Sets and Plural Comprehension

Journal of Philosophical Logic 43 (2-3):517-539 (2014)
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Abstract

The state of affairs of some things falling under a predicate is supposedly a single entity that collects these things as its constituents. But whether we think of a state of affairs as a fact, a proposition or a possibility, problems will arise if we adopt a plural logic. For plural logic says that any plurality include themselves, so whenever there are some things, the state of affairs of their plural self-inclusion should be a single thing that collects them all. This leads to paradoxes analogous to those that afflict naïve set theory. Here I suggest that they are the very same paradoxes, because sets can be reduced to states of affairs. However, to obtain a consistent theoretical reduction we must restrict the usual axiom scheme of Comprehension for plural logic to ‘stratified’ formulas, to avoid viciously circular definitions. I prove that with this modification to the background plural logic, the theory of states of affairs is consistent; moreover, it yields the axioms of the familiar set theory NFU

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Keith Hossack
Birkbeck College

Citations of this work

Plural quantification.Ø Linnebo - 2008 - Stanford Encyclopedia of Philosophy.
Can All Things Be Counted?Chris Scambler - 2021 - Journal of Philosophical Logic 50 (5):1079-1106.
When Do Some Things Form a Set?Simon Hewitt - 2015 - Philosophia Mathematica 23 (3):311-337.
Composition and Identities.Manuel Lechthaler - 2017 - Dissertation, University of Otago

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References found in this work

The Principles of Mathematics.Bertrand Russell - 1903 - Cambridge, England: Allen & Unwin.
Parts of Classes.David K. Lewis - 1991 - Mind 100 (3):394-397.
Introduction to mathematical logic.Elliott Mendelson - 1964 - Princeton, N.J.,: Van Nostrand.
Plural predication.Thomas J. McKay - 2006 - New York: Oxford University Press.

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