Reflexive Logic: Formal Inference Rules for the Terminal Algebra

Translator This paper develops Reflexive Logic, a new system of inference derived from the terminal algebra of the Reflexive Monoid Algebra (RMA).At the final reflexive level, denoted L7, the reflexivization operator R becomes idempotent (R(L7) = L7), producing a structure in which truth, provability, and evaluation coincide.Logical reasoning inside this algebra follows eight internal rules—Reflexive Truth, Reflexive Entailment, Idempotent Substitution, Reflexive Induction, the Fixed-Point Rule, and others—each preserving stability under R.Gödel’s fixed point is reformulated as a theorem of this logic, demonstrating that incompleteness can be viewed as finite convergence rather than contradiction. A new appendix presents a finite simulation in the Lean proof assistant, illustrating how the reflexive operator stabilizes in well-formed systems and fails to do so in unstable ones.Unstable reflexive systems—cyclic, divergent, or chaotic—expose the boundary of self-containment in mathematics and science: they represent open or self-modifying structures where self-reference does not converge.Their treatment through meta-closure is detailed in Generalized Reflexive Closure Algebra (GRCA): Meta–Structure of Reflexive Laws (DOI 10.5281/zenodo.17536867), where the operator Q resolves instability by closing the orbit of R.Together with the finite framework of UF–GRCA, these results show that reflexive instability is not a flaw but the generative mechanism by which mathematics, logic, and science evolve toward self-consistency.