HILOSOPHERS find great difficulty with the problem, How do I find out what is around me? Sometimes I am sure I am awake, but sometimes I wonder if I am dreaming. Psychologists regularly devise puzzles which appear to human sight to be one thing, but in fact are something else.

However, we shall have no trouble whatsoever with these intricate problems. For our purposes, we are only interested in reaching agreement; if all the observers agree, that is enough.

On this definition, does any difficulty remain? Surely it can be agreed without difficulty that the sun rises in the East, and must do so tomorrow and the next day and forever.

The typical result on which agreement can be reached is an observation statement, "This sheep is black," or "This swan is white." If one observer thinks a thing is gold and another thinks it is silver, then it may turn out that the one saw it in daylight and the other in moonlight; the common experience is that observers can reach agreement.

Statements of the kind, "This . . . is . . ." are called singular, they refer only to the one example we are considering. When one person proposes such a proposition, other persons can reply either, "It is indeed," or "True." Proposition P is called "true" if and only if P.

Truth has two criteria: coherence, and correspondence with the observations. An observer can object, "It is not white," or "That is not what I call a swan ¾ it is only a soft toy swan."

When we have agreed on an observation statement, which is to say, a singular proposition, we can go further. We can propose, "Some swans are white." Once the observation has been made, the truth of the particular proposition must follow; one cannot deny the particular proposition without challenging the original observers.

Can we go any further? Can we propose, "Every swan is white?" We can say it, we can even believe it, but we are open to being contradicted ¾ someone, somewhere, may have seen a swan which was not white. Even the agreement of those present is not enough; someone may yet look where no-one has looked before, and find a non-white swan.

"Every swan is . . . " is an example of a universal proposition.. The classical philosophers divided propositions into four kinds: particular and universal, affirmative and negative.

 A: "Every swan is white." E: "No swan is white."

I: "Some swans are white." O: "Some swans are non-white.

(The letters A, I, E, O, come from the words AffIrmo [I affirm], nEgO [I deny]: A and I statements are affirmative [with respect to whiteness,] E and O ones are negative.)

The interesting feature of this Square of Opposition is the relationship between the diagonally opposite corners. The A statement is contrary to the E statement (they cannot both be true) but contradictory to the O statement, and the E statement is contradictory to the I statement: a [true] particular proposition disproves a universal one. (The I and the O statements are sub-contraries; viz. they cannot both be false.) However, there is no example of the opposite, of one proposition proving another. "Every swan is white" does not prove that "some swans are white," because the A proposition leaves open the possibility that there are no swans at all (the only possibility it excludes is that of there being a non-white swan.) "Some swans are white," in contrast, requires that there actually be at least one swan. Particular propositions are "existential," universal ones are not.

We can therefore see that existential propositions are easy to prove ¾ the few observers are enough. But, obviously, particular [existential] propositions are not as informative as universal ones: a universal proposition excludes one of the two (sub-contrary) particulars. We ask, then, whether we can prove any universal proposition?

It is this question that the philosophers have in mind when they say that certainty is out of our reach; they dismiss particular propositions from consideration. Universal propositions ¾ that is, laws, admitting of no exceptions ¾ allow one to argue deductively (if every man is mortal, then . . . ) and this is the essence of intellectual intercourse.

There are universals of which we all know the proofs; an example is Pythagoras’ Theorem, the square on the hypotenuse is equal to . . . . Mathematics consists of laws that can be proved, just with pencil and paper. However, mathematics applies to ideal, or imaginary, triangles and circles and sets and matrices. It appears that these proofs concern only abstractions, not empirical things of which we can make observations.

Thus we see that particular, existential propositions can be proved (by observations) and that universal propositions concerning abstractions can be proved (by argument.) We are left with the class of universal propositions concerning empirics.

This last class ¾ propositions which are universal, and have empirical content ¾ constitutes science. Scientific theories can be falsified (disproved) by just one counter-example.

Notice that science is not "substantiated," let alone "proved", by evidence; evidence can say only "Some swans . . .", science says, "Every swan . . . ." This means that a scientific proposition, a law, is forever in danger of being falsified; it is always possible that someone will devise a new test, and the new test will find an exception to the law. The obvious example is the eclipse of the sun in 1919, when it was found that light rays from stars were deflected by the mass of the sun in accordance with Einstein’s theory, and thus Newton’s law of gravity was falsified.

