Frame Based Formulas for Intermediate Logics

Bezhanishvili, Nick

doi:10.1007/s11225-008-9147-0

Cited by 10 publications

(21 citation statements)

References 24 publications

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“…In order to axiomatize stable si-logics, we recall the theory of frame-based formulas of [7,8]. Although the theory was developed for frames, as was pointed out in [5], dualizing frame-based formulas yields algebra-based formulas that we use here.…”

Section: Stable Superintuitionistic Logicsmentioning

confidence: 99%

“…Let ≤ be a reflexive and transitive relation on the class HA si of subdirectly irreducible Heyting algebras. In [7,8] the class is restricted to (the dual spaces of) finitely generated subdirectly irreducible Heyting algebras, but for our purposes this restriction is not essential. For A, B ∈ HA si , write A < B if A ≤ B and B ≤ A.…”

Section: Stable Superintuitionistic Logicsmentioning

confidence: 99%

“…The following criterion of axiomatizability of si-logics by algebra-based formulas follows from [8,Thm. 3.9] (see also [7,Thm.…”

Section: Stable Superintuitionistic Logicsmentioning

confidence: 99%

“…A similar argument is used in [10,Thm. 11.19] for proving that there are continuum many subframe and cofinal subframe si-logics, as well as in [8,Thm. 3.14] (see also [7,Thm.…”

Section: Stable Superintuitionistic Logicsmentioning

confidence: 99%

“…Indeed, as follows from [11], if L is a si-logic whose corresponding variety V L of Heyting algebras is locally finite (such logics are called locally tabular), then each extension of L is axiomatized over L by Jankov formulas. In fact, L itself is axiomatized over IPC by Jankov formulas [24,7,8].…”

Section: Introductionmentioning

confidence: 99%

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Locally Finite Reducts of Heyting Algebras and Canonical Formulas

Bezhanishvili¹,

Bezhanishvili²

2017

Notre Dame J. Formal Logic

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Abstract. The variety of Heyting algebras has two well-behaved locally finite reducts, the variety of bounded distributive lattices and the variety of implicative semilattices. The variety of bounded distributive lattices is generated by the →-free reducts of Heyting algebras while the variety of implicative semilattices by the ∨-free reducts. Each of these reducts gives rise to canonical formulas that generalize Jankov formulas and provide an axiomatization of all superintuitionistic logics (si-logics for short).The ∨-free reducts of Heyting algebras give rise to the (∧, →)-canonical formulas that were studied in [3]. Here we introduce the (∧, ∨)-canonical formulas, which are obtained from the study of the →-free reducts of Heyting algebras. We prove that every si-logic is axiomatizable by (∧, ∨)-canonical formulas. We also discuss the similarities and differences between these two kinds of canonical formulas.One of the main ingredients of these formulas is a designated subset D of pairs of elements of a finite subdirectly irreducible Heyting algebra A. When D = A 2 , we show that the (∧, ∨)-canonical formula of A is equivalent to the Jankov formula of A. On the other hand, when D = ∅, the (∧, ∨)-canonical formulas produce a new class of si-logics we term stable si-logics. We prove that there are continuum many stable si-logics and that all stable si-logics have the finite model property. We also compare stable si-logics to splitting and subframe si-logics.

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Section: Stable Superintuitionistic Logicsmentioning

confidence: 99%

Section: Stable Superintuitionistic Logicsmentioning

confidence: 99%

“…The following criterion of axiomatizability of si-logics by algebra-based formulas follows from [8,Thm. 3.9] (see also [7,Thm.…”

Section: Stable Superintuitionistic Logicsmentioning

confidence: 99%

“…A similar argument is used in [10,Thm. 11.19] for proving that there are continuum many subframe and cofinal subframe si-logics, as well as in [8,Thm. 3.14] (see also [7,Thm.…”

Section: Stable Superintuitionistic Logicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Locally Finite Reducts of Heyting Algebras and Canonical Formulas

Bezhanishvili¹,

Bezhanishvili²

2017

Notre Dame J. Formal Logic

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Notes on My Scientific Life

de Jongh

2024

Dick De Jongh on Intuitionistic and Provability Logics

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Canonical formulas for k-potent commutative, integral, residuated lattices

2017

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Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Indeed, they provide a uniform and semantic way of axiomatising all extensions of intuitionistic logic and all modal logics above K4. Although the method originally hinged on the relational semantics of those logics, recently it has been completely recast in algebraic terms. In this new perspective, canonical formulas are built from a finite subdirectly irreducible algebra by describing completely the behaviour of some operations and only partially the behaviour of some others. In this paper, we export the machinery of canonical formulas to substructural logics by introducing canonical formulas for k-potent, commutative, integral, residuated lattices (k-CIRL). We show that any subvariety of k-CIRL is axiomatised by canonical formulas. The paper ends with some applications and examples

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Frame Based Formulas for Intermediate Logics

Cited by 10 publications

References 24 publications

Locally Finite Reducts of Heyting Algebras and Canonical Formulas

Locally Finite Reducts of Heyting Algebras and Canonical Formulas

Notes on My Scientific Life

Canonical formulas for k-potent commutative, integral, residuated lattices

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