Is there any alternative study of "probability" where random variables are defined as maps between a sample space and a partially ordered set, rather than a sample space and the real interval $[0,1]$? If it exists, what is this theory called and what are some good references?
In such a setting, one could talk about one event being more likely than another, but perhaps not about the absolute probabilty of an event occurring.
I googled around, but what I found were different concepts involving partially ordered sets and probability (e.g., defining stochastic processes on partially ordered sets to generalize the concept of "time"), but not what I'm talking about.
The term you are looking for is Comparative Probability. I am only familiar with this topic through a brief mention in Appendix A of Probability Theory: The Logic of Science by E.T. Jaynes so cannot provide any information on the details or utility of the subject.