An Equational Axiomatization of Dynamic Negation and Relational Composition

Authors:
Marco Hollenberg
Affiliations:
Department of Philosophy, Utrecht University, Heidelberglaan 8, 3584 CS Utrecht, the Netherlands (E-mail: hollenb@phil.ruu.nl)
Venue:
Journal of Logic, Language and Information
Year:
1997

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Abstract

We consider algebras on binary relations with two main operators: relational composition and dynamic negation. Relational composition has its standard interpretation, while dynamic negation is an operator familiar to students of Dynamic Predicate Logic (DPL) (Groenendijk and Stokhof, 1991): given a relation R its dynamic negation ∼R is a test that contains precisely those pairs (s,s) for which s is not in the domain of R. These two operators comprise precisely the propositional part of DPL.This paper contains a finite equational axiomatization for these ’dynamic relation algebras‘. The completenessresult uses techniques from modal logic. We also lookat the variety generated by the class of dynamic relation algebras and note that there exist nonrepresentable algebras in this variety, ones which cannot be construedas spaces of relations. These results are also proved for an extension to a signature containing atomic tests and union.