Mechanized Metasemantics of Modal Necessity: Valuation Factorization and Modal Non-Collapse

This formalization develops a mechanized metasemantics for modal necessity in Isabelle/HOL. Its core construction factorizes Kripke-style valuation as V = I o delta, where delta maps object-language formulas to syntactic distinctions and I maps such distinctions to possible-worlds truth-conditions. Thus modal evaluation is carried out over interpreted distinctions rather than over valuation alone. The development provides three witnesses. First, a concrete satisfiability witness shows that the abstract interpretive-structure locale is jointly satisfiable. Second, a nontrivial truth-condition witness shows that I0(delta0(Atom 0)) separates the two worlds of the concrete model. Third, a no-collapse witness proves that Dia (Atom 0) w holds while Box (Atom 0) w fails. The development introduces no additional global axioms; its locale assumptions are discharged by an explicit concrete model. Hence the framework is non-vacuous, nontrivial, and non-collapsing. This shows that, in the present framework, modal force is not a feature of bare syntax alone, but is grounded in syntactic distinctions only insofar as they are interpreted as possible-worlds truth-conditions.