Two real, lasting inventions. Many-valued logic is rigorous and generative — fuzzy logic, quantum logic, and modern reasoning-under-uncertainty descend from it. Polish notation is exact and runs in every parser and stack machine.
Non-classical means optional. Dropping bivalence is a deliberate choice of logic suited to certain problems, not a proof that classical logic is wrong. Whether the future is "really" ½ is philosophy, not theorem.
Honest about the man. A giant of the Lwów–Warsaw school, briefly a minister of education in interwar Poland; he survived the war in difficult, much-debated circumstances and ended his life in exile in Dublin. The logic stands on its own.
Set each input to 0 (false), ½ (undetermined), or 1 (true). The connectives follow Łukasiewicz's rules: ¬a = 1−a, a∧b = min, a∨b = max, a→b = min(1, 1−a+b). Watch what happens to the law of the excluded middle (a ∨ ¬a, always true in classical logic) when a value is undetermined — it comes out ½, not 1. Two thousand years of "every statement is true or false" quietly gives way.
P ∨ ¬P is the excluded middle — always true. Here, when P = ½, it is only ½: a statement and its negation can both be undetermined. That is the whole revolution — Łukasiewicz showed you can reason consistently without bivalence. ⚑ Set P and Q to ½ and watch how undetermination propagates; this is the great-grandparent of fuzzy logic (truth as a degree) running in control systems and AI today."I have declared a war upon the universal validity of the principle of the excluded middle." — Jan Łukasiewicz
Łukasiewicz noticed you never need brackets if you write the operator first: instead of (2 + (3 × 4)), write + 2 × 3 4 — scan left to right and the structure is unambiguous. This is Polish notation (prefix). Reverse it — operands first, operator last — and you get RPN, exactly how a stack machine, an HP calculator, the language Forth, and the inside of every compiler evaluate expressions. Type a prefix expression (operators + - *, then numbers) and evaluate it.
The first break from bivalence (1920). Generalised to infinitely many values, it is the root of fuzzy logic, probabilistic and quantum logics — reasoning where "true/false" is too blunt.
Parenthesis-free prefix form. Its reverse, RPN, is the native tongue of stack machines, Forth, and expression evaluation inside compilers and calculators.
Gate kept on. Two honesties. On the logic: three-valued logic is real and fertile, but adopting it is a choice, not a refutation of classical logic — most of mathematics still runs happily on bivalence, and whether the open future is "genuinely" ½ rather than just unknown-to-us is a philosophical stance, not a theorem. Many-valued logic is a powerful tool, not the discovery that Aristotle was wrong. On the man: Łukasiewicz was a founding giant of the Lwów–Warsaw school, one of the most brilliant logic communities ever, and briefly Poland's minister of religious affairs and education. His path through the Second World War — leaving German-occupied Warsaw and ending, after the war, as a professor in Dublin until his death in 1956 — has been the subject of difficult and contested historical discussion; this page renders his mathematics, which stands entirely on its own, and does not launder or litigate the biography. The third truth value and the notation in every parser are his, plainly. ⚑ [[david-hilbert]] ← · next → Tarski (truth itself).