 The first two had already informally been argued for by Brouwer (1908C, 1925E); the third was new. Now suppose that in intuitionistic logic, a proposition may be true \((p\) holds), false \((\neg p\) holds), or have a third truth value \((p'\) holds). Clearly, \(p \rightarrow \neg p'\) and \(\neg p \rightarrow \neg p'\), and therefore \((\neg p \vee p) \rightarrow \neg p'\); but then, by the third of the lemmata above and axiom \ref{GlivAxiom8} in the list, \(\neg p'\). As \(p\) is arbitrary, this means no proposition has the third truth value. (In 1932, Gödel would show that, more generally, intuitionistic propositional logic is no \(n\)-valued logic for any natural number \(n\); see Section 1 of the supplementary document  The Turn to Heyting’s Formalized Logic and Arithmetic.)