What does w ∈ V(p) mean?

Question

I have been recently looking at the Handbook of Modal Logic and have come across the following definition:

M, w ⊨ p iff w ∈ V(p)

I dont understand how w could be an element of V(p) where V assigns a truth value to p. So in other words, what does 'w ∈ V(p)' mean?

Answer 1

The non-modal definition of valuation π is a mapping from propositional letters (e.g. p, q, r) to truth-values (e.g. elements of Bool = {⊤, ⊥}):

Alethic valuation function. π : Prop → Bool.

π(p) is an assignment of a truth-value to the propositional letter p. The modal definition of valuation V that you mentioned is a mapping from propositional letters not to truth-values, but to subsets of W:

Modal valuation function 1. V₁ : Prop → ρ(W).

V₁(p) is not an assignment of a truth-value to p because the truth-value of p depends on the world it's evaluated at. V₁(p) is an assignment to p of that subset of W which includes those worlds that satisfy p.

It is, however, possible to think of V as a mapping from pairs (p, w) ∈ Prop × W to truth-values:

Modal valuation function 2. V₂ : Prop × W → Bool.

Here V₂(p, w) is no longer unary and takes a proposition and a world and gives a truth-value. Now instead of w ∈ V₁(p) we would say the equivalent V₂(p, w) = ⊤.

For contrast, let's consider the Tarski truth-conditions of propositional letter p in those three contexts:

1. (Alethic)       M |= p    iff    π(p) = ⊤.

2. (Modal₁)  M, w |= p    iff    w ∈ V₁(p).

3. (Modal₂)  M, w |= p    iff    V₂(p, w) = ⊤.