Associated Topics || Dr. Math Home || Search Dr. Math

### Propositional Logic: Translating "You cannot ride the roller coaster if ... unless ..."

```Date: 08/06/2009 at 14:51:05
From: Ágost
Subject: Propositional logic -- translating an English sentence

Hi. I have a question regarding an example I read in a math textbook
on translating English sentences using propositional logic.  The
solution of the problem is also explained in the book, but I fail to
understand it.  The problem goes like this:

How can the following English sentence be translated into a
logical expression?

You cannot ride the roller coaster if you are under 4 feet
tall unless you are older than 16 years old.

Solution: Let q represent "You can ride the roller coaster."
Let r represent "You are under 4 feet tall."
Let s represent "You are older than 16 years old."

Then the sentence can be translated to

(r [conjunction] [negation] s) --> [negation] q.

(Unfortunately I can't use proper mathematical signs, so -->
stands for the sign of conditional statement.)

My question is:

The sentence "s" refers to "You are older than 16 years old,"
which in my reading isn't a negation of anything. Still, in the
solution, it is negated.  I fail to understand why "s" has to be
negated.  If you could kindly explain it to me, I would be most
grateful.

Ágost
```

```

Date: 08/06/2009 at 23:27:37
From: Doctor Peterson
Subject: Re: Propositional logic -- translating an English sentence

Hi, Ágost.

I'll do what many people do, and use the caret symbol ^ to
represent conjunction, the tilde symbol ~ for negation, and ->
for the conditional.  Using those conventions, your textbook's

(r ^ ~s) -> ~q

The negation of s lies in the word "unless."

Let's simplify the sentence to

You cannot ride the roller coaster unless you are older
than 16.

This simpler sentence says that you can only ride the roller
coaster if you are older than 16.  If you are NOT older than 16,
you CAN'T ride.  Think about that a bit, and you'll see that
"unless" could be replaced by "if not."

You cannot ride the roller coaster if you are not older
than 16.

Or, changing the order,

If you are not older than 16, then you cannot ride the roller
coaster.

Given that

q = "You can ride the roller coaster"
s = "You are older than 16 years old"

this sentence would translate as

~s -> ~q

Does that help?

Now, technically I disagree with the way the book's answer uses
parentheses.  I'd equate the sentence to

If you are under 4 feet tall, then you cannot ride the roller
coaster if you are not older than 16 years old.

Note that r is the condition for the entire rest of the
sentence.  So I'd write it as

r -> (~s -> ~q)

But I can see how they get their answer, by taking both
conditions together; the sentence as they see it appears to be

If you are under 4 feet tall AND you are not older than 16
years old, you cannot ride the roller coaster.

That is equivalent to my version.

In light of these subtle shades of interpretation, sometimes
it's not entirely clear what the "right" answer is, so don't
worry too much about such minor differences.  But you were
well within your rights to want to understand how "unless"
produced a negative!

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```

```

Date: 08/07/2009 at 03:20:22
From: Ágost
Subject: Thank you (Propositional logic -- translating an English
sentence)

Hi, I would like to thank you for your fast and detailed answer.
It indeed cleared up things.  Best regards, Á
```
Associated Topics:
High School Logic

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search