#### nomadreid

Gold Member

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- An intuitionist logic (excluding the excluded middle) obeys the existence property (why?) and hence is constructive. That implies that admitting more than two truth values eliminates existence proofs, including axioms allowing for existence proofs by some other means than by contradiction. But I do not see the connection.

First, is my assumption that all consistent multi-valued logics obey the principle of explosion from a false proposition correct?

If so, how would one prove that?

(I assume it is, because if not, then by the definition of intuitionist logic by Wolfram Mathworld http://mathworld.wolfram.com/IntuitionisticLogic.html, the frequently made assertion that a fuzzy logic is automatically intuitionist would be false.)

Therefore, the addition of a third truth value assigned to the interpretation of at least one sentence in a theory would, in itself, be incompatible with an axiom which allowed a conclusion preceded by an existence quantifier (and without the assignment of the variable to a constant).

From the websites that I have looked at, the link between a third truth value and the omission of this class of axioms is apparently supposed to be evident. Sorry, but could someone make this link explicit for me?

(It appears to me that an existence proof need not invoke proof by contradiction; for example, the Power Set Axiom and the Axiom of Choice and a lot of the Large Cardinal Axioms are not constructive, as far as I understand. Or is this way off the mark?)

If so, how would one prove that?

(I assume it is, because if not, then by the definition of intuitionist logic by Wolfram Mathworld http://mathworld.wolfram.com/IntuitionisticLogic.html, the frequently made assertion that a fuzzy logic is automatically intuitionist would be false.)

Therefore, the addition of a third truth value assigned to the interpretation of at least one sentence in a theory would, in itself, be incompatible with an axiom which allowed a conclusion preceded by an existence quantifier (and without the assignment of the variable to a constant).

From the websites that I have looked at, the link between a third truth value and the omission of this class of axioms is apparently supposed to be evident. Sorry, but could someone make this link explicit for me?

(It appears to me that an existence proof need not invoke proof by contradiction; for example, the Power Set Axiom and the Axiom of Choice and a lot of the Large Cardinal Axioms are not constructive, as far as I understand. Or is this way off the mark?)