R. Montague, Syntactical treatments of modality with corollaries on reflection principles and finite axiomatizability, Acta Philosophica Fennica, vol. 16 (1963), pp. 153--168.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 Lemmon [65] Myhill [77] 78] Kripke [58] Montague [76], Novikov [79] Mints [73] Kuznetsov and Muravitsky [63] Goldblatt [42] Boolos [25] 26] Shapiro [88] 89] Buss [29] Artemov [5] and many others. However, there were no adequate Godelian provability semantics for S4 found 5 . Moreover, in [76] the problem was announced hopeless. In ....

....Myhill [77] 78] Kripke [58] Montague [76] Novikov [79] Mints [73] Kuznetsov and Muravitsky [63] Goldblatt [42] Boolos [25] 26] Shapiro [88] 89] Buss [29] Artemov [5] and many others. However, there were no adequate Godelian provability semantics for S4 found 5 . Moreover, in [76] the problem was announced hopeless. In this paper we give a complete solution to the problem of provability semantics for S4 (Problem 1 above) as it was understood by Godel in [39] 41] x4. Explicit vs. implicit approaches. The above difficulties with reading S4 modality 2F as 9xProof (x; y) ....

R. Montague, Syntactical treatments of modality with corollaries on reflection principles and finite axiomatizability, Acta Philosophica Fennica, vol. 16 (1963), pp. 153--168.

....the assertion that Consis PA is provable in PA, which is false according to the second Godel incompleteness theorem. In [26] cf. 59] Godel again acknowledged the problem of the provability semantics for S4. This issue was also addressed by Lemmon [44] Myhill [55] 56] Kripke [40] Montague [54], Mints [52] Kuznetsov Muravitsky [43] Goldblatt [27] Boolos [12] 14] Shapiro [62] 63] Buss [17] Artemov [1] and many others. However, the problem of finding an adequate provability semantics for S4 has remained open. A principal difficulty here is caused by the existential quantifier ....

R. Montague. "Syntactical treatments of modality with corollaries on reflection principles and finite axiomatizability", Acta Philosophica Fennica, 16, pp. 153-168, 1963.

....By necessitation, S4 derives 2(2F F ) The latter formula expresses the assertion that Consis PA is provable in PA, which contradicts the Second Godel Incompleteness Theorem. The issue of a provability model for S4 was studied by Godel [18] Lemmon [29] Myhill [39] 40] Kripke [26] Montague [38], Mints [35] Kuznetsov Muravitskii [28] Goldblatt [19] Boolos [9] 10] Shapiro [43] 44] Buss [12] Artemov [1] and many others. However, the problem of a formal provability semantics for S4 has remained open. A principal difficulty here is caused by the existential quantifier over proofs ....

R. Montague. "Syntactical treatments of modality with corollaries on reflection principles and finite axiomatizability", Acta Philosophica Fennica, 16, pp. 153-168, 1963.

....of a language and its semantic representation. If this relation is recursive, paradoxical consequences will ensue. The method I apply is a diagonal argument of the kind that has been current in logic and recursion theory for about forty years. The version that I use is adapted from Montague. [14] If the reasoning is correct, then the project of constructing a comprehensive cognitive semantics is comparable to that of squaring the circle with compass and straightedge. Summary of the argument: if a theory conforming to the psychogogical ideal were correct, and intelligible, an ideal ....

....sentences to thoughts, the thoughts cannot be true and false. If this difficulty could be confined to truth, perhaps we could live with this conclusion and banish truth from psychological semantics. But generalizations of Tarski s theorem on truth erode this alternative. The results of Montague [14] show that the above difficulty with truth also infects the capacity of such semantic theories to deal with certainty, 13 triviality, probability, and idealized knowledge. Where Gn(OE) is the Godel number of OE, Montague showed that there can be no consistent language containing an adequate ....

Montague, R., "syntactical treatments of modality, with corollaries on reflexion principles and finite axiomatizability."In Montague [13], pp. 286-302. 11

....By necessitation, S4 derives 2(2F F ) The latter formula expresses the assertion that Consis PA is provable in PA, which contradicts the Second Godel Incompleteness Theorem. The issue of a provability model for S4 was studied by Godel [13] Lemmon [20] Myhill [27] 28] Kripke [18] Montague [26], Mints [25] Kuznetsov Muravitskii [19] Goldblatt [14] Boolos [7] 8] Shapiro [30] 31] Buss [9] Artemov [1] and many others. However, the problem of a formal provability semantics for S4 has remained open. A principal difficulty here is caused by the existential quantifier over proofs in ....

R. Montague. "Syntactical treatments of modality with corollaries on reflection principles and finite axiomatizability", Acta Philosophica Fennica, 16, pp. 153-168, 1963.

....By necessitation, S4 derives 2(2 ) The latter formula expresses the assertion that Consis PA is provable in PA, which contradicts the second Godel incompleteness theorem. The issue of a provability model for S4 was studied by Godel [16] Lemmon [28] Myhill [35] 36] Kripke [25] Montague [34], Mints [33] Kuznetsov Muravitskii [27] Goldblatt [17] Boolos [9] 10] Shapiro [38] 39] Buss [11] Artemov [1] and many others. However, the problem of a formal provability semantics for S4 has remained open. A principal difficulty here is caused by the existential quantifier over proofs ....

R. Montague. "Syntactical treatments of modality with corollaries on reflection principles and finite axiomatizability", Acta Philosophica Fennica, 16, pp. 153-168, 1963.

.... discussed in [Minsky 1962] The construction of meta circular interpreters has been studied in great detail by Steele and Sussman [1978b] Related studies in the foundations of self descriptive language systems have been carried out by Backus [1973] Brown [1977] Scott [1973] Smullyan [1957] Montague [1963] and Kripke [1975] Hayes has long advocated careful formalization of the control processes of problem solving systems. Towards this end, he has been working for several years on the GOLUX system, a self descriptive theorem prover, and on related representational problems. Hayes 1970, 1971a, ....

Richard Montague, "Syntactical Treatments of Modality, with Corollaries on Reflection Principles and Finite Axiomatizability," Acta Philosophica Fennica, V. 16 (1963), pp. 153-167.

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R. Montague, Syntactical treatments of modality with corollaries on reflection principles and finite axiomatizability, Acta Philosophica Fennica, vol. 16 (1963), pp. 153--168.

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