Forming up Complex logical forms from simple one

Question

This is another problem I have been working from Velleman's How to prove book.

Let P stand for the statement “I will buy the pants” and S for the statement “I will buy the shirt.” What English sentences are represented by the following expressions?

(a) ¬(P ∧ ¬S).

I worked out on this like this:

P ∧ ¬S = I will buy the pant and I will not buy the shirt.
¬(P ∧ ¬S) = ???

Now how to apply inverse to that statement. This confuses me. How do logicians think over this ? One way of figuring this out is by appling De-morgan's theorem. But it hasn't yet been taught in that book so far, so I don't want to think in that term. But even if I apply De morgan's theorem, I haven't been able to reach the correct solution:

On applying De-morgan's theorem, my final solution would look like this:

¬P ∨ S = I will not buy the pant or I will buy the shirt.

But the solution given in the book is this:

I won’t buy the pants without the shirt.

Answer 1

We know that $\,\lnot P \lor S\,$ essentially defines the implication $P\rightarrow S$:

$$\lnot P \lor S \equiv P\rightarrow S$$ This translates to the statement "If I buy the pants, then I'll by the shirt."

But we also know that that an implication is equivalent to its contrapositive: $$P\rightarrow S \equiv \lnot S \rightarrow \lnot P$$

This translates to the statement:

$\quad$ "If I don't buy the shirt, then I won't buy the pants.

$\equiv$ "I won't buy the pants if I don't buy the shirt.

$\equiv$ "I won't buy the pants without the shirt."

This just illustrates that natural language has lots of "different" words to say essentially the same basic logical statements.

Answer 2

The issue here seems to be not so much propositional calculus as ordinary English. It looks to me as if you're perfectly capable of translating propositional formulas, like $\neg(P\land\neg S)$, by rote substitution (of words and phrases for the connectives and propositional variables) into English sentences like "It is not the case that I will buy the pants and I will not buy the shirt." That's fine (except for a possible ambiguity in the English formulation because parentheses aren't available). Unfortunately, although that's a grammatically correct English sentence, it's not something anyone would ordinarily say (except when doing propositional logic exercises). The remaining task is to express the same thing in a form that people would actually use in normal conversation. In general, that task is accomplished by thinking about the situation described by the rote-translation sentence and figuring out how you'd really describe that situation. In the case at hand, it's fairly easy: The $P\land\neg S$ part, translated as "I will buy the pants and I will not buy the shirt," is more colloquially expressed as "I will buy the pants without the shirt," and then you just have to negate that.

The task might actually need some experimentation. The first colloquial form of "I will buy the pants and I will not buy the shirt" that comes to mind might be "I will buy the pants but not the shirt." Negating that leads to rather awkward English, better than the rote substitution but still admitting improvement.

Why should you aim for natural-sounding, colloquial formulations, when the result of rote substitution, though awkward-sounding, is technically correct? My answer to that is that eventually you'll need to do the reverse. You'll be given a colloquial English sentence and you'll need to translate it into logical formulas. That will be much easier if you've already seen, in lots of exercises like the present one, how logical formulas correspond to colloquial sentences, not just to rote substitutions.