Help with 'If, then'- and 'Only if'-sentences in Predicate Logic

Question

So, I have to ask now, because I've spent so much time on these two translations.

My key is

Domain: living things

Px: x is a Pokerplayer

Cx: x is a Chessplayer

Yx: x is Professional

Rx: x is Rich

and the sentences are:

1) "Pokerplayers and Chessplayers are rich if they are professional"

2) "Pokerplayers and Chessplayers are rich only if they are professional"

Now after messing around with using combinations of material conditionals and conjunctions, I ended on only using material conditionals for the first one, like this:

1) ∀x((Px v Cx) --> (Yx --> Rx))

for it seems to be saying that for all living things, if it is a pokerplayer/chessplayer then, if it is professional, then it is rich ...

My question is then, for the second, if I can only "revert"/"switch" the predicates in the main consequent to get the 'only if'-version, like this:

2) ∀x((Px v Cx) --> (Rx --> Yx))

for this seems to be saying that for all living things, if it is a chessplayer/pokerplayer then, if it is rich, then it is profession; which seems to exclude the possibility of any non-professional rich pokerplayers/chessplayers ... But ... doesn't this also allow there to be chessplayer/pokerplayer who are not rich but still professional - since the last material conditional is true even when 'Rx' is false and 'Yx' is true ... Or have I gotten this wrong?

Thank you! (I now notice that I have forgot to use your fancy symbolization tools, but I will try to change it ... I was just in a bit of a hurry to post this question)

Answer 1

Both your translations are correct.

The bare appearance of "if" (stripped of the word "only") usually precedes the antecedent of a conditional.

When "if" is preceded by the qualifier "only", as in "only if", then we need to know that "only if" precedes the consequent of a conditional.

In general: $A\rightarrow B$ says "if $A$ then $B$", but it is also equivalent to "$A$ only if $B$."

In your first statement, the bare "if" correctly precedes the antecedent $P(x)$: we have, in effect,

"$R(x)$, if $P(x)$" $\iff \;P(x)\rightarrow R(x) \iff\;$ "if x is a professional, then x is rich".

In your the second statement, indeed, you have correctly translated the stattement, noting that the consequent of the implication "x is rich only if x is professional" is equivalent to $\;R(x) \rightarrow P(x)$