# A New Look at Cut Elimination for Compositional Truth

## Carlo Nicolai, King's College

In the field of axiomatic theories of truth, conservativity properties of theories are much investigated. Conservativity can be used to argue that, despite the well-known undefinability of truth, there is a sense in which a primitive truth predicate can be reduced to the resources of an underlying mathematical theory that provides basic syntactic structure to truth ascriptions.

Conservativity is typically proved model-theoretically, or proof-theoretically via the elimination of cuts on formulae containing truth (Tr-cuts). The original Tr-cut-elimination argument for the theory of Tarskian, compositional truth CT[B] by Halbach is not conclusive. This strategy has been corrected by Graham Leigh: Leigh supplemented Halbach’s strategy with the machinery of approximations (introduced by Kotlarski, Krajewski and Lachlan in the context of their M-Logic). In the talk we investigate a different, and arguably simpler way of supplementing Halbach’s original strategy. It is based on an adaptation of the Takeuti/Buss free-cut elimination strategy for first-order logic to the richer truth-theoretic context. If successful, the strategy promises to generalize to the type-free setting in a straightforward way. This is joint work with Luca Castaldo.