bart_kosko's picture
Information Scientist and Professor of Electrical Engineering and Law, University of Southern California; Author, Noise, Fuzzy Thinking
Q. E. D. Moments

Everyone should know what proof feels like. It reduces all other species of belief to a distant second-class status. Proof is the far end on a cognitive scale of confidence that varies through levels of doubt. And most people never experience it. 

Feeling proof comes from finishing a proof. It does not come from pointing at a proof in a book or in the brain of an instructor. It comes when the prover himself takes the last logical step on the deductive staircase. Then he gets to celebrate that logical feat by declaring "Q. E. D." or "Quod Erat Demonstrandum" or just "Quite Easily Done." Q. E. D. states that he has proven or demonstrated the claim that he wanted to prove. The proof need not be original or surprising. It just needs to be logically correct to produce a Q. E. D. moment. A proof of the Pythagorean Theorem has always sufficed.

The only such proofs that warrant the name are those in mathematics and formal logic. Each logical step has to come with a logically sufficient justification. That way each logical step comes with binary certainty. Then the final result itself follows with binary certainty. It is as if the prover multiplied the number 1 by itself for each step in the proof. The result is still the number 1. That is why the final result warrants a declaration of Q. E. D. That is also why the process comes to an unequivocal halt if the prover cannot justify a step. Any act of faith or guesswork or cutting corners will destroy the proof and its demand for binary certainty.

The catch is that we can really only prove tautologies.

The great binary truths of mathematics are still logically equivalent to the tautology "1 = 1" or "green is green." This differs from the factual statements we make about the real world — statements such as "Pine needles are green" or "Chlorophyll molecules reflect green light."

These factual statements are approximations. They are technically vague or fuzzy. And they often come juxtaposed with probabilistic uncertainty: "Pine needles are green with high probability." Note that this last statement involves triple uncertainty. There is first the vagueness of green pine needles because there is no bright line between greenness and non-greenness. It is a matter of degree. There is second only a probability whether pine needles have the vague property of greenness. And there is last the magnitude of the probability itself. The magnitude is the vague or fuzzy descriptor "high" because here too there is no bright line between high probability and not-high probability.

No one has ever produced a statement of fact that has the same 100% binary truth status as a mathematical theorem. Even the most accurate energy predictions of quantum mechanics hold only out to a few decimal places. Binary truth would require getting it right out to infinitely many decimal places.

Most scientists know this and rightly sweat it. The logical premises of a math model only approximately match the world that the model purports to model. It is not at all clear how such grounding mismatches propagate through to the model's predictions. Each infected inferential step tends to degrade the confidence of the conclusion as if multiplying fractions less than one. Modern statistics can appeal to confidence bounds if there are enough samples and if the samples sufficiently approximate the binary assumptions of the model. That at least makes us pay in the coin of data for an increase in certainty.

It is a big step down from such imperfect scientific inference to the approximate syllogistic reasoning of the law. There the disputant insists that similar premises must lead to similar conclusions. But this similarity involves its own approximate pattern matching of inherently vague patterns of causal conduct or hidden mental states such as intent or foreseeability. The judge's final ruling of "granted" or "denied" resolves the issue in practice. But it is technically a non sequitir. The product of any numbers between zero and one is again always less than one. So the confidence of the conclusion can only fall as the steps in the deductive chain increase. The clang of the gavel is no substitute for proof.

Such approximate reasoning may be as close as we can come to a Q. E. D. moment when using natural language. The everyday arguments that buzz in our brains hit far humbler logical highs. That is precisely why we all need to prove something at least once — to experience at least one true Q. E. D. moment. Those rare but god-like tastes of ideal certainty can help keep us from mistaking it for anything else.