Labyrinths of Reason (6 page)

Incremental confirmation is common to ironic situations such as the spaceship annihilations. Each prospectors’ information increments a low chance of annihilation (4 percent) to a greater but still low chance (10 percent). Taken together, their information decreases the chance to zero. It is reassuring that such flukes vanish when the chances are higher, when a hypothesis is close to being confirmed in the absolute sense.

You can demonstrate this by playing with the odds a little. Recast the situation, giving the oddsmakers a better handle on the real
state of affairs. For each object, the odds are now 10 percent that it is a natural phenomenon, 80 percent that is a matter spaceship, and 10 percent that it is an antimatter spaceship. Then the oddsmakers must set the chance of annihilation at (80 percent of 10 percent) plus (10 percent of 80 percent)—or 16 percent. Each prospector, upon learning for certain that one of the objects is a matter spaceship, can figure (as before) that the chance of annihilation is 10 percent, the chance that the other object is an antimatter spaceship. Now each prospector’s estimate is
less
than the oddsmakers’. This is as it should be, since they know more than the oddsmakers, and the actual chance happens to be zero.

As this shows, confirmation is only half the story. Evidence may also refute or disconfirm a hypothesis. Philosophers of science, notably Sir Karl Popper, emphasize the role of refutation.

You might think that it’s just a matter of saying a glass is half full or half empty. Actually, there is an asymmetry between confirmation and refutation. It is much easier to refute a generalization than to prove it.

A counterexample is an exception to a putative rule. A white raven is a counterexample to the hypothesis that all ravens are black. A white raven does not merely make the hypothesis less likely. It proves the hypothesis wrong in one fell swoop. Logicians call this
modus tollens
, or “denying the consequent.”

Rarely is the situation so simple in practice. There have been many “counterexamples” to the hypothesis that there is no Loch Ness monster. Every alleged sighting is one. Yet most scientists continue to believe that there is no Loch Ness monster. It is evident that not all supposed counterexamples have enough weight to refute an otherwise confirmed hypothesis.

Most hypotheses on the edges of current knowledge can be tested only in situations where many “auxiliary” hypotheses are tested as well. Auxiliary hypotheses are background assumptions about how the main hypothesis fits into the general body of knowledge; how microscopes, telescopes, and other equipment necessary to test the hypothesis operate; and so on. These auxiliary hypotheses often rule out any quick use of
modus tollens
.

Wesley Salmon cited a neat case of two similar counterexamples leading to rejection of auxiliary and main hypotheses, respectively. Newton’s theory of gravity makes predictions about the future motions
of the planets. In the nineteenth century, these predictions for the orbit of Uranus were found to be slightly, but consistently, wrong.

Some astronomers wondered if the discrepancies might be due to an unknown planet beyond Uranus. Once this planet (Neptune) was discovered in 1846, Newton’s theory was not only removed from doubt but strengthened. Neptune was further evidence for Newton’s theory.

At about the same time, other irregularities were noted in Mercury’s orbit. Astronomers also tried to find a planet near Mercury that might account for the deviation. French amateur astronomer D. Lescarbault reported seeing a planet within Mercury’s orbit in 1859. The planet was accepted as real and named Vulcan by Urbain Jean Leverrier, co-discoverer of Neptune. Subsequent astronomers could not find the planet, though, and it was soon branded a mistake. Mercury continued to depart from its predicted orbit. The deviations were not haphazard but regular, and distinctly different from what Kepler’s laws (founded on Newton’s gravity) predicted.

In this case, the discrepancies were ultimately accepted as evidence that Newton’s theory of gravity is wrong. Mercury’s wobbling orbit was one of the earliest confirmations of Einstein’s general relativity.

The history of Neptune and Vulcan demonstrates two features of counterexamples. First, a counterexample may refute an auxiliary hypothesis rather than the main one. It is important to find out which is at fault. There is usually such ample room for speculation that instant refutations are rare. Second, when a theory is thrown out, it is in favor of a broader theory that makes many of the same predictions as the original. Under typical conditions in the solar system, Einstein’s general relativity predicts gravitational effects all but identical to those of Newton’s simpler theory. The difference turns up only in very intense gravitational fields. Of the planets, Mercury, being closest to the sun, is most subject to these relativistic effects. It alone seems to be out of step with Newton’s laws.

Not only should a new theory account for the successful predictions of the theory it would replace. It should offer new, different predictions of its own. In Karl Popper’s terms, the new theory must
have greater “empirical content.” It must make more testable predictions in more realms of experience than the old theory.

A new theory should be
more
open to possible refutation, not less. If there is one thing that is a dead giveaway for a crank theory, it is that the theory has been modified to restrict its own refutation. An honest hypothesis is open to being disproven. It’s one thing to say, there’s a ghost that appears in the old Miller mansion at the crack of midnight whenever there’s a full moon. That kind of hypothesis is worthy of attention provided there is any reasonable evidence to support it: say, testimony of a few reliable eyewitnesses. Far more typical are ghost stories that restrict refutation: A ghost appears, but never when skeptics are around.

These restrictions usually indicate that a hypothesis has failed the first stages of the confirmation process and is being kept alive by those who wish to believe it regardless of its truth. No one started out believing that

All these provisos were tacked on after confirmation failed to materialize. That doesn’t automatically mean that the modified hypotheses are false, but it is hardly encouraging. If the process of modifying to restrict refutation continues long enough, the ultimate result is the type of hypothesis that Popper sardonically calls “irrefutable.” This may sound good, but think about what it means. It is a hypothesis that cannot possibly be proven false—one so wishy-washy that no possible observation is incompatible with it. That kind of hypothesis doesn’t really say anything.

