But what do you refer to, when you refer to "logic"? Do you perhaps mean boolean logic? Predicate logic? Propositional logic?

There's a common reference class, but no common implementation, when you point at "logic". Each kind of logic is incomplete, but that is fine, because each kind of logic is domain-specific, and you use the kind of logic which is most complete with respect to the problem you're interested in. Which actually raises a point about something you said:

Are there multiple "sciences" all using the name "semiotics"? Does semiotics make any falsifiable claims? Does it make any claims whose meanings can be uniquely determined and that were not claimed before semiotics?

Yes. No. No. The exact same things are true of "logic", however, in its broadest sense. It's only when you get into the specific "sciences" (or domains of logic) that anything interesting is allowed to happen.