In the binary case the usual treatment in statistics is to use the logistic function (and logit) to work with real numbers, then transform back into probability

In the binary case the usual treatment in statistics is to use the logistic function (and logit) to work with real numbers, then transform back into probability space as the last step. This is a little flakey for ordinal numbers, and the usual treatment is to use a learning algorithm to find a mapping from real numbers to ordinal values, either explicitly (if you need a "score") or implicitly. Support vector machines, radial basis functions and neural networks are typically used.