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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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    @ZhenLin 2-valued propositional logic necessarily has only two truth values. Dmitri Bochvar came up with a 3-valued logic which has the same tautologies as classical logic. As I understand it, his logic could get extended to an infinite-valued logic. The catch here is that for his 3-valued logic though the input for his connectives do take on 3 values, the output always takes on two truth values. So, in one sense his logic is 2-valued, but at least arguably in another sense it's not 2-valued. Classical propositional logic necessarily has a set of tautologies or theorems.2013-07-03

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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.