# What does Quine mean by $\;\hat{a}\phi=\;$V?

This is some kind of weird notation. I know how $$\;\hat{a}\;$$ was used in the Principia Mathematica (as an equivalent of $$\;\eta$$-reduction in lambda calculus), but what does it do here $$\;\hat{a}(a=a.\phi)\;$$?

Why is $$\;\hat{a}\phi\;$$ equivalent to a universal set, and why does a $$\;\Lambda\;$$ mean an empty set (if V states for the universal set and according to Quine's "... is either V or $$\Lambda$$") ?

Fragment of W.V.O.Quine's "Reference and Modality"

Let $$\phi$$ a statement that is either True or False.
$$\hat {\alpha}(\alpha = \alpha)$$ is simply $$\{ \alpha \mid \alpha = \alpha \}$$, i.e. the class such that ...
But $$\alpha=\alpha$$ is always true; thus, $$\{ \alpha \mid \alpha = \alpha \} = \text V$$, i.e. the "universe".
Now, the formule $$(\alpha = \alpha) \land \phi$$ has the truth value of $$\phi$$, and thus $$\{ \alpha \mid \alpha = \alpha \land \phi \}$$ is equal to $$\text V$$ or $$\emptyset$$ ($$\Lambda$$) according to the truth value of $$\phi$$.