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Interpretability logic
 Mathematical Logic, Proceedings of the 1988 Heyting Conference
, 1990
"... Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength ..."
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Cited by 33 (9 self)
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Interpretations are much used in metamathematics. The first application that comes to mind is their use in reductive Hilbertstyle programs. Think of the kind of program proposed by Simpson, Feferman or Nelson (see Simpson[1988], Feferman[1988], Nelson[1986]). Here they serve to compare the strength of theories, or better to prove
Modal Matters in Interpretability Logics
, 2004
"... In this paper we expose a method for building models for interpretability logics. The method can be compared to the method of taking unions of chains in classical model theory. Many applications of the method share a common part. We isolate this common part in a main lemma. Doing so, many of our res ..."
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In this paper we expose a method for building models for interpretability logics. The method can be compared to the method of taking unions of chains in classical model theory. Many applications of the method share a common part. We isolate this common part in a main lemma. Doing so, many of our results become applications of this main lemma. We also briefly describe how our method can be generalized to modal logics with a different signature. With the general method, we prove completeness for the interpretability logics IL, ILM, ILM0 and ILW ∗. We also apply our method to obtain a classification of the essential Σ1sentences of essentially reflexive theories. We briefly comment on such a classification for finitely axiomatizable theories. As a digression we proof some results on selfprovers. Towards the end of the paper we concentrate on modal matters concerning IL(All), the interpretability logic of all reasonable arithmetical theories. We prove the modal incompleteness of the logic ILW ∗ P0. We put forward a new principle R, and show it to be arithmetically sound in any reasonable arithmetical theory. Finally we make some general remarks on the logics ILRW and IL(All).
Interpolation and Bisimulation in Temporal Logic
 In Proceedings of WoLLIC'98. Workshop of Logic, Language, Information and Computation
, 1998
"... Building on recent model theoretic results for SinceUntil logics we define an adequate notion of bisimulation and establish general theorems concerning the interpolation property. Using these general results we prove that the basic SUlogic and any SUlogic whose class of frames can be defined ..."
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Building on recent model theoretic results for SinceUntil logics we define an adequate notion of bisimulation and establish general theorems concerning the interpolation property. Using these general results we prove that the basic SUlogic and any SUlogic whose class of frames can be defined by universal Horn formulas have interpolation. In particular, the SUlogic of branching time has interpolation, while linear time fails to have this property. 1 Introduction For many years, modal logic (ML) was viewed as an extension of propositional logic (PL) by the addition of new modalities 3 and 2. Nowadays the picture has changed in many directions. First, modal logic is no longer seen as just an extension of PL but also as a restriction of firstorder logic (FO)  when formulas are interpreted over models, or secondorder logic (SO)  when formulas are considered on frames. Furthermore, 3 and 2 have lost their privileged position as a wide variety of new modalities have been i...
Unary Interpretability Logic
, 1992
"... Let T be an arithmetical theory. We introduce a unary modal operator T to be interpreted arithmetically as the unary interpretability predicate over T. We present complete axiomatizations of the (unary) interpretability principles underlying two important classes of theories. We also prove some basi ..."
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Let T be an arithmetical theory. We introduce a unary modal operator T to be interpreted arithmetically as the unary interpretability predicate over T. We present complete axiomatizations of the (unary) interpretability principles underlying two important classes of theories. We also prove some basic modal results about these new axiomatizations.