Element contents
The Mereology of Classes
Published online by Cambridge University Press: 13 May 2024
Summary
This Element is a systematic study of the question of whether classes are composed of further parts. Mereology is the theory of the relation of part to whole, and we will ask how that relation applies to classes. One reason the issue has received attention in the literature is the hope that a clear picture of the mereology of classes may provide further insights into the foundations of set theory. We will consider two main perspectives on the mereology of classes on which classes are indeed composed of further parts. They, however, disagree as to the identity of those parts. Each perspective admits more than one implementation, and one of the purposes of this work is to explain what is at stake with each choice.
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- Online ISBN: 9781009092241Publisher: Cambridge University PressPrint publication: 06 June 2024
References
Armstrong, David M. 1991. “Classes Are States of Affairs.” Mind 100 (2): 189–201.CrossRefGoogle Scholar
Armstrong, David M. 1995. “Reply to Rosen.” Australasian Journal of Philosophy 73 (4): 626–628.CrossRefGoogle Scholar
Barwise, John, and Moss, Lawrence. 1996. Vicious Circles: On the Mathematics of Circular Phenomena, CSLI-Lecture Notes. Center for the Study of Language and Information at Stanford University.Google Scholar
Bell, John L. 2004. “Whole and Part in Mathematics.” Axiomathes 14 (4): 285–294.CrossRefGoogle Scholar
Bigelow, John. 1993. “Sets Are Haecceities.” In John Bacon, Keith Campbell, and Lloyd Reinhardt (Eds.), Ontology, Causality and Mind, 73–96. Cambridge University Press.Google Scholar
Boolos, George. 1984. “To Be Is to Be a Value of a Variable (or to Be Some Values of Some Variables).” The Journal of Philosophy 81 (8): 430.CrossRefGoogle Scholar
Boolos, George 1985. “Nominalist Platonism.” The Philosophical Review 94 (3): 327–344.CrossRefGoogle Scholar
Burgess, John P. 2004. “E Pluribus Unum: Plural Logic and Set Theory.” Philosophia Mathematica 12(3): 193–221.CrossRefGoogle Scholar
Burgess, John P. 2015. “Lewis on Mereology and Set Theory.” In Barry Loewer and Jonathan Schaffer (Eds.), A Companion to David Lewis, 459–469. John Wiley & Sons, Ltd.CrossRefGoogle Scholar
Button, Tim. 2021. “Level Theory, Part 1: Axiomatizing the Bare Idea of a Cumulative Hierarchy of Sets.” The Bulletin of Symbolic Logic 27 (4): 436–460.CrossRefGoogle Scholar
Caplan, Ben, Tillman, Chris, and Reeder, Pat. 2010. “Parts of Singletons.” Journal of Philosophy 107(10): 501–533.CrossRefGoogle Scholar
Cartwright, Richard L., 2001. “A Question about Sets.” In Byrne, Alex, Stalnaker, Robert, and Wedgwood, Ralph (Eds.), Fact and Value: Essays on Ethics and Metaphysics for Judith Jarvis Thomson, 29–46. Massachusetts Institute of Technology Press.CrossRefGoogle Scholar
Casati, Roberto, and Varzi, Achille C.. 1999. Parts and Places: The Structures of Spatial Representation. Massachusetts Institute of Technology Press.Google Scholar
Cotnoir, Aaron J. 2010. “Anti-Symmetry and Non-Extensional Mereology.” The Philosophical Quarterly 60 (239): 396–405.CrossRefGoogle Scholar
Cotnoir, Aaron J., and Varzi, Achille C.. 2019. “Natural Axioms for Classical Mereology.” The Review of Symbolic Logic 12 (1): 201–208.CrossRefGoogle Scholar
Cotnoir, Aaron J., and Varzi, Achille C. 2021. Mereology. Oxford University Press.CrossRefGoogle Scholar
Eberle, Rolf A. 1970. Nominalistic Systems. Vol. 30. Synthese Library. Springer Science & Business Media.CrossRefGoogle Scholar
Fairchild, Maegan. 2017. “A Paradox of Matter and Form.” Thought: A Journal of Philosophy 6 (1): 33–42.CrossRefGoogle Scholar
Fine, Kit 1999. “Things and Their Parts.” Midwest Studies In Philosophy 23 (1)23: 61–74.CrossRefGoogle Scholar
Fine, Kit 2010. “Towards a Theory of Part.” Journal of Philosophy 107 (11): 559–589.CrossRefGoogle Scholar
Florio, Salvatore, and Linnebo, Øystein. 2021. The Many and the One: A Philosophical Study of Plural Logic. Oxford University Press.CrossRefGoogle Scholar
Forrest, Peter 2002b. “Nonclassical Mereology and Its Application to Sets.” Notre Dame Journal of Formal Logic 43 (2): 79–94.CrossRefGoogle Scholar
Gödel, Kurt. 1947. “What Is Cantor’s Continuum Problem?” The American Mathematical Monthly 54 (9): 515–525.CrossRefGoogle Scholar
Goodman, Jeremy. 2022. “Matter and Mereology.” https://jeremy-goodman.com/MatterMereology.pdf.Google Scholar
