Experience with F S 0 as a Framework Theory

Feferman has proposed a system, FS 0 , as an alternative framework for encoding logics and also for reasoning about those encodings. We have implemented a version of this framework and performed experiments that show that it is practical. Speciically, we describe a formalisation of predicate calculus and the development of an admissible rule that manipulates formulae with bound variables. This application will be of interest to researchers working with frameworks that use mechanisms based on substitution in the lambda calculus to implement variable binding and substitution in the declared logic directly. We suggest that meta-theoretic reasoning, even for a theory using bound variables, is not as diicult as is often supposed, and leads to more powerful ways of reasoning about the encoded theory. x 1 Introduction: why metamathematics? A logical framework is a formal theory that is designed for the purpose of describing other formal theories in a uniform way, and for making the work of building proofs of theorems in those theories easy. Given a description T of a theory in a framework, a proof of a proposition A in that theory becomes a proof of a proposition in the framework theory that the encoding of A has the property`provable in T'. Such a proof is a meta-theorem, and a natural question to explore is whether we can prove more general theorems in the framework theory. Furthermore, how can such metamathematics be used to increase the power and usability of theorem proving systems in practice? x 1.1 The deduction theorem A good example of the advantage that formal metamathematics can ooer is the deduction theorem 13] for a logic presented in a Hilbert style. If the notation`T A means that A is provable