#### Supplement to Actualism

## The Barcan Formula is Logically True

*Proof*: To say that the Barcan Formula is logically true is to
say that every instance of BF is logically true, that is, that every
instance of BF is true_{I}, for every
interpretation **I**. So let ◇∃*x*φ →
∃*x*◇φ be an instance of BF and **I** be an
arbitrary interpretation. To show that this formula is
true_{I}, the definition of
“true_{I}” tells us that we must show that
the formula is true_{I,f} at the actual
world **w**_{0}, for every assignment function **f**.
Because the formula is a conditional, to show it is
true_{I}, we assume, for an arbitrary
assignment **f**, that the antecedent is
true_{I,f} at **w**_{0} and then show
that the consequent is true_{I,f}
at **w**_{0}. So let **f** be an assignment and assume
that the antecedent ◇∃*x*φ of our instance is
true_{I,f} at **w**_{0}. By the
definition of truth at a world (under **I** and **f**) for modal
formulas, it follows that ∃*x*φ is
true_{I,f} at some possible world,
say **w**_{1}. It follows by the definition of truth at a
world for quantified formulas that, for some individual **a** in
the domain of **I**, φ is
true_{I,f[x,a]}
at **w**_{1} and, hence, by the definition of truth at a
world for modal formulas again, that ◇φ is
true_{I,f[x,a]} at
**w**_{0}. So, again, by the definition of truth at a
world for quantified formulas, ∃*x*◇φ is
true_{I,f} at **w**_{0}.