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An Introduction to the Philosophy of Science
By Rudolph Carnap, Martin Gardner Dover Publications, Inc.
Copyright © 1966 Basic Books, Inc.
All rights reserved.
ISBN: 978-0-486-14086-5
CHAPTER 1
The Value of Laws: Explanation and Prediction
THE OBSERVATIONS we make in everyday life as well as the more systematic observations of science reveal certain repetitions or regularities in the world. Day always follows night; the seasons repeat themselves in the same order; fire always feels hot; objects fall when we drop them; and so on. The laws of science are nothing more than statements expressing these regularities as precisely as possible.
If a certain regularity is observed at all times and all places, without exception, then the regularity is expressed in the form of a "universal law". An example from daily life is, "All ice is cold." This statement asserts that any piece of ice—at any place in the universe, at any time, past, present, or future—is (was, or will be) cold. Not all laws of science are universal. Instead of asserting that a regularity occurs in all cases, some laws assert that it occurs in only a certain percentage of cases. If the percentage is specified or if in some other way a quantitative statement is made about the relation of one event to another, then the statement is called a "statistical law". For example: "Ripe apples are usually red", or "Approximately half the children born each year are boys." Both types of law—universal and statistical—are needed in science. The universal laws are logically simpler, and for this reason we shall consider them first. In the early part of this discussion "laws" will usually mean universal laws.
Universal laws are expressed in the logical form of what is called in formal logic a "universal conditional statement". (In this book, we shall occasionally make use of symbolic logic, but only in a very elementary way.) For example, let us consider a law of the simplest possible type. It asserts that, whatever x may be, if x is P, then x is also Q. This is written symbolically as follows:
(x)(Px [contains] Qx).
The expression "(x)" on the left is called a "universal quantifier." It tells us that the statement refers to all cases of x, rather than to just a certain percentage of cases. "Px" says that x is P, and "Qx" says that x is Q. The symbol "[contains]" is a connective. It links the term on its left to the term on its right. In English, it corresponds roughly to the assertion, "If ... then ..."
If "x" stands for any material body, then the law states that, for any material body x, if x has the property P, it also has the property Q. For instance, in physics we might say: "For every body x, if that body is heated, that body will expand." This is the law of thermal expansion in its simplest, nonquantitative form. In physics, of course, one tries to obtain quantitative laws and to qualify them so as to exclude exceptions; but, if we forget about such refinements, then this universal conditional statement is the basic logical form of all universal laws. Sometimes we may say that, not only does Qx hold whenever Px holds, but the reverse is also true; whenever Qx holds, Px holds also. Logicians call this a biconditional statement—a statement that is conditional in both directions. But of course this does not contradict the fact that in all universal laws we deal with universal conditionals, because a biconditional may be regarded as the conjunction of two conditionals.
Not all statements made by scientists have this logical form. A scientist may say: "Yesterday in Brazil, Professor Smith discovered a new species of butterfly." This is not the statement of a law. It speaks about a specified single time and place; it states that something happened at that time and place. Because statements such as this are about single facts, they are called "singular" statements. Of course, all our knowledge has its origin in singular statements—the particular observations of particular individuals. One of the big, perplexing questions in the philosophy of science is how we are able to go from such singular statements to the assertion of universal laws.
When statements by scientists are made in the ordinary word language, rather than in the more precise language of symbolic logic, we must be extremely careful not to confuse singular with universal statements. If a zoologist writes in a textbook, "The elephant is an excellent swimmer", he does not mean that a certain elephant, which he observed a year ago in a zoo, is an excellent swimmer. When he says "the elephant", he is using "the" in the Aristotelian sense; it refers to the entire class of elephants. All European languages have inherited from the Greek (and perhaps also from other languages) this manner of speaking in a singular way when actually a class or type is meant. The Greeks said, "Man is a rational animal." They meant, of course, all men, not a particular man. In a similar way, we say "the elephant" when we mean all elephants or "tuberculosis is characterized by the following symptoms ..." when we mean, not a singular case of tuberculosis, but all instances.
