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.
Okay, so then semiotics is mostly a set of proofs I take it. What, then, are the major assumptions underlying those proofs and which are the most important proofs built on those assumptions?
I spent an hour recently talking with a semiotics professor who was trying to explain semiotics to me. He was very patient, and so was I, and at the end of an hour I concluded that semiotics is like Indian chakra-based medicine: a set of heuristic practices that work well in a lot of situations, justified by complete bullshit.
I learned that semioticians, or at least this semiotician:
When I've read short, simple introductions to semiotics, they didn't say this. They didn't say anything I could understand that wasn't trivial. I still haven't found one meaningful claim made by semioticians, or one use for semiotics. I don't need to read a 300-page tome to understand that the 'C' on a cold-water faucet signifies cold water. The only example he gave me of its use is in constructing more-persuasive advertisements.
(Now I want to see an episode of Mad Men where they hire a semotician to sell cigarettes.)
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?
His notion of "essence" is not the same as Plato's; tokens rather than types have essences, but they are distinct from their physical instantiation. So it's a tripartite Platonism. Semioticians take this division of reality into the physical instantiation, the objective type, and the subjective token, and argue that there are only 10 possible combinations of these things, which therefore provide a complete enumeration of the possible categories of concepts. There was more to it than that, but I didn't follow all the distinctions. He had several different ways of saying "token, type, unbound variable", and seemed to think they were all different.
Really it all seemed like taking logic back to the middle ages.