Abstract
A representation of relation algebras is given by modifying the Routley-Meyer semantics for relevance logic. This semantics is similar to the Kripke semantics for intutionistic logic, but its frames use a ternary accessibility relation instead of a binary one. The representation is foreshadowed in the work of Lyndon as well as by the representation of Boolean algebras with operators by Jónsson and Tarski, but the aim here is to make it explicit and to connect it to the related representations of algebras arising in the study of relevance logic as well as other “substructural logics,” e.g., linear logic. A philosophical interpretation is given of the representation, showing that an element of a relation algebra can be understood as a set of relations, rather than as the intended interpretation as a single relation. It is shown how a Routley-Meyer frame can be represented so its states are relations, and an interpretation is provided in terms of a “relational database”. Connections are mentioned to recent work by Jon Barwise on “information channels” between “sites” and by J. M. Dunn and R. K. Meyer on a ternary frame semantics for combinatory logic.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Allwein, G., and J. M. Dunn 1993 A Kripke semantics for linear logic, The Journal of Symbolic Logic, vol. 58, pp. 514–545.
Anderson, A. R., and N. D. Belnap et al. 1975 Entailment: The logic of relevance and necessity, vol. I, Princeton University Press, Princeton, New Jersey.
Anderson, A. R, N. D. Belnap, and J. M. Dunn et al. 1992 Entailment: The logic of relevance and necessity, vol. II, Princeton University Press, Princeton, New Jersey.
Barwise, J. 1993 Constraints, channels, and the flow of information, Situation theory and its applications (S. Peters and D. Israel, editors), vol. 3, CSLI Lecture Notes, University of Chicago Press, Chicago, pp. 3–27.
Bergmann, G. 1967 Realism, University of Milwaukee Press.
Bialynicki-Birula, A., and H. Rasiowa 1957 On the representation of quasi-Boolean algebras, Bulletin de V Académie Polonaise des Sciences, vol. 5, pp. 259–261.
Birkhoff, G. 1967 Lattice theory, American Mathematical Society, Providence.
Bradley, F. H. 1897 Appearance and reality, Oxford University Press, Oxford.
Brown, C., and D. Gurr 1995 Relations and non-commutative linear logic, Journal of Pure and Applied Algebra, vol. 105, pp. 117–136.
Church, A. 1951 The weak theory of implication, Kontrolliertes Denken, Untersuchgen zum Logikkalkul und zur Logik der Einzelwissenschaften, (A. Menne, A. Wilhelmy, and H. Angsil, editors), Kommissions-Verlag Karl Alber, Munich, pp. 22–37.
Dŏsen, K. 1992 A brief survey of frames for the peirce calculus, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 38, pp. 179–187.
Dunn, J. M. 1966 The algebra of intensional logics, Doctoral Dissertation, University of Pittsburgh, University Microfilms, Ann Arbor; portions relevant to this paper are reprinted in Anderson, Belnap et al. 1975 as §8 and §28.2.
Dunn, J. M. 1982 A Relational Representation of quasi-Boolean algebras, Notre Dame Journal of Formal Logic, vol. 23, pp. 353–357.
Dunn, J. M. 1986 Relevance logic and entailment, Handbook of philosophical logic (D. Gab-bay and F. Guenthner, editors), vol. 3, Reidel, pp. 117–224.
Dunn, J. M. 1991 Gaggle theory: An abstraction of Galois connections and residuation with applications to negation and various logical operations, Logics in AI, Proceedings European workshop JELIA 1990 (J. van Eijck, editor), Lecture Notes in Computer Science 478, Springer-Verlag, Berlin.
Dunn, J. M. 1993a Partial-gaggles applied to substructural logics, Substructural logic (P. Schröder-Heister and K. DOsen, editors), Oxford Press, pp. 63–108.
Dunn, J. M. 1993b Perp and star: Two treatments of negation, Philosophical perspectives: Philosophy of language and logic (J. Tomberlin, editor), vol. 7, pp. 331–357.
Dunn, J. M., and R. K. Meyer 1997 Combinatore and structurally free logic, Logic Journal of the IGPL, vol. 5, pp. 505–537.
