Given the sentence "politicians are likeable only if they are honest", I used the following definitions:
- P(x) = x is a politician
- L(x) = x is likeable
- H(x) = x is honest
I obtained the following translation to predicate logic: $\forall x(P(x) \rightarrow (L(x) \rightarrow H(x)))$
However, I was told that was incorrect because it doesn't capture the relationship between honesty and politicians. My understanding is that because $L(x) \rightarrow H(x)$ is the antecedent of the outer implication, it is relevant only in the context of politicians, so the relationship is still there.
I was told that the correct translation is $\forall x((P(x) \rightarrow L(x)) \rightarrow H(x))$, where the implication with $P(x)$ and $L(x)$ is grouped together. This doesn't make sense to me, since it also seems to say that there are no people who are not politicians who are not honest since if $P(x)$ is false, $P(x) \rightarrow L(x)$ is always true, so $(P(x) \rightarrow L(x)) \rightarrow H(x)$ would be false if $H(x)$ was false. I made this argument to my instructor who said that non-politicians are irrelevant to the original English sentence. However, it seems to me that this translation adds more information than was given in the original sentence.
I'm sorry for such an elementary question, but can anyone provide any guidance for why my interpretation is incorrect, and the provided one is correct?
EDIT: I was also curious if $\forall x((P(x) \ \& \ L(x)) \rightarrow H(x))$ would be valid, since it's logically equivalent to the answer I gave, but I was told that using conjunctions with universal quantifiers is not allowed.