Possibilist Quantification

Abstract

There is a principle of modal logic which is known as the Barcan Formula (see p.9). In symbols it is

$$\forall xL\phi {\text{x}} \equiv L\forall x\phi x$$

((1))

or (equivalently) as the dual for possibility

$$\exists xM\phi x \equiv M\exists x\phi x$$

((2))

(1) means that everything is necessarily φ iff it is necessary that everything is φ and is validated by the semantics introduced in Chapter 1. Nevertheless it is frequently held to be false on the ground that it assumes that exactly the same things exist in all possible worlds. For suppose that we are in world w₁. It may be that all the things which exist in w₁ are φ in every world, but in a world w₂ something which exists there but doesn’t exist in w₁ is not φ.