A general theory of abstraction operators

Philosophical Quarterly 54 (214):105-133 (2004)
I present a general theory of abstraction operators which treats them as variable-binding term- forming operators, and provides a reasonably uniform treatment for definite descriptions, set abstracts, natural number abstraction, and real number abstraction. This minimizing, extensional and relational theory reveals a striking similarity between definite descriptions and set abstracts, and provides a clear rationale for the claim that there is a logic of sets (which is ontologically non- committal). The theory also treats both natural and real numbers as answering to a two-fold process of abstraction. The first step, of conceptual abstraction, yields the object occupying a particular position within an ordering of a certain kind. The second step, of objectual abstraction, yields the number sui generis, as the position itself within any ordering of the kind in question.
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DOI 10.1111/j.0031-8094.2004.00344.x
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References found in this work BETA
Reals by Abstraction.Bob Hale - 2000 - Philosophia Mathematica 8 (2):100--123.
Singular Terms and Arithmetical Logicism.Ian Rumfitt - 2003 - Philosophical Books 44 (3):193--219.
Logic and Existence.Ian Rumfitt - 1999 - Aristotelian Society Supplementary Volume 73 (1):151–180.

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Citations of this work BETA
What Harmony Could and Could Not Be.Florian Steinberger - 2011 - Australasian Journal of Philosophy 89 (4):617 - 639.
What is Neologicism?Bernard Linsky & Edward N. Zalta - 2006 - Bulletin of Symbolic Logic 12 (1):60-99.
A Logic Inspired by Natural Language: Quantifiers As Subnectors.Nissim Francez - 2014 - Journal of Philosophical Logic 43 (6):1153-1172.

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