would have been impossible NOT to have heard about paraconsistent logic! I must say, though, that the topic did not immediately attract my attention at first — at least no more than any other standard logic-related topic (I was happy to devour everything that fell in my hands), or any other topic related to the abstract theory of consequence, or concerning the non-classical jungle out there. After studying FOL, Axiomatic Set Theory, Computability, and after a first semester following a route from Frege to Kripke, however, I dived into Wittgenstein's philosophy of mathematics and was deeply impressed with the latter's shocking aphorisms on the topics of contradiction, (relative) consistency and incompleteness.
Next, in my Master's thesis, I ended up investigating a tool called "possible-translations semantics", which allowed, in principle, for a complicated logic to be interpreted in terms of a suitable combination of much less complex logical scenarios. One particular application for this tool, then, was to provide a computational semantics to paraconsistent logics such as the ones constituting the calculi Cₙ, an infinite hierarchy of logics that had been raising all sorts of problems in the literature for about thirty-five years. Such logics, for instance, were known not to be characterizable by finite matrices; they failed intersubstitutivity of provable equivalents; they resisted standard algebraic treatment. It so happened, though, that it was possible to provide a useful semantics to them by way of a crafty combination of straightforward 3-valued matrices. And that's what I did. As a side effect of that research, I found out how to characterize the logic to which the increasingly weaker calculi Cₙ converged, I ended up reinventing several 3-valued paraconsistent logics from the literature and proving a couple of interesting results concerning their expressive power, and I used computational tools to straighten up lots of little issues that had been around for a while, in particular concerning independence results about certain axiomatic theories for paraconsistent logics.
Next, in my Master's thesis, I ended up investigating a tool called "possible-translations semantics", which allowed, in principle, for a complicated logic to be interpreted in terms of a suitable combination of much less complex logical scenarios. One particular application for this tool, then, was to provide a computational semantics to paraconsistent logics such as the ones constituting the calculi Cₙ, an infinite hierarchy of logics that had been raising all sorts of problems in the literature for about thirty-five years. Such logics, for instance, were known not to be characterizable by finite matrices; they failed intersubstitutivity of provable equivalents; they resisted standard algebraic treatment. It so happened, though, that it was possible to provide a useful semantics to them by way of a crafty combination of straightforward 3-valued matrices. And that's what I did. As a side effect of that research, I found out how to characterize the logic to which the increasingly weaker calculi Cₙ converged, I ended up reinventing several 3-valued paraconsistent logics from the literature and proving a couple of interesting results concerning their expressive power, and I used computational tools to straighten up lots of little issues that had been around for a while, in particular concerning independence results about certain axiomatic theories for paraconsistent logics.
Right after that, I happened to get involved in the organization of the 2nd World Congress on Paraconsistency. That was almost twenty-five years ago... You surely remember, Jean-Yves, when we hit the road to look for a beach village in the state of São Paulo in which this event could happen, right next to the mountains filled with lush tropical forest, right? Was that the beginning of your career as a conference organizer? For me it was the opportunity to get in touch —I literally had to exchange thousands of emails at the occasion— with a huge (and quite exotic) community of non-classical logicians. After compulsively processing hundreds of papers and monographs related to the Brazilian School of Logic, I had this idea of using the conference, in Juquehy, to illustrate thousands of ways in which Newton da Costa's problem of designing maximal paraconsistent fragments of Classical Logic could be solved, and Walter Carnielli used the same conference to talk about our joint "futurological" plans (I had Stanisław Lem in mind) concerning the C-systems: that marked roughly the birth of the Logics of Formal Inconsistency (nowadays often referred to more simply as "LFIs"). But that was a matter to be more properly explored in my PhD.
