Lorenzen’s operative justification of intuitionistic logic

Introduction With his Introduction to Operative Logic and Mathematics, which first appeared in 1955, Paul Lorenzen became an exponent of an approach to the foundations of logic and mathematics, which is both formalistic and intuitionistic in spirit. Formalistic because its basis is the purely syntactical handling of symbols — or “figures”, as Lorenzen preferred to say —, and intuitionistic because the insight into the validity of admissibility statements justifies the laws of logic. It is also intuitionistic with respect to its result, as Heyting’s formalism of intuitionistic logic is legitimatised this way. Along with taking formal calculi as its basis, the notion of an inductive definition becomes fundamental. Together with a theory of abstraction and the idea of transfinitely iterating inductive definitions, Lorenzen devised a novel foundation for mathematics, many aspects of which still deserve attention. When he wrote his Operative Logic, neither a full-fledged theory of inductive definitions nor a proof-theoretic semantics for logical constants was available. A decade later, Lorenzen’s inversion principle was used and extended by Prawitz (1965) in his theory of natural deduction, and in the 1970s, the idea of inversion was used for a logical semantics in terms of proofs by Dummett, Martin-Löf, Prawitz and others. Another aspect which makes Lorenzen’s theory interesting from a modern point of view, is that in his protologic he anticipated certain views of rule-based reasoning and free equality which much later became central to the theory of resolution and logic programming. Lorenzen’s inversion principle in its general form — that is, not in its restricted application in logic — is closely related to principles of definitional reflection in logic programming (Schroeder-Heister 2007). The idea that logical introduction rules are but a special case of rules defining (atomic) propositions was used in a different form in Martin-Löf’s (1971) theory of iterated inductive definitions. Thus there are various interesting points from which we might take a closer look at Operative Logic. Unfortunately, Lorenzen had already lost interest in the subject when issues such as proof-theoretic semantics and resolution-based reasoning became more popular in logic. Within the narrower realm of logic, he had already given up the operative approach in favour of dialogical logic by the end of the 1950s, perhaps motivated by discussions