Evaluation, provably deductive equivalence in Heyting's arithmetic of substitution instances of propositional formulas

Abstract

This paper contains the following results: (i) a theorem of the form: if HA (Heyting's Arithmetic) proves some Σ01 substitution instance of an intuitionistically non valid propositional formula then HA proves a substitution instance of a simpler intuitionistically non-valid formula - unless of course the original formula was - in some appropriate sense - already as simple as possible. The result is shown to be adequate. a proof that De Jongh's Completeness Theorem for arithmetical interpretations of Intuitionistic Propositional Logic is verifiable in HA + con(HA). (iii) a characterization of the closed fragment of the provability logic of HA - this is a solution of Friedman's 35th problem for the case of HA. These results are instances of or corollaries to answers of a common kind of question, which we call the evaluation problem for a certain set of interpretations. A framework is developped to analyze this kind of question.