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I am reviewing some basic propositional logic. The question that I have come across that has given some confusion is Zach blocks e-mails and texts from Jennifer where I am asked to find the negation of this proposition. Here is what they provided as an answer in the book: Zach does not block e-mails from Jennifer, or he does not block texts from Jennifer.

Why couldn't the answer simply be It is not true that Zach blocks e-mails and texts from Jennifer? Why did they have to introduce the disjunction or?

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Of course your answer is fine, indeed clearer. They presumably want to test whether you know how to negate a conjunction and obtain a disjunction. But if this is a test question, surely you should not be expected to be a mind reader. – André Nicolas Dec 11 '12 at 2:00
up vote 6 down vote accepted

What you gave is one possible answer, the book gives another which is logically equivalent, although slightly cleaner statement since it is further reduced. They are the same because of one of De-Morgan's laws; $\neg(P \land Q) \equiv (\neg P) \lor (\neg Q)$

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Zach blocks emails and texts from Jennifer means Zach blocks emails from Jennifer and Zach blocks texts from Jennifer. Negating this gives Zach does not block emails from Jennifer or Zach does not block texts from Jennifer.

It is true that the negation is also It is not true that Zach blocks emails and texts from Jennifer but the point of the exercise is to make one rewrite it in a useful way.

Note that in general, (not (A and B)) is logically equivalent to ((not A) or (not B)).

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Or does "Zach blocks emails and texts from Jennifer" mean "Zach blocks emails and Zach blocks texts from Jennifer"? Don't you love the ambiguities of natural language? (Of course, the translation you give is almost certainly the intended meaning.) – Code-Guru Dec 11 '12 at 4:11
  • $P(x, y)$: x blocks emails from y
  • $Q(x, y)$: x blocks texts from y
  • $z$: Zach
  • $j$: Jennifer

$P(z, j)$: Zach blocks emails from Jennifer;
$Q(z, j)$: Zach blocks texts from Jennifer.

You essentially negated: $P(z, j) \land Q(z, j)$.

So did the text.

The text applied "distribution of negation over conjunction" (one of DeMorgan's Laws):

Start with your translation:

  • "It is not the case that $[(P(z, j)$ and $ Q(z, j)]$" $$\iff \lnot[P(z, j) \land Q(z, j)]$$ $$\iff \lnot P(z, j) \lor \lnot Q(z, j)\tag{DeMorgan's law}$$
  • which is to say: not $P(z, j)$, or, not $Q(z, j)$.

You are correct with your statement that the sentence translates to $\;\;\lnot[P(z,j) \land Q(z, j)]$;
It's logically equivalent to the text's answer: $\;\lnot P(z, j) \lor \lnot Q(z, j)$.

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