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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
The Logic of Provability
, 1997
"... Contents 1. Introduction, Solovay's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 360 2. Modal logic preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 3. Proof of Solovay's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 4. ..."
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Cited by 24 (3 self)
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Contents 1. Introduction, Solovay's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 360 2. Modal logic preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 3. Proof of Solovay's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 4. Fixed point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 5. Propositional theories and Magarialgebras . . . . . . . . . . . . . . . . . . . . . 368 6. The extent of Solovay's theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 369 7. Classification of provability logics . . . . . . . . . . . . . . . . . . . . . . . . . . 371 8. Bimodal and polymodal provability logics . . . . . . . . . . . . . . . . . . . . . 374 9. Rosser orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 10.
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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Cited by 4 (2 self)
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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).
ESSENTIALLY Σ1 FORMULAE IN ΣL
"... Abstract. The essentially Σ1 formulae of ΣL are exactly those which are provably equivalent to a disjunction of conjunctions of ✷ and Σ1 formulae. 1. Introduction and ..."
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Abstract. The essentially Σ1 formulae of ΣL are exactly those which are provably equivalent to a disjunction of conjunctions of ✷ and Σ1 formulae. 1. Introduction and