The Diagonal Lemma Fails in Aristotelian Logic

In the 1950-ies P. F. Strawson proposed the Logic of Presuppositions. In this system sentences with empty subject are considered neither true nor false. Strawson considered only simple sentences with two predicate letters. But the concept can be extended to arbitrary monadic, even polyadic sentences utilizing truth-relevant logic developed by Richard Diaz. It can be shown that self-referential G\"odel's sentence in fact has an empty subject, and can thus be classified by the Logic of Presuppositions as neither true nor false.