Labyrinths of Reason (7 page)

Philosophers have a word for processes requiring an infinity of action:
supertasks
. Some philosophers think that when ascertaining something requires an infinity of actions, it cannot be known at all. Michael Dummett gave this example: “A city will never be built at the North Pole.” To test this, you might hop in a time machine, set it for a given year, and travel to that year to see if a city exists at the North Pole. If not, you set the time machine for a different year, and try again. You could know whether a city will exist at the North Pole at any point in time, but knowing whether it will
ever
be
built is something else again. Knowing that requires knowing an infinity of facts; doing an infinite amount of research.

If the universe is infinite, then “There are no nonblack ravens” is another proposition requiring an infinity of observations. Our genie is capable of empirical supertasks, but we are not. This is really why we confirm from sightings of black ravens and not from failures to sight nonblack ravens. The number of black ravens seen is a way of “keeping score” while actually looking for a counterexample. The more black ravens we have seen, without seeing any nonblack ones, the more confident we feel that there are no nonblack ravens. Nicod’s criterion says that black ravens are a better way of keeping score in the progress of confirmation than are nonblack nonravens. To resolve Hempel’s paradox, we must decide why this is so.

Try a different tack. Categories like “nonravens” and “nonblack things” are unnatural. Most of the time you are first aware that a “thing” is a raven or a herring or a steak knife. You don’t naturally experience objects as “nonravens” or “nonherrings” or “non-steak knives.” Only Hempel’s original formulation (“All ravens are black”) dovetails with how people really think.

Your train of thought
is
quite different with the two versions of the hypothesis. When you see a raven, your thoughts normally run like this:

• (a) Look, there’s a raven.
  

• (b) And it’s black.
  

• (c) So it confirms the statement “All ravens are black.”
  

Connecting a red herring to Hempel’s hypothesis requires a more roundabout stream of consciousness!

• (a) There’s a herring.
  

• (b) It’s red.
  

• (c) Oh, wait, how does that raven paradox go? Yeah, it’s a “nonblack thing” …
  

• (d) … and it’s not a raven.
  

• (e) So it confirms the statement “All nonblack things are nonravens” …
  

• (f) … which is the same as “All ravens are black.”
  

Between steps (a) and (b) in the original formulation—the instant after you realize that the object is a raven, but before you think
about its color—the hypothesis is at risk. In that split second, the raven could be some other color and disprove the statement. The statement “All nonblack things are nonravens” is never really at risk in the second formulation. By the time you reach (c), you have already realized that the object is red (you deduced it was nonblack from the knowledge that it is red) and that it is a herring (you probably knew that all along).

Why is “raven” a reasonable category and “nonraven” not one? Well, ravens share many attributes in common, whereas “nonraven” is just a catchall term for anything that doesn’t qualify. One category is figure and the other is ground. It’s like the joke about a sculptor chiseling away everything that doesn’t look like his subject. Sculptors don’t think that way, and neither do scientists.

There is also a staggering numerical imbalance between the categories. Let’s have one more go at the original idea: that the paradox has something to do with the relative numbers of ravens and nonblack things.

Hempel’s reasoning need
not
lead to a paradox when the number of objects under investigation is clearly finite. Suppose that all that existed in the universe was seven sealed boxes. Unknown to you, five of the boxes contain black ravens; one contains a white raven; and one contains a green crab apple. Then you could reasonably feel that opening a box and finding the crab apple confirms “All ravens are black.” In fact, the speediest way to prove or refute the hypothesis would be to inspect all the nonblack things. There are only two nonblack things vs. six ravens. Of course, this model is artificial. It assumes prior knowledge of the number of things being investigated. You hardly ever know that, at least not at the start of the investigation.

More typical is the case where the original and not the contrapositive hypothesis talks of a knowably finite class of objects. The time, effort, and money required to establish “All ravens are black” is tied to the number of ravens (or the number of nonblack things). According to R. Todd Engstrom of Cornell’s Laboratory of Ornithology, the world population of common ravens is something like half a million. More troublesome is the number of nonblack things. It is astronomical.

