Logic from Distinction: How a Single Non-Identity Forces the Algebra of Inference

We derive the apparatus of formal logic from a single non-identity. The only posit is the floor: there exist x, y in some carrier K with x ≠ y. From this posit, distinguishing predicates, negation, conjunction, disjunction, implication, identity, non-contradiction, modus ponens, and quantification are forced inevitabilities on the algebra of distinguishability that the carrier supplies. The standard menu of formal logics (classical, intuitionistic, linear, modal, quantum, paraconsistent, many-valued) appears not as a list of candidate foundations but as a list of structural choices imposed on the same forced algebra: which truth-value carrier is admitted, whether predicates may be duplicated, whether the distinguishability of distinct cases is total or partial, whether complementary fibers may overlap. The thesis: logic is not posited, not conventional, not empirical; it is the systematic algebra of distinguishability, and distinguishability is what the floor supplies. The honest residue is named: the meta-language used in the derivation is itself bootstrapped from the floor, and we do not pretend to escape that bootstrap; we minimize what the bootstrap posits above the floor to nothing.