Craig's theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras

Maksimova, L. L.

doi:10.1007/bf01670006

Cited by 148 publications

(102 citation statements)

References 10 publications

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“…Intriguingly, the modal logics K4 and S4 do not have uniform interpolation [1,3]. As there are only seven propositional intermediate logics with interpolation [5], the number of intermediate logics with uniform interpolation is necessarily bounded by that number. Ghilardi and Zawadowski showed that there are exactly that many [4].…”

Section: Introductionmentioning

confidence: 99%

Uniform interpolation and sequent calculi in modal logic

Iemhoff

2018

Arch. Math. Logic

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A method is presented that connects the existence of uniform interpolants to the existence of certain sequent calculi. This method is applied to several modal logics and is shown to cover known results from the literature, such as the existence of uniform interpolants for the modal logic K. New is the result that KD has uniform interpolation. The results imply that for modal logics K4 and S4, which are known not to have uniform interpolation, certain sequent calculi cannot exist.

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Section: Introductionmentioning

confidence: 99%

Uniform interpolation and sequent calculi in modal logic

Iemhoff

2018

Arch. Math. Logic

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“…Publications of particular relevance to our discussion include Bacsich [4], Czelakowski and Pigozzi [14], Gabbay and Maksimova [22], Galatos and Ono [24], Kihara and Ono [43,44], Madarasz [45], Maksimova [46][47][48], Montagna [53], Pierce [61], Pigozzi [62], Powell and Tsinakis [63][64][65][66], and Wroński [73,74]. We defer more precise historical and bibliographical details to the appropriate points in the text.…”

Section: Introductionmentioning

confidence: 99%

Amalgamation and interpolation in ordered algebras

2014

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The first part of this paper provides a comprehensive and self-contained account of the interrelationships between algebraic properties of varieties and properties of their free algebras and equational consequence relations. In particular, proofs are given of known equivalences between the amalgamation property and the Robinson property, the congruence extension property and the extension property, and the flat amalgamation property and the deductive interpolation property, as well as various dependencies between these properties. These relationships are then exploited in the second part of the paper in order to provide new proofs of amalgamation and deductive interpolation for the varieties of lattice-ordered abelian groups and MV-algebras, and to determine important subvarieties of residuated lattices where these properties hold or fail. In particular, a full description is given of all subvarieties of commutative GMV-algebras possessing the amalgamation property.

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“…Moreover, it is the only superintuitionistic logic which has both these properties. KC has both finite model property and Craig's interpolation property (as well as stronger Lyndon interpolation property), see [14] and [13]. The weak law of excluded middle is also conservative over intuitionistic logic, meaning we can substitute only propositional variables in it when inferring a formula [13].…”

Section: Introductionmentioning

confidence: 99%

On classical behavior of intuitionistic modalities

Drobyshevich

2014

LLP

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Abstract. We study connections between four types of modal operators  necessity, possibility, un-necessity and impossibility  over intuitionitstic logic in terms of compositions of these modal operators with intuitionistic negation. We investigate which basic compositions, i.e. compositions of the form ¬δ, δ¬ or ¬δ¬, yield modal operators of the same type over intuitionistic logic as over classical logic. We say that such compositions behave classically. We study which modal properties correspond to each basic compositions behaving classically over intuitionistic logic and also prove that KC constitutes the smallest superintuitionistic logic over which all basic compositions behave classically.

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Craig's theorem in superintuitionistic logics and amalgamable varieties of pseudo-boolean algebras

Cited by 148 publications

References 10 publications

Uniform interpolation and sequent calculi in modal logic

Uniform interpolation and sequent calculi in modal logic

Amalgamation and interpolation in ordered algebras

On classical behavior of intuitionistic modalities

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