Right after that, I happened to get involved in the organization of the 2nd World Congress on Paraconsistency. That was almost twenty-five years ago... You surely remember, Jean-Yves, when we hit the road to look for a beach village in the state of São Paulo in which this event could happen, right next to the mountains filled with lush tropical forest, right? Was that the beginning of your career as a conference organizer? For me it was the opportunity to get in touch —I literally had to exchange thousands of emails at the occasion— with a huge (and quite exotic) community of non-classical logicians. After compulsively processing hundreds of papers and monographs related to the Brazilian School of Logic, I had this idea of using the conference, in Juquehy, to illustrate thousands of ways in which Newton da Costa's problem of designing maximal paraconsistent fragments of Classical Logic could be solved, and Walter Carnielli used the same conference to talk about our joint "futurological" plans (I had Stanisław Lem in mind) concerning the C-systems: that marked roughly the birth of the Logics of Formal Inconsistency (nowadays often referred to more simply as "LFIs"). But that was a matter to be more properly explored in my PhD.
Born July 18, 1974, Governador Valadares, Brazil
Born July 18, 1974, Governador Valadares, Brazil
2. How did you further develop your work on paraconsistent logic ?
2. How did you further develop your work on paraconsistent logic ?
OK, so here we were in Juquehy, for the conference, where I met lots of people from all over, and not long after that I had the chance of doing a short research stay in Germany. A couple of months later, with the perspective of returning home and not necessarily receiving a grant to continue working my PhD, I contacted Dirk Batens's group to ask if they would be willing to invite me for a short visit. They told me that they had no funds to dedicate to that enterprise, but that I would in fact be welcome if I were to decide to come on a research job, for a year or two, and to continue my PhD there! So there I went. Life in Ghent was fantastic, and I dedicated lots of time —perhaps too much!— in my first couple of months there to work on this massive paper, "A Taxonomy of C-systems", in which I simply tried to make sense of this whole non-classical tradition in which my logical upbringing had been embedded, in Brazil.
OK, so here we were in Juquehy, for the conference, where I met lots of people from all over, and not long after that I had the chance of doing a short research stay in Germany. A couple of months later, with the perspective of returning home and not necessarily receiving a grant to continue working my PhD, I contacted Dirk Batens's group to ask if they would be willing to invite me for a short visit. They told me that they had no funds to dedicate to that enterprise, but that I would in fact be welcome if I were to decide to come on a research job, for a year or two, and to continue my PhD there! So there I went. Life in Ghent was fantastic, and I dedicated lots of time —perhaps too much!— in my first couple of months there to work on this massive paper, "A Taxonomy of C-systems", in which I simply tried to make sense of this whole non-classical tradition in which my logical upbringing had been embedded, in Brazil.
I was particularly interested in creating an abstract logical environment in which several logical principles related to negation could be stated and distinguished from one another, an environment in which negation itself could and should be understood. You need to understand negation if you want to talk about "contradiction", right? Finding out what was common among so many logics purporting to tame contradictions and control trivialization —paraconsistent logics!— was a huge puzzle. This effort would eventually evolve into a lively research line, with quite a few interesting results: I identified minimal conditions for a connective to deserve being called "negation"; I showed that Stanisław Jaśkowski's logic D2, one of the earliest paraconsistent logics, did not really have a "modal" character and had a consistency connective promptly available upon request; I realized that both the LFIs and the adaptive logics developed in Belgium were based on the same kind of "Derivability Adjustment Theorem" (they differed only on the recipe for recovering the lost "consistency assumption"); I showed that any normal modal logic could be seen as an LFI (all of which fully respecting the property of intersubstitutivity of equivalents) and behind this approach lied the idea that a "true contradiction" was some sort of metaphysical accident (things that were actually true but that could possibly be false).
I was particularly interested in creating an abstract logical environment in which several logical principles related to negation could be stated and distinguished from one another, an environment in which negation itself could and should be understood. You need to understand negation if you want to talk about "contradiction", right? Finding out what was common among so many logics purporting to tame contradictions and control trivialization —paraconsistent logics!— was a huge puzzle. This effort would eventually evolve into a lively research line, with quite a few interesting results: I identified minimal conditions for a connective to deserve being called "negation"; I showed that Stanisław Jaśkowski's logic D2, one of the earliest paraconsistent logics, did not really have a "modal" character and had a consistency connective promptly available upon request; I realized that both the LFIs and the adaptive logics developed in Belgium were based on the same kind of "Derivability Adjustment Theorem" (they differed only on the recipe for recovering the lost "consistency assumption"); I showed that any normal modal logic could be seen as an LFI (all of which fully respecting the property of intersubstitutivity of equivalents) and behind this approach lied the idea that a "true contradiction" was some sort of metaphysical accident (things that were actually true but that could possibly be false).
