Translate a sentence into predicate logic I want to translate this sentence > There exists $a$ such that if for all $b$ different from $a$, $b$ has the propriety $P$ then $a$ has the propriety $Q$ I translated it like this : > $\exists a. [(\forall b. b \neq a \implies P(b)) \implies Q(a)] $ but it looks weird. (mostly because if I have the sentence : > There exists $a$ such that if for all $b$ different from $a$, $b$ has the propriety $P$ then a has the propriety $\neg P$ I would translate it like this : > $\exists a. [(\forall b. b \neq a \implies P(b)) \implies \neg P(a)] $ but it can be read as > $\exists a. \neg P(a)$ ) Am I doing this right or are there mistakes I don't see ?

It indeed looks weird, and that's because _typically_ an existential goes hand in hand with a conjunction ($\land$), rather than a conditional ($\rightarrow$).

But: 'typically' is not the same as 'always'. Indeed, I would say this case is one of those rare exceptions where you do use a conditional. Or at least, your translation coincides exactly with my interpretation of the English sentence as well.