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There is a definition for truth-functional completness for a set of propositional connectives. Is there a definition for truth-functional completness of a set of quantifiers and propositional connectives ?

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There cannot be a direct analogue, because a formula with quantifiers in it can depend on the truth values of infinitely many atomic propositions. There are $2^{2^{\kappa}}$ different possible truth functions of $\kappa$ inputs, and when $\kappa$ is infinite this much larger than the number of possible formulas one can build from any finite set of connectives. So in this sense there is no complete finite set of quantifiers and connectives.

I suppose one could consider complete sets containing (uncountably) infinitely many different quantifiers, but that's not very practical.

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