 Łukasiewicz’s most celebrated achievement was his development of many-valued logics. This revolutionary development came in the context of discussing modality, in particular possibility. To modern logicians, used to the idea of modal logic being grafted onto classical bivalent logic, this may seem odd. But let us consider how Łukasiewicz arrived at the idea. If \(p\) be any proposition, let \(Lp\) notate that it is necessary that \(p\) and \(Mp\) that it is possible that \(p\). The two modal operators are connected by the usual equivalence \(ENLpMNp\). Everyone accepts the implications \(CLpp\) and \(CpMp\). Łukasiewicz supposes one accepts also the converse implications \(CpLp\) and \(CMpp\), as one would from a deterministic point of view. That gives the equivalences \(EpLp\) and \(EpMp\), which effectively collapse modal distinctions. Now add in the idea that possibility is two-sided: if something is possible, then so is its negation: \(EMpMNp\). From these it immediately follows that \(EpNp\), and this is paradoxical in two-valued logic. The way out, as Łukasiewicz portrays it, is to uncollapse the modal distinctions, not by rejecting any of the principles above but by finding a case where \(EpNp\) is true. We entertain the idea of the proposition \(Mp\) being true when \(p\) is neither true nor false. In addition to the truth-values true (1) and false (0), allow then a third value, possible, which we write ‘\(\tfrac{1}{2}\)’, so that when \(p\) is neither true nor false, it is possible, and so is its negation \(Np\), for if \(Np\) were true, \(p\) would be false, and vice versa. If \(Epq\) is true when \(p\) and \(q\) have the same truth-value, then when \(p\) is possible (we write ‘\(\tval{p}\)’ for the truth-value of \(p\), so \(\tval{p}=\tfrac{1}{2}\)) we have