Squares in Fork Arrow Logic

Journal of Philosophical Logic 32 (4):343-355 (2003)
  Copy   BIBTEX

Abstract

In this paper we show that the class of fork squares has a complete orthodox axiomatization in fork arrow logic (FAL). This result may be seen as an orthodox counterpart of Venema's non-orthodox axiomatization for the class of squares in arrow logic. FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs). FAs extend RAs by a binary fork operator and are axiomatized by adding three equations to RAs equational axiomatization. A proper FA is an algebra of relations where the fork is induced by an injective operation coding pair formation. In contrast to RAs, FAs are representable by proper ones and their equational theory has the expressive power of full first-order logic. A square semantics (the set of arrows is U×U for some set U) for arrow logic was defined by Y. Venema. Due to the negative results about the finite axiomatizability of representable RAs, Venema provided a non-orthodox finite axiomatization for arrow logic by adding a new rule governing the applications of a difference operator. We address here the question of extending the type of relational structures to define orthodox axiomatizations for the class of squares. Given the connections between this problem and the finitization problem addressed by I. Németi, we suspect that this cannot be done by using only logical operations. The modal version of the FA equations provides an orthodox axiomatization for FAL which is complete in view of the representability of FAs. Here we review this result and carry it further to prove that this orthodox axiomatization for FAL also axiomatizes the class of fork squares.

Links

PhilArchive



    Upload a copy of this work     Papers currently archived: 93,069

External links

Setup an account with your affiliations in order to access resources via your University's proxy server

Through your library

Similar books and articles

An equational axiomatization of dynamic negation and relational composition.Marco Hollenberg - 1997 - Journal of Logic, Language and Information 6 (4):381-401.
Hyperboolean Algebras and Hyperboolean Modal Logic.Valentin Goranko & Dimiter Vakarelov - 1999 - Journal of Applied Non-Classical Logics 9 (2):345-368.

Analytics

Added to PP
2009-01-28

Downloads
59 (#279,257)

6 months
24 (#121,446)

Historical graph of downloads
How can I increase my downloads?

References found in this work

A completeness theorem in modal logic.Saul Kripke - 1959 - Journal of Symbolic Logic 24 (1):1-14.
Algebraic semantics for modal logics I.E. J. Lemmon - 1966 - Journal of Symbolic Logic 31 (1):46-65.
Multi-dimensional modal logic.Maarten Marx - 1997 - Boston, Mass.: Kluwer Academic Publishers. Edited by Yde Venema.
Algebraic semantics for modal logics II.E. J. Lemmon - 1966 - Journal of Symbolic Logic 31 (2):191-218.

View all 13 references / Add more references