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If a theorem holds for all truth functions of all n-valued logics, in all n-valued logics with quantifiers, n>=2, will it also hold in classical predicate logic where the domain has at least two elements? Does the converse hold?

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What is the difference between a 2-valued logic and classical logic? – Zhen Lin Aug 28 '11 at 2:04
@Zhen I don't know of any difference. That said, I don't want to confuse the fact that say 2-valued logic with quantifiers has domain {T, F} which it quantifies over, while classical predicate logic doesn't necessarily have this domain. – Doug Spoonwood Aug 28 '11 at 2:27
I think you are misinterpreting what it means for a logic to be $n$-valued. – Zhen Lin Aug 28 '11 at 3:37
@Zhen Huh? A 2-valued logic has two truth values, a 3-valued logic has three truth values, and so on. – Doug Spoonwood Aug 28 '11 at 4:06
Yes, but the quantifiers in such logics, if present, are not comparable to the quantifiers of predicate logic. Predicate logic is not a many-valued logic in the same sense. – Zhen Lin Aug 28 '11 at 4:33

Yes, bi-valued Boolean logic is a special case of many-valued logic.

No, e.g. in Lukasevichz logic $p \lor \lnot p$ does not hold but it is correct in bi-valued Boolean logic.

See Petr Hajek's book "Metamathematics of Fuzzy Logic" for more information.

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