R.M. Solovay. Provability interpretations of modal logic. Israel Journal of Mathematics, 28:33--71, 1976. 20  Home/Search   Document Not in Database   Summary   Related Articles   Check

....logics even when these logics are characterized by classes of frames that are not first order definable. For instance, the modal logic K augmented with the McKinsey axiom is captured by the framework presented in [21] Similarly, the provability logic G that admits arithmetical interpretations [25] is treated within the set theoretical framework defined in [5] Both techniques in [21, 5] use a version of classical logic augmented with a theory. Alternatively, G can also be translated into classical logic by first using the translation into K4 defined in [2] and then a translation from K4 ....

R. Solovay. Provability interpretations of modal logics. Israel Journal of Mathematics, 25:287--304, 1976.

....Consequently, a fortiori, every modal logic characterized by a class of frames that is not firstorder definable, is not properly displayable. This includes the well known provability logics G and Grz (for Grzegorczyk) which admit important arithmetical interpretations as logics of provability [Sol76] (see also [Boo93] At first glance, this seems to contradict the fact that DL generalizes Gentzen style calculi since the well known traditional sequent and tableau calculi for these logics [SV80, Lei81, SV82, Val83, Fit83, Avr84, Boo93, Gor99] do enjoy cut elimination. Our contribution By ....

R. Solovay. Provability interpretations of modal logics. Israel Journal of Mathematics, 25:287--304, 1976.

....in this thesis are related to or connected with the provability logic GL. Provability logic GL is one of the normal modal logics, which is obtained from the smallest normal modal logic K by adding Lob s axiom #(#p #p. The name provability logic derives from Solovay s theorem in Solovay [Sol76]. He proved that GL is complete for the formal provability interpretation in Peano arithmetic PA. So, GL has been considered as one of the most important modal logics. Let us briefly explain Solovay s theorem, following Chagrov and Zakharyaschev [CZ97] All syntactical constructions of the ....

R. M. Solovay, Provability interpretations of modal logic, Israel Journal of Mathematics, 25, 1976, pp. 287--304.

....together with a sequence of closed sentences, one for each propositional letter of TGL . Any such T k determines an interpretation ( k of the formulas of TGL where 2 is interpreted by the arithmetic predicate Bew and the interpretation commutes with classical connectives. Soloway s theorem [15] states that TGL if and only if for all k, PA k ( k . In propositional L P we can express the in nite set of equivalences GL k , for every and k, which we may regard as non logical axioms. 18 GIANLUIGI BELLIN AND CARLO DALLA POZZA (iii) Consider the same setting of (ii) for ....

R. Solovay. Provability interpretations of modal logic. Israel Journal of Mathematics,

....semantics of Godel s provability calculus S4, 2. the modal logic of the formal provability predicate Provable(F) It was already clear, however, that 1 and 2 led to essentially different models of Provability, each targeting its own set of applications. Problem 2 was solved by R. Solovay [92] who showed that the modal logic L 4 axiomatized all propositional properties of the formal provability, and by Artemov [4] and Vardanyan [102] who demonstrated that the first order logic of formal provability was not axiomatizable. The issue of provability semantics for S4 was addressed by ....

....of the first order logic of proofs with the standard Godel numbering satisfying special monotonicity restrictions on its Godel numbering. x12. Discussion. 1. There are two provability models each having its own areas of applications: A. The logic L of formal provability ( 25] 26] 91] [92]) with the nonreflexive Lob principle 2(2F F ) 2F . Within this model proofs are represented implicitly by existential quantifiers. The highlights of this model are formalizations of the second Godel incompleteness theorem, Lob theorem and fixed point theorem in the propositional language. L ....

R. Solovay, Provability interpretations of modal logic, Israel Journal of Mathematics, vol. 25 (1976), pp. 287--304.

....[46] 47] also makes use of the format t : F with its informal provability reading. LP may also be regarded as a basic epistemic logic with explicit justifications; a problem of finding such systems was raised by van Benthem in [9] The studies of the logic GL of implicit provability Provable(x) [67], 65] 12] 13] 14] 31] has given vast experience in arithmetical self referential semantics for modal logics. The 39 completeness theorem for LP (Theorem 7.1) could not probably have been obtained without the knowledge accumulated in this area. 11 Acknowledgements This work has benefited ....

