A pluralist justification of deduction

The main conclusion of the thesis is 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. I begin by isolating the essential parts of Dummett’s and Prawitz theory. This allows me to modify it at various places so as to free it from verificationist presuppositions which permeate the original theory. Dummett and Prawitz think that the decision which logic is the justified one goes in favour of intuitionist logic. I show them to be mistaken at two points. First, I show that the meaning of negation cannot be defined proof-theoretically. It follows that the prooftheoretic justification of deduction cannot decide whether negation should be governed by classical or by intuitionist rules. As a consequence, Dummett and Prawitz are left with no good argument against classical logic. I argue that there is also no acceptable amendment of the theory to remedy this. 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 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 wellmotivated position and outline how to defend it against objections that it is incoherent. In a formal chapter 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. The thesis ends with some reflections on the consequences of pluralism for the relation between logic and metaphysics. I argue that what has to be given up is the thought that the proof-theoretic justification of deduction can decide the metaphysical issues between realists and anti-realists.Ph.D. thesis submitted for Philosophy (KCL) on 24 July 2007. Supervisors: Keith Hossack, Mark Sainsbury and Wilfried Meyer-Viol.