 From 1917, Łukasiewicz had been happy with the three-valued logic as formulating adequate notions of modality, with the noted preference for the infinite-valued system as optimally precise. At some time, probably around 1951–52, when he was working on Aristotle’s modal logic, Łukasiewicz changed his mind. There are a number of reasons behind the change of mind but the easiest to identify is Łukasiewicz’s concern that in Ł\(_3\) there are theorems of the form \(L\alpha\), for example \(LCpp\). Why should this be a concern, given that most “standard” modal logics recognize the principle that if \(\alpha\) is a theorem, so is \(L\alpha\)? Łukasiewicz gives two examples to justify the worry. If \({=}ab\) is the proposition that \(a\) is identical with \(b\), then basing identity on the two axioms of self-identity and extensionality