It is unfortunate that our language has this ambiguity, because it is a source of much misunderstanding. Scientists often refer to universal statements—or rather to what is expressed by such statements—as "facts". They forget that the word "fact" was originally applied (and we shall apply it exclusively in this sense) to singular, particular occurrences. If a scientist is asked about the law of thermal expansion, he may say: "Oh, thermal expansion. That is one of the familiar, basic facts of physics." In a similar way, he may speak of the fact that heat is generated by an electric current, the fact that magnetism is produced by electricity, and so on. These are sometimes considered familiar "facts" of physics. To avoid misunderstandings, we prefer not to call such statements "facts". Facts are particular events. "This morning in the laboratory, I sent an electric current through a wire coil with an iron body inside it, and I found that the iron body became magnetic." That is a fact unless, of course, I deceived myself in some way. However, if I was sober, if it was not too foggy in the room, and if no one has tinkered secretly with the apparatus to play a joke on me, then I may state as a factual observation that this morning that sequence of events occurred.
When we use the word "fact", we will mean it in the singular sense in order to distinguish it clearly from universal statements. Such universal statements will be called "laws" even when they are as elementary as the law of thermal expansion or, still more elementary, the statement, "All ravens are black." I do not know whether this statement is true, but, assuming its truth, we will call such a statement a law of zoology. Zoologists may speak informally of such "facts" as "the raven is black" or "the octopus has eight arms", but, in our more precise terminology, statements of this sort will be called "laws".
Later we shall distinguish between two kinds of law—empirical and theoretical. Laws of the simple kind that I have just mentioned are sometimes called "empirical generalizations" or "empirical laws". They are simple because they speak of properties, like the color black or the magnetic properties of a piece of iron, that can be directly observed. The law of thermal expansion, for example, is a generalization based on many direct observations of bodies that expand when heated. In contrast, theoretical, nonobservable concepts, such as elementary particles and electromagnetic fields, must be dealt with by theoretical laws. We will discuss all this later. I mention it here because otherwise you might think that the examples I have given do not cover the kind of laws you have perhaps learned in theoretical physics.
To summarize, science begins with direct observations of single facts. Nothing else is observable. Certainly a regularity is not directly observable. It is only when many observations are compared with one another that regularities are discovered. These regularities are expressed by statements called "laws".
What good are such laws? What purposes do they serve in science and everyday life? The answer is twofold: they are used to explain facts already known, and they are used to predict facts not yet known.
First, let us see how laws of science are used for explanation. No explanation—that is, nothing that deserves the honorific title of "explanation"—can be given without referring to at least one law. (In simple cases, there is only one law, but in more complicated cases a set of many laws may be involved.) It is important to emphasize this point, because philosophers have often maintained that they could explain certain facts in history, nature, or human life in some other way. They usually do this by specifying some type of agent or force that is made responsible for the occurrence to be explained.
In everyday life, this is, of course, a familiar form of explanation. Someone asks: "How is it that my watch, which I left here on the table before I left the room, is no longer here?" You reply: "I saw Jones come into the room and take it." This is your explanation of the watch's disappearance. Perhaps it is not considered a sufficient explanation. Why did Jones take the watch? Did he intend to steal it or just to borrow it? Perhaps he took it under the mistaken impression that it was his own. The first question, "What happened to the watch?", was answered by a statement of fact: Jones took it. The second question, "Why did Jones take it?", may be answered by another fact: he borrowed it for a moment. It seems, therefore, that we do not need laws at all. We ask for an explanation of one fact, and we are given a second fact. We ask for an explanation of the second fact, and we are given a third. Demands for further explanations may bring out still other facts. Why, then, is it necessary to refer to a law in order to give an adequate explanation of a fact?
