 Note the subtle difference between “\(A\) is not intuitionistically true ” and “\(A\) is intuitionistically refutable”: in the first case we know that \(A\) cannot have an intuitionistic proof, the second statement expresses that we have a proof of ¬A, i.e. a construction that derives falsum from any possible proof of \(A\). For the law of trichotomy we have just shown that it is not intuitionistically true. Below it will be shown that even the second stronger form saying that the law is refutable holds intuitionistically. This, however, is not true for all statements for which there exist weak counterexamples. For example, the Goldbach conjecture is a weak counterexample to the principle of the excluded middle, since \(\forall n A(n)\) as above is at present not known to be true or false, and thus we cannot assert \(\forall n A(n) \vee \neg \forall n A(n)\) intuitionistically, at least not at this moment. But the refutation of this statement, \(\neg (\forall n A(n) \vee \neg \forall n A(n))\), is not true in intuitionism, as one can show that for any statement \(B\) a contradiction can be derived from the assumption that \(\neg B\) and \(\neg\neg B\) hold (and thus also from \(B\) and \(\neg B\)). In other words, \(\neg\neg (B \vee \neg B)\) is intuitionistically true, and thus, although there exist weak counterexamples to the principle of the excluded middle, its negation is false in intuitionism, that is, it is intuitionistically refutable.