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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
Interpretability over Peano arithmetic
"... We investigate the modal logic of interpretability over Peano arithmetic (PA). Our main result is an extension of the arithmetical completeness theorem for the interpretability logic ILM ! . This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than sin ..."
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We investigate the modal logic of interpretability over Peano arithmetic (PA). Our main result is an extension of the arithmetical completeness theorem for the interpretability logic ILM ! . This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a theorem answering a question of Orey from 1961. All these results also hold for ZermeloFraenkel set theory (ZF). 1 Introduction Provability logic Provability logic is concerned with the investigation of various metamathematical relations with the aid of modal logic. For instance, provability logic has been used to investigate different notions of provability, interpretability, conservativity, and tolerance. The fundamental connection between modal logic and metamathematics is provided by certain functions called realizations. These are primitive recursive functions ...
Simulation Of Modal Logics With Polymodal Logics
, 2000
"... Introduction In recent years it became important to consider modal logics containing modalities of arity greater then one. In particular, there have been a lot of examples of logics containing binary modal operator, which serve to describe the behavior of dierent styles of so called Labeled Transit ..."
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Introduction In recent years it became important to consider modal logics containing modalities of arity greater then one. In particular, there have been a lot of examples of logics containing binary modal operator, which serve to describe the behavior of dierent styles of so called Labeled Transition Systems, that nd various applications in Computer Science and Linguistics. Among the important examples are Arrow Logic (AL), displaying the dynamical properties of the transitions (see e.g. M. Marx [7]), some embeddings of the categorial logics such as Non Associative LambecCalculus (NL) into the modal logics with binary modalities (see e.g. Kurtonina [6], also J. van Benthem [1]) ; Interpretebility Logic (IL), which can be viewed as an extension of provability logic (see A.Visser [10]), Hybrid Logics, the examples of which can be found in P.Blackburn, J.Seli
Extending ILM with an operator for Σ_1ness
, 2003
"... In this paper we formulate a logic #ILM. This logic extends ILM and contains a new unary modal operator #1 . The formulas of this logic can be evaluated on Veltman frames. We show that #ILM is modally sound and complete with respect to a certain class of Veltman frames. An arithmetical interpretatio ..."
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In this paper we formulate a logic #ILM. This logic extends ILM and contains a new unary modal operator #1 . The formulas of this logic can be evaluated on Veltman frames. We show that #ILM is modally sound and complete with respect to a certain class of Veltman frames. An arithmetical interpretation of the modal formulas can be obtained by reading the #1 operator as formalized #1ness in PA and # as formalized #1conservativity between finite extensions of PA. We show that under this arithmetically interpretation #ILM is sound and complete. The main motivation for formulating #ILM at all is that one counterexample for interpolation in ILM seems to emerge because of the lack of ILM to express #1 ness. We show that #ILM does not have interpolation either. Our counterexample seems to emerge because of the inability of #ILM to express #interpolation[7]. (A formula # # has a #1 interpolant if there exist some # #1 such that PA # and #.) The text of this paper formed the master's thesis of the author at the ILLC, June 2003, under supervision of Prof. Dr. D.H.J. de Jongh. Contents 1
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
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.