PP-2023-04:
Bezhanishvili, Guram and Bezhanishvili, Nick and Moraschini, Tommaso
(2023)
*Degrees of the finite model property: The antidichotomy theorem.*
[Pre-print]

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## Abstract

A classic result in modal logic, known as the Blok Dichotomy Theorem, states that the degree of incompleteness of a normal extension of the basic modal logic K is 1 or the continuum. It is a long-standing open problem whether Blok Dichotomy holds for normal extensions of other prominent modal logics (such as S4 or K4) or for extensions of the intuitionistic propositional calculus IPC (see [11, Prob. 10.5]). In this paper, we introduce the notion of the degree of finite model property (fmp), which is a natural variation of the degree of incompleteness. It is a consequence of Blok Dichotomy Theorem that the degree of fmp of a normal extension of K remains 1 or the continuum. In contrast, our main result establishes the following Antidichotomy Theorem for the degree of fmp for extensions of IPC: each nonzero cardinal κ such that κ is countable or κ is the continuum is realized as the degree of fmp of some extension of IPC. We then use the Blok-Esakia theorem to establish the same Antidichotomy Theorem for normal extensions of S4 and K4. This provides a solution of the reformulation of [11, Prob. 10.5] for the degree of fmp.

Item Type: | Pre-print |
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Report Nr: | PP-2023-04 |

Series Name: | Prepublication (PP) Series |

Year: | 2023 |

Subjects: | Logic Mathematics |

Depositing User: | Nick Bezhanishvili |

Date Deposited: | 13 Jul 2023 21:11 |

Last Modified: | 14 Jul 2023 04:57 |

URI: | https://eprints.illc.uva.nl/id/eprint/2252 |

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