Lexicon of Arguments

 

I 214
Def 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".
>Self-reference, >Predication.
I 216
Classic example for impredicative definition: E.g. What is it for an ordinal number to be finite?

Fin (ON) P [P is inductive & P (0)> P (ON)]

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.
>Induction, >Deduction, >Circular reasoning, >Predicativeness.