Boolean valued model

A traditional model of a language makes every formula of that language either true or false. A Boolean valued model is a generalization in which formulas take on any value in a Boolean algebra.

Specifically, a Boolean valued model of a signature $\mathrm{\Sigma }$ over the language $ℒ$ is a set $𝒜$ together with a Boolean algebra $ℬ$. Then the objects of the model are the functions ${𝒜}^{ℬ}=ℬ\to 𝒜$.

For any formula $\varphi$, we can assign a value $\parallel \varphi \parallel$ from the Boolean algebra. For example, if $ℒ$ is the language of first order logic, a typical recursive definition of $\parallel \varphi \parallel$ might look something like this:

• $\parallel f=g\parallel ={\bigvee }_{f(b)=g(b)}b$

• $\parallel \mathrm{\neg }\varphi \parallel ={\parallel \varphi \parallel }^{\prime }$

• $\parallel \varphi \vee \psi \parallel =\parallel \varphi \parallel \vee \parallel \psi \parallel$

• $\parallel \exists x\varphi (x)\parallel ={\bigvee }_{f\in {𝒜}^{ℬ}}\parallel \varphi (f)\parallel$