# Rewriting Conditionals In Their Well Known Form

The question is,

"Write each of these statements in the form “if p, then q” in English. [Hint:Refer to the list of common ways to express conditional statements.]

a) It snows whenever the wind blows from the northeast.

b)The apple trees will bloom if it stays warm for a week.

c) That the Pistons win the championship implies that they beat the Lakers.

d)It is necessary to walk 8 miles to get to the top of Long’s Peak.

e) To get tenure as a professor, it is sufficient to be world- famous.

f) If you drive more than 400 miles, you will need to buy gasoline.

g)Your guarantee is good only if you bought your CD player less than 90 days ago.

h)Jan will go swimming unless the water is too cold.

I am having a little trouble with c), g), and h).

For c): Presumably, it would appear that this sentence is discussing a championship match, one between the Lakers and Pistons. Hence, I am having difficulty seeing why it has to be written a particular way. Doesn't "If the Pistons win the championship, then they beat the Lakers," and "If the Pistons beat the Lakers, then they win the championship," convey the same meaning?

For g): This is another instance of me not seeing why this conditional statement has to be written any particular way. To me, "If you bought your CD player less than 90 days ago, then your guarantee is good," and "If your guarantee is good, then you bought your CD player less than 90 days ago," convey the same meaning.

For h): I wrote, "If the water is too cold, then Jan won't go swimmming;" however, the answer key says, "If the water is NOT too cold, then Jan will go swimming." Would my answer be acceptable?

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Rearranging the expressions and highlighting "key words":

• c) (The Pistons win the championship) implies (they beat the Lakers).

"p implies q" $\iff$ "$p \rightarrow q$" = "If p then q".

• g) (Your guarantee is good) only if (you bought your CD player less than 90 days ago).

"p only if q" $\iff$ "$p \rightarrow q$" = "If p then q".

• h) (Jan will go swimming) unless (the water is too cold.)

That is: If (it is not the case that the water is too cold), then (Jan will go swimming)
$\iff$ "If (the water is NOT too cold), then (Jan will go swimming).

You can think of "unless" as meaning "if not". Then we have "p unless q" $\iff$ "p, if not q" $\iff$ "if not q, then p".

It might be helpful to review a previous MSE post: Different ways to express "if-then"..

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Does matter where I put that "not," because the two statements, to me, seem to be the same, I just have positioned my "not" in a different place. – Mack Dec 18 '12 at 18:07
Yes, it matters where you put the "not": Notice the key word "if" "p if q" means "if q, then p". So "p unless q" = "p if not q" means "if not q, then p" – amWhy Dec 18 '12 at 18:14
In $p\rightarrow q$: read "if $p$ then $q$...$p$ is the "antecedent" (condition) and $q$ is the "conseqent" (conclusion). The word "if" always immediately precedes the antecedent (the condition), so if you encounter a phrase like "$a$, if $b$", then the antecedent (condition) here is $b$, the consequent being $a$. So it translates to $b \rightarrow a$. The "condition" in $(h)$ is what follows "if": Jan well go swimming if (the water is NOT too cold). So, we can write the equivalent: IF $\lnot$(water is too cold), then Jan will go swimming. – amWhy Dec 18 '12 at 18:56
I should qualify my comment immediately above. When "if" stands alone in a sentence (as opposed to "only if"), it signals the antecedent of a conditional. The exception being "only if": what follows "only if" is the consequent. So "p if q": $q\rightarrow p$, whereas "p only if q": $p\rightarrow q$. – amWhy Dec 18 '12 at 22:00

(c) It’s not discussing a single championship match; rather, it’s talking about the entire post-season tournament. Thus, it’s conceivable that the Pistons could beat the Lakers and later lose to some other team.

(g) It doesn’t actually say that your guarantee is good if you bought your CD player less than $90$ days ago; it just says that if you bought the CD player $90$ days ago or earlier, your guarantee definitely isn’t good. There might be other conditions. For instance, it might be no good if you drove your car over the CD player, even if you bought the CD player yesterday.

(h) No. ‘Jan will go swimming unless the water is too cold’ guarantees that if the water is not too cold, Jan will definitely go swimming. In everyday language it very strongly suggests that if the water is too cold, she won’t go swimming, but in strict logical language it does not actually say this: it could be the first half of the statement Jan will go swimming unless the water is too cold, and maybe even if it is.

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Hmm, I guess I don't know anything about basket-ball, at all. What is a post-season tournament? And is this championship match not between the Lakers and the Pistons? Also, for your explanation on (g), do you mean to say that if you bought the CD player 90 days or $later$, then the guarantee is definitely not good? For (h), I still can't see a distinction between the two answers, sorry. – Mack Dec 18 '12 at 14:50
@EMACK: When the regular professional basketball season is over, the teams with the best records play an elimination tournament to decide the overall winner. The match in the question is between the Lakers and the Pistons, and it is one of the matches in the tournament, but it need not be the championship match, i.e., the match between the last two teams left in the tournament. // No, I meant earlier: $90$ days ago or earlier means $90$ or more days ago. – Brian M. Scott Dec 18 '12 at 15:00
@EMACK: Regarding (h): one statement says that if the water is not too cold, Jan will go swimming. The other makes a stronger statement: if the water is not too cold, Jan will go swimming, and if the water is too cold, Jan will not go swimming. The second statement obviously implies the first; the first does not imply the second. – Brian M. Scott Dec 18 '12 at 15:01
Hmm, I think I am just having difficulty with the semantics of the English language. Do you know of any books that I could read, to improve my understanding of the English language? – Mack Dec 18 '12 at 15:29
@EMACK: Yes, it’s almost certainly much more a problem of language than of understanding the mathematics and logic; unfortunately, I really don’t know of anything to recommend. My best suggestion would be to look at as many examples of problems of this type as you can find, though I realize that this may not be very helpful. – Brian M. Scott Dec 18 '12 at 15:48