Duality and Universal Models for the Meet-Implication Fragment of IPC

Bezhanishvili, Nick; Coumans, Dion; Gool, Samuel J. van; Jongh, Dick de

doi:10.1007/978-3-662-46906-4_7

Cited by 8 publications

(7 citation statements)

References 15 publications

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“…Since f preserves the operations in {0, ∧, →}, f ↾ A c is a Brouwerian semilattice homomorphism. By Lemma 2.4 in [20] (see also [4]) Brouwerian semilattice homomorphisms preserve existing join, hence for all a, b…”

Section: A Birkhoff-like Theorem For Dependence Algebrasmentioning

confidence: 85%

On Intermediate Inquisitive and Dependence Logics: An Algebraic Study

Quadrellaro¹

2021

Preprint

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This article provides an algebraic study of intermediate inquisitive and dependence logics. While these logics are usually investigated using team semantics, here we introduce an alternative algebraic semantics and we prove it is complete for all intermediate inquisitive and dependence logics. To this end, we define inquisitive and dependence algebras and we investigate their model-theoretic properties. We show they are elementary structures axiomatised by Horn sentences and we use our completeness result to state a version of algebraisability for inquisitive and dependence logics. We then focus on finite, core-generated, well-connected inquisitive and dependence algebras: we show they witness the validity of formulas true in inquisitive algebras, and of formulas true in well-connected dependence algebras. Finally, we obtain Representation Theorems for finite, core-generated, well-connected, inquisitive and dependence algebras and we prove some results connecting team and algebraic semantics. * I am very grateful to Fan Yang for many helpful conversations and for her constant support and advice.

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Section: A Birkhoff-like Theorem For Dependence Algebrasmentioning

confidence: 85%

On Intermediate Inquisitive and Dependence Logics: An Algebraic Study

Quadrellaro¹

2021

Preprint

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“…Condition (ii) also implies that no two distinct worlds in the n-universal model for Φ are Φ-equivalent. For modal logics and IPC universal models were thoroughly investigated by a number of authors [14,24,1,23,19] (see [8, §8] and [3, §3] for an overview), and results for fragments of IPC can be found in [15,21,5,7].…”

Section: Universal Modelsmentioning

confidence: 99%

“…This implies that there is a close relationship with the n-canonical model (also known as n-Henkin model). Usually the n-universal model is the "upper part" of the n-Henkin model (see [3,19]), or even, in the case of locally finite fragments, isomorphic to it (see [5,7]). The set of all NNILformulas in an arbitrary number of variables do not have a Lindenbaum-Tarski algebra as such since although they do form a distributive lattice, they are not closed under implication.…”

Section: Universal Modelsmentioning

confidence: 99%

“…(1). In [5] the [∧, →]-fragment of IPC was studied using finite duality for distributive lattices and universal models leading to results about how the universal model for that fragment fits into the overall universal model of IPC, to results about interpolation, and to the relationship of the subframe formulas connected to that fragment with the Jankov-de Jongh formulas. A similar investigation of the NNIL-fragment seems indicated, and should also throw light on the intriguing relationship between those two fragments.…”

Section: Open Problemsmentioning

confidence: 99%

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NNIL-formulas revisited: universal models and finite model property

Ilin

Jongh

Yang

2019

Preprint

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NNIL-formulas, introduced by Visser in 1983-1984 in a study of Σ 1 -subsitutions in Heyting Arithmetic, are intuitionistic propositional formulas that do not allow nesting of implication to the left. The main results about these formulas were obtained in a paper of 1995 by Visser and others. It was shown that NNIL-formulas are exactly the formulas preserved under taking submodels of Kripke models. In the present paper an observation by Bezhanishvili and de Jongh of NNILformulas as the formulas backwards preserved by monotonic maps of Kripke models is applied to construct a universal model for NNIL. This universal model shows that NNIL-formulas are also exactly the ones that are backwards preserved by monotonic maps. The methods developed in constructing the universal model are used in this paper in a new direct proof that these logics have the finite model property.

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“…The meet-implication fragment of intuitionistic propositional logic has been studied by itself in [Cur63,Section 4.C] and more recently in e.g. [BCGJ15,FM14]. Its appeal partly lies in the fact that the category of meet-semilattices is finitely generated, whereas the category of Heyting algebras is not.…”

Section: Introductionmentioning

confidence: 99%

Modal meet-implication logic

Groot¹,

Pattinson²

2022

Logical Methods in Computer Science

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We extend the meet-implication fragment of propositional intuitionistic logic with a meet-preserving modality. We give semantics based on semilattices and a duality result with a suitable notion of descriptive frame. As a consequence we obtain completeness and identify a common (modal) fragment of a large class of modal intuitionistic logics. We recognise this logic as a dialgebraic logic, and as a consequence obtain expressivity-somewhere-else. Within the dialgebraic framework, we then investigate the extension of the meet-implication fragment of propositional intuitionistic logic with a monotone modality and prove completeness and expressivity-somewhere-else for it.

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Duality and Universal Models for the Meet-Implication Fragment of IPC

Cited by 8 publications

References 15 publications

On Intermediate Inquisitive and Dependence Logics: An Algebraic Study

On Intermediate Inquisitive and Dependence Logics: An Algebraic Study

NNIL-formulas revisited: universal models and finite model property

Modal meet-implication logic

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