Yet, first order logic is ‘Generalized Aristotelian Logic,’ but logicians certainly don’t refer to it that way. Should a person’s name be included in a theory, the theory is automatically weakened due to apparent arbitrariness.

Should you truly believe that Bayesian Inference is the one and only way to do logic under uncertainty, call it ‘The Theory of Inference’ or ‘Inference Theory,’ or take a tip from logicians, and call it just plain ‘Inference’. Let other theories of inference compete, in the axiomatic sense, and see if it stands the test of time.

Like all meaningful debates in binary logic, the debates of rational inference will all come down to the interpretation of the conditional, which it so happens is not the extension of material implication. Recall that one way to interpret Godel’s second incompleteness theorem is that material implication does not equal deductive implication. Were they to be equal, then an unprovable proposition would vacuously be provable. Projecting this form of argument to the probabilistic conditional would certainly be the way I would go about assessing an axiomatization of inference.

-Doug