Dummett observed that two different forms of the classical sorites paradox now intervene to show that (i)–(iii) are inconsistent. For on the one hand, by applying mathematical induction – i.e. the principle that for all definite properties \(P(x)\) of natural numbers, we may infer \(\forall x P(x)\) from \(P(0)\) and \(\forall x(P(x) \rightarrow P(x+1))\) – to the predicate \(F(x)\), we may conclude that \(\forall x F(x)\) from (i) and (ii). But then \(F(\tau)\) follows by universal instantiation, in contradiction to (iii). And on the other hand, even without appealing to induction, a contradiction can also be derived in the form of a conditional sorites argument by deriving \(F(\tau)\) as the result of repeatedly applying modus ponens to the series \(F(0), F(0) \rightarrow F(0'), F(0') \rightarrow F(0''), \ldots\) obtained from (ii) by universal instantiation.