Logic and Structure

Dalen, Dirk van

doi:10.1007/978-3-662-08402-1

Cited by 120 publications

(127 citation statements)

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“…In higher order logic, the situation is similar to classical logic [41]. One can define all the connectors and quantifiers from two of them, even in intuitionistic logic.…”

Section: Other Connectors and Quantifiersmentioning

confidence: 92%

Mechanizing common knowledge logic using COQ

Lescanne

2006

Ann Math Artif Intell

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This paper presents a formalization in Coq of Common Knowledge Logic and checks its adequacy on case studies. Those studies allow exploring experimentally the proof-theoretic side of Common Knowledge Logic. This work is original in that nobody has considered Higher Order Common Knowledge Logic from the point of view of proofs performed on a proof assistant. As a matter of facts, it is experimental by nature as it tries to draw conclusions from experiments.

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“…In higher order logic, the situation is similar to classical logic [41]. One can define all the connectors and quantifiers from two of them, even in intuitionistic logic.…”

Section: Other Connectors and Quantifiersmentioning

confidence: 92%

Mechanizing common knowledge logic using COQ

Lescanne

2006

Ann Math Artif Intell

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“…The forcing relation is defined as usual. For the basic results on Kripke models and intuitionistic logic we refer the reader to [6] and [3]. A standard reference on classical model theory is [2].…”

Section: Preliminariesmentioning

confidence: 99%

Preservation theorems for Kripke models

Moniri¹,

Zaare

2009

Mathematical Logic Qtrly

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There are several ways for defining the notion submodel for Kripke models of intuitionistic first-order logic. In our approach a Kripke model A is a submodel of a Kripke model B if they have the same frame and for each two corresponding worlds Aα and Bα of them, Aα is a subset of Bα and forcing of atomic formulas with parameters in the smaller one, in A and B, are the same. In this case, B is called an extension of A. We characterize theories that are preserved under taking submodels and also those that are preserved under taking extensions as universal and existential theories, respectively. We also study the notion elementary submodel defined in the same style and give some results concerning this notion. In particular, we prove that the relation between each two corresponding worlds of finite Kripke models A B is elementary extension (in the classical sense).

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“…That is, a morphism in this category is a classical homomorphism in the sense of [4] and [2]. Let A be an arbitrary small category (in practice, A is often taken to be a small poset category).…”

Section: Kripke Modelsmentioning

confidence: 99%

Kripke submodels and universal sentences

Ellison

Fleischmann

McGinn

et al. 2007

Mathematical Logic Qtrly

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We define two notions for intuitionistic predicate logic: that of a submodel of a Kripke model, and that of a universal sentence. We then prove a corresponding preservation theorem. If a Kripke model is viewed as a functor from a small category to the category of all classical models with (homo)morphisms between them, then we define a submodel of a Kripke model to be a restriction of the original Kripke model to a subcategory of its domain, where every node in the subcategory is mapped to a classical submodel of the corresponding classical model in the range of the original Kripke model. We call a sentence universal if it is built inductively from atoms (including and ⊥) using ∧, ∨, ∀, and →, with the restriction that antecedents of → must be atomic. We prove that an intuitionistic theory is axiomatized by universal sentences if and only if it is preserved under Kripke submodels. We also prove the following analogue of a classical model-consistency theorem: The universal fragment of a theory Γ is contained in the universal fragment of a theory ∆ if and only if every rooted Kripke model of ∆ is strongly equivalent to a submodel of a rooted Kripke model of Γ. Our notions of Kripke submodel and universal sentence are natural in the sense that in the presence of the rule of excluded middle, they collapse to the classical notions of submodel and universal sentence.

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Logic and Structure

Cited by 120 publications

References 0 publications

Mechanizing common knowledge logic using COQ

Mechanizing common knowledge logic using COQ

Preservation theorems for Kripke models

Kripke submodels and universal sentences

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