Proof-Normalisation and Truth by Definition

In this thesis I defend an account of analyticity against some well known objections. I defend a view of analyticity whereby an analytic truth is true by definition, and that logical connectives may be defined by their inference rules. First I answer objections that the very idea of truth-by-definition is metaphysically flawed (things are true because of the world, not definition, it seems). More importantly, I respond to objections that no theory of definitions by inference rules (i.e. implicit definitions) can be given that does not allow spurious definitions (e.g. the `definition' of Prior's connective tonk). I shall argue that demanding normalisation (a.k.a. harmony) of definitional inference rules is a natural and well motivated solution to these objections. I conclude that a coherent account of implicit definition can be given as the basis of an account of analyticity. I then produce some logical results showing that we can give natural deduction rules for complex and interesting logical systems that satisfy a normal form theorem. In particular, I present a deduction system for classical logic that is harmonious (i.e. deductions in it normalise), and show how to extend and enhance it to include strict conditionals and empty reference. Also I discuss two areas where our reasoning and classical logic appear not to match: general conditional reasoning, and reasoning from contradictions. I present a general theory of conditionals (along the lines of Lewis' closest-possible-world account) and I suggest that the logic of conditionals is not entirely analytic. Also, I discuss issues surrounding the ex falso rule and conclude that everything really does follow from a contradiction. Finally I suggest a positive theory of when and how the implicit definitions are made that define our logical language.Mathematical Logic