Universal and existential quantifiers

Question

Are there any examples of a unary predicate $P(x)$ such that the truth value of $P(x)$ remains invariant under exchange of the universal quantifier $\forall$ and the existential quantifier $\exists$?

Thanks.

Answer 1

Thank you for the challenging question. Let us consider only predicates having only the variable x free and let us restrict the discussion to standard first-order logic. Define the universalization of a predicate to be the result of prefixing ‘for every x’ and the existentialization of a predicate to be the result of prefixing ‘for some x’. Using this terminology, the questioner asks: Q1: For which predicates is the universalization logically equivalent to the existentialization? In standard first-order logic, the universalization logically implies the existentialization in every case—because the universes of discourse are required to be non-empty. Thus the question reduces to: Q2: For which predicates is the universalization logically implied by the existentialization? Previous commentators say that this holds for logically true predicates—such as x=x— and for logically false predicates—such as x≠x. There are many points to be made. Here is one. There are predicates that are neither logically true nor logically false whose existentializations logically imply their universalizations. Example:‘for some y, x≠y’.

As far as I know the question Q2 is open.

REQUEST How can this be corrected or otherwise improved?

Answer 2

UNIVERSALIZABLE EXISTENTIALS. Of course, not every existential ‘for some x, Px’ is logically equivalent to the corresponding universal ‘for every x, Px’. But many are. Let us call those that are unversalizable. In fact, there is a one-one function that carries universals into universalizable existentials: let the function U carry the universal ‘for every x, Px’ to the existential ‘for some x, [Px & for every x, Px].

Every universal sentence is logically equivalent to a universalizable existential.

This shows that there are infinitely many universalizable existentials.

But we can generalize. From here it is easy to see that every sentence is logically equivalent to a universalizable existential.

Are there any nice theorems about universalizable existentials? Are there any nice open questions about universalizable existentials?