It is well known that Boolean valued models play significant roles for set-theoretic purposes. But how well-studied are Boolean valued models in a more general setting, as models for random first-order languages?
For example, towards the end of chapter 0 of Bell's "Set Theory: Boolean Valued Models and Independence Proofs", Bell gives the definition of a Boolean-algebra-valued $\mathscr{L}$ structure where $\mathscr{L}$ is a first order language whose sole extralogical symbol is a binary predicate $P$:
A Boolean-algebra valued $\mathscr{L}$ structure is a quadruple $\mathbf{M} = (M, eq, Q, B)$, where $M$ is a non-empty class, $B$ is a complete Boolean algebra and $eq$ and $Q$ are maps from $M \times M \rightarrow H$ satisfying, for all $m, n, m', n'\in M$,
$$ eq(m, m)=1, eq(m, n)=eq(n,m), eq(m,n) \land eq (n, n') \le eq(m, n'), Q(m,n) \land eq(m, m') \le Q (m', n), Q(m,n) \land eq(n, n') \le Q(m, n') $$
For any formula $\phi$ of $\mathscr{L}$ and finite finite sequence $\mathbf{x} = <x_1, ..., x_n>$ of variables of $\mathscr{L}$ containing all the free variables of $\phi$, define for any Boolean-valued $\mathscr{L}$ structure $\mathbf{M}$ a map
$$ ‖\phi‖^{M_x}: M^n \rightarrow B $$
recursively as follows:
$‖x_p = x_q‖^{M_x} = <m_1, ..., m_n> \mapsto eq(m_p, m_q),$
$‖Px_px_q‖^{M_x} = <m_1, ..., m_n> \mapsto Q(m_p, m_q),$
$‖\phi \land \psi‖^{M_x} = ‖\phi‖^{M_x} \land ‖\psi‖^{M_x}$, and similarly for other connectives,
$‖\exists y \phi‖^{M_x} = <m_1, ..., m_n> \mapsto \bigvee_{m \in M} ‖\phi(y/u)‖^{M_{ux}}(m, m_1, ..., m_n)$,
$‖\forall y \phi‖^{M_x} = <m_1, ..., m_n> \mapsto \bigwedge_{m \in M} ‖\phi(y/u)‖^{M_{ux}}(m, m_1, ..., m_n)$
That definition, it seems to me, can be easily generalized to a random first order language with other predicate, constant, or function symbols. And Bells mentions that it can be shown that a formula $\phi$ is $\mathbf{M}$-valid for all $\mathbf{M}$ (a forluma $\phi$ is $\mathbf{M}$-valid just in case $‖\phi‖^{M_x}$ is identically 1) iff $\phi$ is provable in classical first-order logic.
So I'm wondering if there are other interesting results on Boolean-valued models as models for arbitrary first order languages. Or in general, how well-studied is the theory of Boolean valued models, as models for random first order languages? How much of traditional model theory (the theory of 2-valued models of first order languages) can be generalized to Boolean valued models? Are there any books or articles on this topic?
Thanks!
