On superintuitionistic logics as fragments of proof logic extensions

Кузнецов, А. В.; Muravitsky, Alexei

doi:10.1007/bf01881551

Cited by 32 publications

(21 citation statements)

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“…4 Further examples of this include the proof-intuitionistic modal logic of Kuznetsov and Muravitsky [30], which is related to the provability interpretation of modality, and its sublogic studied by Esakia [11], in which the modal connective is interpreted by the dual of the topological derivative operation. These logics have a box-like modality ∆ with ϕ → ∆ϕ as an axiom.…”

Section: Nuclei Under Macneille Completionmentioning

confidence: 99%

Cover semantics for quantified lax logic

Goldblatt¹

2010

Journal of Logic and Computation

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Lax modalities occur in intuitionistic logics concerned with hardware verification, the computational lambda calculus, and access control in secure systems. They also encapsulate the logic of Lawvere-Tierney-Grothendieck topologies on topoi. This paper provides a complete semantics for quantified lax logic by combining the Beth-Kripke-Joyal cover semantics for first-order intuitionistic logic with the classical relational semantics for a "diamond" modality. The main technique used is the lifting of a multiplicative closure operator (nucleus) from a Heyting algebra to its MacNeille completion, and the representation of an arbitrary locale as the lattice of "propositions" of a suitable cover system. In addition, the theory is worked out for certain constructive versions of the classical logics K and S4.An alternative completeness proof is given for (non-modal) first-order intuitionistic logic itself with respect to the cover semantics, using a simple and explicit Henkin-style construction of a characteristic model whose points are principal theories rather than prime saturated ones.The paper provides further evidence that there is more to intuitionistic modal logic than the generalisation of properties of boxes and diamonds from Boolean modal logic.

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Section: Nuclei Under Macneille Completionmentioning

confidence: 99%

Cover semantics for quantified lax logic

Goldblatt¹

2010

Journal of Logic and Computation

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“…(iv) It follows from (ii), (iii) and equation (S1) that in finite Heyting algebras there is n ∈ N such that S (n) (0) = 1. See also Kuznetsov and Muravitsky (1986) and Muravitsky (1990). (v) It follows from the dual categorical equivalence between SH n and SE n and the previous remarks that lpFP is a full subcategory of SE, and lpFP n is a full subcategory of SE n for every n. Moreover, lpFP is dually equivalent to the full subcategory of SH whose objects are finite KMalgebras, and lpFP n is equivalent to the full subcategory of SH n whose objects are finite KM-algebras.…”

Section: Km-algebrasmentioning

confidence: 84%

“…(iii) It was observed in Muravitsky (1990) and also on p. 87 of Kuznetsov and Muravitsky (1986) that the successor exists in every finite Heyting algebra. (iv) It follows from (ii), (iii) and equation (S1) that in finite Heyting algebras there is n ∈ N such that S (n) (0) = 1.…”

Section: Km-algebrasmentioning

confidence: 95%

On products of posets and coproducts of KM-algebras

Castiglioni

Martín

2015

Soft Comput

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“…We recall that the proof-intuitionistic logic KM (=Kuznetsov-Muravitsky [7]) is the Heyting propositional calculus HC enriched by 2 as Prov modality satisfying the following conditions:…”

Section: Introductionmentioning

confidence: 99%