THREE QUESTIONS TO Jc BEALL

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

I first heard about paraconsistent logic (though not by that name) in Amherst MA largely in some throwaway comments by Gary Hardegree about relevance (-ant) logic, and then after seeing draft material of Hardegree's joint book with Mike Dunn. When I saw Mike’s semantics for FDE (which immediately struck me as absolutely natural and beautiful), the idea of gluts — sentences being both true and false (a perspective not required by relevance but obviously screamingly suggested by Mike’s semantics) — sprung forth and excited me. I had previously thought about the dual of gluts (viz., gaps) via Bas van Fraassen’s work (who influenced me in many ways); however, the idea of gluts jumped out as so utterly natural and an obvious way to deal with standard paradoxes. But when I talked with advisors in Amherst about this, I was basically told to firmly lock that idea away and pursue other things, and so I did, never really to explore it much beyond those initial (though very intensive and extensive) notes that I wrote.
I remember that time as if it were this morning — much in the way one vividly remembers a childhood joy, or a lightening bolt of romance, or perhaps especially the simple key to an otherwise difficult proof.

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

It wasn’t hard to achieve further development since, as per my answer to the first question, I firmly and completely locked away my nascent ideas around gluts (and their tie to logic) for much of my graduate career. I simply moved on — though kept the notes! But then I moved to Tasmania (AUS), and met Greg Restall (Melbourne), who kindly allowed me to stay at his (then) house in Sydney so that I could attend the AAL (Australasian Association of Logic) conference — a memorial conference for Richard Sylvan (whom I’d never heard of before meeting Restall). And suddenly the lockbox, in which “my" original idea of a glut-theoretic solution to paradox was hidden, burst open into a world in which the idea was not only taken seriously but already actively pursued by a variety of exceptionally talented paraconsistentists. I was immediately a kid in a candy shop! Truly. Aside from smatterings of Dunn-style relevance logic (largely the mathematics, and a tad of the philosophy), the literature and growing history of paraconsistency was entirely new to me (e.g., mathematician Asenjo’s taking it to be *obvious* that there are gluts in mathematics, while philosophers Routley/Sylvan & Meyer, Mortensen, and Priest *argued* for gluts), and the literature filled my days and nights — all with the extraordinary cooperation of brilliant folks like Greg Restall, Chris Mortensen, Ross Brady, Dom Hyde, Ed Mares, and so (so) many more, not the least of whom was Graham Priest, in addition to those sympathetic with paraconsistency such as Mark Colyvan and Otavio Bueno. I played with ideas feverishly, learning about the fascinating and groundbreaking ideas of the Brazilian logicians, the work of the Aussies and Kiwis, the Europeans and Asian logicians, the Scots (or, at least, The One True Scot), and so much more. I was — and remain — in a wonderland of fun and profound ideas in and around paraconsistency.
Much (though not all) of my work on paraconsistent logic has been in its applications, and specifically its philosophical applications (e.g., /Logical Pluralism/, /Spandrels of Truth/, and recently /Formal Theories of Truth/). I am convinced that there are a great deal of important and fascinating new applications yet to be explored. At the moment, I am applying the work in ways that I hope would’ve made Bob Meyer happy, namely, to traditional christian theories of God. (My book /The Contradictory Christ/ with Oxford is on the shelves in January 2021, and my “sequel” /The Contradictory God/ is going to press with Oxford in 2021.) It's been a lot of fun and extremely interesting to explore traditional problems in theology from my glut- and gap-theoretic perspective, all framed by the, let me say, FDE perspective on logical consequence — God’s perspective on logical consequence, I should note.

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

I’m not an historian, and so will refrain from historical remarks, but the flurry of work following Polish ingenuity around (many-valued) logic, and subsequently in Brazil and Australia and New Zealand and Canada and Japan and USA and still flourishing and blossoming elsewhere (Buenos Aires, Bochum, and so many other places), shows no signs of slowing. This is very good. Very, very good.

Jc Beall as a child

Strangely, traditional places in the USA like Princeton, where exceptional philosophers like John P. Burgess and Hans Halverson work, are places where paraconsistency is known very (very) well but, alas, also remains a very radical thing — stuff for the heretics, as Burgess might put it. I doubt that Princeton is unique in this way, at least among traditional (and traditionally very strong) universities in the world. (Even Notre Dame, where I work, sees paraconsistency as an important development but wildly radical and last-ditch stuff.)
I think that this is unfortunate. By my lights, part of the problem is the wake of old debates, going back to Quine (at the very latest). Somehow, classical logic (i.e., the so-called classical account of logical consequence) became the mainstream account of logical consequence, and now folks talk of “changing logic” and so on. But classical logic (at the hands of folks involving Frege, Russell and the like) is just the entailment behavior of logical vocabulary *in standard mathematics* (i.e., the account of how logical vocabulary behaves in all true mathematical theories — apologies to Chris Mortensen, Zach Weber and others). But why on earth think that that’s how the logical vocabulary behaves in *all* true theories of reality — not just the mathematical ones, but *all* of them? I see no reason to think as much. But somehow this fairly obvious point is lost to the steady beat of the classical-logic drum.
There’s another factor that feeds the current status of paraconsistent logic (and nonclassical logic generally) as “deviant.” Consider the “controversial” question: Should we accept a nonclassical account of logic? How to answer? Obviously, the only way to answer is to *first* fix on an answer to what *logic* is. But here’s where we’ve let things get mushy. With all good intentions the climate in philosophy of logic is to count everything that remotely resembles a deductive relation as “logic.” And so we find ourselves battling the effects of our own “sophistication.” After all, if every relation that remotely resembles a deductive relation is “logic,” well, the question over whether *logic* is nonclassical will be seen as merely terminological — leaving the mainstream account to persist as the standard.
One major challenge, then, is to shake us back into answering basic questions. What is *logic* when we debate whether logic is nonclassical (or just paraconsistent? paracomplete? whathaveyou?)? Why does it matter? Why does logic, so understood, need to be understood by truth-seeking philosophers (of any speciality)?
For what it’s worth, the direction of answer that interests me is one that should strike us all as familiar. LOGIC, on a very familiar and traditional usage, is involved in *all* true theories. Why? How? Well, logical consequence is the formal entailment (or deductive) relation governing the “logical vocabulary,” which, traditionally, is the sparse topic-neutral vocabulary that makes up standard first-order logical vocabulary (sans identity, which is far from topic-neutral). Since that vocabulary is in all true theories (be they about tractors, cats, gods, chemicals, peanuts, cell phones, whatever), and our job as truth-seeking philosophers is to systematically pursue complete true theories, we all need to know how logical consequence works — and then build our theory-specific consequence relations (governing the theory-specific vocabulary) on top of that. On that (familiar and traditional) perspective, the debate over nonclassical logic is the debate over whether the “universal” consequence relation governing logical vocabulary in all true theories is nonclassical. And that’s a debate worth having. (And I’ll bet that the winning relation is paraconsistent. I just don’t know the odds at the moment.)
There’s a little (OK, a great deal) more to say about all that but, alas, no more room to say it. Such is the pattern of life.