How do we know what we know? What is the difference between logical proof and proof beyond reasonable doubt? What is wishful thinking?

Newtonian mechanics are used routinely to predict the orbit of planets and to pilot spacecraft with incredible accuracy. Can we say that Newton was proved correct because he predicted where certain planets would be and when we looked, they were where he said they would be? If Newton’s equations help navigate a Mars probe across space to enter orbit around Mars, doesn’t that validate the theory?

Strangely enough, the accuracy of a prediction does not logically validate the premise upon which it is based. In the case of predicting the positions of planets, the Ptolemaic system was more accurate than one proposed by Kepler even though Ptolemy assumed the Earth is stationary and the sun goes around it. The accuracy of prediction did not validate Ptolemy.

Assuming that a result proves the premise is known as the fallacy of affirmation of the consequent. Proponents of pseudo-sciences such as astrology have a lot of fun with this fallacy. For instance, assume that someone tells you people born under the sign of Leo have aggressive personalities and the next five people you meet who are aggressive all turn out to be Leos. Does that justify a belief in astrology? On a somewhat more sophisticated level, creationists often point out that nothing in science can be proven logically. Therefore the conjecture of Intelligent Design (I can’t bring myself to label it a theory) has just as much a claim to be true as established evolution.

The mirror image of affirmation of the consequent is the fallacy of denial of the antecedent. In this fallacy, one claims that a conclusion is false because the premise is shown to be false. In some sense, this fallacy is even more seductive than affirming the consequent. For instance, ancient people thought that breathing bad air from swamps caused malaria. The logic is “Swamps produce bad air which causes malaria. You hung around a swamp breathing bad air. You have malaria. Therefore the fact that swamps produce bad air is proven.” After we learned that mosquitoes are the vector that carries malaria, we still know that hanging around swamps can be dangerous, but the premise of bad air being the cause is invalidated. The fact that the premise was wrong did not invalidate the conclusion.

Given the nature of these two common logical fallacies, how can we ever learn anything? Books are written about this, and the subject is much too involved to completely discuss here, but to borrow from Ross Perot, “The devil is in the details.” Newtonian mechanics is supported by every observation made over the range of physical parameters that Newton could observe. It predicts new phenomena (like the location of previously undiscovered planets). It is expressed in simple mathematics. Compare this to any predictions of astrology. The first five Leos you meet might all share characteristics. Do all Leos behave that way? Do non-Leos behave this way? If the astrologer is somewhat more sophisticated and defines a “Leo-ness” parameter that predicts logical variations, do matters improve? Are the tenets of astrology reducible to simple mathematical expressions or are the predictions continuously modified by ad hoc rules?

The process of finding scientific laws is at best a messy process, which relies on induction. An observer gathers data and attempts to find meaningful patterns in it. Based on some hypothesized pattern, the observer makes predictions about future observations (or, in the case of astronomy and geology, post-dictions, but that is another story). By a repeated process of investigation and confirmation, the confidence attributed to the hypothesized pattern can by increased or shattered. But there is always that doubt since scientific induction is not the same as logical proof.

The lack of absolute logical proof should not give the pseudo-scientists any comfort. Their claims to equal footing on the pedestal of truth are simply not justified.

In response to the interest my original tutorial generated, I have completely rewritten and expanded it. Check out the tutorial availability through Lockergnome. The new version is over 100 pages long with chapters that alternate between discussion of the theoretical aspects and puzzles just for the fun of it. Puzzle lovers will be glad to know that I included an answers section that includes discussions as to why the answer is correct and how it was obtained. Most of the material has appeared in these columns, but some is new. Most of the discussions are expanded compared to what they were in the original column format.

Tags: fallacy, fallacies, newton, ptolemy, affirmation of the consequent, denial of the antecedent, decision theory