How Would You Translate This Sentence To Predicate Logic?

Question

"There exists an Apple such that for every person, he loves that apple."

I believe the translation is:

$$\exists x(\forall y((\text{Apple}(x) \wedge \text{Person}(y)) \to \text{Loves}(y,x)))$$

would that be correct?

Thanks!

Answer 1

Close but no cigar. The correct answer is:

$$\exists x(\mathrm{Apple}(x) \wedge \forall y(\mathrm{Person}(y) \rightarrow \mathrm{Loves}(y,x)))$$

To see that your answer isn't quite right, consider a universe with one person and no apples. Then the implication is vacuously true (since $\mathrm{Apple}(x)$ is false). So the statement you wrote down becomes

$$∃x∀y \,\mathrm{True}$$

which is vacuously true. So the statement you wrote down is satisfied by a universe in which there are no apples. Therefore, it cannot capture the meaning of a sentence of the form: "There exists an apple such that [whatever]."

Edit. The general principle is this.

1. "For all apples $x$ [whatever]" = "For all $x$, if $x$ is an apple then [whatever]."

2. "There exists an apple $x$ such that [whatever]" = "There exists $x$ such that $x$ is an apple, and [whatever]."