# Sentential Logic Problem

• Aug 3rd 2013, 12:40 PM
Turgus
Sentential Logic Problem
Hi guys,
Having issues understanding the answer provided by the book to the following question:

Question: Let P stand for the statement "I will buy the pants" and S for the statement "I will buy the shirt." What English sentence is represented by the following expression

$\displaystyle \neg(P \land \neg S)$

The books answer is: I won't buy the pants without the shirt.

My answer turns out to be: I will not buy the pants or I will buy the shirt.
To get my answer I did the following:

$\displaystyle \neg(P \land \neg S)$
$\displaystyle \neg P \lor \neg \neg S$ (De Morgan's Law)
$\displaystyle \neg P \lor S$ (Double Negation Law)

Any help would be appreciated! I am only a day into this book and the answer the book provided seems nonsense to me.
Thanks
• Aug 3rd 2013, 01:01 PM
Plato
Re: Sentential Logic Problem
Quote:

Originally Posted by Turgus
Hi guys,
Having issues understanding the answer provided by the book to the following question:
Question: Let P stand for the statement "I will buy the pants" and S for the statement "I will buy the shirt." What English sentence is represented by the following expression
$\displaystyle \neg(P \land \neg S)$
The books answer is: I won't buy the pants without the shirt.

My answer turns out to be: I will not buy the pants or I will buy the shirt.
To get my answer I did the following:
$\displaystyle \neg(P \land \neg S)$
$\displaystyle \neg P \lor \neg \neg S$ (De Morgan's Law)
$\displaystyle \neg P \lor S$ (Double Negation Law)

But your answer has exactly the same meaning as the one in the textbook.

$\displaystyle \neg(P \land \neg S)$ is literally read "It false that I buy the pants and not buy the shirt.
• Aug 3rd 2013, 01:49 PM
Turgus
Re: Sentential Logic Problem
Quote:

Originally Posted by Plato
But your answer has exactly the same meaning as the one in the textbook.

$\displaystyle \neg(P \land \neg S)$ is literally read "It false that I buy the pants and not buy the shirt.

I don't see the relation how my answer has the exact same meaning as the one in the textbook....
The books answer to me is saying:
If I buy the shirt, I will buy the pants.
OR
If I don't buy the shirt, I won't buy the pants.
Isn't that a completely different meaning from my answer?
• Aug 3rd 2013, 02:26 PM
Plato
Re: Sentential Logic Problem
Quote:

Originally Posted by Turgus
I don't see the relation how my answer has the exact same meaning as the one in the textbook....
The books answer to me is saying:
If I buy the shirt, I will buy the pants.
OR
If I don't buy the shirt, I won't buy the pants.
Isn't that a completely different meaning from my answer?

Well, you just don't understand the "if,,,then..."

It is the case that $\displaystyle P \to S \equiv \,\neg P \vee S \equiv \,\neg \left( {P \wedge \neg S} \right)$.

"If I buy the pants then I buy the shirt" is the same as "I don't buy the pants or I buy the shirt" is the same as "it is false that I buy the pants and not buy the shirt".

For me those have the meaning as "I won't buy the pants without the shirt".
• Aug 3rd 2013, 02:30 PM
ChessTal
Re: Sentential Logic Problem
Quote:

Originally Posted by Turgus
I don't see the relation how my answer has the exact same meaning as the one in the textbook....
The books answer to me is saying:
If I buy the shirt, I will buy the pants.
OR
If I don't buy the shirt, I won't buy the pants.
Isn't that a completely different meaning from my answer?

Here is the deal:

The book answer is correct as also yours.

The meaning of the $\displaystyle \neg(P \land \neg S)$ is that:
I will not (buy the pants and will not buy the shirt), i.e:
I will not buy the pants and then i will not buy the shirt also, or:
If i will buy the pants then i will buy the shirt also.

The above correct answer is the same (obviously) with that of the book.

The answer is :
$\displaystyle \neg P \lor S$ And that means either and exactly one of the above 3 is true:
a)I will not buy the pants and will not buy the shirt
b)I will buy the shirt and will buy the pants
c)I will not buy the pants AND will buy the shirt.

Since there are 4 combinations of buying/not buying shirts/pants, the extra option for us to do is:
d)I will buy the pants and not buy the shirt.

Now:
The book answer says that "If i will buy the pants, i will buy the shirt also".
So your a) and c) do not contradict with what the book's answer says. (1)
And of course your b) does also not contradict the book's answer. (2)
Note also that a), b), c), d) are all contradicting to each other. (3)
Note also a)+b)+c)+d) make all our possible cases. (4)
So (1)+(2)+(3)+(4) mean that the book's answer is completely equivalent to yours.
• Aug 3rd 2013, 02:38 PM
Turgus
Re: Sentential Logic Problem
Thanks guy! I understand it now.
• Aug 4th 2013, 12:11 AM
ChessTal
Re: Sentential Logic Problem
Quote:

Originally Posted by Turgus
Thanks guy! I understand it now.

Yes, it's important to understand that an implication of the form A => B means that:
A is true and B is true
A is false and B is true
A is false and B is false.

So the original implication of:
"If I buy the pants then I buy the shirt" , means:
I will buy the pants and i will buy the shirt.
I will not buy the pants and i will buy the shirt.
I will not buy the pants and i will not buy the shirt.