My research interests are in philosophical and formal logic and related areas, such as philosophy of language, philosophy of mathematics, metaphysics and epistemology. I also have an interested in philosophy of mind, and history of philosophy, especially the Ancient Greeks. Reading Plato is one of the most rewarding philosophical experiences. My plan is to read all the dialogues in the order according to their dramaturgic setting, as reconstructed by Catherine Zuckert. In fact, I already started a while ago. Shame the first dialogue on her list is the Laws.
I'm currently putting together a book on negation, denial and falsity and the limits of Dummett's and Prawitz' proof-theoretic semantics. I begin by situating their account in the philosophy of language, in particular Dummett's approach to the theory of meaning. Starting from fairly uncontroversial assumptions, such as that meaning is tied to use and that languages are learnable, I aim to take the reader through steps culled from Dummett's writings all the way to his grand conclusion that classical logic is problematic, as we do not know how to construct a semantic theory for it. I then argue that the meaning of negation cannot be satisfactorily given by proof-theoretic semantics, where the meanings of the logical constants are specified in terms of rules of inference. The primitive assumptions of proof-theoretic semantics is that speakers can follow rules of inference. We add as an additional assumption that speakers have some grasp of negation. This can build on some views Peter Geach has put forward in a different context. However, I argue that this option is not ideal in the context of Dummett's view of language. Alternatively, we could follow a route Huw Price has suggested and assume that there are primitive speech acts of assertion and denial. I argue that for methodological reasons, there is no philosophical mileage in adopting the bilateral logic that Ian Rumfitt has proposed to capture Price's idea in a system of natural deduction. In my view, the best option is to assume speakers have a primitive grasp of truth and falsity. Dummett could be described as trying to avoid negative primitives, and instead define everything in terms of positive primitives, that is truth and following inferences. The option of taking negation as primitive amounts to adopting an additional negative primitive, and the bilateral approach also assumes a positive and a negative notion as basic. The option of taking truth and falsity as equally basic also builds on the insight that we need positive as well as negative primitives. The bilateralist may make a come back, by putting forward an account of how concepts of truth and falsity are derived from more basic speech acts assertion and denial, but this wouldn't be something that needs to be reflected in proof-theory. I'll conclude with some historical reflections on the problem of falsity in Plato.
I'm also interested in epistemological questions concerning the choice of logical primitives, in particular whether negation should be defined in terms of incompatibility.
My main long term future research project is to provide a proof-theoretic semantics for modal operators. This builds on research of my PhD but develops it into a new direction that hasn't had much attention in the literature on proof-theoretic semantics. Philosophically, my account of the semantics for modal operators in terms of the rules of inference governing them promises to provide a satisfactory account of the meaning of modal operators, which does not appeal to possible worlds. Formally, natural deduction for modal logic is a topic that deserves more attention, and I aim to develop new proof systems that fall into the area of natural deduction rather than sequent calculi.
I wrote my Ph.D. thesis in King's College in London, under the supervision of Keith Hossack, Mark Sainsbury and Wilfried Meyer-Viol. In my Ph.D. thesis I argued that, rather than deciding disputes over the validity of logical laws between classical and intuitionist logicians, Dummett's and Prawitz' proof-theoretic justification of deduction entails a pluralism in which both logics have their place. Dummett and Prawitz think that the decision which logic is the justified one goes in favour of intuitionist logic. I argue that they are mistaken at two points. First, the meaning of negation cannot be defined proof-theoretically and should be taken to be a primitive notion. Hence the proof-theoretic justification of deduction cannot decide whether negation should be governed by classical or by intuitionist rules. I argue that there also is no amendment that stays true to the spirit of the proof-theoretic justification of deduction and would succeed where the original theory foundered. Secondly, Dummett and Prawitz only consider deductions made from sets of hypotheses, but there is at least one other reasonable way of collecting them, which is used in the system R of relevance logic. I conclude that the proof-theoretic justification of deduction commits us to accepting at least classical, intuitionist and relevance logic. Because this logical pluralism is a consequence of the proof-theoretic justification of deduction, I argue that it is a well-motivated position and outline how to defend it against objections that it is incoherent to accept more than one logic. In a formal theory I specify the general forms of rules of inference and general methods for determining elimination/introduction rules for logical constants from their introduction/elimination rules. On this basis I re-define Dummett's and Prawitz' notions of harmony and stability in a formally precise way and provide generalised procedures for removing maximal formulas from deductions. The result is a general framework for proving normalisation theorems for a large class of logics, which is important for the delineation of the range of correct logics. The thesis ends with some reflections on the formalisation of logical concepts and Carnap's Principle of Tolerance.
The Philosophy of Pumpkin