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 ᵎ x ᵎ for some set ᵎ) 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: 92,197

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

The logic of Peirce algebras.Maarten De Rijke - 1995 - Journal of Logic, Language and Information 4 (3):227-250.
An equational axiomatization of dynamic negation and relational composition.Marco Hollenberg - 1997 - Journal of Logic, Language and Information 6 (4):381-401.
Arrow logic and infinite counting.Ágnes Kurucz - 2000 - Studia Logica 65 (2):199-222.
Cylindric modal logic.Yde Venema - 1995 - Journal of Symbolic Logic 60 (2):591-623.
Rectangular games.Yde Venema - 1998 - Journal of Symbolic Logic 63 (4):1549-1564.
A System of Dynamic Modal Logic.Maarten de Rijke - 1998 - Journal of Philosophical Logic 27 (2):109 - 142.
A system of dynamic modal logic.Maarten de Rijke - 1998 - Journal of Philosophical Logic 27 (2):109-142.
Event, state, and process in arrow logic.Satoshi Tojo - 1999 - Minds and Machines 9 (1):81-103.
An axiomatization of s13.Michael J. Carroll - 1978 - Philosophia 8 (2-3):381-382.
T × W Completeness.Franz Kutschervona - 1997 - Journal of Philosophical Logic 26 (3):241-250.
An axiomatization of full computation tree logic.M. Reynolds - 2001 - Journal of Symbolic Logic 66 (3):1011-1057.

Analytics

Added to PP
2011-05-29

Downloads
35 (#458,712)

6 months
10 (#275,239)

Historical graph of downloads
How can I increase my downloads?

Citations of this work

No citations found.

Add more citations

References found in this work

No references found.

Add more references