The Equivalence of the Disjunction and Existence Properties for Modal Arithmetic

In a modal system of arithmetic, a theory S has the modal disjunction property if whenever S¬□φ∨□ψ, either S¬□φ or S¬□ψ. S has the modal numerical existence property if whenever S¬∃x□φ(x), there is some natural number n such that S¬□φ(n). Under certain broadly applicable assumptions, these two properties are equivalent