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The Interpolation Theorem for IL and ILP
 Uppsala University
, 1998
"... In this article we establish interpolation for the minimal system of interpretability logic IL. We prove that arrow interpolation holds for IL and that turnstile interpolation and interpolation for the modality easily follow from this. Furthermore, these properties are extended to the system ILP. ..."
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In this article we establish interpolation for the minimal system of interpretability logic IL. We prove that arrow interpolation holds for IL and that turnstile interpolation and interpolation for the modality easily follow from this. Furthermore, these properties are extended to the system ILP. The related issue of Beth Definability is also addressed. As usual, the arrow interpolation property implies the Beth property. From the latter it follows via an argumentation which is standard in provability logic, that IL has the fixed point property. Finally we observe that a general result of Maksimova [11] for provability logics can be extended to interpretability logics, implying that all extensions of IL have the Beth property. Keywords Interpretability Logic, Interpolation Properties, Beth Property, Fixed Point Property. 1 Introduction 1.1 Some History Interpretability logics are extensions of provability logics introduced by Visser in [15]. There the modal logics IL, ILM and ILP a...
Interpolation, Definability and Fixed Points in Interpretability Logics
 Advances in Modal Logic, Volume 2, Lecture Notes
, 2000
"... this article we study interpolation properties for ..."
No Escape from Vardanyan’s Theorem
, 2003
"... Vardanyan’s Theorem states that the set of PAvalid principles of Quantified Modal Logic, QML, is complete Π 0 2. We generalize this result to a wide class of theories. The crucial step in the generalization is avoiding the use of Tennenbaum’s Theorem. ..."
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Vardanyan’s Theorem states that the set of PAvalid principles of Quantified Modal Logic, QML, is complete Π 0 2. We generalize this result to a wide class of theories. The crucial step in the generalization is avoiding the use of Tennenbaum’s Theorem.
Problems in the Logic of Provability
, 2005
"... 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 ..."
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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
WHAT IS THE RIGHT NOTION OF SEQUENTIALITY?
"... Abstract. In this paper we give an informally semirigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion msequentiality as the notion that precisely captures the intuitions behind sequentiality. ..."
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Abstract. In this paper we give an informally semirigorous explanation of the notion of sequentiality. We argue that the classical definition, due to Pudlák, is slightly too narrow. We propose a wider notion msequentiality as the notion that precisely captures the intuitions behind sequentiality. The paper provides all relevant separating examples minus one. 1.
THE ARITHMETICS OF A THEORY
"... Abstract. In this paper we study the interpretations of a weak arithmetic, like Buss ’ theory S1 2, in a given theory U. We call these interpretations the arithmetics of U. We develop the basics of the structure of the arithmetics of U. We study the provability logic(s) of U from the standpoint of t ..."
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Abstract. In this paper we study the interpretations of a weak arithmetic, like Buss ’ theory S1 2, in a given theory U. We call these interpretations the arithmetics of U. We develop the basics of the structure of the arithmetics of U. We study the provability logic(s) of U from the standpoint of the framework of the arithmetics of U. Finally, we provide a deeper study of the arithmetics
Towards the Interpretability Logic of x,arianne B. Kalsbeek Logic Group
, 1991
"... Among the different interpretability logics corresponding to (classes of) arthmetical theories, the interpretability logic of Itp+EXP (to which we will refer as ILeXp), takes a special place. Though we have no explicit axiomatization for ILeXp, we do have a complete description of the theory. Visser ..."
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Among the different interpretability logics corresponding to (classes of) arthmetical theories, the interpretability logic of Itp+EXP (to which we will refer as ILeXp), takes a special place. Though we have no explicit axiomatization for ILeXp, we do have a complete description of the theory. Visser shows, in [VIS], that relative interpretability
The many faces of interpretability
"... Abstract. In this paper we discus work in progress on interpretability logics. We show how semantical considerations have allowed us to formulate nontrivial principles about formalized interpretability. In particular we falsify the conjecture about the nature of the interpretability logic of all re ..."
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Abstract. In this paper we discus work in progress on interpretability logics. We show how semantical considerations have allowed us to formulate nontrivial principles about formalized interpretability. In particular we falsify the conjecture about the nature of the interpretability logic of all reasonable arithmetical theories. We consider this an interesting example of how purely semantical considerations give new nontrivial facts about syntactical and arithmetical notions. In addition we give some apparatus that allows us to push ‘global ’ semantical properties into more ‘local ’ syntactical ones. With this apparatus, the rather wild behavior of the different interpretability logics are nicely formulated in a single notion that expresses their differences in a uniform way. This paper consists of three parts. We start of by giving a short introduction to interpretability logics. In the second part we discuss how a careful analysis of the modal semantical behaviour of interpretability logics lets us formulate nontrivial interpretability principles. In the third part we present a semantical bookkeeping tool which pushes ‘global ’ semantical considerations into ‘local’ syntactical ones. The hope is that this machinery will provide a general and uniform treatment of the bewildering field of modal interpretability logics. 1