Labyrinths of Reason (5 page)

Paradox is thus a much deeper and more universal concept than the ancients would have dreamed. Rather than an oddity, it is a mainstay of the philosophy of science. Paradoxes are both appealing and haunting. There is a subversive joy in seeing logic tumble like a house of cards. All the well-known paradoxes of confirmation theory and epistemology were conceived more or less in the spirit of intellectual play. In few other fields is it possible for the interested nonexpert to sample so much of the true flavor of the field and have fun doing it. How we know—the interplay of induction and deduction, of ambiguity and certainty—is the theme of the paradoxes to follow.

T
HE BEST-KNOWN modern paradox of confirmation was proposed by German-born American philosopher Carl G. Hempel in 1946. Hempel’s “paradox of the ravens” deals with induction, the drawing of generalizations. It is a mischievous reaction to those who think that science may be resolved into a cookbook scientific method.

Hempel imagined a birdwatcher trying to test the hypothesis “All ravens are black.”
¹

The conventional way of testing that theory is to
seek out ravens and check their color. Every black raven found confirms (provides evidence for) the hypothesis. On the other hand, a single raven of any color other than black disproves the hypothesis on the spot. Find even one red raven and you need look no further: The hypothesis is wrong.

All are agreed on the above. Hempel’s paradox begins with the claim that the hypothesis may be restated as “All nonblack things are nonravens.” Logic tells us that this is entirely equivalent to the original hypothesis. If all ravens are black, then certainly anything which isn’t black can’t be a raven. This rewording is known as a
contrapositive
, and the contrapositive of any statement is identical in meaning.

“All nonblack things are nonravens” is a lot easier to test. Every time you see something that
isn’t
black, and it turns out
not
to be a raven, this restated hypothesis is confirmed. Instead of looking for ravens on damp, inaccessible moors, you need only look for nonblack things that aren’t ravens.

A blue jay is sighted. It’s nonblack and it’s not a raven. That confirms the contrapositive version of the hypothesis. So does a pink flamingo, a purple martin, and a green peacock. Of course, a nonblack thing doesn’t even have to be a bird. A red herring, a gold ring, a blue lawn elf, and the white paper of this page also confirm the hypothesis. The birdwatcher does not have to stir from his easy chair to gather evidence that all ravens are black. Wherever you are right now, your visual field is filled with things that confirm “All ravens are black.”

Now clearly this is ridiculous. There is yet a further absurdity. To see it, suppose that you want to dismiss the paradox by saying, all right, evidently the blue jay or the red herring
does
confirm “All ravens are black” to some infinitesimal degree. If you could summon up a magic genie, capable of examining all the nonblack things in the world in the blink of an eye, and if that genie found that not one of those nonblack things was a raven, that assuredly would prove that there are no nonblack ravens—that all ravens are black. Maybe it is not so incredible that a red herring could confirm the hypothesis.

Don’t get too cozy with this resolution. It is easy to see that that
same red herring
also
confirms “All ravens are white.” The contrapositive of the latter is “All nonwhite things are nonravens,” and the herring, being a nonwhite thing, confirms it. An observation cannot confirm two mutually exclusive hypotheses. Once you admit such a patent contradiction, it is possible to “prove” anything. The red herring confirms that the color of all ravens is black, and also that that color is white; ergo:

Black is white. QED.

Reasonable assumptions have led to resounding contradiction.

To scientists, Hempel’s paradox is more than a puzzle. Any hypothesis has a contrapositive, and confirming instances of the contrapositive are often very easy to find. Something is certainly wrong. But what?

Hempel’s raven is a good introduction to the perils and puzzles of confirmation. Of all the major paradoxes to be discussed, it is among the most nearly resolved. It will be worthwhile to back up a bit before coming to the resolution, though, to discuss the background of the paradox.

To put it in as few words as possible, confirmation is the search for truth. It is the mainspring of science, and more than that, it is something we do every day of our lives.

Analyzing confirmation is almost like analyzing sneezing: We know what it is, but it is usually so automatic that it is hard to say exactly how it is done. The paradoxes of confirmation probably owe a lot to this shared set of subconscious expectations. These expectations can lead us astray.

As you remember from high school, there is a “scientific method” that goes roughly as follows. You form a hypothesis—a guess about how the world works. Then you try to test it through observation or experiment. The evidence you gather either confirms the hypothesis or refutes it. Like much of what you learn in high school, this is true while leaving important things unsaid.

Most useful hypotheses are generalizations. Hempel’s paradox plays off a bit of common sense called “Nicod’s criterion” after philosopher Jean Nicod. To put it in terms of black ravens, this says that (a) sighting a black raven makes the generalization “All ravens are black” more likely; (b) sighting a nonblack raven disproves the statement; and (c) observations of black nonravens and nonblack
nonravens are irrelevant. A black bowling ball or a blue lawn elf cannot tell us anything about the color of ravens. Nicod’s criterion is behind all scientific inquiry, and if something is wrong with it, we are in serious trouble indeed.

The sighting of a black raven furnishes evidence in favor of the hypothesis that all ravens are black, but of course does not
prove
that the hypothesis is right. No single observation can do that. Sightings of black ravens, in the absence of ravens of any other color, increase your confidence (reasonably enough) that all ravens are black.