R. Solovay, "Provability interpretations of modal logic", Israel Journal of Mathematics, 25, pp. 287-304, 1976.

....property: KS 1 : S n 2A if and only if KS 1 : S n A (47) Theorem 7.2 ( Che80] Exercise 4.61) Property (47) holds for the modal systems K, KD, KT, K4, KDB, KD4, KTB, KT4 and KT5. Theorem 7.3 Property (47) holds for the modal system KG. Proof By Solovay s completeness theorem (see [Sol70]) an L(2) wff A is provable in KG if and only if, for every realization Phi, A Phi is a theorem of PA. Hence if KG 2A then by Solovay s completeness theorem, for each realization Phi PA Bew(dA Phi e) which implies that PA A Phi and again by Solovay s completeness theorem KG ....

R. Solovay. Provability Interpretations of Modal Logic. Israel Journal of Mathematics, 25:287--304, 1970.

....logic GL is formulated axiomatically using Lob s Axiom, rather than the Rule. Between the work of Godel and Lob, every modal formula provable in GL is valid all translations into arithmetic yield sentences provable in PA. Solovay proved the remarkable result that the converse is also true [11], and thus GL is exactly the modal counterpart of PA provability. Solovay s proof is much too complex to be even sketched here. Su#ce it to say that it involves a di#cult fixpoint construction, using either the second recursion theorem of recursion theory, or Godel s fixedpoint result for PA. Any ....

Solovay, R. M. Provability interpretations of modal logic. Israel Journal of Mathematics 25 (1976), 287--304.

....a fortiori, every modal logic characterized by a class of frames that is not first order definable, is not properly displayable. This includes the well known provability logics G 3 and Grz (for Grzegorczyk) which admit important arithmetical interpretations as logics of provability [Sol76] (see also [Boo93] At first glance, this seems to contradict the fact that DL generalizes Gentzen style calculi since the well known traditional sequent and tableau calculi for these logics [SV80, Lei81, SV82, Val83, 3 Also called GL (for Godel and Lob) KW, K4W, PrL. 1 Fit83, Avr84, Boo93, ....

R. Solovay. Provability interpretations of modal logics. Israel Journal of Mathematics, 25:287--304, 1976.

....on an implicit definition of C. This has been, in fact, the main effort of this work. The connection between nonmonotonic reasoning and a logic in which fixed points are characterizable, namely the modal logic G, has been early investigated by (Doyle, 1980) The modal logic G (Boolos, 1979; Solovay, 1976) is the logic in which the notion of provability in Peano Arithmetic is interpreted. For this reason the characterization of fixed points goes through implicit definability which, as we discussed above, means that there exists a unique solution to the fixed point equation; this commitment to ....

....: A 1 A 2 ) and A 1 ; A 2 2 Phi. 4. 2 A discussion on Provability, Consistency and the logic G There is a well established area of pure logic where modalities have been used to interpret the notions of provable and consistent in PA, the first order 17 Peano Arithmetic (Bernardi, 1975; Solovay, 1976; Boolos, 1979; Smorynski, 1985) Following Godel s procedure for numbering theorems of PA, a first order provability predicate can be constructed. Once the unary predicate of provability is interpreted as a modal operator 2, according to Solovay s translation, the modal counterpart of the ....

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Solovay, R. (1976). Provability interpretations of modal logic. Israel Journal Math., 25:287--304.

....that Default logic can have a pure logical characterization just in case the logic, in which a default theory is embedded, enjoys the definability of fixed points, that is theorems of the form ( above. It should be observed that following results from Solovay and Montague (see (Montague, 1963; Solovay, 1976), logics enjoying definability of fixed points cannot have among their axiom schemata and rules of inferences both T and Necessitation. As a corollary of the previous theorems we can give the following, Corollary 1 Let hW; Di be a default theory, Tr 1 hW;Di and Tr 2 hW;Di (p) the two ....

Solovay, R. (1976). Provability interpretations of modal logic. Israel Journal Math., 25:287--304.

....while any modal logic in , KD4Z KD45:1 can embed both default and autoepistemic logic. On the other hand, the second issue deals with the following question to which extent negation by failure can be simulated in the modal language itself We explore this point following the idea realised by [Ber75, Sam75, Sol76] for representing provability in Peano arithmetic with the modal logic G. Already McDermott and Doyle speculated in their paper (see the Discussion in [MD80] that the modal logic G could be applicable in improving their nonmonotonic logic, but they never carried through in proving this ....

