logic - natural language to symbolic form

Question

How to translate the following natural-language statements to symbolic form:

• "If everybody complains, no one will get help"
  • Is this correct? $\forall x C(x) \rightarrow \lnot (\forall y H(y))$, where $C(x)$ is "x is complaining" and $H(y)$ is "y gets help"
• "Everybody loves somebody sometime"
  • "Everybody loves somebody" becomes $\forall x \exists y L(x,y)$, where $L(x,y)$ is "x loves y". But how to translate sometime?

PS: this question is a follow-up to an old question.

Answer 1

I think your first one is incorrect; as you have written, you are saying that if everyone complains, then not everyone gets help (that is, at least one person does not get help). It should probably be $$\left( \forall x Cx\right) \longrightarrow \left( \forall y\left(\neg Hy\right)\right)$$ or equivalently, $$\left( \forall x Cx\right) \longrightarrow \neg\left(\exists y Hy\right).$$

For the second, I think you need a ternary relation, something like $L(x,y,t)$ to mean “$x$ loves $y$ at time $t$”. Then you could write $$\forall x\exists y\exists t L(x,y,t).$$

Answer 2

I would symbolize the first statement like this:

$$\forall x(Cx) \to \neg \exists y(Hy)$$

I believe your symbolization is incorrect because $\neg (\forall y Hy)$ is equivalent to $\exists y \neg Hy$ ("There exists a y such that y does not get help" You're looking for "Nobody gets help").

Now, I would symbolize "Everybody loves someone" as

$$\forall x \exists y(Lxy)$$

which is pretty much exactly how you symbolized it.

As for "sometime," I'm not exactly sure how you would symbolize the time in this context. Perhaps you can use temporal logic. This would require you to be pretty familiar with modal logic, though. It depends on how you interpret "sometime," which, in my opinion, is pretty ambiguous.