Hereditarily Structurally Complete Superintuitionistic Deductive Systems

Abstract

Propositional logic is understood as a set of theorems defined by a deductive system: a set of axioms and a set of rules. Superintuitionistic logic is a logic extending intuitionistic propositional logic $$\mathsf {Int}$$ . A rule is admissible for a logic if any substitution that makes each premise a theorem, makes the conclusion a theorem too. A deductive system $$\mathsf {S}$$ is structurally complete if any rule admissible for the logic defined by $$\mathsf {S}$$ is derivable in $$\mathsf {S}$$ . It is known that any logic can be defined by a structurally complete deductive system—its structural completion. The main goal of the paper is to study the following problem: given a superintuitionistic logic L, is the structural completion of L hereditarily structurally complete? It is shown that, on the one hand, there is continuum many of such logics, including $$\mathsf {Int}$$ , and many of its standard extensions. On the other hand, there is continuum many superintutitionistic logics structural completion of which is not hereditarily structurally complete (the Medvedev and Kreisel–Putnam logics are notable examples). It is observed that the class of hereditarily structurally complete superintuitionistic consequence relations does not have the smallest element and it contains continuum many members lacking the finite model property. The following statement is instrumental in obtaining negative results: if a Lindenbaum algebra of formulas on one variable is finite and has more than 15 elements, then a structural completion of such a logic is not hereditarily structurally complete