What question did this study set out to answer?

The study aims to redefine truth values as states of topological convergence and unify various forms of inference within a dynamical framework.

June 17, 2026Open Access

Dynamic Logic: Truth as a Topological Convergence State

Key Points

The study aims to redefine truth values as states of topological convergence and unify various forms of inference within a dynamical framework.
Proposed a new criterion for a logical system based on topological charge.
Derived a phase coupling equation to describe inference dynamics.
Demonstrated the resolution of paradoxes using the sine function.
Identified that I=1 signifies truth, I=-1 signifies falsehood, while fractional and fluctuating I indicate fuzzy and indeterminate truth values respectively.
Unified the forms of inference under a single dynamic mechanism, showing clear operational definitions for each.
Repositioned Gödel's incompleteness theorems as resulting from discrete formal systems, relating them to failures in topological convergence.

Abstract

Abstract Classical logic, founded on the binary true/false distinction, faces four major predicaments: continuous truth values, detachment from actual thought, self-referential paradoxes, and Gödel incompleteness. Starting from the phase closure condition of Generative Axiom 4, this paper proposes that: truth values are states of topological convergence, inference is a phase coupling process, and paradoxes are the dynamics of self-referential feedback.The central discovery is that the topological charge I is the sole criterion of a logical system—I=1 is true, I=-1 is false, fractional I is fuzzy, and fluctuating I is indeterminate. This criterion is self-sufficiently established within this paper through the operational definition of the flip operation, the rigorous derivation of the phase coupling equation, and the intrinsic guarantee of topological convergence. The four forms of inference are unified under the same dynamical mechanism: deduction is the propagation of phase locking, induction is multi-body coupled phase synchronization, abduction is inverse gradient flow source tracing, and analogy is modular automorphism mapping. If the phase closes, inference is determinate; if the phase does not close, inference is indeterminate. This paper further proves that Russell's paradox and the liar paradox are automatically resolved within the sine function—the self-referential term sin(γ-γ)=0 transforms a static contradiction into uniform rotation. Gödel's incompleteness theorems are repositioned as a property of discrete formal systems, manifesting as a failure of topological convergence under emergent continuity. Logic, mathematics, and physics are unified within the generative framework: logic is the locking and fluctuation of the topological charge, mathematics is the invariant of the topological charge, and physics is the projection of the topological charge. Keywords: dynamic logic; topological convergence; phase coupling; topological charge; truth value; deductive inference; inductive inference; abductive inference; analogical inference; paradox; Gödel's incompleteness theorems

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