Aristotle asked: is "there will be a sea-battle tomorrow" true, or false? Neither — not yet. In 1920 Łukasiewicz gave that "not yet" a value of its own, ½, and built a working logic on three truths. The law of the excluded middle — either P or not-P — quietly stops being a law.
source Jan Łukasiewicz, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic — archive.org/details/aristotlessyllog0000ukas. Rendered, not quoted.
In De Interpretatione ch. 9, Aristotle worried about the sea-battle: a statement about a contingent future seems neither true nor false now, or the future would be fixed. Łukasiewicz took that seriously and added a third value — ½, "possible / undetermined" — beside true (1) and false (0).
The same man wrote the definitive modern reading of Aristotle's syllogistic — so the spine that began with two-valued term logic closes, in his hands, on three-valued logic.
Values are 0, ½, 1. The connectives, chosen so classical logic is the special case with no ½:
¬a = 1 − a a ∧ b = min(a,b) a ∨ b = max(a,b)a → b = min(1, 1 − a + b) a ↔ b = 1 − |a − b|
Feed only 0s and 1s and every table collapses to Boole's. The ½ is the new territory.
Every cell is computed from the formulas above — Łukasiewicz's, not looked up. Try a=½ with ∨¬: "P or not-P" comes out ½, not 1. The excluded middle is gone.
table for →
excluded middle a ∨ ¬a:
Two classical laws stop holding the moment ½ appears:
Excluded middle a ∨ ¬a: at a=½ it is ½, not 1 — a contingent statement is not forced true-or-false. Non-contradiction a ∧ ¬a: at a=½ it is ½, not 0.
What survives: a → a = 1 always (identity), and everything classical when ½ is banned. Classical logic is exactly Ł3 with the middle value forbidden — the two-valued world is a fragment of the three.
Sphere ① forced everything into true-or-false. Here the undetermined gets a seat — and the seat is a trit. Map {0, ½, 1} to {−, 0, +} and the third logic value is the balanced-ternary zero: the value that is neither, the pivot.
So the logic spine and the number spine meet: Łukasiewicz's "possible" and World I's center stone 0 are the same idea — the middle that a two-valued world had to pretend away. The fold closes on the trit.
Recomputes the full Ł3 implication table and checks it against the known values, confirms a→a=1 and that excluded middle is ½ at a=½. Then corrupt the implication.
Kleene's implication is a real rival — it differs from Łukasiewicz's at exactly ½→½ (½, not 1). Swap it in and the audit, comparing to Łukasiewicz's table, turns red.
"Three-valued logic is fuzzy logic." Cut. Fuzzy logic (Zadeh, 1965) uses a continuum [0,1] of degrees; Ł3 has exactly three values. (Łukasiewicz did later generalize to infinitely-many with Tarski — a different system.)
"½ means probability 0.5." Cut. ½ is "undetermined / possible", not a 50% chance. A coin's outcome has a probability; the sea-battle statement has no truth value yet. Different things.
"There is one third value." Kept, corrected. Ł3 is one choice of tables; the adversary lists the rivals. This sphere runs Łukasiewicz's specific system and names it.
The strongest case against, stated fairly:
And classical logic plus probability handles most working needs; many-valued logic stayed a specialty until fuzzy control and hardware gave it uses. The honest claim: Ł3 is a consistent, complete logic that takes Aristotle's sea-battle at its word — one coherent way to give the undetermined a name, not the only one. The engine runs Łukasiewicz's; the wall marks that it is a choice.