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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.

Thanks in advance,

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 
answer looks like this:

  (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

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