Express $V(\diamond \alpha)$ set theoretically in terms of $V(\alpha)$

Question

I am reading Modal Logic. While going through the basics of the subject I am having problem in a place. Please help me.

Say we are dealing with a frame $(W,R)$ and defined a model $M$ using a valuation map $V$ that maps any well formed formula (wff) to a subset of $W$. We define for any wff $\alpha$, $V(\alpha) =\{w\in W \mid M,w \vDash \alpha\}$

Now I can easily see that $V(\lnot \alpha),V(\alpha \land \beta)$ can easily be written in terms of $V(\alpha),V(\beta)$ like $V(\lnot \alpha)=V(\alpha)^c,V(\alpha \land \beta)= V(\alpha) \cap V(\beta)$

But I want to write $V(\diamond \alpha)$ in terms of $V(\alpha)$ where I am stuck.

I see that $V(\diamond \alpha)=\{w \in W \mid \exists v \in W \text{ with } wRv \text{ and }M,v \vDash \alpha \}=\{w \in W \mid \exists v \in W \text{ with } wRv \text{ and }v\in V(\alpha) \}$

But I am not able to express it set theoretically in terms of $V(\alpha)$

Perhaps I am missing something trivial. I am very new in this area and I apologise for not making a great effort from my end.

Please help me to solve this problem. I will be very grateful. Thnx in advance.

Answer 1

Note the possibility and necessity operators aren't truth functional. In other words, the so-called principle of compositionality (also known as Frege's principle) that the meaning of complex sentence depend exclusively of the meaning of its compounds does not hold.

This implies that, no matter how we try, we can't fully build a truth table for those operators:

$$\begin{array} {|c|} \hline P & \Diamond P & \Box P& \\ \hline 1 & 1 & ?& \\ \hline 0 & ? & 0 \\ \hline \end{array}$$ (And even so, I need to assuming reflexivity to make this table)

We usually express this saying that the modal operators of necessity and possibility are intensional. Opposed to extensional operators, to determine the meaning of $\Diamond$ and $\Box$ more than the truth value of their operands are needed, and so, here comes alternative semantics e.g. the possible word semantics, the usual formal semantics a modal logic is given.

Finally, answering your question

So is it not possible to express set theoretically in terms of $V(α)$?

No, as long as your above definition of $V(α)$ attempts to grasp the meaning of $\alpha$ extensionally. Why don't you try extending it to include accessibility?