 Another suggestion might be to conclude that the paradox depends upon an instance of the principle of Excluded Middle, that either \(R\) is a member of \(R\) or it is not. This is a principle that is rejected by some non-classical approaches to logic, including intuitionism. However it is possible to formulate the paradox without appealing to Excluded Middle by relying instead upon the Law of Non-contradiction. We do so as follows: Given the definition of \(R\) it follows that \(R \in R \equiv{\sim}(R \in R)\). So \(R \in R \supset{\sim}(R \in R)\). But we also know that \(R \in R \supset R \in R\). So \(R \in R \supset(R \in R \wedge{\sim}(R \in R))\). But by the Law of Non-contradiction we know that \({\sim}(R \in R \wedge{\sim}(R \in R))\). So by modus tollens we conclude that \({\sim}(R \in R)\). At the same time we also know that since \(R \in R \equiv{\sim}(R \in R)\), it follows that \({\sim}(R \in R) \supset R \in R\), and thus that \(R \in R\). So we can deduce both \(R \in R\) and its negation using only intuitionistically acceptable methods. 