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 Maarten Maartensz:    Philosophical Dictionary | Filosofisch Woordenboek                      

 A - Abstraction - in logic

 

Abstraction in logic: In logic, there are several subtle and mathematically worked out approaches to and uses of abstraction. Two noteworthy ones are abstraction in set theory and lambda-abstraction.

1. Abstraction in set theory

Intuitively, sets are the extensions of properties or relations, which can be dealt with and thought of as one, if only by having one term for them. Thus, for example {x: x is human} is the set (or class or collection) of things that have the property of being human. Here the braces indicate that what is contained in them is a set (or class or collection). The part without the braces is the abstraction of "the things x that are human", obtained from the formula "x is human" by fronting it with "x:" that is pronounced as "the x such that".

This abstraction is often referred to as "the property" - here the "the property of being human", and the reason, apart from intuition and linguistic usage, is especially to be found in the following theorem of set theory (that is here stated with some detail left out):

a e {x: F(x)} iff F(a) or for relations: (a,b) e {xy: R(x,y)} iff R(a,b)

That is: For an arbitrary property F, any a is an element of the set of things that are F iff a is F, and similary for relations.

This gives great facility and manipulative ease in moving from statements to properties or relations and sets of elements having these properties or relations, and the other way, from sets of elements that satisfy a certain abstraction to a statement asserting this for given elements that do.
 


See also: Abstraction, Such that


Literature:

Carnap, Halmos, Quine, Stegmüller
 

 Original: Sep 5, 2004                                                Last edited: 15 September 2007.   Top