Hamkins, Joel David, and Kikuchi, Makoto. 2016. “Set-Theoretic Mereology.” Logic and Logical Philosophy 25 (3): 285–308.Google Scholar
Hellman, Geoffrey. 1989. Mathematics without Numbers: Towards a Modal-Structural Interpretation. Clarendon Press.Google Scholar
Hellman, Geoffrey 1996. “Structuralism without Structures.” Philosophia Mathematica 4 (2): 100–123.CrossRefGoogle Scholar
Horsten, Leon. 2016. “Absolute Infinity in Class Theory and in Theology.” In Boccuni, Francesca and Sereni, Andrea (Eds.), Objectivity, Realism, and Proof, Boston Studies in the Philosophy and History of Science, 318: 103–122. Springer International Publishing.CrossRefGoogle Scholar
Horsten, Leon, and Welch, Philip. 2016. “Reflecting on Absolute Infinity.” Journal of Philosophy 113 (2): 89–111.Google Scholar
Hovda, Paul. 2009. “What Is Classical Mereology?” Journal of Philosophical Logic 38 (1): 55–82.CrossRefGoogle Scholar
Incurvati, Luca. 2020. Conceptions of Set and the Foundations of Mathematics. Cambridge University Press.CrossRefGoogle Scholar
Jacinto, Bruno, and Cotnoir, Aaron J.. 2019. “Models for Hylomorphism.” Journal of Philosophical Logic 48 (5): 909–955.CrossRefGoogle Scholar
Johnston, Mark. 2006. “Hylomorphism.” The Journal of Philosophy 103 (12): 652–698.CrossRefGoogle Scholar
Kanamori, Akihiro. 2003. “The Empty Set, the Singleton, and the Ordered Pair.” The Bulletin of Symbolic Logic 9 (3): 273–298.CrossRefGoogle Scholar
Koslicki, Kathrin. 2008. The Structure of Objects. Oxford University Press on Demand.CrossRefGoogle Scholar
Leśniewski, Stanisław. 1927. “O Podstawach Matematyki [On the Foundations of Mathematics].” Przeglad Filozoficzny 30: 164–206.Google Scholar
Leśniewski, Stanisław 1999. “Foundations of the General Theory of Sets. i.” Filozofia Nauki 7 (3–4): 173–208.Google Scholar
Lewis, David 1993. “Mathematics Is Megethology.” Philosophia Mathematica 1 (1): 3–23.CrossRefGoogle Scholar
Maddy, Penelope. 1983. “Proper Classes.” The Journal of Symbolic Logic 48 (1): 113–139.CrossRefGoogle Scholar
McCarthy, Timothy. 2015. “A Note on Unrestricted Composition.” Thought: A Journal of Philosophy 4 (3): 202–211.CrossRefGoogle Scholar
McDaniel, Kris. 2009. “Structure-Making.” Australasian Journal of Philosophy 87 (2): 251–274.CrossRefGoogle Scholar
McGee, Vann. 1997. “How We Learn Mathematical Language.” The Philosophical Review 106 (1): 35–68.CrossRefGoogle Scholar
Menzel, Christopher. 2014. “Wide Sets, ZFCU, and the Iterative Conception.” Journal of Philosophy 111 (2): 57–83.CrossRefGoogle Scholar
Mormann, Thomas. 2012. “On the Mereological Structure of Complex States of Affairs.” Synthese 187 (2): 403–418.CrossRefGoogle Scholar
Mormann, Thomas 2013. “Heyting Mereology as a Framework for Spatial Reasoning.” Axiomathes 23 (1): 137–164.CrossRefGoogle Scholar
Oliver, Alex, and Smiley, Timothy. 2006. “What Are Sets and What Are They For?” Philosophical Perspectives 20 (1): 123–155.CrossRefGoogle Scholar
Oliver, Alex, and Smiley, Timothy 2018. “Cantorian Set Theory.” The Bulletin of Symbolic Logic 24 (4): 393–451.CrossRefGoogle Scholar
Rosen, Gideon. 1995. “Armstrong on Classes as States of Affairs.” Australasian Journal of Philosophy 73 (4): 613–625.CrossRefGoogle Scholar
Russell, Jeffrey Sanford. 2016. “Indefinite Divisibility.” Inquiry 59 (3): 239–263.CrossRefGoogle Scholar
Shapiro, Stewart. 1991. Foundations without Foundationalism: A Case for Second-Order Logic. Vol. 17. Clarendon Press.Google Scholar
Tarski, Alfred. 1983. “Foundations of the Geometry of Solids.” In Corcoran (Ed.), John, Logic, Semantics, Metamathematics, 24–29. Hackett.Google Scholar
Urbaniak, Rafal. 2014. Leśniewski’s Systems of Logic and Foundations of Mathematics. Springer.CrossRefGoogle Scholar
Uzquiano, Gabriel. 2003. “Plural Quantification and Classes.” Philosophia Mathematica 11 (1): 67–81.CrossRefGoogle Scholar
Uzquiano, Gabriel 2015a. “Varieties of Indefinite Extensibility.” Notre Dame Journal of Formal Logic 56 (1): 147–166.CrossRefGoogle Scholar
Uzquiano, Gabriel 2015b. “A Neglected Resolution of Russell’s Paradox of Propositions.” The Review of Symbolic Logic 8 (2): 328–344.CrossRefGoogle Scholar
Uzquiano, Gabriel 2018. “Groups: Toward a Theory of Plural Embodiment.” Journal of Philosophy 115 (8): 423–432.CrossRefGoogle Scholar
Zermelo, Ernst. 1930. “Über Grenzzahlen und Mengenbereiche: Neue Untersuchungen über die Grundlagen der Mengenlehre.” Fundamenta Mathematicae 16: 29–47.CrossRefGoogle Scholar
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