Inferential Semantics for First-Order Logic: Motivating Rules of Inference from Rules of Evaluation
Neil W. Tennant · 2009
I am grateful to Tim, as my teacher in logic, for emphasizing that its main notions are relational: ‘ϕ is deducible from ∆ ’ and ‘ϕ is a logical consequence of ∆’. The notions ‘ϕ is a theorem ’ and ‘ϕ is logically true’ are special cases. In the early 1970s Tim gave a formative series of lectures emphasizing how proofs are to be understood as perfected arguments, in Aristotle’s sense. The present discussion of verification and falsification is fully in the inferentialist spirit of Tim’s emphases. The aim is to render even the notions ‘ϕ is true ’ and ‘ϕ is false ’ as essentially relational and inferential. A sentence’s truth-value is determined relative to collections of rules of inference that constitute an interpretation. Moreover, truthmakers and falsity-makers are themselves proof-like objects, encoding the inferential process of evaluation involved. The inference rules involved in the determination of truth-value are almost identical to those involved in securing the transmission of truth from premises to conclusion of a valid argument. We shall see how smoothly one can ‘morph ’ the former into the