Mr. Nance,

My student has a question on Exercise 4 number 14.  Her answer for was ~C ⊃ S instead of S ⊃ ~C. Can the statements “I will go swimming only if the water is not cold” be considered logically equivalent to “If the water is not cold, I will go swimming”?

Also, how can I explain the difference between “If the water is not cold I will go swimming” and “I will go swimming unless the water is cold”?

Thank you!

Let me answer your second question first. You don’t need to explain the difference between “If the water is not cold then I will go swimming” and “I will go swimming unless the water is cold” because these are logically equivalent. The rule is this: p unless q is equivalent to if not q then p. You used this rule correctly. The statement “You will starve unless you eat sometime” means “If you do not eat sometime, then you will starve.”

But the statement “I will go swimming only if the water is not cold” is not logically equivalent to “If the water is not cold, then I will go swimming” because there might be other restrictions to swimming. The statement form p only if q is NOT equivalent to if q then p. The form p if q (without the “only”) IS equivalent to if q then p, but the “only” switches the order. The rule is this: p only if q is equivalent to if p then q. For example, the statement “He is a father only if he is a man” (which is true) is not equivalent to “If he is a man then he is a father” (which can be false). It is equivalent to “If he is a father then he is a man” (which is true).

In summary:
p unless q = if not q then p
p only if q = if p then q
Blessings!