∼(∃x)(Px • Qx) ≡ (∀x)(Px ⊃ ∼Qx)
-Consider assuming '∼(∃x)(Px • Qx)' for a conditional proof of one direction of the biconditional in the above logical truth. Which of the following propositions is a legitimate second step in that proof?
A) (∀x) ∼ (Px • Qx)
B) Px ⊃ ∼Qx
C) Px • Qx
D) Pc • Qc
E) ∼(∀x)(Px ⊃ ∼Qx)

Question

Quizplus · Accepted Answer

The negation of an existential quantifier (∃x) can be converted into a universal quantifier (∀x) with the negation of the predicate, turning ∼(∃x)(Px • Qx) into (∀x)∼(Px • Qx). This is a direct application of the logical rules concerning quantifiers and negation.

∼(∃X)(Px • Qx) ≡ (∀X)(Px ⊃ ∼Qx)