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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Abstract
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 Mathematics |
| 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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