# 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

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

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:

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

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

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

$\neg(P \land \neg S)$ 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:
OR
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:
OR
Isn't that a completely different meaning from my answer?

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

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

"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:
OR
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 $\neg(P \land \neg S)$ 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.

$\neg P \lor S$ 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

Since there are 4 combinations of buying/not buying shirts/pants, the extra option for us to do is:

Now:
The book answer says that "If i will buy the pants, i will buy the shirt also".
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: