Interpolation theorems in modal logics and amalgamable varieties of topological Boolean algebras

Maksimova, L. L.

doi:10.1007/bf01673502

Cited by 68 publications

(40 citation statements)

References 7 publications

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“…As it was shown in [14], in modal logics the Property B1 is equivalent to the Craig Interpolation Property that is satisfied rather seldom [8]. The properties IPD and B2 are independent in N E(K) [13].…”

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confidence: 80%

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Interrelation of algebraic, semantical and logical properties for superintuitionistic and modal logics

Maksimova

1999

Banach Center Publ.

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We consider the families L of propositional superintuitionistic logics (s.i.l.) and N E(K) of normal modal logics (n.m.l.). It is well known that there is a duality between L and the lattice of varieties of pseudo-boolean algebras (or Heyting algebras), and also N E(K) is dually isomorphic to the lattice of varieties of modal algebras. Many important properties of logics, for instance, Craig's interpolation property (CIP), the disjunction property (DP), the Beth property (BP), Hallden-completeness (HP) etc. have suitable properties of varieties as their images, and many natural algebraic properties are in accordance with natural properties of logics. For example, a s.i.l. L has CIP iff its associated variety V (L) has the amalgamation property (AP); L is Hallden-complete iff V (L) is generated by a subdirectly irreducible Heyting algebra. For any n.m.l. L, the amalgamation property of V (L) is equivalent to a weaker version of the interpolation property for L, and the superamalgamation property is equivalent to CIP; L is Hallden-complete iff V (L) satisfies a strong version of the joint embedding property.Well-known relational Kripke semantics for the intuitionistic and modal logics seems to be a more natural interpretation than the algebraic one. The categories of Kripke frames may, in a sense, be considered as subcategories of varieties of Heyting or modal algebras. We discuss the question to what extent one may reduce problems on properties of logics to consideration of their semantic models. Preliminaries.A language of propositional modal logic contains the connective → and the propositional constant ⊥, and an unary modal operation 2. Other connectives

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confidence: 80%

“…In particular, the Hallden Property, DP and PVS are undecidable in the class of superintuitionistic logics; CIP, IPD, HP are undecidable in N E(K4). At the same time, CIP and IPD are decidable in N E(S4) and in the family of all superintuitionistic logics [7,8].…”

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confidence: 99%

Interrelation of algebraic, semantical and logical properties for superintuitionistic and modal logics

Maksimova

1999

Banach Center Publ.

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“…Furthermore, there are only finitely many amalgamable varieties of topoboolean algebras [16,17] and finitely many amalgamable varieties of Heyting algebras [18]. On the contrary, there exists a continuum of amalgamable varieties of diagonalizable algebras [19].…”

Section: Lemma 25mentioning

confidence: 93%

“…All amalgamable varieties of epistemic algebras were described in [16,17]. We can prove that simple epistemic algebras are exactly modal algebras A = (A, ∧, ¬, , P), where P = and Px = ⊥ for x = .…”

Section: Lemma 25mentioning

confidence: 98%

A weak form of interpolation in equational logic

Maksimova

2008

Algebra Logic

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The notions of a weak interpolation property and of weak amalgamation are introduced. It is proved that in varieties with the congruence extension property, the weak interpolation property is equivalent to the weak amalgamation property. In turn, weak amalgamability of a variety is equivalent to amalgamability of a class of finitely generated simple algebras in this variety.In the present paper, varieties of so-called bounded algebras are investigated. This name derives from the notion of bounded lattices having a greatest element and a least element ⊥. Various classes of bounded algebras are tied with logic. For example, classical logic is interpreted via Boolean algebras, modal logic via modal algebras, and intuitionistic logic via Heyting algebras. In a two-valued Boolean algebra, the constant is identified with a truth value "true," and the constant ⊥ with a truth value "false." In many logical calculi, such as classical, intuitionistic, etc., the inconsistency of a theory is equivalent to the absurdity ⊥ being derivable in that theory. In this event the theory becomes trivial, i.e., all formulas are derivable in it. Under the passage to algebraic semantics, investigations of the calculi reduce to studying suitable classes of algebras, and often varieties of algebras, i.e., classes that are defined by some sets of identities. In varieties of algebras associated with logics, inconsistency is frequently associated with derivability of the identity ⊥ = . The sentence "contradiction implies everything" has an algebraic interpretation as the statement ⊥ = ⇒ ∀x∀y (x = y). The algebras satisfying this condition are said to be bounded.The interpolation theorem, proved in [1] for classical predicate logics, has different equivalent formulations which become nonequivalent in intuitionistic and modal logics. Here we consider a weak interpolation property for varieties of bounded algebras. Some results concerning weak interpolation in modal logics were obtained in [2]. These results may be naturally re-worded in the language of varieties of modal algebras. In the present paper we show that part of the results in [2] can well be extended to arbitrary varieties of bounded algebras.

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“…Note than in [21, section V] the Kripke frame, consisting of two possible worlds, is presented for V2. In [23] Maksimova considers all normal extensions of modal logic S4 with the Craig interpolation property. From this it follows that modal logic V2 is the single normal extension of modal logic S5 with the Craig interpolation property (between S5 and C 2 ).…”

Section: Logic Trmentioning

confidence: 99%

Four-Valued Logics BD and DM4: Expansions

Karpenko

2017

B Sect Log

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The paper discusses functional properties of some four-valued logics which are the expansions of four-valued Belnap’s logic DM4. At first, we consider the logics with two designated values, and then logics defined by matrices having the same underlying algebra, but with a different choice of designated values, i.e. with one designated value. In the preceding literature both approaches were developed independently. Moreover, we present the lattices of the functional expansions of DM4.

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Interpolation theorems in modal logics and amalgamable varieties of topological Boolean algebras

Cited by 68 publications

References 7 publications

Interrelation of algebraic, semantical and logical properties for superintuitionistic and modal logics

Interrelation of algebraic, semantical and logical properties for superintuitionistic and modal logics

A weak form of interpolation in equational logic

Four-Valued Logics BD and DM4: Expansions

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