## Journal of Symbolic Logic

### Products of ‘transitive’ modal logics

#### Abstract

We solve a major open problem concerning algorithmic properties of products of ‘transitive’ modal logics by showing that products and commutators of such standard logics as K4, S4, S4.1, K4.3, GL, or Grz are undecidable and do not have the finite model property. More generally, we prove that no Kripke complete extension of the commutator [K4,K4] with product frames of arbitrary finite or infinite depth (with respect to both accessibility relations) can be decidable. In particular, if 𝒞₁ and 𝒞₂ are classes of transitive frames such that their depth cannot be bounded by any fixed n <ω, then the logic of the class { 𝔉₁ × 𝔉₂ | 𝔉₁ ∈ 𝒞₁, 𝔉₂∈ 𝒞₂ } is undecidable. (On the contrary, the product of, say, K4 and the logic of all transitive Kripke frames of depth ≤ n, for some fixed n <ω, is decidable.) The complexity of these undecidable logics ranges from r.e. to co-r.e. and Π₁¹-complete. As a consequence, we give the first known examples of Kripke incomplete commutators of Kripke complete logics.

#### Article information

Source
J. Symbolic Logic, Volume 70, Issue 3 (2005), 993-1021.

Dates
First available in Project Euclid: 22 July 2005

https://projecteuclid.org/euclid.jsl/1122038925

Digital Object Identifier
doi:10.2178/jsl/1122038925

Mathematical Reviews number (MathSciNet)
MR2155277

Zentralblatt MATH identifier
1103.03020

#### Citation

Gabelaia, David; Kurucz, Agi; Wolter, Frank; Zakharyaschev, Michael. Products of ‘transitive’ modal logics. J. Symbolic Logic 70 (2005), no. 3, 993--1021. doi:10.2178/jsl/1122038925. https://projecteuclid.org/euclid.jsl/1122038925

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