Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-26T04:19:24.529Z Has data issue: false hasContentIssue false
  • English
  • Français

THE LOGIC OF COMPARATIVE CARDINALITY

Part of: Set theory Algebraic logic General logic

Published online by Cambridge University Press:  22 June 2020

YIFENG DING ,
MATTHEW HARRISON-TRAINOR  and
WESLEY H. HOLLIDAY
YIFENG DING
Affiliation:
GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA94720, USAE-mail: [email protected]
MATTHEW HARRISON-TRAINOR
Affiliation:
SCHOOL OF MATHEMATICS AND STATISTICS, VICTORIA UNIVERSITY OF WELLINGTON, 6140NEW ZEALANDE-mail: [email protected]
WESLEY H. HOLLIDAY
Affiliation:
DEPARTMENT OF PHILOSOPHY AND GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE, UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA94720, USAE-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and complementation of sets, but also about the relative size of sets. We completely axiomatize such reasoning under the Cantorian definition of relative size in terms of injections.

Keywords

cardinality comparisonsqualitative probabilityBoolean algebrafields of sets

MSC classification

Primary: 03B60: Other nonclassical logic
Secondary: 03E10: Ordinal and cardinal numbers 03G05: Boolean algebras
Type
Articles
Information
The Journal of Symbolic Logic , Volume 85 , Issue 3 , September 2020 , pp. 972 - 1005
Check for updates
Copyright
© The Association for Symbolic Logic 2020

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

REFERENCES

Burgess, J. P., Axiomatizing the logic of comparative probability . Notre Dame Journal of Formal Logic , vol. 51 (2010), no. 1, pp. 119–126.CrossRefGoogle Scholar
Cantone, D., A survey of computable set theory . Le Matematiche , vol. 43 (1988), no. 1, 2, pp. 125–194.Google Scholar
Cantor, G., Gesammelte Abhandlungen , Georg Olms, Hildesheim, 1962.Google Scholar
de Finetti, B., La ‘logica del plausible’ secondo la concezione di Polya, Atti della XLII Riunione, Societa Italiana per il Progresso delle Scienze , 1949, pp. 227–236.Google Scholar
del Cerro, L. F. and Herzig, A., A modal analysis of possibility theory , Symbolic and Quantitative Approaches to Uncertainty (Kruse, R. and Siegel, P., editors), Lecture Notes in Computer Science, vol. 548, Springer, Berlin, 1991.Google Scholar
Dubois, D. and Prade, H., Possibility Theory , Plenum Press, New York, 1988.CrossRefGoogle Scholar
Ferro, A., Omodeo, E. G., and Schwartz, J. T., Decision procedures for elementary sublanguages of set theory. I. Multi-level syllogistic and some extensions . Communications on Pure and Applied Mathematics , vol. 33 (1980), no. 5, pp. 599–608.CrossRefGoogle Scholar
Galilei, G., Dialogues Concerning Two New Sciences , Dover, New York, 1954, p. 1638.Google Scholar
Gärdenfors, P., Qualitative probability as an intensional logic . Journal of Philosophical Logic , vol. 4 (1975), no. 2, pp. 171–185.CrossRefGoogle Scholar
Jech, T., Set Theory , Springer, Berlin, 2003.Google Scholar
Kraft, C. H., Pratt, J. W., and Seidenberg, A., Intuitive probability on finite sets . The Annals of Mathematical Statistics , vol. 30 (1959), no. 2, pp. 408–419.CrossRefGoogle Scholar
Mancosu, P., Measuring the size of infinite collections of natural numbers: Was Cantor’s theory of infinite number inevitable? The Review of Symbolic Logic , vol. 2 (2009), no. 4, p. 2009.CrossRefGoogle Scholar
Moss, L. S., Syllogistic logic with cardinality comparisons , J. Michael Dunn on Information Based Logics , (Bimbo, K., editor), Springer, New York, 2016, pp. 391–415.CrossRefGoogle Scholar
Moss, L. S. and Raty, C., Reasoning about the sizes of sets: Progress, problems and prospects, Proceedings of the Fourth Workshop on Bridging the Gap between Human and Automated Reasoning (Schon, C., editor), CEUR Workshop Proceedings, 2018, pp. 33–39.Google Scholar
Moss, L. S. and Topal, S., Syllogistic logic with cardinality comparisons, on infinite sets . The Review of Symbolic Logic , vol. 13 (2018), pp. 1–22.CrossRefGoogle Scholar
Papadimitriou, C. H., On the complexity of integer programming . Journal of the ACM (JACM) , vol. 28 (1981), no. 4, pp. 765–768.CrossRefGoogle Scholar
Scott, D., Measurement structures and linear inequalities . Journal of Mathematical Psychology , vol. 1 (1964), pp. 233–247.CrossRefGoogle Scholar
Segerberg, K., Qualitative probability in a modal setting . Proceedings of the Second Scandinavian Logic Symposium , vol. 63 (1971), pp. 341–352.CrossRefGoogle Scholar