Confirmation is trickier than it appears. You might think that the more confirming evidence for a hypothesis, the more likely it is to be true. Not necessarily. It is possible for two confirming observations to prove a hypothesis
false
. That is what happens in the following thought experiment, inspired by philosopher Wesley Salmon.

Suppose that some of the planets in the universe are made of matter and some of antimatter (as has been speculated). Matter and antimatter look exactly alike. There is no way of telling, by examining a distant star in a telescope, whether it is made of matter or antimatter. Even the star’s light gives nothing away, for the photon is its own antiparticle, and an antimatter star shines with the same kind of light as a regular star. The only thing is, when antimatter touches regular matter—BOOM!!! Both are annihilated in a tremendous explosion.

This unfortunate fact makes interstellar contact hazardous. A spaceship from planet X chances upon a spaceship from planet Y. They radio messages to each other (radio waves are made of photons too and are neither matter nor antimatter). Computers on board the ships decipher the alien languages and establish diplomatic relations. The two spaceships agree to dock and exchange goodwill ambassadors. Everything is fine until the last moment. Then the rockets make contact and BOOM!!!—or not, depending on the composition of planets X and Y. Whenever one is matter and the other antimatter, both ships are blown to smithereens. (There’s no explosion if
both
spaceships are antimatter.)

One day, astronomers here on Earth report that they’ve sighted two tiny points of light that may be spaceships approaching each other. They’re not sure the objects are spaceships, but on the basis of past experience the astronomers can say that for each point of
light there is a 30 percent chance it is a spaceship and a 70 percent chance it is an irrelevant natural phenomenon. It is also known from past experience that any pair of spaceships that approach closely always do dock. All the other species in the galaxy seem oblivious to the matter/antimatter problem, and have to learn the hard way.

So the big question is: Will they blow up or not? Oddsmakers in Las Vegas start accepting ghoulish bets on whether there will be an annihilation. The oddsmakers reason like this: It is known that two-thirds of the planets in the universe are made of matter and one-third are antimatter. Thus for each point of light there is a 70 percent chance it is a natural phenomenon of no interest here; a 20 percent chance it is a spaceship made of matter; and a 10 percent chance that it is an antimatter spaceship.

Call the two points of light A and B. An annihilation can occur in one of two mutually exclusive ways. Either object A is a matter spaceship and B is an antimatter spaceship, or A is an antimatter spaceship and B is a matter spaceship. The chance of the first case is 20 percent of 10 percent, or 2 percent. The odds of the second case is 10 percent of 20 percent; again, 2 percent. Since these two possibilities are mutually exclusive, the total chance of an annihilation is 2 percent plus 2 percent: 4 percent.

The oddsmakers set the payoffs to bettors based on this calculated probability. Now suppose that a space prospector, returning home to Earth, grazes object A in a trillion-to-one freak accident. The prospector learns that object A
is
a spaceship and is made of ordinary matter (this from the fact that there was no explosion). Arriving on Earth, the prospector finds out about the possible annihilation and the Las Vegas book on it.

The prospector would do well to exploit his “inside information” and bet on annihilation. He knows for a fact that object A is a spaceship, whereas everyone else thinks it is probably (70 percent chance) just an asteroid or some other natural body. Given that A is a spaceship of ordinary matter, there is a 10 percent chance of annihilation, since that is the chance that object B is a spaceship and made of antimatter. The oddsmakers have set the chance at 4 percent, but the prospector, with his more complete knowledge, can set the chance at 10 percent.

Fine. Now what if another space prospector had an identical accident with object B, and also determined that it is a spaceship made of matter? This second prospector could of course use the same reasoning to arrive at the same conclusion: that the chance of
annihilation has been boosted from 4 percent to 10 percent.
But
the combined information of the two prospectors actually rules out an annihilation entirely. They have determined that the spaceships are both made of the same kind of matter as Earth, and that means that the chance of annihilation is a big fat zero!

Two items of evidence (the prospectors’ collisions with the spaceships) each confirm the hypothesis that there will be an annihilation, even though the observations together refute it. I prefer to call this an irony rather than a paradox, for there is no doubt that such strange turns of affairs can exist. The probability calculations of the oddsmakers, of the prospectors, and of us, aware of both prospectors’ experiences, are sound. These peculiar situations have been studied intensively by confirmation theorists.

The peculiarity is partly semantic. The verb “confirm” is used in two ways. In everyday speech, we almost always use “confirm” in the
absolute
sense, to mean that something is clinched; established beyond reasonable doubt. “The boss confirmed that Sandra got the raise” means that, whatever the doubts beforehand, it is now just about 100 percent certain that Sandra got the raise.

Hardly any experiment provides absolute confirmation of a hypothesis. Scientists and confirmation theorists often use “confirm” in the
incremental
sense. To confirm incrementally is to “provide evidence for” or “increase the probability of.” We speak of probabilities because confirmation of a generalization is always tentative.

You can incrementally confirm a hypothesis that was, and still is, unlikely to be true. We would not say “The boss confirmed that Sandra got the raise” to mean that some equivocal comment of the boss’s has upped the chance of Sandra’s getting the raise from 15 percent to 18 percent. But that type of confirmation is typical of scientific research.