Jech really glosses over class forcing. I cannot find a good reference online. I have two questions about it.
1) Jech says, "As for the forcing relation in general, we cannot generally define $||\varphi ||$ because $\sum X$ does not generally exist if $X \subset B$ is a class [where $B$ is the class-sized boolean algebra]. However, we can still define $p \Vdash \varphi$ using the formulas in Theorem 14.7 [the Properties of Forcing theorem]." What does he mean by this?
2) In order for the quotient structure of a boolean-valued model $\mathfrak{A}$ to be a 2-valued model in which, where $F$ is an ultrafilter on the boolean algebra $B$, $\mathfrak{A}/F \models \varphi([a_1],\ldots,[a_n]) \Leftrightarrow ||\varphi(a_1,\ldots,a_n)|| \in F$, we need fullness. Jech says that $||\varphi||$ can't even be defined for class-forcing, let alone that fullness hold, so... how do we do anything?
Maybe I just need a reference for class-forcing that doesn't completely gloss over it. Thanks.
One particular reference for class forcing would be Sy Friedman's "Fine Structure and Class Forcing", as well as his handbook chapter on the topic of class forcing.
But you should also read the following paper by Peter Holy, Regula Krapf, Philipp Luecke, Ana Njegomir and Philipp Schlicht: Class forcing, the forcing theorem and Boolean completions.