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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.

Agent Baron
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  • $a \text{ only if } b \dashv \vdash a \implies b$, if that helps. – GFauxPas Jan 26 '15 at 18:19
  • See this post : "$A$ only if $B$" must be symbolized as : $A \rightarrow B$. Thus : "It is a sunny day only if it is a rainy day" is : "if it is a sunny day, then it is a rainy day". – Mauro ALLEGRANZA Jan 26 '15 at 20:25
  • If we want to use predicate logic, we must have something like : $\forall x (S(x) \rightarrow R(x))$, where the domain of quantification is the set $D$ of days. – Mauro ALLEGRANZA Jan 26 '15 at 20:47
  • Is that the right statement? Are you sure you don't want: "It is always a sunny day only if it is never a rainy day". – Graham Kemp Jan 27 '15 at 02:55

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"It is always a sunny day only if it is a rainy day."

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."

$$\forall x \big(D(x)\to S(x)\big) \to \neg\exists x \big(D(x)\wedge R(x)\big)$$

Graham Kemp
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    I added some more information regarding the structure of the sentence. I hope that helps clear the air a bit. – Agent Baron Jan 27 '15 at 04:24