Atomic Systems in Proof-Theoretic Semantics: Two Approaches

Abstract

Atomic systems are systems of rules containing only atomic formulas. In proof-theoretic semantics for minimal and intuitionistic logic they are used as the base case in an inductive definition of validity. We compare two different approaches to atomic systems. The first approach is compatible with an interpretation of atomic systems as representations of states of knowledge. The second takes atomic systems to be definitions of atomic formulas. The two views lead to different notions of derivability for atomic formulas, and consequently to different notions of proof-theoretic validity. In the first approach, validity is stable in the sense that for atomic formulas logical consequence and derivability coincide for any given atomic system. In the second approach this is not the case. This indicates that atomic systems as definitions, which determine the meaning of atomic sentences, might not be the proper basis for proof-theoretic validity, or conversely, that standard notions of proof-theoretic validity are not appropriate for definitional rule systems.