The Omnisyndetic Framework Volume I

Overview

This volume sets out a closed first-order logic whose primitives are contrast and return. From the base split of true from false follows a working ladder: contrast → distinction → condition → one-way licence → return → truth. Truth occurs at return when a licence withstands opposition and its rival fails. The text develops axioms, lemmas, and theorems of contrast and distinction, with complete audit-ready proofs. Diagonal self-address is barred, so Curry-style fixed-point constructions cannot arise. Liar forms remain unresolved wherever no admissible licence exists. Yablo's sequence appears as a chain that cannot close, so no return is possible. Gödel incompleteness is recovered as the absence of a reachable licence.

Alongside the formal calculus stands the Syndetic Contrast Algebra, a visual companion that renders licences and returns as directed structure. Each section has three parts: the visual algebra; a sutured dialogue between the Seeker and the Challenger in Platonic form; and the formal proof sets in first-order syntax.

The framework establishes a structural ontology - relation first, closure earned, identity read through contrast. As a diagnostic layer it translates across formal languages and computing bases, allowing pre-return evaluation of what code can run, what can resolve, and how. In practice, admissible codes are classified, licences verified, and returns recorded before any statement is issued. The foundational geometry set used across the Omnisyndetic project is stated and proved here, making this the logical and geometric base of the Framework.