Translate a sentence into predicate logic

Question

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 ?

Answer 1

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.

Answer 2

I think your brackets are in the wrong place. Should be

$\exists a. [\forall b. b \neq a \implies (P(b) \implies Q(a))]$

Alternatively you could write this as:

$\exists a. [\forall b. (b = a) \lor (P(b) \implies Q(a))]$