The following predicate symbols are given:
$J(x)$ translates to "$x$ is a serious jogger";
$R(x)$ translates to "$x$ is a race";
$T(x,y)$ translates to "$x$ takes part in race $y$".
How do we translate:
"One can only be a serious jogger if one takes part in some race"
I am having difficulty translating the word "one" which appears twice in the sentence above. Critique me on the following two efforts, i.e. how do they translate differently.
$\forall x \, (J(x) \to \exists y \, (R(y) \land T(x,y)))$
vs
$\exists x \,\exists y \, (J(x) \land R(y) \land T(x,y))$
I am not a native English speaker but in my opinion
means that "taking part in some race" is a necessary condition to "be a serious jogger", for every individual $x$. Therefore, the logic form of the sentence above is \begin{align} \forall x \, (J(x) \to \exists y \, (R(y) \land T(x,y))). \end{align} The "One" at the beginning of the sentence refers to a generic individual, so it is translated by the universal quantifier. "Some race" is clearly translated by the existential quantifier.