I tried to give a detailed answer, but this stupid forum keeps on giving a database error

Although you said that it sounded like it might be a biconditional, you did not actually translate it

into FOL using a biconditional but you stated ExEy (Something ->

Something else)â€¦ chances are that if you have an Ex(P->Q) then it is probably wrong..

Here is my take on it, but donâ€™t take my word for it though..

1)

Since Running (Q) is a necessary condition for Winning (P), we must write it in the form P->Q

(P is a sufficient condition for Q but Q is a necessary condition for P, ie, winning is a sufficient

reason for running, but running is a necessary condition to win the race)

2)

Using the term â€œOneâ€, we have to assume that the speaker is generalizing, thus talking about

something that is true of everybody (every person in the domain), thus we will have to use a

universal quantifier (Ax)

3)

Since we want to talk about Early Bird as a race, we have to identify its free variable (earlybird)

in our FOL. We donâ€™t want to introduce it for the first time inside the scope of any of our

quantifiers, so in this case it will be best if we state Race(earlybird) first and once we have done

that, continue writing the rest of our sentence, thus we start our sentence with

Race(earlybird) ^

4)

So, we can write Race(earlybird) ^ Ax(Wins(x,earlybird) ->.........)

5)

What gets me is the part â€œOne runs 30km or more every weekâ€.

Does one write Ey(Runs(x,y) ^ Â¬ LessThan(y,30))

or does one write Ay(Runs(x,y) -> Â¬ LessThan(y,30)).

Runs(x,y) already means x Runs y km every week.

If we look at the Universal form, we may interpret it as saying â€œNo kilometres that one runs

every week are less than 30â€.

If we look at the Existential form, we may interpret it as saying â€œSome kilometres that one runs

every week are not less than 30â€

From these interpretations, it would seem that we have to go the Universal route, thus

6)

Race(earlybird) ^ Ax(Wins(x,earlybird) -> Ay(Runs(x,y) -> Â¬ LessThan(y,30)))

Is that the correct way of translating the particular sentence from English to FOL? I donâ€™t knowâ€¦

the study guide doesnâ€™t give enough guidance so I cannot say yes with absolute certainty