Quaternionic Logic: A G-Lattice Unifying Boolean and Fuzzy Frameworks

A layered ontology of 72 semantic primitives across six algebraic layers—Boolean, Fuzzy, Ordinal, Modal, Trivalent, and Probabilistic—was previously shown to discriminate real knowledge domains from pseudoscience controls. However, the six algebras remained independent systems with no formal unification. We construct a single quaternionic operator O = 0, 1, +, i, j, k from which all six algebras admit dimensional projections—four (Boolean, Gödel fuzzy, Gödel–Kripke modal, and Kleene trivalent) as exact recoveries, two (ordinal and probabilistic) as interpretive correspondences. The operator assigns to each concept a four-dimensional coordinate (r, i, j, k) ∈ 0, 1⁴, where r encodes crystallised truth, i encodes potentiality, j encodes verifiability, and k encodes recursive selection. Five axioms (K1–K5) characterise k, and its irreducibility is proven: no function of (r, i, j) can reproduce k. Logical connectives on V = 0, 1 × −1, 1³ form a bounded residuated lattice with involutive negation. An SU (2) group acts on truth values by conjugation, defining context transformations that preserve truth while rotating epistemic character. The central result is a commutation theorem: lattice connectives commute with the group action if and only if they depend exclusively on the real component. Boolean logic emerges as the context-free (group-invariant) special case. The combined structure is a G-lattice: a residuated lattice equipped with a non-abelian group action. Additional results include: a cancellative–generative classification of nine algebras of opposition, four diagnosable pathologies as axis collapses, and a liquid zone (all four axes positive) shown to be open, convex, and path-connected. Completeness reduces to G∼ (Gödel logic with involutive negation). All results are verified computationally at machine precision (~10⁻¹⁶) over 50, 000 test cases.