In inductive logic/probability theory a set of betting rates is called coherent if it's not open to a sure-loss contract (called a Dutch Book).
Interestingly you can prove that:
(1) A set of betting rates is coherent iff it satisfies the rules of probability.
If I understand your question correctly, you are asking if it's possible to prove a similar sort of result for rules of logic: So give somehow an analogous definition of coherence for a set of assignments of truth values to some set of sentences, and then prove that such an assignment is coherent iff it satisfies (in some sense) the rules of logic.
"Rules of logic" here should refer to an axiomatization of logic, since in the case of probability it's similarly the axioms that betting rates have to satisfy to avoid a Dutch book. The obvious idea would be to define coherence as follows:
Df. A set of truth assignments to some sentences is coherent iff there is a truth valuation that gives the same truth-values to the sentences as your assignments do.
(The above definition means that coherence guarantees that your truth assignments are not open to a sure error, i.e. you could be right in your assignments of truth values.)
Define a set S of sentences as follow: if your assignment assigns 1 to a formula F, then F∈S, if your assignment assigns 0 to a formula F then not-F∈S. Let's call this set S the set determined by the truth assignments.
It's now easy to show that a set of truth assignments to some sentences is coherent iff the set S determined by those truth assignments is satisfiable.
Now since the soundness and completeness theorems tell us that a set S is satisfiable iff S is consistent (relative to the rules/axiomatization of logic), the analogue of theorem (1) is as follows:
(1)* A set of truth assignments is coherent iff the set determined by the truth assignments is consistent (relative to an axiomatization of logic)
You can think of this as a version of the soundness and completeness theorems.