#### Supplement to Actualism

## 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
**w**_{0}. To do this, we assume that the antecedent is
true_{I} at **w**_{0} and then show that the
consequent is true_{I} at **w**_{0}. So
assume that the antecedent of the 5 axiom, namely, ◊φ, is
true_{I} at **w**_{0}. It follows, by the
definition of truth, that φ is true_{I} at some
possible world, say **w**_{1}. Now to show that
□◊φ is true_{I} at **w**_{0},
we need to show that ◊φ is true_{I} at all
possible worlds. So pick an arbitrary possible world,
say **w**_{2}. Note that ◊φ is
true_{I} at **w**_{2}, since φ is
true_{I} at **w**_{1}. But
since **w**_{2} was chosen arbitrarily, it follows that
◊φ is true_{I} in all possible worlds. So
□◊φ is true_{I} at the actual world.