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Definition of a three place relation-called satisfaction, which holds between a formula, a model, and an assignment of values to variables.
We cannot give a direct inductive definition of truth. Truth is a relation that holds between sentences and models. But the matrix of a quantified sentence typically won't be a sentence. For example, is a sentence, but its matrix
is not. Thus an inductive truth definition defined solely in terms of sentences couldn't explain why
would be true in a model, for there are no sentential subformulae for such a definition to refer to.
An indirect Approach.
So instead, we're going to proceed indirectly by defining a three place relation-called satisfaction-which holds between a formula, a model, and an assignment of values to variables. Given a model , an assignment of values to variables in
(or more simply, an assignment in
) is a function
from the set of variables to
. Assignments are a technical aid that tells us what the free variables stand for. By making use of assignment functions, we can inductively interpret arbitrary formulae in a natural way, which will make it possible for us to define the concept of truth for sentences.
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