Pertun Broberg, AntonLeigh, Graham E.2025-03-262025-03-262025-03-26https://hdl.handle.net/2077/85862We explore the proof theory of subsystems of the Friedman–Sheard axiomatic theory of truth FS given by restricting the use of the necessitation and co-necessitation rules. It is known that removing either rule completely (while keeping the other) does not diminish the arithmetic theorems. We present variations of this result for the non-arithmetic theorems of FS. In particular, it is proved that FS has the same consequences as the extension of its subtheory with only necessitation by either a single application of co-necessitation or the schema of transfinite induction up to the ordinal ϕ20.engText