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For the universe of discourse consisting of people, the following natural language translations are intuitive

$\forall$ : Everyone

$\exists$ : Someone

$\neg \forall$ : No one

And what about $\neg \exists$? Not-someone? This seems to be the analogy of the $O$ proposition in propositional logic (which I still don't truly grasp why the predicate is distributed). Specifically, consider the following with $L(x, y)$ as $x$ loves $y$:

$\forall x \forall y L(x,y) \equiv \forall x \neg \exists y \neg L(x,y)$

The left side translates to "everyone loves everyone." But the best I can do for the right side is "everyone does not love not-someone." Does anyone have a more fluid translation than: "For all x there is no y such that x does not love y?"

doctorjay
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1 Answers1

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$\lnot \forall$ means "not everyone". For example, Let the domain of discourse be all natural numbers; let the predicate $P(x)$ mean "x is a prime number". Clearly, $\forall x(P(x))$ is false, because it claims every natural number is prime. So we can truthfully negate that sentence: $\lnot \forall x(P(x))$ which means not all natural numbers are prime. That isn't equivalent to "no natural numbers are prime".

$\lnot \exists$ means "there does not exist anyone", or "nobody".

So the right hand side can be translated as: "For every person $x$, there isn't a person $y$ that $x$ does not love."

amWhy
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  • both these comments provide a reasonable and similar answer, I am intrigued by the fact that in order to translate into something resembling natural language, we have to place the object of the sentence before the predicate and use a helping verb. Speaks more to a flaw of natural language than to a difficulty with logical syntax – doctorjay May 26 '17 at 15:21