The Boolean Tile Basis: A Fractal Foundation for Propositional and Quantified Logic

Abstract

The sixteen two-variable Boolean connectives form a complete, self-contained generative alphabet in which propositional variables, logical operators, and constants inhabit the same categorical level. This paper develops the Boolean Tile Basis: a framework in which each discovered. In the symmetric {-1, 1} basis, these attractors acquire a spectral interpretation via the Walsh-Hadamard Transform: each tile’s Fourier structure determines its fractal geometry, with XOR generating the Sierpiński triangle at Hausdorff dimension log(onto geometric axis collapse: ∀ is AND-pooling, ∃ is OR-pooling, and the three orders of quantification exhaust the three degrees of freedom of logical space. The paper extends the fractal images of 3)/log(2) — a result grounded in the carry-free isomorphism (XOR = addition mod 2, Sierpiński = the carry-free locus, Lucas’s theorem closes the triangle). The n-variable XOR IFS has Hausdorff dimension n−1; at n=3 this is exactly 2, producing a complete surface whose axial projections fill the unit square. The framework extends to many-valued logic: Łukasiewicz operators take clean geometric form in {-1, 1}, and the self-equivalence paradox generates a provably chaotic tent map with Lyapunov exponent log 2. Logical quantification maps formal systems introduced in St. Denis and Grim (1997) by formalising their algebraic necessity.connective corresponds to a 2×2 binary tile, NOR is the sole generator from which all 15 others are constructible, and recursive self-application of any tile produces a unique IFS fractal attractor. The connection to fractal geometry is not contingent: a 2×2 tile is a quadtree node by definition, a quadtree under MSB-first labelling generates Morton path codes natively, and a Morton-addressed quadtree under a fixed Boolean rule is an IFS by structure. The fractal images of Boolean operators are therefore logically necessary, not

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