HOW CAN WE TELL?

The last example shows that we cannot always tell the difference; the tangle of effects that become new causes, and of the many causes that have the same effect is unknown and difficult to penetrate. The correlation is there, but not easy to interpret.

So the ultimate criterion is a good understanding of the mechanism that leads from cause to effect. "Good understanding" can be tested by predicting correctly and by finding quantitative laws which are accurately obeyed by the mechanism.

For example, the process by which literacy (or for that matter, a contagious disease) spreads through a population is well understood.

The rate of increase is proportional to two factors that work against each other. It is proportional to the fraction of persons who are already literate, because they teach others to read and write (or transmit the disease to them); but it is also proportional to the fraction of the remaining illiterates (or uninfected) -- the rate of increase declines as there are fewer illiterates left to be taught (or healthy people to be infected).

Rates of increase can be related in differential equations, whose solution blooms out into the full relation between the variables such as literacy vs. time. In this case it is a "logistic curve," which plots as a straight line on semilogarithmic paper when the ratio of literates to illiterates is plotted against time. (Semilogarithmic paper has an x scale with increments corresponding to equal differences, but the increments on the y scale correspond to equal multiples.) When this is tested against the historical record, the quantitative agreement is very good, giving reasonable assurance that the process of learning (or contagion) of the many from the few is the cause-effect mechanism, and that the agreement is not merely a correlation without causality.