David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
Renata P. de Freitas, Jorge P. Viana, Mario R. F. Benevides, Sheila R. M. Veloso & Paulo A. S. Veloso
Journal of Philosophical Logic 32 (4):343-355 (2003)
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.
|Keywords||arrow logic fork algebra modal logic relation algebras|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
Paulo A. S. Veloso, Renata P. De Freitas, Petrucio Viana, Mario Benevides & Sheila R. M. Veloso (2007). On Fork Arrow Logic and Its Expressive Power. Journal of Philosophical Logic 36 (5):489 - 509.
Similar books and articles
W. J. Blok (1979). An Axiomatization of the Modal Theory of the Veiled Recession Frame. Studia Logica 38 (1):37 - 47.
Maarten De Rijke (1998). A System of Dynamic Modal Logic. Journal of Philosophical Logic 27 (2):109 - 142.
Roger D. Maddux (1989). Nonfinite Axiomatizability Results for Cylindric and Relation Algebras. Journal of Symbolic Logic 54 (3):951-974.
Yde Venema (1998). Rectangular Games. Journal of Symbolic Logic 63 (4):1549-1564.
Yde Venema (1995). Cylindric Modal Logic. Journal of Symbolic Logic 60 (2):591-623.
Ágnes Kurucz (2000). Arrow Logic and Infinite Counting. Studia Logica 65 (2):199-222.
Maarten De Rijke (1995). The Logic of Peirce Algebras. Journal of Logic, Language and Information 4 (3):227-250.
Marco Hollenberg (1997). An Equational Axiomatization of Dynamic Negation and Relational Composition. Journal of Logic, Language and Information 6 (4):381-401.
Paulo A. S. Veloso, Renata P. de Freitas, Petrucio Viana, Mario Benevides & Sheila R. M. Veloso (2007). On Fork Arrow Logic and its Expressive Power. Journal of Philosophical Logic 36 (5):489 - 509.
Renata P. De Freitas, Jorge P. Viana, Mario R. F. Benevides, Sheila R. M. Veloso & Paulo A. S. Veloso (2003). Squares in Fork Arrow Logic. Journal of Philosophical Logic 32 (4):343 - 355.
Added to index2009-01-28
Total downloads6 ( #204,725 of 1,101,771 )
Recent downloads (6 months)5 ( #59,646 of 1,101,771 )
How can I increase my downloads?