The formalization of interpretability, Studia Logica, 50, pp. 81--106.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Schroder [50] A fragment of this calculus was axiomatized by Alfred Tarski [52] Tarski s axiomatization, in a slightly altered form, became the definition of relation algebras [6] 25] 27] For further introductory and historical material on relation algebras see [6] 23] 24] 27] 32] [33], and [54] This section contains just enough basic definitions and results for the applications given later. Most of the material in this section can be found in [6] or [27] Definition 23. A relation algebra is an algebraic structure of the form where hA; i is a Boolean algebra, ....

, The origin of relation algebras in the development and axiomatization of the calculus of relations, Studia Logica 50 (3/4) (1991), 421--455.

....recognize reasoning about more complex notions like interpretability where arithmetic can be shown to reason adequately, and also to strengthen Solovay s results directly. Extensive overviews on the subject can be found in Boolos [1993b] and Smory nski [1985] a short history in Boolos and Sambin [1991]. Let us proceed somewhat farther in formulating Solovay s theorems, and call an arithmetic realization of the language of modal logic (see section 2) into the language of the arithmetic theory T ( Sigma 1 sound and extending I Sigma 1 , sometimes I Delta 0 ) a mapping that commutes with the ....

....be valid in all transitive models of ZF , which happens to be a proper extension of L (see Boolos [1993b] If one is somewhat careful, Solovay s construction can be adapted to show that Solovay s theorems hold for all (r.e. sound) extensions of I Delta 0 EXP (de Jongh, Jumelet and Montagna [1991]) The two theorems formulated earlier in this section can be similarly generalized. However, the most intriguing question, whether 488 G. Japaridze and D. de Jongh Solovay s theorems hold for essentially weaker theories, such as Buss s S 1 2 or even S 2 , remains open. This problem was ....

[Article contains additional citation context not shown here]

The formalization of interpretability, Studia Logica, 50, pp. 81--106.

....well as the one in the language of 4, actually coincides with L. Using the uniform version of Solovay s theorem, Smory nski [1985] showed that CS is the minimal bimodal provability logic, i.e. it coincides with PRL T;U for a certain pair of finite extensions T; U of Peano arithmetic. Beklemishev [1992] showed that there is even a pair of provability predicates for Peano arithmetic itself for which the corresponding bimodal provability logic coincides with CS. Such predicates can be called independent in the sense that they know as little about each other as is possible in principle. It should ....

....In his review of these and some consecutive papers Beklemishev [1993b] rightly remarked that the changing of the orders of the proofs in Guaspari Solovay style interferes with the order induced by the function g and makes some of the results somewhat less clear than one might wish. Montagna [1992] applied the results on provable fixed points in a study of metamathematical rules, i.e. rules like Pr T (p ff q) ff that can be considered as realizations of modal logical rules (in case: 2A=A) He classified these rules into two types: rules giving only polynomial speed up in proofs in ....

[Article contains additional citation context not shown here]

The logic of linear tolerance, Studia Logica, 51, pp. 249--277.

....The two facets mentioned above were, one might say, integrated by showing that propositional reasoning about the formalized provability predicate is decidable and can be adequately described in arithmetic itself. And in the same period the de Jongh Sambin fixed point theorem (see Sambin [1976], Smory nski [1978,1985] was proved for modal logical systems with the provability interpretation in mind. Since that time the main achievements have been to show that similar results mostly fail for predicate logic, to recognize reasoning about more complex notions like interpretability where ....

....B A(B) and L C A(C) then L B C . 4.3. Theorem. Existence of fixed points) If p occurs only boxed in A(p) then there exists a formula B, not containing p and otherwise containing only variables of A(p) such that L B A(B) After the original proofs by de Jongh and Sambin (see Sambin [1976], Smory nski [1978,1985] and, for the first proof of uniqueness, Bernardi [1976] many other, different, proofs have been given for the fixed point theorems, syntactical as well as semantical ones, the latter e.g. in Gleit and Goldfarb [1990] It is also worthwhile to remark that theorem 4.3 ....

[Article contains additional citation context not shown here]

The uniqueness of the fixed-point in every diagonalizable algebra, Studia Logica, 35, pp. 335--343. G. Boolos

....recognize reasoning about more complex notions where arithmetic can be shown to reason adequately, like interpretability, and also to strengthen Solovay s results directly. Extensive overviews on the subject can be found in Boolos [1993b] and Smory nski [1985] a short history in Boolos and Sambin [1991]. Let us proceed somewhat farther in formulating Solovay s theorems. Let us call an arithmetic realization of the language of modal logic (see section 2) into the language of the arithmetic theory T ( Sigma 1 sound and extending I Sigma 1 , sometimes I Delta 0 ) a mapping that commutes with ....

