Bulletin of Symbolic Logic

Erdős graphs resolve Fine’s canonicity problem

Robert Goldblatt, Ian Hodkinson, and Yde Venema

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We show that there exist 20 equational classes of Boolean algebras with operators that are not generated by the complex algebras of any first-order definable class of relational structures. Using a variant of this construction, we resolve a long-standing question of Fine, by exhibiting a bimodal logic that is valid in its canonical frames, but is not sound and complete for any first-order definable class of Kripke frames (a monomodal example can then be obtained using simulation results of Thomason). The constructions use the result of Erdős that there are finite graphs with arbitrarily large chromatic number and girth.

Article information

Bull. Symbolic Logic, Volume 10, Issue 2 (2004), 186-208.

First available in Project Euclid: 26 April 2004

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Zentralblatt MATH identifier

Boolean algebras with operators modal logic random graphs canonical extension elementary class variety


Goldblatt, Robert; Hodkinson, Ian; Venema, Yde. Erdős graphs resolve Fine’s canonicity problem. Bull. Symbolic Logic 10 (2004), no. 2, 186--208. doi:10.2178/bsl/1082986262. https://projecteuclid.org/euclid.bsl/1082986262

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