A theory that has been falsified is false for evermore, but it is not therefore useless. Newton’s law of gravity (masses exert an attraction varying as the product of the masses and the inverse square of the distances) is false, but every space probe relies upon this false theory: until our vehicles attain speeds close to that of light, Newton’s law is good enough.

Not only is science never verified, it is not even probable. On the contrary, a scientific theory is valued only as it is IMprobable, as it excludes many outcomes and includes only few; Darwin’s theory of evolution, which can, it seems, explain whatever kind of life may be discovered, is hardly science at all. It has been shown that a scientific theory is not probable, as follows: Consider Newton’s laws, which predict the tides and the eclipses and the fall of shot and the trajectory of missiles ¾ we may be sure that these Laws have a probability of at least one-half. But Einstein’s theory works in every case where Newton’s Laws work, and certain others where Newton’s Laws do not ¾ it must have a higher probability, say 0.51. Then the probability of either Newton’s Laws or Einstein’s theory being true must be 1.01. But no probability can conceivably be greater than unity. Thus Newton’s Laws cannot have a probability of even one-half . . . .

Notice that one cannot test a theory until one has a theory; making a guess is logically prior to making an experiment. It was impossible that anyone should find electro-magnetic radiation ¾ radio ¾ until Herz had predicted that it should have certain effects; the theory allowed an experiment to be done. Science is essentially an inventive, creative activity; it is not a mere process of measuring things accurately and arranging the results in order. On the contrary, science advances when the existing laws are disproved, when it is learned that things are not as was supposed (always assuming that the mind of man is equal to proposing a new theory to embrace the new results.)

In fact, theories can be invented without limit (although most of the inventions will no doubt be trivial, e. g. that Newton’s laws are different in the immediate vicinity of a star, as an explanation of the precession of the orbit of Mercury) ¾ for any set of data, one can always invent another theory. How, then, does science deal with the numerous theories? Arbitrarily: it chooses always the "simplest," the most general, the one that is most improbable and subject to the most tests.

Restricting science to falsifiable laws is necessary, as we have discussed, for intellectual intercourse, but it has two incidental effects. It has the result that science progresses quickly, because false theories can be detected easily; and it means that people agree with science, because anyone who disagrees with any theory can make a test and see whether it fails. (Mystics, and those who have experienced para-psychological phenomena, object that scientists fail to attend to their reports. However, for the result to be accepted as a falsification, the result of the test must be one that other people can repeat ¾ outcomes that occur only for some observers are not scientific evidence.)

What, then, of the experiments of which we hear every day, those that prove that there is a connection between smoking and getting cancer, or between eating X and getting cancer, or even between taking Y and getting rid of cancer? These are claimed to be true at the 95% level, or the 99% level, or ¾ in the case of para-normal phenomena ¾ at the 99.999% level.

These are the results of "controlled experiments", where the experimenter takes two groups of subjects, who appear to be indistinguishable for the purpose of the test, applies the experimental procedure to one group, but only pretends to apply it to the other group. (This may be done only historically, for instance by searching the records of a hospital for several patients with similar prognoses, and then comparing the ones who were given Z to those who were not.) The result of such an experiment is a correlation between the controlled variable and the outcome (possibly, of course, we find a negative correlation ¾ the treatment makes the patients worse.) A correlation is something other than correspondence, it is a correspondence with some proportion of exceptions (a correlation of 0.8 would have 20% of exceptions.)

The results of these controlled experiments are not laws; they do not say "every swan is . . .," only "some swans are. . ." or "most swans are . . . ." Thus they are only existential truths; they prove that the correlation exists.

These existential propositions are, or may be, true in a very strong sense. If a number of experiments have shown that no correlation exists between, say, race and intelligence, then one experiment showing that it does exist is not accepted as a disproof; rather, there is argument as to whose results are to be accepted. Critics are not choosing between one theory and another, but only between one researcher and another.

Just as a scientific theory which is false may yet be useful, so an existential theory, one subject to exceptions, may also be useful, i.e. a guide for action. An insurance company is content to know that most women live longer than most men; the company does not mind being wrong sometimes, as long as the money they lose when they are wrong is less than the money they save when they are right. (It is important to distinguish organizations that only need to be right more often than not from those that are required to be right as matter of course; one cannot afford to lose even the occ