I am trying to figure out how to express the sentence “not all rainy days are cold” using predicate logic. This is actually a multiple-choice exercise where the choices are as follows:
(A) $\forall d(\mathrm{Rainy}(d)\land \neg\mathrm{Cold}(d))$
(B) $\forall d(\neg\mathrm{Rainy}(d)\to \mathrm{Cold}(d))$
(C) $\exists d(\neg\mathrm{Rainy}(d)\to\mathrm{Cold}(d))$
(D) $\exists d(\mathrm{Rainy}(d)\land \neg\mathrm{Cold}(d))$
I am really having a hard time understanding how to read sentences correctly when they are in predicate logic notation. Can someone give me a hint on how to do this and also how to approach the problem above?
Best Answer
Think about the positive statement first (of which your statement is the negation). That is, consider the following statement: "All rainy days are cold."
Use the following notation:
$P(d):$ The day is rainy.
$Q(d):$ The day is cold.
Thus, we may represent the positive statement as follows: $$ \forall d(P(d)\to Q(d)).\tag{1} $$ The statement you are considering is the negation of $(1)$; that is, you are considering the statement, "Not all rainy days are cold." Thus, you need to negate $(1)$: $$ \neg[\forall d(P(d)\to Q(d))] \equiv \exists d\neg[P(d)\to Q(d)]\equiv \exists d[P(d)\land \neg Q(d)]. $$ Thus, the answer to your question is D.