Given an IF formula \ varphi, associate, to each universal quantifier ( \ forall v / V ) occurring in it, a new function symbol g _ v ( a " Kreisel function " ).
42.
The restriction to formulas in prenex form containing only existential quantifiers does not limit the generality of the theorem, because formulas can be converted to prenex form and their universal quantifiers can be removed by Herbrandization.
43.
In particular, it can be expressed as formula consisting of a number of leading universal quantifiers followed by a quantifier-free body ( these formulas are at level \ Pi ^ 0 _ 1 of the arithmetical hierarchy ).
44.
Perhaps the simplest problem for alternating machines to solve is the quantified Boolean formula problem, which is a generalization of the Boolean satisfiability problem in which each variable can be bound by either an existential or a universal quantifier.
45.
Assigning Boolean semantics to classical predicate calculus requires that the model be a complete Boolean algebra because the universal quantifier maps to the infimum operation, and the existential quantifier maps to the supremum; this is called a Boolean-valued model.
46.
Under a Montagovian approach, the indefinite " a donkey ", which is assumed to be inherently an existential quantifier, ends up becoming a universal quantifier, an unwelcome result because the change in quantificational force cannot be accounted for in any principled way.
47.
The set of formulae to check for validity is here supposed to contain no free variables; this is not a limitation as free variables are implicitly universally quantified, so universal quantifiers over these variables can be added, resulting in a formula with no free variables.
48.
Finally, we would like, for reasons of technical convenience, that the " prefix " of ? ( that is, the string of quantifiers at the beginning of ?, which is in normal form ) begin with a universal quantifier and end with an existential quantifier.
49.
If a value of n is fixed, the set of p for which there is such a corresponding open U is itself an open set ( being a union of open sets ), and the universal quantifier on n corresponds to the ( countable ) intersection of these sets.
50.
These symbols can be used by the rule for universal quantifiers, so that \ forall y . \ gamma ( y ) can generate \ gamma ( c ) even if c was not in the original formula but is a Skolem constant created by the rule for existential quantifiers.
