THREE QUESTIONS TO J. MICHAEL DUNN


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

I am a kind of accidental paraconsistent logician. I chose to major in philosophy at Oberlin after taking a course in logic. I became interested in the various set theoretical paradoxes such as the Russell Paradox. But I was more interested in how to avoid them rather than to live with them. (It never even occurred to me that this last was an option.) I explored various formalizations of set theory: ZF, MK, NF, ML. etc. Then I went to the University of Pittsburgh for graduate study, and was introduced to relevance logic by Nuel Belnap (who eventually was my Ph.D. advisor). Nuel’s own Ph.D. supervisor was Alan Ross Anderson at Yale. In my last year at “Pitt” Alan moved there and served on my dissertation committee. So I have the two patriarchs of relevance logic in my academic ancestry.
I first became interested in contradictions not implying everything because of the requirement on a propositional relevance logic that if a sentence A implies a sentence B, then they must share a propositional variable. (p & ~p) -> q obviously fails that requirement. I cannot remember when I first heard of “paraconsistent logics” per se. I think I first came across the idea of paraconsistent logics from the work of Newton da Costa. Florencio Asenjo was teaching in the math department at Pitt. I took three courses from him, but he modestly never mentioned his pioneering work on the “Logic of Antinomies” -- which anticipated Graham Priest’s system Logic of Paradox LP.
My own work on paraconsistent logics began in my 1966 dissertation with a 4-valued semantics for the first-degree entailment fragment of the system R of relevant implication.

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

I am not now nor have I ever been a dialetheist. When I developed this semantics I was too timid too say that a sentence might be both true and false, so instead spoke of a sentence giving both definite positive information and negative information about a topic. But I was invited to speak in a joint symposium of the APA in 1969 on “Natural Language vs. Formal Language” and I got up my nerve and developed a semantics in which sentences could be assigned either T, F, both T and F, or neither. I said:
''Do not get me wrong -- I am not claiming that there are sentences which are in fact both true and false. I am merely pointing out that there are plenty of situations where we suppose, assert, believe, etc., contradictory sentences to be true, and we therefore need a semantics which expresses the truth conditions of contradictions in terms of the truth values that the ingredient sentences would have to take for the contradictions to be true.''
(A copy of the manuscript on which my talk was based, along with a copy of my cv, can be found here)

Michael Dunn, at the time of his PhD. It will soon be published in the book series Logic PhDs with a presentation by Katalin Bimbó

I shortly after connected the 4-valued semantics to Richard Jeffrey’s coupled truth-trees and published the “relevant” part of my APA-ASL talk in 1976 as "Intuitive Semantics for First-Degree Entailments and Coupled Trees." I went on to develop "A Kripke-Style Semantics for R-Mingle Using a Binary Accessibility Relation," as a modification of the Kripke-Grzegorczyk semantics for intuitionistic logic, but allowing sentences to be both true and false, and requiring preservation of falsity as well as the usual truth preservation. In the late 1970’s I developed a way of looking at this 3-valued semantics in terms of homomorphic images, and also explored Robinson’s Arithmetic inspired by Robert Meyer’s Relevant Peano Arithmetic. In 1979 in a paper with Robert Meyer and Richard Routley we showed that Curry’s Paradox (which uses only the implication and not negation) was provable in the standard relevance logics. This destroyed for me the hope that an interesting relevant (paraconsistent) set theory could be developed, because the underlying logic would have to be too weak to be useful. In the 1990s and early 2000s I explored a number of general definitions of negation which allowed for paraconsistency.

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

I see paraconsistent logic(s) as a tool for reasoning. Sometime while an undergraduate student at Oberlin (perhaps influenced by the spirit of their earlier student Quine), I became a Pragmatist. In the last few years I have actually articulated this into a view of “Logics as tools.” Perhaps this is because I gradually converted from philosophy to computer science (or more accurately I added the latter to the former). So I literally see the evolution of paraconsistent logics as a kind of survival of the fittest tool for the job at hand.
Anyway, the main motivation that I see for paraconsistent logic comes from the Internet and its World Wide Web. Belnap was prescient in motivating what has become known as the Belnap-Dunn 4-valued logic by talking of an unstructured database in which various people might enter inconsistent data. This is before the World Wide Web. And now not only do we have this enormous unstructured database, but we have “bots” roaming it for information. Assuming that these bots will perform inferences, I see paraconsistent logic as an essential tool to prevent “Explosion.” Ideally this would be combined with other tools for reasoning and learning such as probability and statistics, deep learning and other machine learning techniques. My most recent work on paraconsistent logics expands the 4-values to in effect a 4-valued subjective probability, using what I call the “The Opinion Tetrahedron.” This is an expansion of Audun Jøsang’s Opinion Triangle into 3 dimensions. Perhaps the most theoretically challenging problem for paraconsistent logics has to do with incorporating belief revision and defeasible reasoning. Ed Mares and Katalin Bimbó have respectively advanced these topics in the context of relevance logic.
There are other possible applications of paraconsistent reasoning in computer science, for example in cloud computing where updates have to be done more or less “simultaneously” across a number of computers. The updates are not truly simultaneous, and so there is the danger of conflicting information.