Boolean valued model

A traditional model of a languagePlanetmathPlanetmath makes every formulaMathworldPlanetmathPlanetmath of that language either true or false. A Boolean valued model is a generalizationPlanetmathPlanetmath in which formulas take on any value in a Boolean algebraMathworldPlanetmath.

Specifically, a Boolean valued model of a signaturePlanetmathPlanetmathPlanetmath Σ over the language is a set 𝒜 together with a Boolean algebra . Then the objects of the model are the functions 𝒜=𝒜.

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

  • f=g=f(b)=g(b)b

  • ¬ϕ=ϕ

  • ϕψ=ϕψ

  • xϕ(x)=f𝒜ϕ(f)

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