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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

$\exists x \,\exists y \, (J(x) \land R(y) \land T(x,y))$

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  • $\begingroup$ No need to translate "can". It is "A only if B", i.e. $A \to B$ $\endgroup$ Commented Apr 20, 2020 at 11:34
  • $\begingroup$ How did you get your second expression, $\exists x \exists y \ldots$? $\endgroup$
    – Jam
    Commented Apr 20, 2020 at 11:37
  • $\begingroup$ My issue is with translating "one" and I thought it might be possible to translate "one" with $\exists$ as in "there is at least one". So "at least one" serious Jogger,x, and at least one race (translated from "some race"),y. $\endgroup$
    – Tightrope
    Commented Apr 20, 2020 at 11:40
  • $\begingroup$ I see. I think the statement is supposed to use "one" as in "someone", i.e., anyone. So the stronger quantifier, $\forall$ would be more appropriate than $\exists$. $\endgroup$
    – Jam
    Commented Apr 20, 2020 at 11:51
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    $\begingroup$ @Jam - agreed. Often my problem with translating these sentences comes down to the possible ambiguity of the English meaning. $\endgroup$
    – Tightrope
    Commented Apr 20, 2020 at 12:17

1 Answer 1

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I am not a native English speaker but in my opinion

One can only be a serious jogger if one takes part in some race

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.

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