modal_duality
plain-language theorem explainer
Modal duality equates necessity of a property p at configuration c with the negation of possibility of its negation. Researchers formalizing modal logic over recognition state spaces, where necessity encodes cost-minimizing futures, would cite this equivalence. The proof is a direct biconditional argument that applies the definitions of □ and ◇ in each direction.
Claim. Let $p$ be a predicate on configurations and $c$ a configuration. Then necessity of $p$ at $c$ holds if and only if it is not possible that the negation of $p$ holds at some accessible future from $c$.
background
Configurations are points in recognition state space, each carrying a positive real value, a time coordinate in ticks, and a bound on the absolute logarithm of the value. ConfigProp is an abbreviation for predicates on these configurations. Modal necessity □p at c means p holds for every y in the set of possible futures P(c) from c, while possibility ◇q at c means there exists some y in P(c) such that q(y). The module develops modal operators over this simplified state space, importing LawOfExistence and the ILG configuration structure for physical parameters.
proof idea
The term proof uses constructor to split the biconditional. The forward direction assumes □p at c, takes a witness y with ¬p(y), and derives a contradiction by applying the necessity assumption. The reverse direction assumes the negation of possibility of negation, fixes an arbitrary future y, and shows p(y) by reductio: supposing ¬p(y) yields an immediate witness contradicting the assumption.
why it matters
This supplies the standard duality between necessity and possibility in the modal layer of Recognition Science. It reinforces the reading of □ as cost-forced outcomes under the J-cost framework and the LawOfExistence. No downstream theorems are recorded, leaving open how the duality interacts with the forcing chain T5-T8 or the Recognition Composition Law.
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