The sentence is asking you to prove that you're lucky enough to look under 18. That's illogical! (Especially as what they want proof of is that you are *over* 18 anyway!)
My impression is that if you are lucky enough that you look like you are under 18 that you will have to prove that you look like you are under 18. Too much beer can lead people to come up with totally illogical sentences.
Nicely put, anyway, but unfortunately the idea goes astray.
To me, "logical sentence" refers to the grammar, which is fine, despite a missing not-so-important comma. The idea behind the sentence, however, is loopy, if not illogical.
Like Ron and Mr. Trilby said, you can simply add a dis- in front of "prove" to make the sentence logical.
I saw it in TESCO!
Not really logical, because ID is required to prove a person is of legal age not to prove a person isn't of legal age. It is an interesting sign though, and perhaps humor is intended.
To me, it's clear. If you are lucky to be over 18 and still to look under 18, you will have to prove that you are over 18, for it's taken for granted that if you look under 18, you are under 18.
I've seen this sign in a ABC store window. TOTALLY ILLOGICAL!
It's generally true that if you look like underaged kid you'll have to prove that you are 18 or above showing your ID card.
So... it's logically correct. The wrong thing is IF YOU ARE LUCKY ENOUGH - on the one hand it's nice to look younger, but on the other hand always walking with the ID card in your pocked when going for a bottle of vodka or cigarettes... This is NOT LUCK. :D So... that's the ILLOGICAL part.
It's perfectly logical if you make one intended assumption. The first part of the sentence (the part before the missing comma) implies that you are over 18 and lucky to look under 18. As opposed to the other people that look under 18 that are under 18 (they aren't lucky, they are normal). Once that quite simple assumption is made, the second half of the sentence makes perfect sense. Prove that you are lucky. The only way to prove that you are lucky is to prove that you are over 18. Otherwise you would be one of those non-lucky (normal) people that look their age.
Logical, but ambiguous.
Deliberate ambiguity of the sign is to humor the customer. I don't think there was a single person who seriously tryed to prove being lucky enough to look under 18.