Consider the following well-formed formulae:
I. ~forall x (P(x))
II. ~exists x (P(x))
III. ~exists x (~P(x))
IV. exi...

Question

Consider the following well-formed formulae:
I.   ~forall x (P(x))
II.  ~exists x (P(x))
III. ~exists x (~P(x))
IV.  exists x (~P(x))

Which of the following pairs are logically equivalent?

A. I and III
B. I and IV
C. II and III
D. II and IV

Accepted Answer

B. I and IV are logically equivalent: both express 'not all x satisfy P', i.e., ~forall x P(x) = exists x ~P(x). Using the quantifier negation laws: ~forall x P(x) is equivalent to exists x ~P(x). So formula I equals formula IV — they both say 'there exists an x for which P(x) is false'. Formula II (~exists x P(x)) = forall x ~P(x) which equals formula III negated. Formula III equals forall x P(x) (double negation). So I and IV are equivalent.

Consider the following well-formed formulae: I. ~forall x (P(x)) II. ~exists x (P(x)) III. ~exists x (~P(x)) IV. exists x (~P(x)) Which of the following pairs are logically equivalent? A. I and III B. I and IV C. II and III D. II and IV