The answer is that fact explanations are really law explanations in disguise. When we examine them more carefully, we find them to be abbreviated, incomplete statements that tacitly assume certain laws, but laws so familiar that it is unnecessary to express them. In the watch illustration, the first answer, "Jones took it", would not be considered a satisfactory explanation if we did not assume the universal law: whenever someone takes a watch from a table, the watch is no longer on the table. The second answer, "Jones borrowed it", is an explanation because we take for granted the general law: if someone borrows a watch to use elsewhere, he takes the watch and carries it away.
Consider one more example. We ask little Tommy why he is crying, and he answers with another fact: "Jimmy hit me on the nose." Why do we consider this a sufficient explanation? Because we know that a blow on the nose causes pain and that, when children feel pain, they cry. These are general psychological laws. They are so well known that they are assumed even by Tommy when he tells us why he is crying. If we were dealing with, say, a Martian child and knew very little about Martian psychological laws, then a simple statement of fact might not be considered an adequate explanation of the child's behavior. Unless facts can be connected with other facts by means of at least one law, explicitly stated or tacitly understood, they do not provide explanations.
The general schema involved in all explanation of the deductive variety can be expressed symbolically as follows:
1. (x) (Px [contains] Qx)
2. Pa
3. Qa
The first statement is the universal law that applies to any object x. The second statement asserts that a particular object a has the property P. These two statements taken together enable us to derive logically the third statement: object a has the property Q.
In science, as in everyday life, the universal law is not always explicitly stated. If you ask a physicist: "Why is it that this iron rod, which a moment ago fitted exactly into the apparatus, is now a trifle too long to fit?", he may reply by saying: "While you were out of the room, I heated the rod." He assumes, of course, that you know the law of thermal expansion; otherwise, in order to be understood, he would have added, "and, whenever a body is heated, it expands". The general law is essential to his explanation. If you know the law, however, and he knows that you know it, he may not feel it necessary to state the law. For this reason, explanations, especially in everyday life where common-sense laws are taken for granted, often seem quite different from the schema I have given.
At times, in giving an explanation, the only known laws that apply are statistical rather than universal. In such cases, we must be content with a statistical explanation. For example, we may know that a certain kind of mushroom is slightly poisonous and causes certain symptoms of illness in 90 per cent of those who eat it. If a doctor finds these symptoms when he examines a patient and the patient informs the doctor that yesterday he ate this particular kind of mushroom, the doctor will consider this an explanation of the symptoms even though the law involved is only a statistical one. And it is, indeed, an explanation.
Even when a statistical law provides only an extremely weak explanation, it is still an explanation. For instance, a statistical medical law may state that 5 per cent of the people who eat a certain food will develop a certain symptom. If a doctor cites this as his explanation to a patient who has the symptom, the patient may not be satisfied. "Why", he asks, "am I one of the 5 per cent?" In some cases, the doctor may be able to provide further explanations. He may test the patient for allergies and find that he is allergic to this particular food. "If I had known this", he tells the patient, "I would have warned you against this food. We know that, when people who have such an allergy eat this food, 97 per cent of them will develop symptoms such as yours." That may satisfy the patient as a stronger explanation. Whether strong or weak, these are genuine explanations. In the absence of known universal laws, statistical explanations are often the only type available.
In the example just given, the statistical laws are the best that can be stated, because there is not sufficient medical knowledge to warrant stating a universal law. Statistical laws in economics and other fields of social science are due to a similar ignorance. Our limited knowledge of psychological laws, of the underlying physiological laws, and of how those may in turn rest on physical laws makes it necessary to formulate the laws of social science in statistical terms. In quantum theory, however, we meet with statistical laws that may not be the result of ignorance; they may express the basic structure of the world. Heisenberg's famous principle of uncertainty is the best-known example. Many physicists believe that all the laws of physics rest ultimately on fundamental laws that are statistical. If this is the case, we shall have to be content with explanations based on statistical laws.
(Continues...)
Excerpted from An Introduction to the Philosophy of Science by Rudolph Carnap, Martin Gardner. Copyright © 1966 Basic Books, Inc.. Excerpted by permission of Dover Publications, Inc..
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