Say that space is ‘gunky’ if every part of space has a proper part. Traditional theories of gunk, dating back to the work of Whitehead in the early part of last century, modeled space in the Boolean algebra of regular closed (or regular open) subsets of Euclidean space. More recently a complaint was brought against that tradition in Arntzenius (2008) and Russell (2008): Lebesgue measure is not even finitely additive over the algebra, and there is no countably additive measure on the algebra. Arntzenius advocated modeling gunk in measure algebras instead—in particular, in the algebra of Borel subsets of Euclidean space, modulo sets of Lebesgue measure zero. But while this algebra carries a natural, countably additive measure, it has some unattractive topological features. In this paper, we show how to construct a model of gunk that has both nice rudimentary measure-theoretic and topological properties. We then show that in modeling gunk in this way we can distinguish between finite dimensions, and that nothing in lost in terms of our ability to identify points as locations in space.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Arntzenius, F. (2008). Gunk, topology, and measure. In D. Zimmerman (Ed.) , Oxford studies in metaphysics (Vol. 4, 225–247). Oxford: Oxford University Press.
Arntzenius, F., & Hawthorne, J. (2005). Gunk and continuous variation. The Monist, 88(4), 441–465.
Balbiani, P., Tinchev, T., Vakarelov, D. (2007). Modal logics for region-based theories of space. Fundamenta Informaticae, 81, 29–82.
Bennett, B. (1996). Modal logics for qualitative spatial reasoning. Journal of the IGPL, 4(1), 23–45.
Birkhoff, G. (1936). Order and the inclusion relation. In Comptes rendus du Congrès international des mathématiciens. Oslo.
Bricker, P. (2016). Composition as a kind of identity. Inquiry, 59(3), 264–294.
Clarke, B. (1981). A calculus of individuals based on ‘connection’. Notre Dame Journal of Formal Logic, 22, 204–218.
Cohn, A G, & Varzi, A. (2003). Mereotopological connection. Journal of Philosophical Logic, 32(4), 357–390.
de Laguna, T. (1922). Point, line and surface as sets of solids. Journal of Philosophy, 17, 449–461.
De Vries, H. (1962). Compact spaces and compactifications: an algebraic approach. Assen: Van Gorcum.
Dimov, G, & Vakarelov, D. (2006a). Contact algebras and region-based theory of space: Proximity approach i. Fundamenta Informaticae, 74(2,3), 209–249.
Dimov, G, & Vakarelov, D. (2006b). Contact algebras and region-based theory of space: Proximity approach ii. Fundamenta Informaticae, 74(2,3), 251–282.
Dimov, G, & Vakarelov, D. (2006c). Topological representation of precontact algebras. In W. MacCaull, M. Winter, I. Duntsch (Eds.) Relational methods in computer science. RelMiCS 2005, Lecture Notes in Computer Science, vol 3929. Berlin: Springer.
Gerla, G. (1995). Pointless Geometries. In Buekenhout, F. (Ed.) Handbook of incidence geometry: buildings and foundations, chapter 18: Elsevier.
Givant, S, & Halmos, P. (2009). Introduction to Boolean algebras. New York: Springer.
Grzegorczyk, A. (1960). Axiomatizability of geometry without points. Synthese, 12, 228–235.
Gruszczyński, R. (2016). Non-standard theories of space. Nicolaus Copernicus University Scientific Publishing House.
Cohn, A.G., & Hazarika, S.M. (2001). Qualitative spatial representation and reasoning: an overview. Fundamenta Informaticae, 46(1–2), 1–29.
Lewis, D. (1986). On the plurality of worlds. Oxford: Blackwell.
Lewis, D. (1991). Parts of classes. Oxford: B. Blackwell.
Randell, D A, Cui, Z, Cohn, AG. (1992). A spatial logic based on regions and connection. In Proceedings of the third international conference on principles of knowledge representation and reasoning (pp. 165–176).
Roeper, P. (1997). Region-based topology. Journal of Philosophical Logic, 26 (3), 251–309.
Russell, J. (2008). The structure of gunk: adventures in the ontology of space. In Oxford studies in metaphysics (Vol. 4 pp. 248–274). Oxford: Oxford University Press.
Scott, D. (2009). Mixing modality and probability, lecture notes. UC Berkeley, Logic Colloquium.
Shapiro, S, & Hellman, G. (2018). Varieties of continua: from regions to points and back. Oxford: Oxford University Press.
Stell, J. (2000). Boolean connection algebras: anew approach to the Region-Connection Calculus. Artificial Intelligence, 122(1–2), 111–136.
Skyrms, B. (1993). Logical atoms and combinatorial possibility. The Journal of Philosophy, 90(5), 219–232.
Tarski, A. (1956). Foundations of the geometry of solids. In Woodger, J.H. (Ed.) Logic, semantics, and metamathematics (pp. 24–29). Oxford: Clarendon Press.
Vakarelov, D. (2007). Region-based theory of space: algebras of regions, representation theory, and logics. In D. Gabbay, M. Zakharyaschev, S. Goncharov (Eds.) , Mathematical problems from applied logic (Vol. II, pp. 267–348). New York: Springer.
Vakarelov, D, Düntsch, I, Bennett, B. (2001). A note on proximity spaces and connection based mereology. In C. Welty, & B. Smith (Eds.) , Proceedings of the international conference on formal ontology in information systems (pp. 139–150).
Vakarelov, D, Dimov, G, Düntsch, I, Bennett, B. (2002). A proximity approach to some region-based theories of space. Journal of Applied Non-Classical Logics, 12(3), 527–559.
Whitehead A. (1929). Process and reality. New York: Macmillan.
Zimmerman, D. (1996). Could extended objects be made out of simple parts? An argument for “atomless gunk”. Philosophy and Phenomenological Research, 56(1), 1–29.
We thank Achille Varzi and two anonymous referees for very helpful comments.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Lando, T., Scott, D. A Calculus of Regions Respecting Both Measure and Topology. J Philos Logic 48, 825–850 (2019). https://doi.org/10.1007/s10992-018-9496-8
- Point-free space