Combinatorial Semantics: Semantics for Frequent Validity

In ordinary first–order logic, a valid inference in a language L is one in which the conclusion is true in every model of the language in which the premises are true. To accommodate inductive/uncertain/probabilistic/nonmonotonic inference, we weaken that demand to the demand that the conclusion be true in a large proportion of the models in which the relevant premises are true. More generally, we say that an inference is [p,q] valid if its conclusion is true in a proportion lying between p and q of those models in which the relevant premises are true. If we include a statistical variable binding operator “%” in our language, there are many quite general (and useful) things we can say about uncertain validity. A surprising result is that some of these things may conflict with Bayesian conditionalization.