One day it is discovered that there is a monster in Loch Ness. There’s just one monster; sonar equipment has established that
there are no more of its kind. You want to test the hypothesis “All Loch Ness monsters are green.” You approach the monster in a submarine, switch on the searchlights, and look out the porthole. The monster is green. Since there are no more Loch Ness monsters, the statement “All Loch Ness monsters are green” is thereby proven.

Here a single test of a hypothesis holds a lot of weight. There is only one chance for a nongreen monster to disprove the hypothesis. Taking the contrapositive seems even more ridiculous here than with the ravens. The contrapositive is “All nongreen things are non-Loch Ness monsters.” Imagine going around and assigning a number to every nongreen thing in the world. Nongreen thing #42,990,276 is a blue lawn elf. Is it a non-Loch Ness monster? Yes! It supports the hypothesis….

This is a woefully roundabout approach. Still supposing that there is just one Loch Ness monster and thus one potential counterexample, the chance that that arbitrary nongreen thing #42,990,276 is going to disprove the hypothesis is no greater than
1/N
, where
N
is the number of nongreen things. There might be something like 10
⁸⁰
atoms in the observable universe (which is the number written by putting 80 zeros after a “1”). There are at least that many nongreen objects. You might even claim that abstractions like numbers qualify as nongreen objects. Then the number is infinite.

This reasoning, anticipated by Hempel in his original musings in the 1940s, is very tempting. Possibly, a red herring
does
confirm “All ravens are black”—but only to an infinitesimal degree, because there are so many nonblack things. Checking the color of ravens is simply a more efficient way of confirming the hypothesis. In this vein, philosopher Nicholas Rescher estimated the costs of examining a statistically significant sample of ravens and nonblack objects. Rescher put the research tab at $10,100 for the ravens vs. $200
quadrillion
for nonblack objects!

There remains the dilemma of how a red herring can confirm “All ravens are black”
and
“All ravens are white.” You can try to picture it as being something like the mathematics of infinitesimals. The confirmation provided by a red herring for “All ravens are black” is on the order of 1/infinity. The “infinity” in the denominator refers to the infinity of nonblack objects, of which the red herring is one. Since the herring is also a nonwhite object, it should confirm “All ravens are white” to an identical degree of 1/infinity.
One divided by an infinite quantity is defined to be an infinitesimal, a number greater than zero but smaller than any regular fraction.

Does infinitesimal confirmation make the conflict any more palatable? We would be saying that a red herring confirms both “All ravens are black” and “All ravens are white,” but only to an infinitesimal degree.

A small truth is still a truth; a small he is yet a lie; and a contradiction is still a contradiction, even on an infinitesimal scale. The only out is to admit that the confirmation in both cases is precisely zero—as plain horse sense demands. Why, then, isn’t a confirming instance of a hypothesis a confirming instance of its contrapositive?

Sometimes one paradox suggests the resolution of another. Paul Berent’s paradox of the 99-foot man is another demonstration of the fallibility of Nicod’s criterion. Say you subscribe to the reasonable belief: “All human beings are less than 100 feet tall.” Everyone you’ve ever seen is a confirming instance of this hypothesis. Then one day you go to the circus and see a 99-foot-tall man. Surely you leave the circus
less
confident that all people are less than 100 feet tall. Why? The 99-foot man is yet another confirming instance.

There are two sources of this paradox. First, we don’t always say what we mean. Sometimes the words we use imperfectly express the (often vague) hypothesis in our heads.

Chances are, you meant that no human being attains fantastic height; height an order of magnitude or more greater than the average. The precise figure of 100 feet was not vital. It was pulled out of the air as an example of the great height that you thought was definitely out of the question.

Had you been using the metric system, you might have said, “All human beings are less than 30 meters tall.” Thirty meters comes to 98. 43 feet, so the 99-foot man
would
be a counterexample to the 30-meter hypothesis. One feels that what you meant by saying “All human beings are less than 100 feet tall” is partially violated by the 99-foot man. It is like obeying the letter but not the intent of the law.

