Thread: Sentential Logic Problem

1. Sentential Logic Problem

Hi there,
I would be extremely appreciative to anyone that could help/explain this reasoning a little to me.

Q. Let P stand for the statement "I will buy the pants" and S for the statement "I will buy the shirt." What English sentencecs are represented by the following expression

$\neg(\mbox{P}\wedge\neg\mbox{S})$

correct answer from book [How to Prove It, 2006, Velleman] is...
I won't buy the pants without the shirt.

Now, to me, this doesn't make any sense.

I would interpret the part inside the parenthesis to be:
I will buy the pants but not the shirt.

Now, outside, it's the negation of that entire statement, which requires a double negative on the shirt part, and from an english viewpoint doesn't make much sense to me.

Only other way I can interpret this myself is:
I will not buy both the pants and not the shirt.

Thanks for any help.

2. Hello lannett

Welcome to Math Help Forum!
Originally Posted by lannett
Hi there,
I would be extremely appreciative to anyone that could help/explain this reasoning a little to me.

Q. Let P stand for the statement "I will buy the pants" and S for the statement "I will buy the shirt." What English sentencecs are represented by the following expression

$\neg(\mbox{P}\wedge\neg\mbox{S})$

correct answer from book [How to Prove It, 2006, Velleman] is...
I won't buy the pants without the shirt.

Now, to me, this doesn't make any sense.

I would interpret the part inside the parenthesis to be:
I will buy the pants but not the shirt.

Now, outside, it's the negation of that entire statement, which requires a double negative on the shirt part, and from an english viewpoint doesn't make much sense to me.

Only other way I can interpret this myself is:
I will not buy both the pants and not the shirt.

Thanks for any help.
I'm afraid I agree with the answer in the book, but you need to juggle things about a bit first.

The proposition 'I won't buy the pants without the shirt' can be re-written 'If I do not buy the shirt, then I won't buy the pants'. This is symbolised as:

$\neg S \Rightarrow \neg P$

which can be re-written without an implication sign as

$\neg(\neg S) \lor \neg P$ (draw up a truth table if you haven't seen this before)

which, by De Morgan's Law, is logically equivalent to

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