boolean.cppm File Reference

Formal definition of the Boolean system 𝔹 within the numeric ontology. More...

#include <concepts>
#include <functional>
#include <type_traits>
#include <utility>
import dedekind.numbers:scalars;
import dedekind.sets;
import dedekind.sequences;
import dedekind.order;
import dedekind.category;
import dedekind.algebra;

Include dependency graph for boolean.cppm:

Namespaces
namespace	dedekind::numbers

Detailed Description

Formal definition of the Boolean system 𝔹 within the numeric ontology.

Copyright 2026 The Dedekind Authors Licensed under the Apache License, Version 2.0.

@partition :numbers @dependency :algebra, :topology, :cardinalities, :scalars

numbers_boolean__Booleans

This partition defines the Boolean species—the simplest non-trivial numerical system. It establishes the formal mapping between logical truth values and the algebraic structure of a semiring.

The Boolean system is modeled as an Idempotent Commutative Semiring over the set {0, 1}. In this mapping:

• Addition (⊕) is represented by Logical OR (∨).
• Multiplication (⊗) is represented by Logical AND (∧).
• The Additive Identity (0) is False (⊥).
• The Multiplicative Identity (1) is True (⊤).

This structure is uniquely characterized by its cardinality |S| = 2 and its idempotency property (x ⊕ x = x), which distinguishes it from the additive properties of ℕ or ℝ.

Wikipedia: Boolean algebra, Semiring, Idempotency

Copyright

2026 The Dedekind Authors Licensed under the Apache License, Version 2.0.

Note

"It is not of the essence of mathematics to be conversant with the ideas of number and quantity." – George Boole, An Investigation of the Laws of Thought (1854)