Natural deduction as higher-order resolution

Lawrence C. Paulson
1986 The Journal of Logic Programming  
An interactive theorem prover, Isabelle, is under development. In LCF, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause. Resolution gives both forwards and backwards proof, supporting a large class of logics. Isabelle has been used to prove theorems in Martin-Lofs constructive type theory. Quantifiers pose several difficulties: substitution, bound variables,
more » ... tion. Isabelle's representation of logical syntax is the typed X-calculus, requiring higherorder unification. It may have potential for logic programming. Depth-first subgoaling along inference rules constitutes a higher-order PROLOG. a BACKGROUND At least seven interactive theorem provers use the LCF framework. They differ primarily in what logic is used for conducting proofs. A new theorem prover, Isabelle, is intended to unify these diverging paths. A recurring theme will be the relationship between syntax and semantics. We compute by means of syntax but think in terms of semantics. A system of inference rules is a syntactic codification of semantic concepts, and must be shown to respect them. Sets and functions are semantic; formal type theories and Zermelo-Fraenkel set theory are syntactic. The ultimate semanticnotion is truth, faintly approximated by theorems of formal logic. Regarding an axiom system as holy writ blurs the distinction. Martin-Lof [22] discusses the evolution of formal logic from an intuitionistic viewpoint. There are many distinct semantic viewpoints, and accordingly
doi:10.1016/0743-1066(86)90015-4 fatcat:d2jkwnbuyzeo7ko2ymi6qvxvpm