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It is just $\neg \exists x \neg P(x)$? Which says there is no $x$ which makes $P(x)$ false?

user3064097
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    Yes, you have it right. – André Nicolas Jan 28 '14 at 00:07
  • Suggestion: IMO, often the best way to understand logic is through "non-mathematical" examples. So, say your statement $\forall x P(x)$ is "every student passed this course". Using the word "some", you can rephrase this as, "it's not true that some student failed this course" - a bit clunky as far as the English goes, but it illustrates the logic. In symbols, $\neg\exists x,\neg P(x)$. – David Jan 28 '14 at 00:22
  • @David: Horrible suggestion. These examples obscure the correct interpretation and meaning of what you try to convey. I have seen many a student fail to grasp the intuitive idea behind $\forall x\exists y$ being inequivalent to $\exists y\forall x$, because they focused on the "non mathematical examples". The best way is to stick to the definitions, until the definitions stick with you. – Asaf Karagila Jan 29 '14 at 04:04
  • @Asaf, your students must be very different from mine! In my experience, if you give students say algebraic or geometric examples then they tend to focus on the algebra or geometry instead of the logic. Basic logic is about the meanings of simple words like "all" and "and" and "if", and essentially, students know what these mean. Sure, some things need to be cleared up, for example the meaning of "only if" and the fact that in mathematics "all" really means "ALL", but I don't think this invalidates the basic approach. Building on what students already know is always pedagogically sound. – David Jan 29 '14 at 04:13
  • @Asaf, also I don't understand the relevance of your comments about order of quantifiers, since this was an issue neither in the question nor in my comment. – David Jan 29 '14 at 04:13
  • @David: Natural language is a harsh mistress. The whole point and inherent difficulty of the mathematical language, and of the purpose of logic in formalization of mathematics, is to remove the ambiguity in natural language. Working with the definition, and the definition alone is a sure way not to make mistakes, and to eventually understand much better. The example was just an example for why "non mathematical" examples can be bad. – Asaf Karagila Jan 29 '14 at 04:20
  • @Asaf, IMHO there is so much that you have said that is true but misapplied, that I'm not sure it's suitable for a comment thread. Happy to continue the discussion by chat or whatever - I'm fairly new on this forum so don't entirely know what is available. – David Jan 29 '14 at 04:26

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Yes. This is a consequence of duality (deMorgan) laws:

$a \wedge b$

...law of double negation...

$= \neg(\neg(a \wedge b))$

...duality law...

$= \neg(\neg{a} \vee \neg{b})$

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