PP-2020-22: The nerve criterion and polyhedral completeness of intermediate logics

PP-2020-22: Adam-Day, Sam and Bezhanishvili, Nick and Gabelaia, David and Marra, Vincenzo (2020) The nerve criterion and polyhedral completeness of intermediate logics. [Pre-print]

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We investigate a recently-devised polyhedral semantics for intermediate logics, in which formulas are interpreted in n-dimensional polyhedra. An intermediate logic is polyhedrally complete if it is complete with respect to some class of polyhedra. We provide a necessary and sufficient condition for the polyhedral-completeness of a logic. This condition, which we call the Nerve Criterion, is expressed in terms of the so-called ‘nerve’ of a poset, a construction which we employ from polyhedral geometry.
The criterion allows for the investigation of the polyhedral completeness phe- nomenon using purely combinatorial methods. Utilising it, we show that there are continuum many intermediate logics that are not polyhedrally-complete. We also provide a countably infinite class of logics axiomatised by the Jankov-Fine formulas of ‘starlike trees’, which includes Scott’s Logic, all of which are polyhedrally-complete.

Item Type: Pre-print
Report Nr: PP-2020-22
Series Name: Prepublication (PP) Series
Year: 2020
Subjects: Logic
Depositing User: Nick Bezhanishvili
Date Deposited: 17 Sep 2020 11:10
Last Modified: 17 Sep 2020 11:10
URI: https://eprints.illc.uva.nl/id/eprint/1759

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