I am using logics (propositional, predicate, modal) to model one domain, but there are variables that have non-boolean domains, these variables are degrees (it is sufficient that they are degrees, there is no need to have full arithmetic. But they are not categories, because the sequence of degrees can be important). If variable v can have n different values of degrees then v can be modelled with n propositional variables and relevant additional constraints (e.g. that only one variable v1, ..., vn can be true): v=1 => p1=true, v=2 => p2=true, etc.
But maybe there is some syntactic sugar of maybe specialised logic available for handling such cases.
In general - is it possible in logic to model (encode) principle "quantitative changes lead to qualitative changes"?
Łukasiewicz logic and other fuzzy logics are relevant, I think. In the standard semantics for these logics, the truth-value are real numbers. Łukasiewicz logic can be used to formalise Ulam's game, which may also be of interest.