## Notre Dame Journal of Formal Logic

### Bimodal Logics with a “Weakly Connected” Component without the Finite Model Property

Agi Kurucz

#### Abstract

There are two known general results on the finite model property (fmp) of commutators $[L_{0},L_{1}]$ (bimodal logics with commuting and confluent modalities). If $L$ is finitely axiomatizable by modal formulas having universal Horn first-order correspondents, then both $[L,\mathbf{K}]$ and $[L,\mathbf{S5}]$ are determined by classes of frames that admit filtration, and so they have the fmp. On the negative side, if both $L_{0}$ and $L_{1}$ are determined by transitive frames and have frames of arbitrarily large depth, then $[L_{0},L_{1}]$ does not have the fmp. In this paper we show that commutators with a “weakly connected” component often lack the fmp. Our results imply that the above positive result does not generalize to universally axiomatizable component logics, and even commutators without “transitive” components such as $[\mathbf{K3},\mathbf{K}]$ can lack the fmp. We also generalize the above negative result to cases where one of the component logics has frames of depth one only, such as $[\mathbf{S4.3},\mathbf{S5}]$ and the decidable product logic $\mathbf{S4.3}\!\times\!\mathbf{S5}$. We also show cases when already half of commutativity is enough to force infinite frames.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 58, Number 2 (2017), 287-299.

Dates
Received: 9 May 2014
Accepted: 14 October 2014
First available in Project Euclid: 9 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1489028416

Digital Object Identifier
doi:10.1215/00294527-3870247

Mathematical Reviews number (MathSciNet)
MR3634982

Zentralblatt MATH identifier
06751304

#### Citation

Kurucz, Agi. Bimodal Logics with a “Weakly Connected” Component without the Finite Model Property. Notre Dame J. Formal Logic 58 (2017), no. 2, 287--299. doi:10.1215/00294527-3870247. https://projecteuclid.org/euclid.ndjfl/1489028416

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