Hostname: page-component-848d4c4894-5nwft Total loading time: 0 Render date: 2024-05-08T19:03:32.559Z Has data issue: false hasContentIssue false
  • English
  • Français

THE LOGIC OF PARTITIONS: INTRODUCTION TO THE DUAL OF THE LOGIC OF SUBSETS

Published online by Cambridge University Press:  26 February 2010

DAVID ELLERMAN
DAVID ELLERMAN*
Affiliation:
Department of Philosophy, University of California/Riverside
*
*DEPARTMENT OF PHILOSOPHY, UNIVERSITY OF CALIFORNIA/RIVERSIDE, 4044 MT. VERNON AVE., RIVERSIDE, CA 92507. E-mail: david@ellerman.org
Get access Rights & Permissions [Opens in a new window]

Abstract

Modern categorical logic as well as the Kripke and topological models of intuitionistic logic suggest that the interpretation of ordinary “propositional” logic should in general be the logic of subsets of a given universe set. Partitions on a set are dual to subsets of a set in the sense of the category-theoretic duality of epimorphisms and monomorphisms—which is reflected in the duality between quotient objects and subobjects throughout algebra. If “propositional” logic is thus seen as the logic of subsets of a universe set, then the question naturally arises of a dual logic of partitions on a universe set. This paper is an introduction to that logic of partitions dual to classical subset logic. The paper goes from basic concepts up through the correctness and completeness theorems for a tableau system of partition logic.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 3 , Issue 2 , June 2010 , pp. 287 - 350
Copyright
Copyright © Association for Symbolic Logic 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

BIBLIOGRAPHY

Awodey, S. (2006). Category Theory. Oxford: Clarendon Press.CrossRefGoogle Scholar
Birkhoff, G. (1948). Lattice Theory. New York: American Mathematical Society.Google Scholar
Boole, G. (1854). An Investigation of the Laws of Thought on which are Founded the Mathematical Theories of Logic and Probabilities. Cambridge: Macmillan and Co.CrossRefGoogle Scholar
Britz, T., Mainetti, M., & Pezzoli, L. (2001). Some operations on the family of equivalence relations. In Crapo, H., and Senato, D., editors. Algebraic Combinatorics and Computer Science: A Tribute to Gian-Carlo Rota. Milano: Springer, pp. 445–459.CrossRefGoogle Scholar
Church, A. (1956). Introduction to Mathematical Logic. Princeton: Princeton University Press.Google Scholar
Dubreil, P., & Dubreil-Jacotin, M.-L. (1939). Théorie algébrique des relations d’équivalence. Journal de Mathématique Pures at Appliquées, 18, 63–95.Google Scholar
Ellerman, D. (2006). A theory of adjoint functors with some thoughts on their philosophical significance. In Sica, G., editor. What is Category Theory? Milan: Polimetrica, pp. 127–183.Google Scholar
Ellerman, D. (2009). Counting distinctions: On the conceptual foundations of Shannon’s information theory. Synthese, 168, 119–149.CrossRefGoogle Scholar
Finberg, D., Mainetti, M., & Rota, G.-C. (1996). The logic of commuting equivalence relations. In Ursini, A., and Agliano, P., editors. Logic and Algebra. New York: Marcel Dekker, pp. 69–96.Google Scholar
Fitting, M. C. (1969). Intuitionistic Logic, Model Theory, and Forcing. Amsterdam: North-Holland.Google Scholar
Grätzer, G. (2003). General Lattice Theory (second edition). Boston: Birkhäuser Verlag.Google Scholar
Gödel, K. (1933). Zur intuitionistischen arithmetik und zahlentheorie. Ergebnisse eines mathematischen Kolloquiums, 4, 34–38.Google Scholar
Haiman, M. (1985). Proof theory for linear lattices. Advances in Mathematics, 58, 209–242.CrossRefGoogle Scholar
Kung, J. P., Rota, G.-C., & Yan, C. H. (2009). Combinatorics: The Rota Way. New York: Cambridge University Press.CrossRefGoogle Scholar
Lawvere, F. W. (1986). Introduction. In Lawvere, F. W., and Schanuel, S., editors. Categories in Continuum Physics (Buffalo 1982) LNM 1174. Berlin: Springer-Verlag, pp. 1–16.Google Scholar
Lawvere, F. W. (1991). Intrinsic co-Heyting boundaries and the Leibniz rule in certain toposes. In Carboni, A., Pedicchio, M., and Rosolini, G., editors. Category Theory (Como 1990) LNM 1488. Berlin: Springer-Verlag, pp. 279–297.Google Scholar
Lawvere, F. W., & Rosebrugh, R. (2003). Sets for Mathematics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Lawvere, F. W., & Schanuel, S. (1997). Conceptual Mathematics: A First Introduction to Categories. New York: Cambridge University Press.Google Scholar
Mac Lane, S. (1971). Categories for the Working Mathematician. New York: Verlag.CrossRefGoogle Scholar
Mac Lane, S., & Moerdijk, I. (1992). Sheaves in Geometry and Logic: A First Introduction to Topos Theory. New York: Springer.Google Scholar
Ore, O. (1942). Theory of equivalence relations. Duke Mathematical Journal, 9, 573–627.Google Scholar
Restall, G. (2000). An Introduction to Substructural Logics. London: Routledge.CrossRefGoogle Scholar
Rockafellar, R. (1984). Network Flows and Monotropic Optimization. New York: John Wiley.Google Scholar
Smullyan, R. M. (1995). First-Order Logic. New York: Dover.Google Scholar
Stanley, R. P. (1997). Enumerative Combinatorics, Vol. 1. New York: Cambridge University Press.CrossRefGoogle Scholar
van Dalen, D. (2001). Intuitionistic logic. In Goble, L., editor. The Blackwell Guide to Philosophical Logic. Oxford: Blackwell, pp. 224–257.Google Scholar
Whitman, P. (1946). Lattices, equivalence relations, and subgroups. Bulletin of the American Mathematical Society, 52, 507–522.Google Scholar
Wilson, R. J. (1972). Introduction to Graph Theory. London: Longman.Google Scholar