Girard, J.-Y. 1987 Linear logic, Theoretical Computer Science, vol. 50, pp. 1–102.
Gyssens, M., L. V. Saxton, and D. van Gucht 1993 Tagging as an alternative to object creation, Proceedings of the conference on query processing in object-oriented, complex-object and nested relation databases.
Jónsson, B., and A. Tarski 1951-52 Boolean algebras with operators, American Journal of Mathematics, vols. 73-74, pp. 891–939, 127–162.
Kraegeloh, K.-D., and P. C. Lockemann 1976 Hierarchies of database languages: An example, Information Systems, vol. 1, pp. 79–90.
Lambek, J. 1958 The mathematics of sentence structure, American Mathematical Monthly, vol. 65, pp. 153–172.
Lyndon, R. C. 1950 The representation of relation algebras, Annals of Mathematics, ser. 2, vol. 51, pp. 707–729.
Maddux, R. 1991 The origin of relation algebras in the development of the axiomatization of relations, Studia Logica, vol. 91, pp. 421–455.
Meyer, R. K. 1979 A Boolean valued semantics for R, Research Paper no. 4, Logic group, Research School of Social Sciences, Australian National University, Canberra.
Meyer, R. K., M. Bunder, and L. Powers 1991 Implementing the ‘fool’s model’ for combinatory logic, Journal of Automated Reasoning, vol. 7, pp. 597–630.
Meyer, R. K., and M. A. McRobbie 1982 Multisets and relevant implication, Australasian Journal of Philosophy, vol. 60, pp. 265–281.
Meyer, R. K., and R. Routley 1972 Algebraic analysis of entailment, Logique et Analyse, n.s., vol. 15, pp. 407–428.
Meyer, R. K., and R. Routley 1973 Classical relevant logics (I), Studia Logica, vol. 32, pp. 51–66.
Meyer, R. K., and R. Routley 1974 Classical relevant logics (II), Studia Logica, vol. 33, pp. 183–194.
Orlowska, E. 1992 Relational proof systems for relevant logics, The Journal of Symbolic Logic, vol. 57, pp. 1425–1440.
Pratt, V. 1991 Action logic and pure induction, Logics in AI, Proceedings of the European workshop JELIA 1990 (J. van Eijck, editor), Lecture Notes in Computer Science 478, Springer-Verlag, Berlin.
Routley, R., and R. K. Meyer 1972 The semantics of entailment, II-III, Journal of Philosophical Logic, vol. 1, pp. 53–73, 192–208.
Routley, R., and R. K. Meyer 1973 The semantics of entailment, Truth, syntax and modality (H. Leblanc, editor), North-Holland, Amsterdam, pp. 199–243.
Scott, D. S. 1976 Data types as lattices, SIAM Journal of Computing, vol. 5, pp. 522–587.
Schröder-Heister, P., and K. DOsen 1993 (editors), Substructural logic, Oxford University Press, Oxford.
Stone, M. 1936 The theory of representations for Boolean algebras, Transactions of the American Mathematical Society, vol. 40, pp. 37–111.
Stone, M. 1937 Topological representations of distributive lattices and Brouwerian logics, Časopsis pro Pěstovaní Matematiky a Fysiky, vol. 67, pp. 1–25.
Tarski, A. 1941 On the calculus of relations, The Journal of Symbolic Logic, vol. 6, pp. 73–89.
Tarski, A., and S. Givant 1985 A formalization of set theory without variables, American Mathematical Society Colloquium Publications, vol. 41, Providence.
Van Benthem, J. 1991 Language in action: Categories, lambdas and dynamic logic, Studies in Logic and the Foundations of Mathematics, vol. 130, North-Holland, New York.
Ward, M., and R. P. Dilworth 1939 Residuated lattices, Transactions of the American Mathematical Society, vol. 45, pp. 335–354.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Dunn, J.M. (2001). A Representation of Relation Algebras Using Routley-Meyer Frames. In: Anderson, C.A., Zelëny, M. (eds) Logic, Meaning and Computation. Synthese Library, vol 305. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0526-5_3
Download citation
DOI: https://doi.org/10.1007/978-94-010-0526-5_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-3891-1
Online ISBN: 978-94-010-0526-5
eBook Packages: Springer Book Archive