The CORRECT formula for the sentence "not all Rainy days are Cold" is:

A. forall d (~Rainy(d) - Cold(d))
B. exists d (...

Question

The CORRECT formula for the sentence "not all Rainy days are Cold" is:

A. forall d (~Rainy(d) -> Cold(d))
B. exists d (Rainy(d) ^ Cold(d))
C. ~forall d (Rainy(d) ^ Cold(d))
D. exists d (Rainy(d) ^ ~Cold(d))

Accepted Answer

D. exists d (Rainy(d) ^ ~Cold(d)) 'All Rainy days are Cold' translates to forall d (Rainy(d) -> Cold(d)). To say 'not all' means there exists at least one day that is Rainy but not Cold. Applying quantifier negation and the negation-of-implication rule gives exists d (Rainy(d) ^ ~Cold(d)), which is option D.

The CORRECT formula for the sentence "not all Rainy days are Cold" is: A. forall d (~Rainy(d) -> Cold(d)) B. exists d (Rainy(d) ^ Cold(d)) C. ~forall d (Rainy(d) ^ Cold(d)) D. exists d (Rainy(d) ^ ~Cold(d))