THREE QUESTIONS TO Luis Estrada Gonzalez

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

Like many young logic students committed to the Boole-Frege-Russell logic (BFR henceforth), I first read about violations of the principle of non-contradiction (PNC) in Quine’s Philosophy of Logic when I was an undergrad. Like many young logic students committed to BFR as the best theory of validity, I was convinced by Quine that any attempt at showing counterexamples to the PNC —or many other principles in the vicinity— was born from confusion. It was only after realizing that Subalternation —the argument from “Every S is P” to “Some S is P”— is invalid in BFR that I started my way into other logics, and my estrangement from Quine’s views.
Nonetheless, after some time of exposure, one internalizes BFR and trying to argue against it becomes really difficult. Priest’s writings, in particular Doubt Truth to be a Liar and In Contradiction, were liberating. They helped me to see how an articulate worldview not based on BFR might look like. Then, the paraconsistent attack and, in fact, all previous attacks on BFR, seemed insufficient to me. The development of paraconsistent logic was momentous, sure, but that was only the tip of the iceberg to me. What if logicality has place not only for contradictions, but for everything, and even everything at once? This led me to entertain the thought that, in logic, there is room for the ideas that everything is possible and that everything is true.
Chris Mortensen had already explored the idea that everything is possible. I tried to improve his arguments and expanding them to an explicit consideration of trivialism. Mortensen supported me from the very beginning, and, in a sense, he supervised remotely part of my PhD. He replied very kindly my emails and guided me through some complexities in his work, in particular on paraconsistent category theory. John Lane Bell defended the idea that toposes are mathematical universes and that the laws holding through all of them are the true invariant laws of mathematics. The standard view is that those laws are the intuitionistic ones. Mortensen (with Peter Lavers) showed how to dualize toposes in a way that the laws holding at them are rather paraconsistent. Thus, the invariant laws of mathematics are not intuitionistic, but at most those at the intersection of intuitionistic logic and its dual. But then I identified other logical assumptions, not necessitated by category theory, that were at the core of the idea that intuitionistic logic has a privileged status in mathematics. Doing without those assumptions would weaken even more the logic holding at all mathematical universes. This fitted well with my idea that there are no laws holding in all cases, and with my sympathy towards Béziau’s initial ideas on universal logic and axiomatic emptiness.


Born March 27, 1983 in Durango, Mexico

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

When I got hired by UNAM, I decided to deepen on the idea of axiomatic emptiness in logic and on the philosophical foundations of a universal logic project. I did not plan to work on paraconsistent logic. Nonetheless, soon after that there happened two turning points in my career that took me back to thinking about contradictions and their logic.
The first one was meeting and working with María del Rosario Martínez-Ordaz. For some reason, she stopped doing logic after her BA studies and devoted herself to the philosophy of science. I was not a huge fan of that field, but she got me interested in it; I was lucky enough to be considered by her to supervise her doctoral dissertation. Priest and others identified three kinds of theoretical contradictions: those internal to a theory, those between theories, and those between theory and observations. María was right that the last kind was understudied within the formal approaches, so she decided to tackle them.
However, I was never comfortable with the idea that the paraconsistent tools are a mere palliative for the lack of resources to use BFR, a temporal substitute for it. If I were to work in philosophy of science at all, I would definitely follow the path indicated by Richard Sylvan in Ultralogic as universal? Trying to re-do the whole of our cognitive enterprises from scratch once one has been convinced that a logic different from BFR would be a better theory of validity, at least to do philosophy, maths and science, seems crazy. But one cannot simply pretend that one has not find serious reasons to change things, and if things need be changed radically, let that be. (And I meant that not only for logic.)



Luis with a good bunch of members of the Logician’Liberation League in 2019

What one cannot pretend is that this can be done by a single person. Zach Weber has been doing an amazing job in continuing Sylvan’s ultramodal agenda, but it is difficult to find people who can take the whole project seriously. Again, I have been very lucky. With Fernando Cano-Jorge we have recently taken seriously Sylvan’s idea that going ultramodal could affect our understanding of some limitative results in physics. And Sylvan was right: going ultramodal blocks the reasoning behind the infamous inequality at the core of Bell’s theorem. Perhaps results like this one could help in convincing more people to give Sylvan’s project a chance.
The second turning point was the workshop on connexive logic organized by Heinrich Wansing and Hitoshi Omori in UNILOG 2015. It renewed my interest in connexive logics, which I had approached once from a very reck, paracomplete perspective. Many of the logics they develop are not merely paraconsistent but outrightly inconsistent, that is, they are logics with inconsistent theorems. That fascinated me because it meant to me that a new portion of the logical landscape was there to be explored. Connexive logics, and contra-classical logics in general, quickly lead to inconsistency even with weak logical assumptions, so the interplay with paraconsistency is almost unavoidable there. Most of my time nowadays is devoted to think about inconsistency, other forms of contra-classicality, their interactions and the consequences of all that for our conception of logic. I have a bunch of wonderful students —Elisángela Ramírez-Cámara, Christian Romero-Rodríguez, Sandra D. Cuenca, Miguel Ángel Trejo-Huerta— working on those topics and they are doing really well. In addition, I co-supervise with Marek Nasieniewski the dissertation of Ricardo Nicolás-Francisco, who wants to apply the discussive approach to some inconsistent logics, including some belonging to the connexive brand.

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

As I have said, I do not consider paraconsistent logic a mere temporal tool to be used whilst the conditions to finally use BFR are given. Like any (family of) logic(s), paraconsistent logic suggests its own pictures of logicality, rationality, science, our very lives and the overall surrounding world. I do not know whether those pictures are true, but I do not want to delactose them to prevent offending the good logical souls, I prefer exploring them just at face value.
In answering the first question, I said that understanding that a contradiction does not entail any other proposition was merely the tip of the iceberg. Heinrich Wansing and his collaborators are making a really strong case for inconsistent logics. But we should be ready to give the next step too and learn to live with logics where something is both valid and invalid and other theories at the “meta” levels. The coexistence of several brands of paraconsistency, especially those that tolerate contradictions but prefer to avoid them in the long run and those whose embrace them even at the meta-theoretical level, will be a worth seeing scenario. I am sure that we will discuss these topics during the next World Congress of Paraconsistency to be held in 2024 in Oaxaca, Mexico!
I enjoy logic; I think that it is full of fun and that one can have a wonderful time thinking about it. Nevertheless, the main challenge right now, and not only for paraconsistent logic, is to face the fact that doing logic, like most —if not all— our activities, is both an ethical and a political act. The way I do logic tries to reflect not only that I like it a lot, but also my deep conviction that things, in the most diverse spheres, can be different, actually better, and that we should fight for the difference and the better with all the resources at our hands, and the chance of teaching logic to the youth is one of the most powerful resources. Doing logic in the way I do it tries to reflect my deepest conviction that there are many ways of thinking, reasoning and being in the world, that everything is (logically) possible. Contrary to what the Vienneses said, and no matter how much one tries to escape, the slopes of logic are neither icy nor detached from the duty of doing better in this world of us.