Below is a question comes from the book How to Prove It written by Daniel J. Velleman.

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?

$\lnot(P \land \lnot S)$

The given answer is, “I won't buy the pants without the shirt.“ I don't understand the answer. Can someone explain? Or maybe the answer is wrong?

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    $\begingroup$ It represents "if I will buy the pants, then I will buy the shirt." Oops... perhaps, but not necessarily (the part in quotes is half a joke). $\endgroup$ – Doug Spoonwood Jun 23 '13 at 2:43

Remember that, by DeMorgan's Laws, $$\lnot (A\land B)\iff(\lnot A\lor\lnot B),$$ so $$\lnot (P\land \lnot S)\iff \lnot P\lor(\lnot\lnot S)\iff\lnot P\lor S.$$ Our goal is to describe what it means for $\lnot (P\land \lnot S)$ to be true. So, let's agree that it is true, and see what we conclude.

If you don't buy the pants, then $\lnot P$ is true, and hence so is $\lnot P\lor S$, regardless of what $S$ is. However, if you do buy the pants, then $\lnot P$ is false, and in this case, the only way $\lnot P \lor S$ can be true is if $S$ is true, i.e. if you buy the shirt.

Thus, to say that $\lnot (P\land \lnot S)$ is true is to say that maybe you won't buy the pants, maybe you will; but if you do, you'll definitely also buy the shirt.

  • $\begingroup$ Thanks for the detail explanation! $\endgroup$ – Dave Clifford Jun 22 '13 at 9:17
  • $\begingroup$ A minor point is that the question appears on p. 14 of Velleman (2nd ed.), whereas DeMorgan's laws are not mentioned until p. 20. $\endgroup$ – J W Dec 30 '14 at 14:46

Let's work from the inside out:

  • $S$ means: "I will buy the shirt."
  • $\neg S$ means: "I won't buy the shirt."
  • $P \land \neg S$ means: "I will buy the pants and I won't buy the shirt." Another way to say this is: "I will buy the pants but not the shirt." (or, equivalently, "I will buy the pants without the shirt.").
  • $\neg(P \land \neg S)$ means: "I won't buy the pants without the shirt."

Due to the ambiguity of the English language, there will certainly be multiple "translations" and no unique "answer". Later on, you might learn that: $$ \neg(P \land \neg S) \quad \equiv \quad \neg P \lor S \quad \equiv \quad P \rightarrow S $$ Thus, a (perhaps more intuitive) valid translation of $\neg(P \land \neg S)$ would be: "If I buy the pants, then I must also buy the shirt."

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    $\begingroup$ Can you give other translation for ¬(P∧¬S) other than 'I won't buy the pants without the shirt' and 'If I buy the pants, then I must also buy the shirt.' Assuming one doesn't know ¬(P∧¬S)≡¬P∨S≡P→S, how can he reason the answer to be so? $\endgroup$ – Dave Clifford Jun 22 '13 at 9:32
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    $\begingroup$ @DaveClifford Other translations include "It is not the case that I will buy the pants but not the shirt." and "Either I will buy the shirt or I won't buy the pants (or I will do both and buy the shirt without the pants)." In general, to see why these statements are true, it helps to construct a truth table (that is, consider all possible combinations of truth values of $P$ and $S$ and see if a proposed English translation is consistent with whatever Boolean value that the expression evaluates to). $\endgroup$ – Adriano Jun 22 '13 at 9:48
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    $\begingroup$ Oh, the translation 'it is not the case that I will buy the pants but not the shirt' makes sense to me a lot! Thanks! $\endgroup$ – Dave Clifford Jun 22 '13 at 10:09
  • $\begingroup$ You might want to note that you don't necessarily need the intermediate step of "¬P∨S" to prove the equivalence (¬(P∧¬S)≡(P→S)) here, at least in some logical calculi, where it holds. $\endgroup$ – Doug Spoonwood Jun 23 '13 at 2:49
  • $\begingroup$ For those reading Velleman in order without glancing ahead, note that truth tables are introduced in the following section (1.2). $\endgroup$ – J W Dec 30 '14 at 14:56

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