Defining Measures in a Mereological Space (an exploratory paper)

Giuseppina Barbieri, Giangiacomo Gerla

DOI: http://dx.doi.org/10.12775/LLP.2021.005

Abstract


We explore the notion of a measure in a mereological structure and we deal with the difficulties arising. We show that measure theory on connection spaces is closely related to measure theory on the class of ortholattices and we present an approach akin to Dempster’s and Shafer’s. Finally, the paper contains some suggestions for further research.


Keywords


connection structures; measures; mereological space; mereology; region-based theories of space

Full Text:

PDF

References


Arntzenius, F., “Gunk, topology, and measure”, pages 225–247 in D. Zimmerman (ed.), Oxford Studies in Metaphysics, vol. 4, Oxford: Oxford University Press, 2008. Also: “Gunk, topology and measure”, pages 327–343, Chapter 16, in D. DeVidi, M. Hallett and P. Clark (eds.), Logic, Mathematics, Philosophy: Vintage Enthusiasms. Essays in Honour of John L. Bell, vol. 75 of series “The Western Ontario Series in Philosophy of Science”, Springer, 2011. DOI: https://doi.org/10.1007/978-94-007-0214-1_16

Arntzenius, F., Space, Time, and Stuff, Oxford: Oxford University Press, 2012. DOI: https://doi.org/10.1093/acprof:oso/9780199696604.001.0001

Barbieri, G., and G. Gerla, “Measures in Euclidean point-free space” (in progress).

Biacino, L., and G. Gerla, “Connection structures”, Notre Dame Journal of Formal Logic 32, 2 (1991): 242–247. DOI: https://doi.org/10.1305/ndjfl/1093635748

Clarke, B., “A calculus of individuals based on connnection”, Notre Dame Journal of Formal Logic 22, 3 (1981): 204–218.

Clarke, B., “Individuals and points”, Notre Dame Journal of Formal Logic 26, 1 (1985): 61–75. DOI: https://doi.org/10.1305/ndjfl/1093870761

Dempster, A.P., “Upper and lower probabilities induced by a multivalued mapping”, Ann. Math. Stat. 38 (1967): 325–339.

Dempster, A.P., “A generalization of Bayesian inference”, Journal of the Royal Statistical Society, Series B 30 (1968): 205–247.

Gerla, G., and R. Gruszczyński, “Point-free geometry, ovals, and half-planes”, Rev. Symb. Log. 10, 2 (2017): 237–258. DOI: https://doi.org/10.1017/S1755020316000423

Gruszczyński, R., and A. Pietruszczak, “The relations of supremum and mereological sum in partially ordered sets”, pages 105–122 in C. Calosi and P. Graziani (eds.), Mereology and the Science, Parts and Wholes in the Contemporary Scientific Context, vol. 371 of Synthese Library, Springer, 2014. DOI: https://doi.org/10.1007/978-3-319-05356-1_6

Gruszczyński, R., and A. Varzi, “Mereology then and now”, Logic and Logical Philosophy 24 (2015): 409–427. DOI: https://doi.org/10.12775/LLP.2015.024

Horn, A., and A. Tarski, “Measures in Boolean algebras”, Transactions of the American Mathematical Society 64, 3 (1948): 467–497. DOI: https://doi.org/10.1090/S0002-9947-1948-0028922-8

Lando, T., and D. Scott, “A calculus of regions respecting both measure and topology”, Journal of Philosophical Logic 14 (2019): 825–850. DOI: https://doi.org/10.1007/s10992-018-9496-8

Leśniewski, S., “On the foundations of mathematics”, Translated from the Polish and with an introduction by Vito F. Sinisi, Topoi 2, 1 (1983): 3–52.

Pietruszczak, A., Metamereology, Toruń: The Nicolaus Copernicus University Scientific Publishing House, 2018. DOI: https://doi.org/10.12775/3961-4

Pietruszczak, A., Foundations of the Theory of Parthood. A Study of Mereology, vol. 54 of series “Trends in Logic”, Springer International Publishing, 2020. DOI: https://doi.org/10.1007/978-3-030-36533-2

Roeper, P., “Region-based topology”, Journal of Philosophical Logic 26 (1997): 251–309. DOI: https://doi.org/10.1023/A:1017904631349

Russell, J., “The structure of gunk: Adventures in the ontology of space”, pages 248–274 in Oxford Studies in Metaphysics, vol. 4, Oxford: Oxford University Press, 2008.

Shafer, G., A Mathematical Theory of Evidence, Princeton University Press, 1976. DOI: https://doi.org/10.2307/j.ctv10vm1qb

Tarski, A., “Les fondaments de la géométrie des corps”, pages 29–33 in Księga Pamiątkowa Pierwszego Polskiego Zjazdu Matematycznego, suplement to Annales de la Société Polonaise de Mathématique, Kraków, 1929.

Whitehead, A.N., An Enquiry Concerning the Principles of Natural Knowledge, Cambridge University Press, 1919.

Whitehead, A.N., The Concept of Nature, Cambridge University Press, 1920. DOI: https://doi.org/10.1017/CBO9781316286654

Whitehead, A.N., Process and Reality, New York: The Macmillan Co., 1929.








ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

Partnerzy platformy czasopism