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.