A Calculus of Reasonings — The Principle of Naturality, Part II

Two reasonings can start from incompatible premises and still share points in their foundations that neither party sees: unnoticed consensus. Finding them means comparing every claim of each party with every claim of the others and following each coincidence to the end. No reader can do that by hand. This work builds an object on which that comparison is computed. Each party's reasoning is a tree: its premise and what it derives. A natural-language inference model judges every pair of claims from different parties, and a certificate marks the pairs that entail each other. On these, five matrix operations (reduction, threshold, product, closure, count) and one minimum cut compute: what each party is committed to, and by which path; what all parties hold at once; what accepting a proposal would cost each one; what disowning a claim would take; which proposal all parties hold most evenly. Each answer carries the path that produces it. Proved: every reading exists, is unique and does not depend on the order of computation. The cost of disowning a claim is a minimum cut (Menger). Measured on a documented conflict (Nile/GERD) and a fictional control: chains by topic and three-party triangles exceed chance, but the overall size of the common ground does not. On the Nile, African Union mediation is a more balanced bridge than the case's own proposal, and a proposal foreign to the problem gets no support. The error enters through the inference model, not the computation, so better models improve the results without changing any definition. The calculus does not say what is true. It shows where the parties coincide without knowing it and what each option costs. It supports decisions and is meant to be used under professional supervision.