How to model degree variable in logic (new type of modal logic?)?
What semantics of the logical operators would you like to have? Two of the best studied semantics for degrees are (propositional) Łukasiewicz logic and Gödel-Dummet logic. The latter enforces the domain to be a linear order which would allow you to model them via the naturals. It is also axiomatizable, since it is equivalent to intuitionistic propositional logic with the axiom of linearity (A =&gt; B) \/ (B =&gt; A). 

It also has a hyper-sequent calculus which is sound and complete (have a look at e.g. A. Avron "A Simple Proof of Completeness and Cut-elimination for Propositional Gödel Logic"), which you could implement in Prolog.