There is another root of the paradox. Have the hypothesis be the substance of a running bet you have with a friend. If ever a 100-foot-or-taller person turns up, you lose and owe your friend dinner at a fancy restaurant. The hypothesis is propounded not out of intellectual curiosity but solely to formalize the bet. Only the exact terms
of the wager count. The 99-foot man is close but no cigar. He poses no threat whatsoever of deciding the wager against you.

You would still feel that the 99-foot man hurts the chances of your hypothesis being right. This is because you know many facts about human growth and variation that allow you to deduce an increased likelihood of a 100-foot person from the fact of the 99-foot man. Nearly every human attribute recurs (even to a greater degree) eventually. The 99-foot man demonstrates that it is genetically and physically possible for a person to attain a height of about 100 feet.

Now imagine that you find a way to test your hypothesis without acquiring any nonessential information. At the busiest part of Fifth Avenue, you place a sensor in the sidewalk that detects whenever anyone walks over it. A hundred feet above the sidewalk sensor is an electric eye. When someone steps on the sensor, the electric eye determines whether a beam of light 100 feet above the sidewalk has been broken by a tall pedestrian. A recording device keeps track of the total pedestrian traffic and the 100-foot-or-taller pedestrian traffic.

You check the meter to see the results. The readout is “0/310,628”—meaning that 310,628 pedestrians have passed, none (o) of whom was 100 feet tall. Each of the 310,628 pedestrians is a confirming instance of the hypothesis. Each confirms the hypothesis to a precisely equal degree. It would be ridiculous to say that some of the pedestrians provided more confirmation than others when all you know of the pedestrians is that they are shorter than 100 feet.

If it so happened that the 99-foot man crossed Fifth Avenue and was one of the people counted, he would confirm the hypothesis as much as anyone else, in your state of ignorance. Thanks to him, the meter reads “0/310,628” rather than “0/310,627,” and you are slightly more confident for it.

Clearly, it is the additional information (that the man is 99-feet tall, and what you know about human variation) that converts a simple confirming instance into one that effectively disconfirms.

Philosopher Rudolf Carnap suggested that there is a “requirement of total evidence.” In inductive reasoning, it is necessary that you use all available information. If you know nothing of the 99-foot man and only look at the meter readings, then he is a valid confirming instance. When you know more, he’s not.

The requirement of total evidence has occasioned much soul-searching in the scientific community because it addresses much of the research arena of biochemistry, astronomy, physics, and other fields. The way we investigate genes or subatomic particles is more
akin to the pedestrian traffic meter than simple observation. We do not meet RNA or quarks face to face; rather, we pose an exact question and learn the answers from machines.

Nothing is wrong with this, provided we do not limit our knowledge-gathering unnecessarily. If we are ignorant of other factors, and necessarily so, then we can generalize only from the information that is available. However, the more complete the information gathered, the more effective we are in making generalizations.

Let’s recap. Science deals mostly in generalizations: “All X’s are Y.” Only through generalization can we compress our sensory experience into manageable form.

Generalizations are concealed negative hypotheses: “There is no such thing as a non-Y X”; or “The above rule has no exceptions.” A generalization’s contrapositive corresponds to the identical negative hypothesis.

In an infinite universe, proving a negative hypothesis is a supertask. (If the universe is merely finite but very big, proving a negative hypothesis is a herculean labor so close to a supertask as to make no difference.) We are incapable of supertasks, and have reason to be suspicious of knowledge attainable only through supertasks anyway.

Instead we establish generalizations through confirming instances: “X’s that are Y”; black ravens in Hempel’s example. This can never rigorously prove a generalization; only disprove it (through a counterexample: a nonblack raven). Tallying sightings of black ravens is a way of keeping score on how well established the hypothesis is. We feel that each black raven represents another instance in which the hypothesis was truly at risk of being disproved and came through unscathed. We do not feel that nonblack nonravens (confirming instances of the contrapositive) hold the same—or any—weight. And the puzzle of the ravens is to give a legitimate reason for this empirical instinct.