How to express the statement "not all rainy days are cold" using predicate logic? 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?

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 $(1)$: $$ \
eg[\forall d(P(d)\to Q(d))] \equiv \exists d\
eg[P(d)\to Q(d)]\equiv \exists d[P(d)\land \
eg Q(d)]. $$ Thus, the answer to your question is **D**.