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### Translating Logic Statements

```Date: 10/11/2005 at 16:08:41
From: Diane
Subject: Logic

In a statement like "Joe will go to the movies only if Sam wins the
race", how is it written as a conditional (if...then) statement?

I think it is written as "If Sam wins the race then Joe will go to
the movies." However, several textbooks have it as "If Joe goes to
the movies then Sam wins the race." I am confused over the switch of
ideas from if to then.

```

```
Date: 10/11/2005 at 18:48:32
From: Doctor Achilles
Subject: Re: Logic

Hi Diane,

Thanks for writing to Dr. Math.

Translating "only if" is one of the most difficult things to get from
English to symbolic logic.  Your textbooks are correct, the sentence:

"Joe will go to the movies only if Sam wins the race."

Is logically equivalent to:

"If Joe goes to the movies, then Sam wins the race"

First, let's look at the second sentence:

"If Joe goes to the movies, then Sam wins the race"

There are 4 possibilities:

Joe goes to the movies and Sam wins the race
In this case, the sentence is TRUE.

Joe doesn't go to the movies and Sam wins the race
In this case, the sentence is TRUE.

Joe doesn't go to the movies and Sam doesn't win the race
In this case, the sentence is TRUE.

Joe goes to the movies and Sam doesn't win the race
In this case, the sentence is FALSE.

Ok, now let's go back to the original sentence:

"Joe will go to the movies only if Sam wins the race"

What this sentence says is that the ONLY situation in which Joe goes
to the movies is when Sam wins the race.  So this sentence will be
FALSE if Joe goes to the movies and Sam doesn't win the race.

If Sam does win the race, Joe doesn't have to go to the movies, he can
or he can stay home.  But he is only ALLOWED to go if Sam wins.

Here's another way of looking at this problem.  Would you agree that:

"Joe will go to the movies only if Sam wins the race"

Is equivalent to:

"If Sam doesn't win the race, then Joe won't go to the movies"?

Notice that the sentence:

"If Sam doesn't win the race, then Joe won't go to the movies"

Is the contrapositive of:

"If Joe goes to the movies, then Sam wins the race"

Contrapositives are logically equivalent--for more on this see:

Truth of the Contrapositive
http://mathforum.org/library/drmath/view/63215.html

So:

"If Joe goes to the movies, then Sam wins the race"

Is equivalent to:

"If Sam doesn't win the race, then Joe won't go to the movies"

And therefore is equivalent to:

"Joe will go to the movies only if Sam wins the race"

Hope these explanations help.  If you have other questions or you'd

- Doctor Achilles, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Logic

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