Proof Theoretic Semantics Blocks Pathological Self-Reference

The system uses proof-theoretic semantics, where the meaning of a statement is determined entirely by its inferential role within a theory. A theory T consists of a finite set of basic statements together with everything that can be derived from them using the inference rules. The statements derivable in this way are the theorems of T. A statement is true in T exactly when T proves it. A statement is false in T exactly when T proves its negation. Some statements are neither true nor false in T. These are the non-well-founded statements: statements whose inferential justification cannot be grounded in a finite, well-founded proof structure. This includes self-referential constructions such as Gödel-type sentences.