None of this would have been unearthed, anyway, had I not gotten a research grant in Lisbon, to work with Carlos Caleiro and his group — and finally finish my PhD thesis. Sooner or later, anyway, I also came to study the paraconsistent logics arising from David Nelson's approach, I investigated applications of paraconsistent logics to databases, I studied several different kinds of proof systems (tableaux, sequents, resolution) and algebraic characterizations for paraconsistent logics, I designed paraconsistent logics that contained a consistency connective but no definable classical negation… It's been a big adventure. Most important of all, and this is what I am most proud of, this whole project turned out to be really fruitful to pursue, and for people from all over the world! As for me, I keep returning to paraconsistent grounds, every so often. I have traveled quite a bit to lecture on my research. Interestingly, more often than not and unless someone really asks me to, I choose to talk about issues unrelated to paraconsistency. And most of my papers are not on paraconsistent logic at all. But, judging from citation records, it seems paraconsistency still remains rather popular?
None of this would have been unearthed, anyway, had I not gotten a research grant in Lisbon, to work with Carlos Caleiro and his group — and finally finish my PhD thesis. Sooner or later, anyway, I also came to study the paraconsistent logics arising from David Nelson's approach, I investigated applications of paraconsistent logics to databases, I studied several different kinds of proof systems (tableaux, sequents, resolution) and algebraic characterizations for paraconsistent logics, I designed paraconsistent logics that contained a consistency connective but no definable classical negation… It's been a big adventure. Most important of all, and this is what I am most proud of, this whole project turned out to be really fruitful to pursue, and for people from all over the world! As for me, I keep returning to paraconsistent grounds, every so often. I have traveled quite a bit to lecture on my research. Interestingly, more often than not and unless someone really asks me to, I choose to talk about issues unrelated to paraconsistency. And most of my papers are not on paraconsistent logic at all. But, judging from citation records, it seems paraconsistency still remains rather popular?
3. How do you see the evolution and further challenges for paraconsistent logic ?
3. How do you see the evolution and further challenges for paraconsistent logic ?
Non-classical logics are often born out of dissatisfaction, right? The logical paradoxes and the nonchalant behavior of the material implication, for instance, troubled intuitionistic logicians, modal logicians and relevance logicians, to different degrees and with different end results. Other "deviant" logics had even more structural worries, and cared, for instance, about the consumption of resources. Paraconsistent logics gave a great contribution to the literature in directly focusing attention on the behavior of negation. While intuitionistic logicians saw negation as just some sort of abbreviation for an implicative formula, modal logicians pretty much abandoned strict implication and ended up worrying about non-alethic kinds of affirmation, relevance logicians obsessed over reasoning steps given by a single implication, and linear logicians did not even remember to include a connectival operation simulating negation in their mustard watches, we find paraconsistent folks as sort of the first ones to really worry about understanding the fundamental notion of negation, in particular, and the fundamental phenomenon of opposition, in general. No serious comprehensive study of negation, nowadays, can afford to leave paraconsistency aside (yet some still do!).
Non-classical logics are often born out of dissatisfaction, right? The logical paradoxes and the nonchalant behavior of the material implication, for instance, troubled intuitionistic logicians, modal logicians and relevance logicians, to different degrees and with different end results. Other "deviant" logics had even more structural worries, and cared, for instance, about the consumption of resources. Paraconsistent logics gave a great contribution to the literature in directly focusing attention on the behavior of negation. While intuitionistic logicians saw negation as just some sort of abbreviation for an implicative formula, modal logicians pretty much abandoned strict implication and ended up worrying about non-alethic kinds of affirmation, relevance logicians obsessed over reasoning steps given by a single implication, and linear logicians did not even remember to include a connectival operation simulating negation in their mustard watches, we find paraconsistent folks as sort of the first ones to really worry about understanding the fundamental notion of negation, in particular, and the fundamental phenomenon of opposition, in general. No serious comprehensive study of negation, nowadays, can afford to leave paraconsistency aside (yet some still do!).
