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 Mailing address: Department of PhilosophyUniversity of Alberta 2–40 Assiniboia Hall Edmonton, AB, T6G 2E7 Canada
Email address:
Erdös number: Three
Gödel number: ∞ (None, really.)
Membership:
Association for Symbolic
Logic
(ASL)
Academic degrees:
PhD (philosophy and cognitive
science)
Latest update on: January 13th, 2012 |
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Research My main research area is logic. I am especially interested in nonclassical logics, including relevance and substructural logics, modal logics, combinatory logics and λ-calculi. Nonclassical logics were first introduced as a solution to philosophical problems such as the problem of implication and entailment. Later on, new logics emerged in many disciplines from mathematics and computer science to linguistics. Some of my research interests — beyond logic and the philosophy of logic — are connected to those areas. I have a keen interest in the philosophy of mathematics, in particular, in the foundations of mathematics. I also actively follow developments in the philosophy of computer science and informatics, especially, concerning questions related to theoretical computer science — such as algorithmics, programming language design, information theory, complexity theory and cyber security. Earlier, I worked on some questions in the semantics of natural languages, primarily, using modal (tense) logics and discourse representation theory. Some of my research results concern semantics for various nonclassical logics (e.g., calculi for combinatory logic). J. Michael Dunn and I wrote a book on the relational semantics of nonclassical logical calculi. These kinds of semantics generalize Kripke’s possible worlds semantics for normal modal logics, and have been the preferred sort of semantics for decades. A few years ago, we worked out representations for Kleene logic and action logic, which are closely related to dynamic logic. We also collaborated on defining a four-valued semantics for the minimal substructural logic. A more precise characterization of classes of structures for various nonclassical logics leads straightforwardly to topological frames. I proved topological duality theorems for ortho- and De Morgan lattices and for the algebra of the logic of entailment. These duality theorems provide plenty of insights into the logics themselves, because interpretations are maps in a category. Combinatory logics and λ-calculi are tied to other nonclassical logics in various ways, one of which is the Curry–Howard correspondence. Certain combinatory terms are equivalent to proofs various nonclassical logics; therefore, the question of decidability turns into the question of inhabitation. Combinatory logic in itself is an elegant and powerful formalism that can represent computable functions — just as Turing machines and Markov algorithms can. I completed the writing of a book on combinatory logic, which is to appear this summer. The proof theory of nonclassical logics sometimes gets tricky. The original classical sequent calculus is a seemingly simple proof system, but in reality, the proofs are tightly controlled — as evidenced by the cut theorem. Some well-known relevance logics cannot be formalized as extensions of the associative Lambek calculus, which causes various “technical complications.” I succeeded in defining sequent calculi for some of these relevance logics together with proving the cut theorem for them. The purely inductive proofs that I used to show the cut theorem for the single cut rule yield a purely inductive cut proof for LK, Gentzen's original sequent calculus for classical logic — without a detour through mix. Currently, the main research project I am working on, in collaboration with J. Michael Dunn within my SSHRC SRG, is the problem of the decidability of the implicational fragment of ticket entailment. We are fairly certain that we have solved this problem, which was open for 50 years, and we are in the process of producing papers that present and explain our results. Obviously, I believe that logic — together with its many connections and offsprings — is an exciting and thriving field of research.
Teaching In the winter term of the 2011/12 academic year I teach Phil 367: Introduction into the Philosophy of Mathematics and Phil 120: Symbolic Logic 1. In the fall term of the 2011/12 academic year I taught Phil 220: Symbolic Logic 2 and Phil 421: Modal Logic together with Phil 522: Topics in Logic.
Selected publications
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