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"


See The Notation in Principia Mathematica.

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 $\phi$.


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