If Socrates is probably a man, he’s probably mortal.
How do you extend classical logic to reason with uncertain propositions, such as the statement above? Suppose we agree to represent degrees of plausibility with real numbers, larger numbers indicating greater plausibility. If we also agree to a few axioms to quantify what we mean by consistency and common sense, there is a unique system that satisfies the axioms. The derivation is tedious and not well suited to a blog posting, so I’ll cut to the chase: given certain axioms, the inevitable system for plausible reasoning is probability theory.
There are two important implications of this result. First, it is possible to develop probability theory with no reference to sets. This renders much of the controversy about the interpretation of probability moot. Instead of arguing about what a probability can and cannot represent, one could concede the point. “We won’t use probabilities to represent uncertain information. We’ll use ‘plausibilities’ instead, derived from rules of common sense reasoning. And by the way, the resulting theory is identical to probability theory.”
The other important implication is that all other systems of plausible reasoning — fuzzy logic, neural networks, artificial intelligence, etc. — must either lead to the same conclusions as probability theory, or violate one of the axioms used to derive probability theory.
See the first two chapters of Probability Theory by E. T. Jaynes (ISBN 0521592712) for a full development. It’s interesting to note that the seminal paper in this area came out over 60 years ago. (Richard Cox, “Probability, frequency, and reasonable expectation”, 1946.)