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Using this glossary:

$Gx: x \text{ is green}$

$Hx: x \text{ is heavy}$

$Rx: x \text{ is red}$

My textbook says that the sentence "All red things heavy, but some green things aren't" is translated to: $$\forall{x} (Rx \implies Hx) \land \exists{x} (Gx \land \lnot{Hx})$$

But on the doing the exercise I couldn't figure out why it was translated this way. Isn't it more obvious to translate it this way?

$$\forall{x} (Rx \implies Hx) \land \exists{x} (Gx \implies \lnot{Hx})$$

I would like to know why the former is correct and the latter is wrong.

Vivek Joshy
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1 Answers1

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$\exists x (Gx \implies \neg Hx)$ can be translated to: there is some thing, where if it is green, it is not heavy.

This does not assert the existence of anything green. If there is some red object (heavy or not), this statement is vacuously true.

Whereas $\exists x (Gx \land \neg Hx)$ can be translated to: there is some thing which is green and not heavy.

This asserts the existence of a green object that is not heavy.

player3236
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  • Can you explain why it doesn't assert the existence of something green? Isn't it the case that ∃ means "at least one"? So wouldn't there be at least one green thing? – Vivek Joshy Oct 25 '20 at 07:26
  • $\exists$ means "at least one". So at least one thing satisfies $(Gx \implies \neg Hx)$. It doesn't say anything about $Gx$ being true. – player3236 Oct 25 '20 at 08:03
  • I see. Thanks I understand now. – Vivek Joshy Oct 25 '20 at 08:05