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 $\mathcal{L}$ is a set $\mathcal{A}$ together with a Boolean algebra $\mathcal{B}$. Then the objects of the model are the functions ${\mathcal{A}}^{\mathcal{B}}=\mathcal{B}\to \mathcal{A}$.
For any formula $\varphi $, we can assign a value $\parallel \varphi \parallel $ from the Boolean algebra. For example, if $\mathcal{L}$ is the language of first order logic, a typical recursive definition of $\parallel \varphi \parallel $ might look something like this:

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

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

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

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$\parallel \exists x\varphi (x)\parallel ={\bigvee}_{f\in {\mathcal{A}}^{\mathcal{B}}}\parallel \varphi (f)\parallel $
Title  Boolean valued model 

Canonical name  BooleanValuedModel 
Date of creation  20130322 12:51:08 
Last modified on  20130322 12:51:08 
Owner  Henry (455) 
Last modified by  Henry (455) 
Numerical id  8 
Author  Henry (455) 
Entry type  Definition 
Classification  msc 03C90 
Classification  msc 03E40 
Defines  Booleanvalued model 