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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?2011-08-28
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    @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.2011-08-28
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    I think you are misinterpreting what it means for a logic to be $n$-valued.2011-08-28
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    @Zhen Huh? A 2-valued logic has two truth values, a 3-valued logic has three truth values, and so on. http://en.wikipedia.org/wiki/Many-valued_logic2011-08-28
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    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.2011-08-28
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    @Zhen I'm not talking about the quantifiers in 3-valued predicate logic, for example. Rather, I'm talking about quantifiers in "3-valued propositional logic with quantifiers". A statement like "for all p, p" in 2-valued propositional logic with quantifiers, quantifies over {T, F} and basically means the conjunction of "F" and "T", which is false. In an "n-valued logic with quantifiers", you quantify over the truth set. Sorry, if I misunderstood you.2011-08-28
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    Yes, that is precisely what I mean: quantifiers over the truth values are incomparable to the quantifiers in predicate logic. It occurs to me that you haven't stated what you mean by ‘all’ $n$-valued logics. What counts and what doesn't count?2011-08-28
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    Thanks, that cleared it up quickly I think. n-valued logics are truth-operational, while the predicates of predicate logic don't necessarily satisfy closure. So, the basic answer is "no", I think.2011-08-28
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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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