A Modal Extension of Weak Generalisation Predicate Logic

We introduce a new axiomatic system of modal logic, BM, extending classical first order logic by adding the binary modal symbol “▹” intended to simulate the metamathematical provability predicate “⊢” of classical logic. We demonstrate via examples how BM can be used to write equational proofs of first order classical theorems, and show that this ability hinges on a “conservation result”: BM proves A ▹ B for classical A and B iff A ⊢ B holds classically. We introduce appropriate Kripke semantics with respect to which we prove BM is sound and complete. As a corollary we prove the above mentioned conservation result.