....rule of KD4Z is that ff is a KD4Z theorem iff 2ff is a KD4Z theorem. Note that this equivalence implies the decidability of ff as far as 2 acts as a provability operator. For a discussion on the role of KD4Z as a provability logic and its relation with the classical provability logic G = KW ([Boo79, Sol76]) see [ACP96a] The other new modal logic that we deal with is KD45:n. It extends KD4 with the n euclideaness axiom. It says that if a formula ff can ever be proved then it will certainly before the step n 1 but, differently from Z, it says nothing about the truth of ff. Obviously KD45:n ....

R.M. Solovay. Provability interpretations of modal logic. Israel Journal Math., 25:287--304, 1976.

....when we are interested not only to know that a certain statement A is valid, but also have to keep track on some evidences of its validness: 2 p A may stand for p is a proof of A , p is a program which computes A , A has a proof of the complexity p , etc. The language of the provability logic ([3]) with the provability operator 2 only, where 2A stands for A is provable , can not do this job. However, labeled modalities alone fail to express some key principles of provability, e.g. the fact that a set of theorems is closed under modus ponens; it can easily be done with the use of the ....

....is complete with respect to the entire class of standard proof predicates. 2.1 System B Gamma 2.1 Definition. For some technical reasons we consider an auxiliary system B Gamma which has the same axioms as B and rules R1 and R2. Axioms A1 A3 came from the Godel Lob provability logic GL (cf.[3], 8] 9] together with rules R1 and R2. So B Gamma derives all GL tautologies. Axiom A4 came from the system P ( 1] A5 and A6 reflect how 2 and 2 p i ( Delta) match together. 2.2 Example. B Gamma 2 p A 2A for every A 2 L . Indeed, 1. 2 p A A (Axiom A4) 2. 22 p A 2A (From 1. ....

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R. M. Solovay, "Provability interpretations of modal logic," Israel Journal of Mathematics, vol. 25, pp. 287--304, 1976.

....of propositional variables. The problem in provability logic, L, that brought us on the trail of the semantic types was the computation of the exactly provable formulas 2 in the fragment L 1 1 (see Chapter 5) According to Solovay s theorem on provability interpretations for formulas of L [Solovay 76] the theorems of L are those modal formulas that are provable in Peano arithmetic (PA) under arbitrary arithmetical interpretations (interpreting 2 as the formalized provability predicate in PA) If we fix the arithmetical interpretation of one or more of the propositional variables, the ....

....= n m (l) Proof. Obvious, as Th n m (k) Th n m (l) a Chapter 5 Exactly provable L formulas 5. 1 Introduction In this chapter we will study the exactly provable formulas in fragments of provability logic L (GL in [Boolos 93] PRL in [Smory nski 85] According to Solovay s theorem [Solovay 76] on provability interpretations the theorems of the provability logic L are precisely those modal formulas that are provable in PA under arbitrary arithmetical interpretations (interpreting 2 as the formalized provability predicate in PA) The logic L is also known to be the logic of the ....

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R. Solovay, `Provability interpretations of modal logic', Israel Journal of Mathematics 25 287--304 (1976).

....of Proofs Sergei Artemov Steklov Mathematical Institute, Vavilov str. 42, 117966 Moscow, Russia. e mail: art log.mian.su Tyko Stra en y University of Berne, IAM, Langgassstr. 51, CH 3012 Berne. e mail: strassen iam.unibe.ch Abstract Propositional Provability Logic was axiomatized in [7]. This logic describes the behaviour of the arithmetical operator y is provable . The aim of the current paper is to provide propositional axiomatizations of the predicate x is a proof of y by means of modal logic, with the intention of meeting some of the needs of computer science. 1 ....

.... y is provable . The aim of the current paper is to provide propositional axiomatizations of the predicate x is a proof of y by means of modal logic, with the intention of meeting some of the needs of computer science. 1 Introduction The Propositional Provability Logic GL was axiomatized in [7]. This logic describes the behaviour of the arithmetical operator y is provable by means of modal logic. Although some properties of this logic are relevant for computer science (e.g. various forms of Godel s incompleteness theorem for consistency proofs in databases) GL is rather a ....

