Boole took Aristotle's words and wrote them as equations. Let 1 be everything, 0 be nothing, and a class be a symbol. Then logic obeys arithmetic — with one strange extra law, x·x = x, whose only solutions are 0 and 1. That single equation is why thought is two-valued, and why a wire has two states.
source George Boole, An Investigation of the Laws of Thought (1854) — archive.org/details/investigationofl01bool. Rendered, not quoted.
Boole's move: a class (all cats, all black things) becomes a symbol obeying algebra. Writing 1 for the universe and 0 for the empty class:
x·y = things that are both x and y (intersection). 1−x = everything that is not x. From "and", "not", and these two constants, all of logic follows.
"Or" he wrote as x + y — but only when x and y are disjoint. That restriction is a real defect; see the graveyard.
Ordinary algebra has x·x = x². Boole's logic adds the index law: a class intersected with itself is just itself —
x · x = x ⟺ x·(1−x) = 0
Solve it: x(1−x)=0 forces x = 0 or x = 1. Nothing between. Boole called it "the fundamental law of thought" — it is the law of non-contradiction (nothing is both x and not-x), and it is why the whole system is binary.
| a | b | a·b |
|---|
Every value is computed from Boole's operations on 0 and 1 — the table is generated, never stored. There are exactly 16 two-input functions; each is one column of four bits.
The dictionary that turns a syllogism into an equation:
| logic | algebra |
|---|---|
| the universe / nothing | 1 / 0 |
| x and y | x·y |
| not x | 1 − x |
| All X are Y | x·(1−y) = 0 |
| No X are Y | x·y = 0 |
"All X are Y" as x(1−y)=0 means: there is nothing that is X and not-Y. Aristotle's syllogism is now a system of equations you can solve.
The previous sphere ended on a gap: Aristotle's 24 valid moods vs Boole's 15, the difference being existential import. Here is where it resolves.
"All X are Y" = x(1−y)=0 is true whenever x = 0 — when the class X is empty. Aristotle's logic could not say that; Boole's constant 0 can. The nine "weakened" syllogisms fail precisely because a "some" conclusion needs x ≠ 0, which the algebra refuses to assume.
The syllogism did not disappear — it became solvable. That is the whole point of the FOLD: each sphere is the next one's premise.
Re-checks Boole's laws over every input right now, then lets you break one.
If AND stops meaning "both", then x·x = x and x(1−x) = 0 collapse — the audit will catch it and turn red. Computed, not painted.
"Boole's + is inclusive OR." Cut. Boole's addition was exclusive/partial — x + y was only defined for disjoint classes. Jevons (1864) fixed it to the inclusive OR we use; the engine above uses the modern inclusive form and says so.
"x² = x is a typo for x² = 1." Cut. It is exactly x² = x (idempotence); its roots are 0 and 1, not ±1. That is the binary result — verified in the audit.
"Boole invented binary computers." Kept, corrected. Boole gave the algebra; Shannon (1938) connected it to switching circuits. The two-state wire is downstream, not Boole's claim.
The strongest case against, stated fairly:
And Boole's own + was defective (partial), so the pristine "algebra of logic" is a later reconstruction, not quite his 1854 text. The engine is honest about using the modern inclusive operations — the law that survives untouched is x·x = x, and it is checked live.