Skolemization

Skolemization eliminates existential quantifiers by replacing each existentially quantified variable with a Skolem constant or Skolem function.

In effect, we are saying ``If there exists (at least) one, give the algebraic name a to it.'' Having named the existential variable, we can eliminate the quantifier.

In general, an existential variable is replaced by a Skolem function of all the universal variables to its left. (A Skolem constant is a function of no variables.)

Each Skolem constant or function that is introduced must be a new one, distinct from any constant or function symbol that has been used already.

Example: &exist x &forall y &forall z &exist w P(x, y, z, w)
This is Skolemized as P(a, y, z, f(y, z)). &exist x has no universals to its left, so it is Skolemized as a constant, a. &exist w has universals y and z to its left, so it is Skolemized as a function of y and z.

After Skolemizing, universal quantifiers are eliminated; all remaining variables are understood to be universally quantified.