The Kripke–Wittgenstein Paradox Through the Lens of TNA: Rule-Following as Failure of Local Closure

The Kripke-Wittgenstein rule-following paradox demonstrates that no finite set of behavioral facts can uniquely determine the rule followed by a system, as the same input-output history can be interpreted as infinitely many distinct functions. This paper reformulates the paradox as an instance of Failure of Local Closure under the Theory of Axiomatic Necessity (TNA), arguing that operational dynamics (N₀) are structurally insufficient to determine the admissibility structure that legitimizes a rule. By analyzing the distinction between addition and Kripke's "quus" function, we show that a structurally external selector (N₁) is required to collapse the space of possible interpretations into a single realized rule without generating infinite regress.