Philosophy Lexicon of Arguments

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Field, Hartry
 
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Impredicativeness I 214
Definition impredicative/Field: completely impredicative properties: are not at all derived from previously available properties. - In particular, there is no property to be a property. - Quasi-impredicative: also allows "property to be a property".
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I 216
Classic example for impredicative definition: E.g. What is it for an ordinal number to be finite? - Fin (OZ) P [P is inductive & P (0)> P (OZ)] - whereby P is inductive is defined as:
b [P(b) > P(b + 1)] - ((s) All successors have the same property (to be a number)). - The invalid objection against the impredicative definition (VsImpredicativity) is that one cannot know that a given number, e.g. 2 is finite because, in order to show this, we must be able to show that 2 has every inductive property of 0 - to show that 2 is finite, we must show first that exactly this 2 is finite (circular). - Solution/Field: the solution is simple: if finiteness is an inductive property, then 2 is finite. - No circle.

Fie I
H. Field
Realism, Mathematics and Modality Oxford New York 1989

Fie II
H. Field
Truth and the Absence of Fact Oxford New York 2001

Fie III
H. Field
Science without numbers Princeton New Jersey 1980


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Ed. Martin Schulz, access date 2017-03-26