IPhone troubles - not sure whether it is black listed or not?

So recently my father's iPhone 5 has not been working, meaning it is not receiving or sending calls and texts. I have called up the carrier service and they have said that the phone has been blacklisted (much to my dad's astonishment though I can't say I was severely surprised). So I looked this information up and have found that this is the case with blacklisted phones but you can still use 3G and there's no signal bars.

HOWEVER! It has 3G AND the signal bars are all full! Though people have said that the bars don't mean a thing (though whether or not they are pulling my leg I'm still not sure about), I'm still not sure what to think about the situation. It is all a little confusing and this question is really just to see another's perspective of the situation. I'd just like someone to confirm whether this would still be the case with a blacklisted phone or not.

The phone is blacklisted. That means it can't register to the carrier's network to make or receive calls. Period.

The signal bars merely show that there's an antenna for the carrier nearby (or more than one) that - if the phone was registered and authenticated on the service provider's network - it could use.

If there are no bars, there's no usable signal, there's nothing the phone can connect with to the service carrier's network.

Since the phone "is not receiving or sending calls and texts" and the service carrier "said that the phone has been blacklisted", assume the phone is blacklisted and it won't work until the blacklisting is removed.

You could use it for Wi-Fi only, like an ipod touch. Nothing through celltowers.