The 5 Axiom is Logically True

Proof: To see that the 5 axiom is true in every interpretation, pick an arbitrary interpretation I. To show that a conditional sentence is true_I, the definition tells us that we must show that it is true_I at the actual world w₀. To do this, we assume that the antecedent is true_I at w₀ and then show that the consequent is true_I at w₀. So assume that the antecedent of the 5 axiom, namely, ◊φ, is true_I at w₀. It follows, by the definition of truth, that φ is true_I at some possible world, say w₁. Now to show that □◊φ is true_I at w₀, we need to show that ◊φ is true_I at all possible worlds. So pick an arbitrary possible world, say w₂. Note that ◊φ is true_I at w₂, since φ is true_I at w₁. But since w₂ was chosen arbitrarily, it follows that ◊φ is true_I in all possible worlds. So □◊φ is true_I at the actual world.