A lattice of normal modal logics

Maksimova, L. L.; Рыбаков, Владимир В.

doi:10.1007/bf01463150

Cited by 87 publications

(50 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is well-known that superintuitionistic logics are in close relation with normal extensions of the modal logic S4 (see, for instance, [2,17]). It is evident that the Halldencompleteness of a modal logic L extending S4 implies the Hallden-completeness of the superintuitionistic fragment of L. It was noted in [1] that some stronger property, namely, so-called Principle of Variable Separation holds in a superintuitionistic logic L, whenever some modal companion of L is Hallden-complete.…”

mentioning

confidence: 99%

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

Maksimova

1999

Banach Center Publ.

View full text Add to dashboard Cite

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

show abstract

mentioning

confidence: 99%

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

Maksimova

1999

Banach Center Publ.

View full text Add to dashboard Cite

show abstract

“…The mappings ρ : NExtS4 −→ NExtInt and τ : NExtInt −→ NExtS4 were defined in [3]. It is well known that the former mapping is a lattice epimorphism and the latter is an embedding; see [3].…”

Section: Preliminariesmentioning

confidence: 99%

“…It is well known that the former mapping is a lattice epimorphism and the latter is an embedding; see [3]. Another mapping, σ : NExtInt −→ NExtGrz, defined by the equality σL = Grz ⊕ τ L for any L ∈ NExtInt, where Grz is Grzegorczyk logic, is an isomorphism; see [1] and [2].…”

Section: Preliminariesmentioning

confidence: 99%

“…The lattice operations are the set intersection ∩ as meet and the deduction closure ⊕ as joint. Other logics from NExtInt and NExtS4 will also appear in the sequel.The mappings ρ : NExtS4 −→ NExtInt and τ : NExtInt −→ NExtS4 were defined in [3]. It is well known that the former mapping is a lattice epimorphism and the latter is an embedding; see [3].…”

mentioning

confidence: 99%

“…Another mapping, σ : NExtInt −→ NExtGrz, defined by the equality σL = Grz ⊕ τ L for any L ∈ NExtInt, where Grz is Grzegorczyk logic, is an isomorphism; see [1] and [2]. This, along with inequalities obtained in [3], implies thatfor any logic M ∈ NExtS4. Thus it can be suggested that any M ∈ NExtS4 is equal to M * ⊕ τ ρM for some logic M * ⊆ Grz.…”

mentioning

confidence: 99%

See 2 more Smart Citations

On modal components of the S4-logics

Muravitsky

EPiC Series in Computing

View full text Add to dashboard Cite

We consider the representation of each extension of the modal logic S4 as sum of two components. The first component in such a representation is always included in Grzegorczyk logic and hence contains "modal resources" of the logic in question, while the second one uses essentially the resources of a corresponding intermediate logic. We prove some results towards the conjecture that every S4-logic has a representation with the least component of the first kind. PreliminariesWe consider the intuitionistic propositional logic Int and modal propositional logic S4, both defined with the postulated rule of substitution, along with the lattices of their normal consistent extensions, NExtInt (intermediate logics) and NExtS4 (S4-logics), respectively. The lattice operations are the set intersection ∩ as meet and the deduction closure ⊕ as joint. Other logics from NExtInt and NExtS4 will also appear in the sequel.The mappings ρ : NExtS4 −→ NExtInt and τ : NExtInt −→ NExtS4 were defined in [3]. It is well known that the former mapping is a lattice epimorphism and the latter is an embedding; see [3]. Another mapping, σ : NExtInt −→ NExtGrz, defined by the equality σL = Grz ⊕ τ L for any L ∈ NExtInt, where Grz is Grzegorczyk logic, is an isomorphism; see [1] and [2]. This, along with inequalities obtained in [3], implies thatfor any logic M ∈ NExtS4. Thus it can be suggested that any M ∈ NExtS4 is equal to M * ⊕ τ ρM for some logic M * ⊆ Grz. Indeed, for M * one can always take M ∩ Grz; see [4]. Furthermore, we have the following.LetAn unspecified element of L will be denoted by τ . Then any M ∈ NExtS4 can be represented as M = M * ⊕ τ , where M * ⊆ Grz, i.e. ρM * = Int. In this representation of M , the first term, M * , is called the modal component of M and the second term, τ , is its assertoric (or superintuitionistic) component (or τ -component). Such a representation of M we call a τ -representation.It has been noticed [4] that the assertoric component of M is uniquely determined by M and equals τ ρM , but its modal component may vary. Given an S4-logic M , the modal components of M constitute a dense sublattice of NExtS4 with the top element M ∩ Grz. This on-going research aims at proving the conjecture: Every S4-logic has a least modal component. Examples of the modal components of some S4-logicsBelow one can see different situations related to modal components of some S4-logics.

show abstract

The algebraic significance of weak excluded middle laws

Lávička

Moraschini

Raftery

2022

Mathematical Logic Qtrly

View full text Add to dashboard Cite

For (finitary) deductive systems, we formulate a signature‐independent abstraction of the weak excluded middle law (WEML), which strengthens the existing general notion of an inconsistency lemma (IL). Of special interest is the case where a quasivariety K$\mathsf {K}$ algebraizes a deductive system ⊢. We prove that, in this case, if ⊢ has a WEML (in the general sense) then every relatively subdirectly irreducible member of K$\mathsf {K}$ has a greatest proper K$\mathsf {K}$‐congruence; the converse holds if ⊢ has an inconsistency lemma. The result extends, in a suitable form, to all protoalgebraic logics. A super‐intuitionistic logic possesses a WEML iff it extends KC$\mathsf {KC}$. We characterize the IL and the WEML for normal modal logics and for relevance logics. A normal extension of sans-serifSsans-serif4$\mathsf {S4}$ has a global consequence relation with a WEML iff it extends sans-serifSsans-serif4.sans-serif2$\mathsf {S4.2}$, while every axiomatic extension of sans-serifRsans-serift$\mathsf {R^t}$ with an IL has a WEML.

show abstract

A lattice of normal modal logics

Cited by 87 publications

References 8 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

On modal components of the S4-logics

The algebraic significance of weak excluded middle laws

Contact Info

Product

Resources

About