Distributed Relation Logic

Logic and Logical Philosophy 26 (1):19-61 (2017)


We extend the relational algebra of Chin and Tarski so that it is multisorted or, as we prefer, typed. Each type supports a local Boolean algebra outfitted with a converse operator. From Lyndon, we know that relation algebras cannot be represented as proper relation algebras where a proper relation algebra has binary relations as elements and the algebra is singly-typed. Here, the intensional conjunction, which was to represent relational composition in Chin and Tarski, spans three different local algebras, thus the term distributed in the title. Since we do not rely on proper relation algebras, we are free to re-express the algebras as typed. In doing so, we allow many different intensional conjunction operators. We construct a typed logic over these algebras, also known as heterogeneous algebras of Birkhoff and Lipson. The logic can be seen as a form of relevance logic with a classical negation connective where the Routley-Meyer star operator is reified as a converse connective in the logic. Relevance logic itself is not typed but our work shows how it can be made so. Some of the properties of classical relevance logic are weakened from Routley-Meyer’s version which is too strong for a logic over relation algebras.

Download options


    Upload a copy of this work     Papers currently archived: 72,694

External links

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

Through your library


Added to PP

8 (#1,013,851)

6 months
1 (#388,311)

Historical graph of downloads
How can I increase my downloads?

References found in this work

The Theory of Representations for Boolean Algebras.M. H. Stone - 1936 - Journal of Symbolic Logic 1 (3):118-119.
Logics for Classes of Boolean Monoids.Gerard Allwein, Hilmi Demir & Lee Pike - 2004 - Journal of Logic, Language and Information 13 (3):241-266.

Add more references

Citations of this work

No citations found.

Add more citations

Similar books and articles

Boolean Deductive Systems of BL-Algebras.Esko Turunen - 2001 - Archive for Mathematical Logic 40 (6):467-473.
Lower Level Connections Between Representations of Relation Algebras.György Serény - 1986 - Bulletin of the Section of Logic 15 (3):123-125.
Groups and Algebras of Binary Relations.Steven Givant & Hajnal Andréka - 2002 - Bulletin of Symbolic Logic 8 (1):38-64.
Relation Algebras of Every Dimension.Roger D. Maddux - 1992 - Journal of Symbolic Logic 57 (4):1213-1229.
Hyper-Archimedean BL-Algebras Are MV-Algebras.Esko Turunen - 2007 - Mathematical Logic Quarterly 53 (2):170-175.
Topological Duality for Nelson Algebras and its Application.Andrzej Sendlewski - 1984 - Bulletin of the Section of Logic 13 (4):215-219.
Undecidable Semiassociative Relation Algebras.Roger D. Maddux - 1994 - Journal of Symbolic Logic 59 (2):398-418.
An Equational Axiomatization of Dynamic Negation and Relational Composition.Marco Hollenberg - 1997 - Journal of Logic, Language and Information 6 (4):381-401.
Binary Relations and Permutation Groups.Hajnal Andréka & Ivo Düntsch - 1995 - Mathematical Logic Quarterly 41 (2):197-216.
On the Homogeneous Countable Boolean Contact Algebra.Ivo Düntsch & Sanjiang Li - 2013 - Logic and Logical Philosophy 22 (2):213-251.
Maharam algebras.Boban Veličković - 2009 - Annals of Pure and Applied Logic 158 (3):190-202.