# How to translate English phrase "One can..." into FOL

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

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

• No need to translate "can". It is "A only if B", i.e. $A \to B$ Commented Apr 20, 2020 at 11:34
• How did you get your second expression, $\exists x \exists y \ldots$?
– Jam
Commented Apr 20, 2020 at 11:37
• 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. Commented Apr 20, 2020 at 11:40
• 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 $\exists$.
– Jam
Commented Apr 20, 2020 at 11:51
• @Jam - agreed. Often my problem with translating these sentences comes down to the possible ambiguity of the English meaning. Commented Apr 20, 2020 at 12:17

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.