Strengthening Consistency Results in Modal Logic

Many treatments of epistemological paradoxes in modal logic proceed along the following lines. Begin with some enumeration of assumptions that are individually plausible but when taken together fail to be jointly consistent (or at any rate fail to stand to reason in some way). Thereupon proceed to propose a resolution to the emerging paradox that identifies one or more assumptions that may be comfortably discarded or weakened and that in the presence of the remaining assumptions circumvents the troubling inconsistency defining the paradox [11] (cf. Chow [8] and de Vos et al. [16]). Typical among such assumptions are logical standards expressed in the form of inference rules and axioms pertaining to knowledge and belief, such as axiom scheme K -- that is to say, the distributive axiom scheme of the form K( ϕ ψ) (K ϕ K ψ). The choice of precisely which assumptions to temper can, at times, have an element of arbitrariness to it, especially when the choice is made from among several independent alternatives underpinning distinct resolutions in the absence of clear criteria or compelling grounds for distinguishing among them.