Proof-theoretic Semantics and Tactical Proof

The use of logical systems for problem-solving may be as diverse as in proving theorems in mathematics or in figuring out how to meet up with a friend. In either case, the use is typically through proof-search. A general framework supporting proof-search and its mechanization is the theory of tactics and tacticals; indeed, tactics provide both a uniform theory within which proof-search can be studied and a practical technology underpinning automated reasoning tools such as proof assistants. What is the semantics of a tactical system? What justifies tactical proofs as logical arguments? Proof-theoretic semantics is an approach to meaning centered exclusively on inference and proof. In this paper, we use proof-theoretic validity in the Dummett-Prawitz tradition to give a semantics for the theory of tactical proof in its full generality. This semantics is justified by showing that it arises naturally from the established relationship between tactics and logic, as well as being implicit in much of the literature on proof theory.