Title: Axiomatic Semantics
Abstract: It is frequently said that Gödel’s discovery of coding formal languages in
arithmetic demonstrates that many mathematical theories can reason about their
own metatheory. This is undoubtably true if by ‘metatheory’ we have in mind
only properties of purely syntactic concepts such as syntax and provability.
But what about semantic concepts like truth and logical validity? Or even ones
lacking formal definition such as justification and ideal provability?
This tutorial will examine the possibilities and limitations for formal
representations of these concepts, and the connection between the expressive
resources of a theory and deductive (i.e., proof-theoretic) strength. Our
primary focus will be on formal representations of truth, motivated by the
question: how close can a formal theory come to recognising its own
correctness?