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POLYHEDRAL COMPLETENESS OF INTERMEDIATE LOGICS: THE NERVE CRITERION

Part of: Boolean algebras (Boolean rings) General logic Ordered sets Distributive lattices Polytopes and polyhedra

Published online by Cambridge University Press:  14 November 2022

SAM ADAM-DAY Open the ORCID record for SAM ADAM-DAY [Opens in a new window] ,
NICK BEZHANISHVILI ,
DAVID GABELAIA Open the ORCID record for DAVID GABELAIA [Opens in a new window]  and
VINCENZO MARRA Open the ORCID record for VINCENZO MARRA [Opens in a new window]
SAM ADAM-DAY*
Affiliation:
MATHEMATICAL INSTITUTE UNIVERSITY OF OXFORD OXFORD, UK
NICK BEZHANISHVILI
Affiliation:
INSTITUTE FOR LOGIC, LANGUAGE AND COMPUTATION UNIVERSITY OF AMSTERDAM AMSTERDAM, THE NETHERLANDS E-mail: N.Bezhanishvili@uva.nl
DAVID GABELAIA
Affiliation:
A. RAZMADZE MATHEMATICAL INSTITUTE OF I. JAVAKHISHVILI TBILISI STATE UNIVERSITY TBILISI, GEORGIA E-mail: gabelaia@gmail.com
VINCENZO MARRA
Affiliation:
DIPARTIMENTO DI MATEMATICA ‘FEDERIGO ENRIQUES’ UNIVERSITÀ DEGLI STUDI DI MILANO MILAN, ITALY E-mail: Vincenzo.Marra@unimi.it
*
E-mail: adamday@maths.ox.ac.uk
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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. The first main result of this paper is 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 Alexandrov’s notion of the nerve of a poset. It affords a purely combinatorial characterisation of polyhedrally complete logics. Using the Nerve Criterion we show, easily, that there are continuum many intermediate logics that are not polyhedrally complete but which have the finite model property. We also provide, at considerable combinatorial labour, a countably infinite class of logics axiomatised by the Jankov–Fine formulas of ‘starlike trees’ all of which are polyhedrally complete. The polyhedral completeness theorem for these ‘starlike logics’ is the second main result of this paper.

Keywords

Polyhedral semanticsNerve Criterionnervestarlike treepolyhedronintermediate logicmodal logicJankov–Fine formulatriangulationrational polyhedronbarycentric subdivisiongraded posetgeometric realisationp-morphism

MSC classification

Primary: 03B55: Intermediate logics
Secondary: 03B45: Modal logic (including the logic of norms) 06A07: Combinatorics of partially ordered sets 06D20: Heyting algebras 52B05: Combinatorial properties (number of faces, shortest paths, etc.)
Type
Article
Information
The Journal of Symbolic Logic , Volume 89 , Issue 1 , March 2024 , pp. 342 - 382
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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