Halldén-completeness by gluing of Kripke frames.
Lloyd Humberstone, J.F.A.K. van Benthem · Notre Dame Journal of Formal Logic · 1983
We give in this paper a sufficient condition, cast in semantic terms, for Hallden-completeness in normal modal logics, a modal logic being said to be Hallden-complete (or Ήallden-reasonable') just in case for any disjunctive formula provable in the logic, where the disjuncts have no propositional variables in common, one or other of those disjuncts is provable in the logic.It is clear from this definition that the weakest normal modal logic, the system K, is not Hallden-complete, in view of the provability in K of the formula αl v OT, since neither disjunct is a theorem of K and we may rewrite 1 as p Λ ~p and T as q v ~q.(Notation and terminology are as in [5] some historical and philosophical remarks on Hallden-completeness may be found in [3].)Thus the Hallden-complete normal logics are either extensions of K + • 1 (the 'Absurd' system) or of K + 0 T (the system D).Amongst the latter systems are such familiar modal logics as T, S4, and S5, already known to be Hallden-complete.There are several proofs in the literature for these systems and other isolated cases of systems in this spectrum, though these have an ad hoc appearance about them in the sense that they tend to exploit rather specific properties of the individual systems concerned (see, e.g., [4]).We point to a common semantic principle which brings some order into the situation, and see how far it takes us in general.It turns out (Theorem 2, below) that a very simple semantic condition is sufficient for Hallden-completeness amongst the extensions of D. P-morphic fusionThe main theorem of the paper requires first a definition.