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R. M. Solovay, "Provability interpretations of modal logic," Israel Journal of Mathematics, vol. 25, pp. 287--304, 1976.

....even when these logics are characterized by classes of frames that are not first order definable. For instance, the modal logic K augmented with the McKinsey axiom is captured by the framework presented in [Ohl93] Similarly, the provability logic G 3 that admits arithmetical interpretations [Sol76] is treated within the settheoretical framework defined in [dMP95] Both techniques in [Ohl93, dMP95] use a version of classical logic augmented with a theory. Alternatively, G can also be translated into classical logic by first using the translation into K4 defined in [BH94] and then a ....

R. Solovay. Provability interpretations of modal logics. Israel Journal of Mathematics, 25:287--304, 1976.

....We proceed with the Solovay construction and define the Solovay function h(t) and the arithmetical formulas l = j for the model K 0 and for the usual Godel proof predicate Proof (x; y) and put (S i ) jS i l = j ] i = i: The following Solovay Lemma holds: 2.8 Lemma. [5] 1. PA 0 l n ; 2. l = 0 is true, but each of the theories PA l = i is consistent for i = 0; 1 : n; 3. PA l = i Provable(p l 6= i q) i = 1; 2; n; 4. PA l = i :Provable(p l 6= j q) i = 0; 2; n; i OE j; 5. PA l = i Provable(p l 6= j q) i = 1; 2; ....

....of the injectivity of Prf (k; pD q) is false for all k 2i 1. Thus PA :9x p i Prf (x; pD q) and PA l = j :B . Now we proceed with different inductions on formulas, first for k 0, and then for k = 0. Let 1 k n. The induction step in case of is straightforward (cf.[5]) The induction step in case B j 2H: we proceed with the standard Solovay argument. If k fl 2H, then for all j k j fl H; for all j k PA l = j H ; PA kOEj l = k H ; PA Provable(p kOEj l = k q) Provable(pH q) y by 2.8 (4) PA l = k k6OEj ....

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R. M. Solovay, "Provability interpretations of modal logic," Israel Journal of Mathematics, vol. 25, pp. 287--304, 1976.

....logics even when these logics are characterized by classes of frames that are not first order definable. For instance, the modal logic K augmented with the McKinsey axiom is captured by the framework presented in [21] Similarly, the provability logic G 1 that admits arithmetical interpretations [25] is treated within the set theoretical framework defined in [5] Both techniques in [21, 5] use a version of classical logic augmented with a theory. Alternatively, G can also be translated into classical logic by first using the translation into K4 defined in [2] and then a translation from K4 ....

R. Solovay. Provability interpretations of modal logics. Israel Journal of Mathematics, 25:287--304, 1976.

....models. For each system of this class of logics soundness and completeness are proved. Moreover, some principles of the Basic Logic of Proofs, mainly concerning fixed points, are investigated. 1 Introduction The Propositional Provability Logic GL for Peano Arithmetic PA was axiomatized in [8]. GL describes the behaviour of the arithmetical operator A is provable by means of modal logic. Since GL is decidable, one has an elegant and efficient tool for studying subjects centered around Godel s incompleteness theorems, e.g. Lob s theorem, substitutions, fixed points and formalizations. ....

....of Proofs with syntactical models and then investigate some basic properties, mainly concerning fixed points. In the remaining of this section, a summary of the definitions of the Basic Logic of Proofs is given. Most definitions are in accordance with those of the classical Provability Logic ([8]) Next, appropriate axiom systems are presented. And finally, the syntactical models are defined. Supported by the Union Bank of Switzerland (UBS SBG) and by the Swiss Nationalfonds (projects 21 27878.89 and 20 32705.91) In section 2, the soundness and completeness of the proof systems with ....

R. M. Solovay, "Provability interpretations of modal logic," Israel Journal of Mathematics, vol. 25, pp. 287--304, 1976.

....R [ f(0; w) w 2 Wg; S 0 0 = S 1 [ f(1; w) w 2 Wg and for each w 2 W , S 0 w = Sw . Observe that hW 0 ; R 0 ; fS 0 w gw2W 0 i is a finite Veltman frame. Following the traditional way of arithmetical completeness proofs, we are going to embed this frame into T by means of a Solovay [9] style function g : W 0 and sentences Limw (w 2 W 0 ) which assert that w is the limit of g. This function will be defined in such a way that the following basic lemma holds: Lemma 3 a) T proves that g has a limit in W 0 , i.e. T W fLim r : r 2 W 0 g. b) If w 6= u, then T ....

