This paper is concerned with the `logical structure' of arithmetical theories. We survey results concerning logics and admissible rules of constructive arithmetical theories. We prove a new theorem: the admissible propositional rules of Heyting Arithmetic are the same as the admissible propositional rules of Intuitionistic Propositional Logic. We provide some further insights concerning predicate logical admissible rules for arithmetical theories. Key words: Intuitionistic Logic, Heyting Arithmetic, Kripke models, admissible rules MSC codes: Primary: 03F25, 03F30, Secondary: 03-02, 03B20, 03F50, 03F40 Contents 1 Introduction 3 2 Theories and Logics 3 2.1 Propositional Logics of Theories . . . . . . . . . . . . . . . . . . 4 2.2 Predicate Logics of Theories . . . . . . . . . . . . . . . . . . . . . 5 2.3 A Brief History of de Jongh's Theorem . . . . . . . . . . . . . . . 7 2.4 Markov's Principle and Church's Thesis . . . . . . . . . . . . . . 9 2.5 Exactness and Extension . . . . ...

In this note we compare propositional logics for closed substitutions and propositional logics for open substitutions in constructive arithmetical theories. We provide a strong example where these logics diverge in an essential way. We prove that for Markov’s Arithmetic, i.e. Heyting’s Arithmetic plus Markov’s principle plus Extended Church’s Thesis, the logic of closed and the logic of open substitutions are the same. 1

In the first part of the paper we discuss some conceptual problems related to the notion of proof. In the second part we survey five major open problems in Provability Logic as well as possible directions for future research in this area. 1

Abstract. We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property. Dedicated to Petr Hájek, on the occasion of his 70th Birthday 1. Preface The three authors of this paper have enjoyed Petr Hájek’s acquaintance since the late eighties, when a lively community interested in the metamathematics of arithmetic shared ideas and traveled among the beautiful cities of Prague, Moscow, Amsterdam, Utrecht, Siena, Oxford and Manchester. At that time, Petr Hájek and Pavel Pudlák were writing their landmark book Metamathematics of First-Order Arithmetic [HP91], which Petr Hájek tried out on a small group of eager graduate students in Siena in the months of February and March 1989. Since then, Petr Hájek has been a role model to us in many ways. First of all, we have always been impressed by Petr’s meticulous and clear use of correct notation, witness all his different types of dots and corners, for example in the Tarskian ‘snowing’-snowing lemmas [HP91]. But also as a human being, Petr has been a role model by his example of living in truth, even in averse circumstances [Hav89]. The tragic story of the Logic Colloquium 1980, which was planned to be held in Prague and of which Petr Hájek was the driving force, springs to mind [DvDLS82]. Finally, we were moved by Petr’s open-mindedness when coming to terms with a situation that turned out to look disconcertingly unlike the ‘standard model ’ 1. Therefore, in this paper, we would like to pay homage to Petr Hájek. Unfortunately, we cannot hope to emulate his correct use of dots and corners. Instead, we do our best to provide some pleasing non-standard models and non-classical arithmetics. 2.

This paper is concerned with the `logical structure ' of arithmetical theories. We survey results concerning logics and admissible rules of constructive arithmetical theories. We prove a new theorem: the admissible propositional rules of Heyting Arithmetic are the same as the admissible propositional rules of Intuitionistic Propositional Logic. We provide some further insights concerning predicate logical admissible rules for arithmetical