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?


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    $\begingroup$ Note: this question is cross-posted on MSE. $\endgroup$ – Hanul Jeon Mar 9 at 16:29
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    $\begingroup$ If I remember correctly, Rasiowa and Sikorski, in their book "Mathematics of Metamathematics" use Boolean-valued models to prove the completeness theorem for first-order logic (in the case of a countable vocabulary). The idea is that a theory has a canonical model with values in its Lindenbaum algebra, and then a quotient by a suitably generic ultrafilter produces a 2-valued model. $\endgroup$ – Andreas Blass Mar 9 at 18:21
  • $\begingroup$ @HanulJeon Yes. Sorry about that! I initially meant to post on overflow but accidentally posted on MSE instead. Shall I delete the one on MSE? $\endgroup$ – Severine Climacus Mar 9 at 20:02
  • $\begingroup$ Somebody may correct me, but I think that general Boolean-valued models are somehow closely related to to sheaf models on Stone spaces. I also think that probability algebra–valued models are also closely related to what are called 'randomizations' in continuous logic. $\endgroup$ – James Hanson Mar 9 at 20:09
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    $\begingroup$ Continuous logic (in the sense of Chang and Keisler) where the predicates take values in Stone spaces should be equivalent to ordinary discrete first-order logic (possibly without equality). You can always decompose a Stone space–valued predicate into a family of $\{0,1\}$-valued predicates. $\endgroup$ – James Hanson Mar 9 at 20:24

In 1970th there was more interest on this topic. See for example;

Some aspects of Boolean-valued model theory

Filter Constructions in Boolean Valued Model Theory

On Chang's omitting types theorem in Boolean valued model theory

Eastern model theory for Boolean-valued theories

Also as you mentioned there are connections with sheaves, see

Sheaves and Boolean Valued Model Theory.

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    $\begingroup$ Thanks so much! Any idea why the interest on this topic gradually dies out? $\endgroup$ – Severine Climacus Mar 10 at 21:05
  • $\begingroup$ You are welcome. I don't know about it, but this usually happens to many other topics as well, for example, ``model theoretic forcing''. $\endgroup$ – Mohammad Golshani Mar 12 at 8:42
  • $\begingroup$ See also Boolean Types in Dependent Theories. $\endgroup$ – Mohammad Golshani Mar 12 at 8:44

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