R.M.Solovay, Provability interpretations of modal logic. Israel Journal of Mathematics 25 (1976), pp. 287-304.

.... [7] D1) if T 8xA then T 8xBew T (pAhxiq) D2) T 8x(Bew T (p(A B)hxiq) Bew T (pAhxiq) Bew T (pBhxiq) D3) T 8x(Bew T (pAhxiq) Bew T (pBew T (pAhxiq)hxiq) where we write pAhxiq for a term with a free variable x disquoting any occurrence of x in A (see Section 2) Solovay, [9], showed that the original propositional versions of the derivability conditions identify all the valid modal schematic properties of Bew T (the other modal axiom, the formalization of Lob s theorem, is derivable from (D1) D3) using the diagonalization lemma) Although the first order ....

Robert Solovay. Provability interpretations of modal logic. Israel Journal of Mathematics, 25:287--304, 1976.

....8 maximal ones of particular interest. 1 Introduction This paper discusses Magari algebras (often called diagonalizable algebras) over a finite number of generators. Magari algebras are the algebras corresponding to the provability logic L (GL in [2] PRL in [11] According to Solovay s theorem [12] on provability interpretations the theorems of the provability logic L are precisely those modal formulas that are provable in PA under arbitrary arithmetical interpretations (interpreting 2 as the formalized provability predicate in PA) Here, we are concerned with finitely generated Magari ....

.... 1 ; Delta Delta Delta ; p n for which there is a sequence of arithmetical sentences A 1 ; Delta Delta Delta ;A n such that an L formula is an L consequence of T iff is a theorem of PA in the arithmetical interpretation in which the atomic formula p i is interpreted as A i (see e.g. [12], 2] or [11] Written out: T axiomatizes an arithmetically interpreted theory: f j T L g = f j PA (A 1 ; Delta Delta Delta ; A n )g: The faithfully interpretable propositional theories T in L n (i.e. L restricted to the language of p 1 ; Delta Delta Delta ; p n ) are ....

R. Solovay, Provability Interpretations of Modal Logic, Israel Journal of Mathematics 25, 287--304, 1976.

....on T as well as on the chosen formalization of the proof predicate P roof T . We will be a bit ambiguous in this respect. When talking about the provability logic of a certain theory, we will always assume that a not to unusual proof predicate is fixed in advance. The famous article of Solovay [Solovay 76] may well be seen as the starting point of provability logic. In this paper Solovay proves that the the provability logic of PA is the logic now known as L or GL, consisting of the principles K;K4 and L, the tautologies of classical propositional logic and the rules of necessitation (A=2A) and ....

R. Solovay. Provability interpretations of modal logic. Israel Journal of mathematics 25, 1976.

....sentences, in a way that preserves logical form and maps modalities on formalizations of metamathematical relations. The so called arithmetical completeness theorems (ACTs) have played an extremely important role in provability logic ever since the first ones were showed by Solovay in 1976 [Sol76]. These theorems connect some notion of realization with a system of modal logic L and may be phrased as follows: Let A be a modal formula. Then every realization translates A into a true (provable) arithmetical sentence iff A is provable in L. Thus, ACTs tell us exactly what metamathematical ....

R.M. Solovay. Provability interpretations of modal logic. Israel Journal of Mathematics, 25:287--304, 1976.

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R.M. Solovay. Provability interpretations of modal logic. Israel Journal of Mathematics, 28:33--71, 1976. 20

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R. Solovay. Provability Interpretations of Modal Logic. Israel Journal of Mathematics, 25:287--304, 1970.

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Robert M. Solovay. Provability interpretations of modal logic. Israel Journal of Mathematics, 25:287--304, 1976. 10

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R. Solovay, Provability interpretations of modal logic, Israel Journal of Mathematics, vol. 25 (1976), pp. 287--304.

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R.M. Solovay. Provability interpretations of modal logic. Israel Journal Math., 25:287--304, 1976.

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R.M. Solovay. Provability interpretations of modal logic. Israel Journal of Mathematics, 25:287--304, 1976.

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