There was a disagreement in my college class regarding what the following statement would be in a predicate wff format:
It is always a sunny day only if it is a rainy day.
Where D(x) is "x is a day", S(x) is "x is sunny", and R(x) is "x is rainy".
Is there anyone who can help settle this?
UPDATE: I notice some people may be tempted to suggest alterations to the sentence. I agree with this, as it is an awkwardly written sentence. What makes its current structure significant is that the sentence is quoted directly from a question in a textbook my class was discussing.
To paraphrase: "If anything is a day and it is sunny, then it must be a day and it is rainy." $$\forall x \Big((D(x)\wedge S(x))\to (D(x) \wedge R(x))\Big)$$
Though we might be tempted to simplify: "if anything is a day, then if it is sunny, then it must be rainy". $$\forall x \Big(D(x)\to \big(S(x))\to R(x)\big)\Big)$$
$$\forall x \in D: \big(S(x))\to R(x)\big)$$
However, I suspect that something was missing from the original statement.
Perhaps it was : "It is always a sunny day only if it is never a rainy day."
$$\forall x \big(D(x)\to S(x)\big) \to \neg\exists x \big(D(x)\wedge R(x)\big)$$