# Definability of Sets of Truth Assignments

I have some questions about under what conditions a set of truth assignments is the model of some set of sentences.

To be more precise, suppose I'm dealing with only propositional logic. Let $K$ be a set of truth assignments, i.e. those maps assigning truth values to propositional variables.

For any set $\Sigma$ of sentences, let $Mod(\Sigma):=\{v:v\ \mbox{is a truth assignment and v satisfies$\Sigma$}\}$.

Definition. Let $K$ be a set of truth assignments. Call $K$ definable if $K=Mod(\Sigma)$ for some set $\Sigma$ of sentences; if such $\Sigma$ can be chosen finite (or equivalently, containing a single sentence), then $K$ is finitely definable.

I can prove that, in a language with only finitely many propositional variables, any set of truth assignments is definable. Since in such language there are only finitely many truth assignments, what I proved can be rephrased as: any finite set of truth assignments is definable. Now my questions are:

1. In a language with denumerably many propositional variables, there are continuum many truth assignments. Does what I proved still hold in this case? Is any set of truth assignments, or any finite set of truth assignments, definable? Or finitely definable?
2. Is there a criterion for a set of truth assignments to be definable or finitely definable in a general setting? What would happen if we work in a language with uncountably many propositional variables?

• Being finitely definable implies there are only finitely many variables used in $S$. Thus, any valid truth assignment for $S$ will remain one if we flip the truth value of any of the remaining variables from true to false or vice versa. If this is not the case with a set of truth assignments (i.e. no finite subset of the variables has this property), it is not finitely definable.
• Then, for each variable in the assignment, we'll add an implication of the form "distinguishing sentence $\Rightarrow$ variable" (or $\neg$ variable, depending on what the assignment tells us). Thus, we've essentially added (infinite) implication saying that the distinguishing sentence determines the rest of the truth assignment completely.