The proposition that “ESP exists, but it is so iffy that even the best psychics may do no better than chance in controlled experiments”—which is essentially what some ESP apologists have said—is beyond refutation. You might ask, “How would the world be any different if ESP
didn’t
exist?”

Why can’t scientists give poorly supported hypotheses the benefit
of the doubt? The main reason is that many, many hypotheses can be devised to account for any fixed body of data. If we say, “Okay, ESP exists, because no experiment has ruled it out” (which is true), we would have to allow a multitude of equally unrefuted hypotheses. In the end it is a desire for simplicity that leads scientists to accept only those hypotheses that can be confirmed. Indeed, says Popper, the aim of science should be to try to eliminate as many hypotheses as possible with new data.

The basics of confirmation in place, let’s return to Hempel’s paradox with this added perspective. The first thing that concerns most people hearing the paradox for the first time is this business about the contrapositive. “Nonblack things” and “nonravens” are awkward constructions. Is “All nonblack things are nonravens” really equivalent to “All ravens are black”? If it’s
not
, there is no paradox.

Here is a good way to see that they
are
logically equivalent. Forget about our human and imperfect attempts at knowledge. Pretend that we have at our service a genie who can ascertain any and all
specific
facts instantly. In other words, the genie can determine any of Hume’s “matters of fact”—the direct, sensory results of any observation, without any interpretation, interpolation, or editorial comment.

Also like Hume, the genie claims that it doesn’t quite understand generalizations. So if you want to know whether a statement such as “All ravens are black” is true, you have to explain it to the genie as an aggregate of individual observations. You have to tell the genie exactly what he should do to determine if Hempel’s hypothesis is right or wrong.

It may come as a surprise that observations of black ravens are virtually irrelevant to the
ultimate
truth or falsity of “All ravens are black.” This flatly contradicts the foregoing discussion, but remember we are now talking about the genie and not humans. The genie is going to determine the final, cosmic truth of the statement, not merely find evidence to support it. Observations of black ravens can neither prove nor disprove the statement.

Suppose the genie found a black raven. Would that prove that all ravens are black? Of course not. Suppose the genie found a million black ravens. Would that prove it? No; there could still be ravens of other colors. The statement “All swans are white” was supported
by all available evidence until the discovery of Australia. There are black swans in Australia.

Suppose that the universe is infinite and there is an infinity of other planets so similar to Earth that they have black ravens on them and that the genie thereby finds an infinite number of black ravens. Would
that
prove it? No; for the same reason. At this point the genie would rightly get impatient with us, for evidently no amount of black ravens will settle anything. Looking for black ravens is a wild-goose chase.

Think about it, and you will realize that the crux of the matter is nonblack ravens. The only way Hempel’s statement can be
wrong
is for there to be a raven somewhere that isn’t black. The only way the statement can be
right
is for there to be no such raven. To decide ultimate truth or falsity, the genie must search for nonblack ravens. If he finds even one, the statement is irretrievably false. If he searches the entire universe—everywhere a nonblack raven could possibly be—and finds none, then Hempel’s statement is unimpeachably true.

In a pragmatic sense, “All ravens are black” only
seems
to be talking about black ravens. When you translate it into an operational definition for the genie, it really says: “There is no such thing as a nonblack raven.”

Now let’s have the genie test the contrapositive statement, “All nonblack things are nonravens.” This is another pie-in-the-sky generalization incomprehensible to the genie. We explain: “The only way ‘All nonblack things are nonravens’ can be wrong is for there to be at least one nonblack raven. The only way it can be right is by the complete absence of nonblack ravens everywhere.”

This is just how we explained the original statement. What you must do to prove or refute “All ravens are black” is identical to what you must do to prove or refute “All nonblack things are nonravens.” That is strong grounds for asserting that the two statements are equivalent.

You might object that there is one slight difference. Does not the truth of “All ravens are black” imply that there is at least one black raven?

Take the hypothesis “All centaurs are green.” The genie, looking for nongreen centaurs, would find none and report the statement true. Of course, there are no centaurs of any description. It sounds funny to say the statement is true, then.

This point is again one of semantics. Logicians
do
allow that statements such as “All centaurs are green” and “If X is a centaur,
X is green” are true. For various reasons it is most convenient to do so. Hence, to a logician, there is no distinction whatsoever between a statement and its contrapositive.

You are free to dissent and insist that there must be at least one green centaur for the statement to be true. Doing so creates this slight asymmetry between Hempel’s original hypothesis and the contrapositive: With the original, you have to tell the genie to make sure there is at least one black raven before reporting the statement to be true. With the contrapositive, the genie must find at least one nonblack nonraven (like a red herring). I do not think that this significantly alters the essential equivalence of the statements. Finding the obligatory black raven or red herring is but a formality; the genie’s real task in either case is making sure there are no nonblack ravens.

A “negative hypothesis” is one that claims that something doesn’t exist. It is extremely difficult to prove negative hypotheses. (“Never say never.”) It is one thing for a genie to check every place a nonblack raven could be and thus prove that there is no such thing. It is something else for us humans.

You set off on a raven-hunting expedition, see lots of black ravens, and don’t find any nonblack ravens. At length you start to get sick of the whole business. All your friends say you’ll
never
find a nonblack raven. When is it okay to call it quits and stop looking?

As a practical matter, you do quit sooner or later. Thereafter you feel pretty confident that there are no nonblack ravens. This does not begin to prove that all ravens are black in a logically rigorous sense, though. To do that, you would indeed have to check everywhere in the universe that a raven might be. That is obviously an unreasonable requirement.