.... to be valid in all transitive models of ZF , which happens to be a proper extension of L (see Boolos [1993b] If one is somewhat careful, Solovay s construction can be adapted to show that Solovay s theorems hold for all (r.e. sound) extensions of I Delta 0 EXP (de Jongh, Jumelet and Montagna [1991]) The two theorems formulated earlier in this section can be similarly generalized. However, the most intriguing question, whether Solovay s theorems hold for essentially weaker theories, such as Buss s S 1 2 or even S 2 , remains open. This problem was thoroughly investigated by Berarducci and ....

[Article contains additional citation context not shown here]

The formalization of interpretability, Studia Logica, 50, pp. 81--106.

....recognize reasoning about more complex notions where arithmetic can be shown to reason adequately, like interpretability, and also to strengthen Solovay s results directly. Extensive overviews on the subject can be found in Boolos [1993b] and Smory nski [1985] a short history in Boolos and Sambin [1991]. Let us proceed somewhat farther in formulating Solovay s theorems. Let us call an arithmetic realization of the language of modal logic (see section 2) into the language of the arithmetic theory T ( Sigma 1 sound and extending I Sigma 1 , sometimes I Delta 0 ) a mapping that commutes with ....

.... to be valid in all transitive models of ZF , which happens to be a proper extension of L (see Boolos [1993b] If one is somewhat careful, Solovay s construction can be adapted to show that Solovay s theorems hold for all (r.e. sound) extensions of I Delta 0 EXP (de Jongh, Jumelet and Montagna [1991]) The two theorems formulated earlier in this section can be similarly generalized. However, the most intriguing question, whether Solovay s theorems hold for essentially weaker theories, such as Buss s S 1 2 or even S 2 , remains open. This problem was thoroughly investigated by Berarducci and ....

[Article contains additional citation context not shown here]

On the proof of Solovay's theorem, Studia Logica, 50, pp. 51--70. D. H. J. de Jongh and F. Montagna

....well as the one in the language of 4, actually coincides with L. Using the uniform version of Solovay s theorem, Smory nski [1985] showed that CS is the minimal bimodal provability logic, i.e. it coincides with PRL T;U for a certain pair of finite extensions T; U of Peano arithmetic. Beklemishev [1992] showed that there is even a pair of provability predicates for Peano arithmetic itself for which the corresponding bimodal provability logic coincides with CS. Such predicates can be called independent in the sense that they know as little about each other as is possible in principle. It should ....

....In his review of these and some consecutive papers Beklemishev [1993b] rightly remarked that the changing of the orders of the proofs in Guaspari Solovay style interferes with the order induced by the function g and makes some of the results somewhat less clear than one might wish. Montagna [1992] applied the results on provable fixed points in a study of metamathematical rules, i.e. rules like Pr T (p ff q) ff that can be considered as realizations of modal logical rules (in case: 2A=A) He classified these rules into two types: ones giving only polynomial speed up in proofs in ....

[Article contains additional citation context not shown here]

The logic of linear tolerance, Studia Logica, 51, pp. 249--277.

....formalized version of the second incompleteness theorem, i.e. if it is provable in PA that PA is consistent, then PA is inconsistent, is provable in PA itself. The area here called the logic of provability arose in the seventies when, as one might say, these two facets were integrated by Solovay [1976], who showed that propositional reasoning about the formalized provability predicate is decidable and can be adequately described in arithmetic itself. In the same period the de Jongh Sambin fixed point theorem (Sambin [1976] Smory nski [1978,1985] was proved for modal logical systems with this ....

....when, as one might say, these two facets were integrated by Solovay [1976] who showed that propositional reasoning about the formalized provability predicate is decidable and can be adequately described in arithmetic itself. In the same period the de Jongh Sambin fixed point theorem (Sambin [1976], Smory nski [1978,1985] was proved for modal logical systems with this interpretation in mind. Since that time the main achievements have been to show that similar results mostly fail for predicate logic, to recognize reasoning about more complex notions where arithmetic can be shown to reason ....

[Article contains additional citation context not shown here]

The uniqueness of the fixed-point in every diagonalizable algebra, Studia Logica, 35, pp. 335--343. G. Boolos

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The formalization of interpretability, Studia Logica, 50, pp. 81--106.

No context found.

The logic of linear tolerance, Studia Logica, 51, pp. 249--277.

No context found.

The uniqueness of the fixed-point in every diagonalizable algebra, Studia Logica, 35, pp. 335--343. G. Boolos

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H. Deutsch. The completeness of S. Studia Logica, 38(2):137-147, 1979.

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