Australasian Journal of Logic 12 (3):159-176 (2015)

John Wigglesworth
University of York
In this paper, we explore the idea that sets depend on, or are grounded in, their members. It is said that a set depends on each of its members, and not vice versa. Members do not depend on the sets that they belong to. We show that the intuitive modal truth conditions for dependence, given in terms of possible worlds, do not accurately capture asymmetric dependence relations between sets and their members. We extend the modal truth conditions to include impossible worlds and give a more satisfactory account of the dependence of a set on its members. Focusing on the case of singletons, we articulate a logical framework in which to evaluate set-theoretic dependence claims, using a normal first-order modal logic. We show that on this framework the dependence of a singleton on its single members follows from logic alone. However, the converse does not hold.
Keywords set theory  metaphysical dependence
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DOI 10.26686/ajl.v12i3.2131
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References found in this work BETA

On the Plurality of Worlds.David Lewis - 1986 - Wiley-Blackwell.
Modal Logic as Metaphysics.Timothy Williamson - 2013 - Oxford, England: Oxford University Press.
Ontological Dependence.Tuomas E. Tahko & E. J. Lowe - 2020 - Stanford Encyclopedia of Philosophy.
Monism: The Priority of the Whole.Jonathan Schaffer - 2010 - Philosophical Review 119 (1):31-76.

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Remarks on Ontological Dependence in Set Theory.Thomas Macaulay Ferguson - 2016 - Australasian Journal of Logic 13 (3):41-57.

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