# What does predicative mean?

I'm reading this page https://plato.stanford.edu/entries/type-theory/ where the words predicative and impredicative are used in this context:

Notice that for defining the predicate R, we have used an impredicative existential quantification on predicates. It can be shown that the predicative version of Frege’s system is consistent

What does predicative mean?

Edit: I actually found the terms expanded on later in the page

In order to further motivate this hierarchy, here is one example due to Russell. If we say Napoleon was Corsican. we do not refer in this sentence to any assemblage of properties. The property “to be Corsican” is said to be predicative. If we say on the other hand Napoleon had all the qualities of a great general we are referring to a totality of qualities. The property “to have all qualities of a great general” is said to be impredicative.

A definition is said to be impredicative if it generalizes over a totality to which the entity being defined belongs. Otherwise the definition is said to be predicative.

Consider the following definition :

Let n be the least natural number such that n cannot be written as the sum of at most four cubes.

It is an impredicative definition, because it generalizes over all natural numbers, including n itself.

Predicate R, used in the Fregean version of Russell's Paradox, is defined as follows :

R(x) iff ∃P [x = the "extension" of P and ¬P(x)]:

this definition is impredicative because the definition uses an existential quantification on predicates (the totality to which the predicate R belongs).

• In the article you linked, the predicate "is prime" is called predicative even though it needs to reference the totality of all natural numbers in order to exclude all but two of them as divisors. Is that a mistake or am I making a mistake? – Mark Oct 14 '18 at 20:14
• @Mark - (3) does not define a number n : numbers are already "existing" at this point. It defines a property of numbers : Prime(n). – Mauro ALLEGRANZA Oct 14 '18 at 20:20
• I see now, since it does not generalize over all properties, primeness predicative – Mark Oct 14 '18 at 20:32