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 $\Sigma$ 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 $\mathcal{A}^{\mathcal{B}}=\mathcal{B}\rightarrow\mathcal{A}$ .

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

•

$\lVert f=g\rVert=\bigvee_{f(b)=g(b)}b$
•

$\lVert\neg\phi\rVert=\lVert\phi\rVert^{\prime}$
•

$\lVert\phi\vee\psi\rVert=\lVert\phi\rVert\vee\lVert\psi\rVert$
•

$\lVert\exists x\phi(x)\rVert=\bigvee_{f\in\mathcal{A}^{\mathcal{B}}}\lVert\phi% (f)\rVert$

Title	Boolean valued model
Canonical name	BooleanValuedModel
Date of creation	2013-03-22 12:51:08
Last modified on	2013-03-22 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	Boolean-valued model