There is and there will always be a future, of course, in studying well-behaved (sub)classes of (paraconsistent) systems or investigating the properties of particular (paraconsistent) logics. This is one perennial source of challenges. Not exactly my cup of tea, but sometimes it has to be done. Some fifteen years ago, in Melbourne, at the 4th World Congress on Paraconsistency, the stakes were high about the end of that very series of events, as paraconsistent logics were expected to become too commonplace to deserve separate attention. When ideas are internalized into a culture, one often stops talking about them as if they were still in flesh, as if they were still hurting. There seems to be something, though, in paraconsistency that keeps feelings running high. Part of it might be due to dialetheism, with which paraconsistency is still often confused. If, instead of talking about "true contradictions", one were simply to admit that there might be models or situations in which accepting a sentence A does not commit one to rejecting the sentence not-A (dually to what happens on intuitionistic grounds, in which rejecting a sentence A does not always force not-A to be accepted), and in particular situations in which accepting A and not-A does not commit one to accepting all other sentences from a given language, would such kind of self-restraint still generate excited reactions? So, what if a certain connective perfectly qualifies as a negation but does not behave as a contrary-forming operator? Defending the existence of such kinds of models or connectives is a (seemingly quite natural) way of going paraconsistent. Maybe nothing to kill or die for, however.
There is and there will always be a future, of course, in studying well-behaved (sub)classes of (paraconsistent) systems or investigating the properties of particular (paraconsistent) logics. This is one perennial source of challenges. Not exactly my cup of tea, but sometimes it has to be done. Some fifteen years ago, in Melbourne, at the 4th World Congress on Paraconsistency, the stakes were high about the end of that very series of events, as paraconsistent logics were expected to become too commonplace to deserve separate attention. When ideas are internalized into a culture, one often stops talking about them as if they were still in flesh, as if they were still hurting. There seems to be something, though, in paraconsistency that keeps feelings running high. Part of it might be due to dialetheism, with which paraconsistency is still often confused. If, instead of talking about "true contradictions", one were simply to admit that there might be models or situations in which accepting a sentence A does not commit one to rejecting the sentence not-A (dually to what happens on intuitionistic grounds, in which rejecting a sentence A does not always force not-A to be accepted), and in particular situations in which accepting A and not-A does not commit one to accepting all other sentences from a given language, would such kind of self-restraint still generate excited reactions? So, what if a certain connective perfectly qualifies as a negation but does not behave as a contrary-forming operator? Defending the existence of such kinds of models or connectives is a (seemingly quite natural) way of going paraconsistent. Maybe nothing to kill or die for, however.
In my view, the future history of paraconsistency will only benefit from researchers adopting the right kind of attitude. More tolerance and less partisanship is needed, that's for sure. Trying, for instance, to defend that there is a correct philosophical stance to be adopted in doing a certain kind of study —talking about "the philosophy of the C-systems", for instance— may happen to put an unnecessary weight on your product. It becomes a burden, rather than a welcome merchandise. We need to establish bridges and dialogues, rather than be part of a cult. We need to break the resistance of those "who don't know and don't want to know". While, to the best of my knowledge, there is no World Congress on Intuitionism and no World Congress on Relevance, we are, as we speak, on the eve of the 7th World Congress of Paraconsistency, that will soon happen in Oaxaca. I am looking forward to the day in which we will all look backward to evaluate the significance of this event!
In my view, the future history of paraconsistency will only benefit from researchers adopting the right kind of attitude. More tolerance and less partisanship is needed, that's for sure. Trying, for instance, to defend that there is a correct philosophical stance to be adopted in doing a certain kind of study —talking about "the philosophy of the C-systems", for instance— may happen to put an unnecessary weight on your product. It becomes a burden, rather than a welcome merchandise. We need to establish bridges and dialogues, rather than be part of a cult. We need to break the resistance of those "who don't know and don't want to know". While, to the best of my knowledge, there is no World Congress on Intuitionism and no World Congress on Relevance, we are, as we speak, on the eve of the 7th World Congress of Paraconsistency, that will soon happen in Oaxaca. I am looking forward to the day in which we will all look backward to evaluate the significance of this event!
João Marcos, as a child