A NONTRANSITIVE THEORY OF TRUTH OVER PA

David Ripley has argued extensively for a nontransitive theory of truth by dropping the rule of Cut in a sequent calculus setting in order to get around triviality caused by paradoxes such as the Liar. However, comparing his theory with a wide range of classical approaches in the literature is problematic because formulating it over an arithmetical background theory such as Peano Arithmetic is non-trivial as Cut is not eliminable in Peano Arithmetic. Here we make a step towards closing this gap by providing a suitable restriction of the Cut rule, which allows for a nontransitive theory of truth over Peano Arithmetic that is proof-theoretically as strong as the strongest known classical theory of truth.