In his Three Grades of Modal involvement, Quine uses a notation that is not familiar for me. Does someone know how he intends to be using them?
What 'p' represents is a statement, hence true or false (and devoid of free 'x'). If 'p' is true, then the conjunction '$x=\Lambda.p$' is true of one and only one object x, viz., the empty class $\Lambda$; whereas if 'p' is false the conjunction '$x=\Lambda.p$' is true of no object x whatever. The class $\hat{x}(x=\Lambda.p$) therefore, is the unit class $\iota\Lambda$ or $\Lambda$ itself according as 'p' is true or false. Moreover, the equation: $$\hat{x}(x=\Lambda.p)=\iota\Lambda $$ is by the above considerations, logically equivalent to 'p'.
Does someone know what he means by a conjunction of a statement and a set--'$x=\Lambda.p$'--and what '$\hat{x}$' stands for? from Three Grades of Modal Involvement
$x=\Lambda.p$ means $(x=\Lambda).p$, the conjunction of two statements. For any statement $\phi(x)$ about $x$, the notation $\hat x\,\phi(x)$ means what we'd nowadays write as $\{x:\phi(x)\}$.