October 16th, 2010

“The true logic for this world is the calculus of probabilities.” So sayeth James Clerk Maxwell, and a mighty truth it is. There are no absolute truths in the real world, merely likelihoods. We cannot say of a certainty that the earth is round, only that the probability that it is round is 99.9999....%. After all, how do we know that we are not the butt of some cosmic practical joke perpetrated by an immensely powerful alien civilization that has rigged our environment to make us think that our world is round when it is really flat? Every statement we make about our world must be couched in probabilistic terms. This applies at both ends of the logical spectrum. Just as it is highly likely (but not absolutely certain) that the earth is round, it is also highly unlikely (but not absolutely certain) that it is flat. In regarding the world, we must always heed the probabilities.

Yet such thinking can be unnecessarily tedious. Sure, the planet might be flat, but situations in which this possibility is meaningful are vanishingly rare. So we have come up with a logical approximation that offers great utility: the boolean simplification. We reduce much of the world to simple black-and-white, yes-or-no statements. The earth is round -- period. We don’t worry about the ridiculously tiny probability that it is flat -- we sweep that microscopic detail under the rug and embrace the simpler boolean statement.

The boolean simplification is really only the most extreme of simplifications. It is illuminating to think of this matter in terms of the number of bits we use to approximate the truth. The boolean approximation is one bit deep: yes or no, 1 or 0. We could go one bit deeper and express some truths in two bits, allowing us four degrees of truth: 0, 1, 2, and 3. This allows us to avoid the difficulty of statements such as “All humans are men and none are women” by instead saying that “Half of humans are men and half are women”. We can add more bits to get more resolution, like so:

3 bits: I weigh one-eighth of a ton (pretty far off the mark)

4 bits: I weigh one-sixteenth of a ton (much closer, but still off)

5 bits: I weigh 3/32’s of a ton (still bad)

6 bits: I weigh 5/64’s of a ton (getting closer)

7 bits: I weigh 9/128’s of a ton (closer)

8 bits: I weigh17/256’s of a ton (now we’re getting really close)

and so on. More bits of resolution give us more accuracy in our assessment of the truth of the world. At the extreme, we’d use an enormous number of bits of resolution, to arrive at statements like this:

I weigh 133.451769323 pounds

But in practice, we seldom need so much accuracy, so we can get away with using a limited number of bits. Indeed, some situations require more bits and some situations require fewer bits -- and in some situations, a single bit (yes or no) is entirely sufficient to our needs. Do you want ketchup with your hamburger -- yes or no? We seldom scruple to worry about whether we want a few molecules of ketchup or a few gallons: we just decide “yes” or “no”.

Thus, the boolean simplification is handy and practical. However, it is only an approximation, and many people lose sight of this fact in ways that can lead them into error. The most common form of this is the wild leap from multi-bit truth to boolean falsehood. I was inspired to write this essay by reading a discussion of the old “nature versus nurture” argument. A lot of people get hot under the collar about the notion that our genetic makeup might affect our behavior. Their mistake is most easily expressed in their phrase “genetic determinism”, a patently boolean concept: our behavior must be either controlled by our genes or independent of our genes. The concept is absurdly boolean: it should be obvious to any reasonable person that genes do not control every aspect of our behavior, and it should be equally obvious that our genetic makeup does to some degree influence our behavior. It is not some cultural accident that men just happen to be more promiscuous than women: that difference is attributable to obvious genetic factors. The role of genetic factors in human behavior is not a yes-or-no, black-or-white phenomenon: it is a matter of degree. We can argue about the range and intensity of genetic influences on human behavior, but to take a black-and-white approach and deny such influences is absurdly boolean.