Mathematics Boolean Algebra From Rs Boolean Algebra Cert

Boolean algebra certificate

Boolean algebra is the mathematics of true and false, with operations like AND, OR, and NOT. The smallest interesting Boolean algebra has just two truth values, but the next one, on three bits, has 2^3 = 8 elements. A Boolean algebra on n bits has 2^n elements, and the three-bit version is the one that appears throughout digital logic and set theory.

The BooleanAlgebraCert declaration is a small machine-checked certificate. It records three facts: the set of five canonical Boolean operations (AND, OR, NOT, NAND, NOR) has exactly 5 members, the three-bit algebra has exactly 8 atoms, and those 8 atoms equal 2^3. Each fact is proved by direct computation, and the whole certificate contains no unfinished proofs and no added axioms. The certificate itself is a structure that bundles these three proofs together.

The five operations are not arbitrary: NAND and NOR are the universal gates from which all other Boolean functions can be built, and the five together form the standard basis taught in every logic course. The number 8 = 2^3 is the size of the power set of a three-element set, which is why the Boolean lattice on three bits has eight atoms. These are classical facts, independent of any framework.

In Recognition Science, the certificate acts as a bridge. The framework's recognition lattice on three bits, written Q₃ = {0,1}³, is itself a Boolean algebra, so the classical structure and the framework's structure coincide on this point. The certificate does not derive Boolean algebra from recognition; it verifies that the three-bit recognition lattice satisfies the defining properties of a Boolean algebra.

What the certificate does not claim matters as much as what it proves. It does not claim that Boolean algebra is the only structure on three bits, nor that the five operations are the only possible ones. It does not claim that the framework's recognition cycle forces Boolean algebra to exist. The certificate is a compatibility check, not a derivation.

theorem boolOpCount : Fintype.card BoolOp = 5 := by decide

theorem atomCount_eq_8 : atomCount = 8 := by decide

theorem atoms_eq_2cubeD : atomCount = 2 ^ 3 := rfl

What this page does not claim

The certificate does not derive Boolean algebra from recognition principles. The certificate does not claim the five operations are the only possible Boolean operations.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/BooleanAlgebraFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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