A. Chagrov and M. Zakharyaschev, Modal logic, Oxford Science Publications, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....presented in for instance [4] as it happens, however, we could manage to actually reduce the correspondence result to the classical case. This reduction is based on semantic ideas and related to the G odel translation of intuitionistic logic into the modal logic S4, cf. Chagrov and Zakharyaschev [10] for an overview. When it comes to canonicity for Sahlqvist sequents, our proof method takes advantage of developments within the algebraic theory of canonical extensions of algebraic lattice expansions (that is, lattices expanded with further operations) This eld originates with the seminal ....

....Sambin and Vaccaro [49] for details. Our result will be based on a reduction to this classical case; the heart of our proof is a translation from distributive modal logic to classical boolean modal logic, analogous to the well known G odel translation from intuitionistic logic to modal logic, cf. [10]. The semantic intuition behind this translation is simply that by considering the ordering of a distributive modal frame F as just another binary relation on the domain, we may treat F as a frame for classical (poly )modal logic. This idea already has a history in the literature on ....

Chagrov, A. and Zakharyaschev, M., Modal logic, Oxford Logic Guides 35, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1997.

.... Heyting and Kolmogorov made Brouwer s definition of intuitionistic truth explicit, though informal, by introducing what is now known as the Brouwer Heyting Kolmogorov (BHK) semantics ( 48] 49] 52] The BHK semantics is widely recognized as the intended semantics for intuitionistic logic ([30], 31] 38] 56] 67] 71] 94] 95] 97] 98] 101] 104] Its description uses the unexplained primitive notions of construction and proof (Kolmogorov used the term problem solution for the latter) It stipulates that ffl a proof of AB consists of a proof of A and a proof of B, ....

....Int itself. Here is the list of major known classical semantics for intuitionistic logic 2 . 1. Algebraic semantics (Birkhoff, 1935, 24] 2. Topological semantics (Stone, 1937; Tarski, 1938, 69] 2 Comprehensive surveys of these and other semantics for intuitionistic logic can be found in [30], 85] 95] 4 SERGEI N. ARTEMOV 3. Realizability semantics (Kleene, 1945, 51] 4. Beth models (1956, 22] 5. Dialectica Interpretation (Godel, 1958, 40] 6. Curry Howard isomorphism (1958, 32] 7. Medvedev s logic of problems (1962, 71] 8. Kripke models (1965, 59] 9. ....

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A. Chagrov and M. Zakharyaschev, Modal logic, Oxford Science Publications, 1997.

....In 1932 Kolmogorov ( 16] gave an informal description of the calculus of problems in classical mathematics and conjectured that it coincides with intuitioinistic propositional logic Int. Kleene realizability [15] Medvedev finite problems [23] and its variants ( 36] 37] are regarded (cf. 34] [10], 36] 37] as formalizations of Kolmogorov s calculus of problems. However, they give only necessary conditions for Int, each of them realizes some formulas not derivable in Int. In 1933 Godel ( 12] defined Int on the basis of the notion of proof in a classical mathematical system, where ....

....of S4 admits a decoding via LP as a statement about specific proofs. Let k(F ) denote a translation of an intuitionistic formula F into the plain modal language which puts the prefix 2 in front of all atoms and implications in F . It is well known that Int F iff S4 k(F ) see, for example, [10]) 3.5 Corollary. Realization of intuitionistic logic) For any Int formula F Int F , LP (k(F ) r for some realization r: 3.6 Corollary. Arithmetical completeness of Int. Int F iff there is a realization r and constant specification CS such that k(F ) r CS valid. Kolmogorov s ....

A. Chagrov and M. Zakharyaschev, Modal Logic, Oxford Science Publications, 1997.

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A. Chagrov and M. Zakharyaschev, Modal logic, Oxford Science Publications, 1997.

....logic. 1 In 1931 34 Heyting and Kolmogorov made Brouwer s definition of intuitionistic truth explicit, though informal, by introducing what is now known as Brouwer Heyting Kolmogorov (BHK) semantics. BHK semantics is widely recognized as the intended semantics for intuitionistic logic ([18], 19] 20] 24] 37] 47] 50] 72] 73] 74] 75] 76] BHK semantics gives an informal explanation of the truth of intuitionistic connectives. A statement is true if it has a proof, and a proof of a logically compound statement is given in terms of the proofs of its components. The description ....

....a proof of A, would return a proof of . This semantics was partially introduced by Heyting [29] clauses for conjunction and disjunction) and by Kolmogorov [34] clauses for implication and negation) The above formulation of BHK semantics appeared in [30] For further comments one may consult [18], 20] 24] 69] 72] 73] 74] The natural problem of formalizing BHK semantics and establishing the completeness of Int with respect to this semantics remained open until recently despite a long history of studies in this area (see section 3 of this paper) To be sure, there are many models of ....

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A. Chagrov and M. Zakharyaschev, Modal Logic, Oxford Science Publications, 1997.

....the relationship between proofs and types, and subsumes the calculus, modal calculus and combinatory logic. Introduction The intended meaning of the intuitionistic logic was informally explained first in terms of operations on proofs due to Brouwer, Heyting and Kolmogorov (cf. 48] 49] [13]) This interpretation is widely known as the BHK semantics of intuitionistic logic. However, despite some similarities in the informal description of the functions assigned to the intuitionistic connective, the Brouwer Heyting semantics and the Kolmogorov semantics have fundamentally different ....

....Technical Report CFIS 98 06, Cornell University y 627 Rhodes Hall, Cornell University, Ithaca NY, 14853 U.S.A. email:artemov hybrid.cornell.edu and Moscow University, Russia. 1 Kleene realizability [22] Medvedev finite problems [33] and its variants ( 50] 51] are regarded (cf. 48] [13], 50] 51] as formalizations of Kolmogorov s calculus of problems. However, they give only necessary conditions for Int, each of them realizes some formulas not derivable in Int. A formalization of the BHK semantics suggested by Kreisel in [25] turned out to be based on an inconsistent theory ....

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A. Chagrov and M. Zakharyaschev, Modal Logic, Oxford Science Publications, 1997. 56

....modal language and (ii) is invariant with respect to the choice of a proof system is realized by a proof polynomial. Introduction The intended meaning of the intuitionistic logic was informally explained first in terms of operations on proofs due to Brouwer, Heyting and Kolmogorov (cf. 43] 44] [12]) This interpretation is widely known as the BHK semantics of intuitionistic logic. However, despite some similarities in the informal description of the functions assigned to the intuitionistic connective, the Heyting semantics and the Kolmogorov semantics have fundamentally different ....

....We call this sort of interpretation of Int classical BHK semantics. Classical realizabilities: Kleene realizability [19] function realizability [20] modified realizability [24] Medvedev s calculus of finite problems [31] and its variants, give conditions necessary but not sufficient for Int(cf.[12], 45] 46] 47] Each of them realizes some formulas Department of Mathematics, Cornell University, email:artemov math.cornell.edu and Moscow University, Russia. not derivable in Int. A formalization of the BHK semantics suggested by Kreisel in [23] turned out to be based on an inconsistent ....

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A. Chagrov and M. Zakharyaschev, Modal Logic, Oxford Science Publications, 1997.

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