Finite axiomatizability in Łukasiewicz logic
type: Item
instance of: scholarly article
title: Finite axiomatizability in Łukasiewicz logic
zbMATH Open document ID: 1248.03043
author: Daniele Mundici
published in: Annals of Pure and Applied Logic
publication date: 2011-09-22T00:00:00Z
review text: Using the deep relationship between Łukasiewicz logic, MV-algebras, and piecewise linear geometry, the main result the paper proves is a classification of (equivalent classes of) finitely axiomatisable theories in Łukasiewicz propositional logic by means of weighted abstract simplicial complexes. Recall that an abstract simplicial complex is a pair \((V,\Sigma)\) where \(V\) is a finite nonempty set and \(\Sigma\) is a set of subsets of \(V\) whose union is \(V\) and such that every subset of an element of \(\Sigma\) is again in \(\Sigma\). A \textit{weighted abstract simplical complex} \((V,\Sigma,\omega)\) is an abstract simplicial complex \((V,\Sigma)\) together with a function \(\omega\) mapping \(V\) to the positive integers. The main steps of the proofs are: 1) The fact that the Lindenbaum algebra of every (theory consisting of a) \(1\)-satisfiable formula \(\theta\) in \(n\) propositional variables is the algebra of \(n\)-variate McNaughton functions restricted to \(\mathrm{Mod}(\theta) = \{v \in [0,1]^n \mid v(\theta) = 1\}\), where truth-assignments have been identified with points in the unit hypercube. 2) The fact that for each nonempty subset \(X \subseteq [0,1]^n\), \(X\) is a rational polyhedron (i.e., a finite union of compact convex polyhedra with rational vertices) iff \(X = \mathrm{Mod}(\psi)\) for some formula \(\psi\). 3) The lemma that proves that a theory \(\Theta\) is finitely axiomatisable iff \(\Theta = \mathrm{Th}(P) = \{\psi \mid v(\psi) = 1, \forall v \in P\}\), for \(P\) some rational polyhedron. 4) The theorem that proves that the map \(P \mapsto \mathrm{Th}(P)\) determines a bijective correspondence between \(\mathbb{Z}\)-homeomorphism classes of rational polyhedra in some \(n\)-cube, and equivalence classes of finitely axiomatisable theories (where two polyhedra \(P\), \(Q\) are said to be \(\mathbb{Z}\)-homeomorphic if there is a piecewise linear homeomoprhism \(\eta : P \to Q\) that is onto and such that each linear piece of both \(…
reviewed by: Stefano Aguzzoli
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 06D35
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 5949324
zbMATH Keywords: Łukasiewicz logic
zbMATH Keywords: MV-algebra
zbMATH Keywords: finitely axiomatizable theory
zbMATH Keywords: finitely presented algebra
zbMATH Keywords: rational polyhedron
zbMATH Keywords: simplicial complex
MaRDI profile type: Publication
full work available at URL: j.apal.2011.06.026
OpenAlex ID: W2034881471
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DOI: 10.1016/J.APAL.2011.06.026
description: scientific article; zbMATH DE number 5949324