On theories with the general disjunction property
type: Item
instance of: scholarly article
title: On theories with the general disjunction property
zbMATH Open document ID: 1177.03038
published in: Algebra Universalis
publication date: 2007-09-03T00:00:00Z
review text: A theory \(T\) has the disjunction property iff, for all atomic sentences \(\varphi\) and \(\psi\), \(T\vdash\varphi\vee\psi\) implies \(T \vdash\varphi\) or \(T\vdash\psi\). \(T\) has the instantiation property iff, for any collection \(\{\theta_i(x):i\in I\}\) of atomic sentences, \(T\vdash \exists\bigwedge_i\theta_i (x)\) implies the existence of a variable-free term \(t\) such that \(T\vdash \bigwedge_i\theta_i(t)\). Both properties hold for Horn clause theories. The paper defines, for arbitrary theories \(T\) and sets of formulae \(\Phi\), a so-called general disjunction property relative to \(\Phi\) which simultaneously generalizes the disjunction resp. the instantiation property (in the case of countable conjunctions). The main result is a characterization of theories \(T\) having the general disjunction property in terms of existence of a model \({\mathcal A}\) of \(T\) having an expansion \({\mathcal A}'\) by countably many new constants such that the validity of certain existential sentences (in the expanded language) in \({\mathcal A}'\) implies the provability (in the original language) of a certain universal sentence from \(T\). Note that the author uses the same symbol for provability and satisfaction. As an application, the author shows that the theory of Boolean algebras with at least one atom, while not being equivalent to a Horn theory, nevertheless admits a Horn axiomatization relative to the set of all negations of atomic formulae.
Mathematics Subject Classification ID: 03C05
Mathematics Subject Classification ID: 08B20
zbMATH DE Number: 5186254
zbMATH Keywords: Horn theory
zbMATH Keywords: disjunction property
zbMATH Keywords: instantiation property
author: Miodrag Kapetanović
MaRDI profile type: Publication
full work available at URL: s00012-007-1992-6
OpenAlex ID: W2094567724
DOI: 10.1007/S00012-007-1992-6
description: scientific article; zbMATH DE number 5186254