Claudio Gutierrez
Translated from Spanish by Inés Gutiérrez
Claudio Gutierrez
121. From Analogy to Generalization
The analogy argument can be considered as a prediction operation that takes us from the proven truth of certain singular propositions to the predicted truth of another singular proposition. For instance, from the assertion that x, y, .... z (different teaspoons of a remedy) have the properties X, Y, .... Z in common, and w (another teaspoon of the same remedy) has the properties X, Y, ...., to the assertion that w also has property Z. That is, if w resembles x, y, .... z in some attributes, it will resemble them in this one also.
It is important to point out that the conclusion reached is singular, i.e. refers to a particular individual of the universe: this teaspoon of liquid I have in front of me. But on the other hand, since we could apply prediction by analogy to any other teaspoon, and also to that other one, we may as well summarize all these predictions into a single general one and assert that, for any individual of the universe, it is true that it is enough for it to have all properties X, Y, .... for it also to have property Z. That is to say, we can state a universal proposition as our conclusion, and not simply a proposition about a specific individual.
At bottom, then, there is no difference between getting the universal conclusion, applied to all individuals, and having got the singular conclusion applied to an indeterminate one. However, by going all the way to the universal conclusion, what we may call the inductive leap –jumping from the proven to the not-yet proven–, becomes more apparent and, for the same reason, makes us conscious of the serious risk of going astray. Just as the induction “from singular to singular” is germane to the fallacy of illegitimate instantiation, the induction “from singular to universal” is germane to the fallacy of illegitimate generalization (see sección 99). From the proof of one or several cases, moving to the assertion of all cases, a risky step in deed! We may only run this risk if enough guarantees are provided –in the context where the proof is performed– that we are not incurring that deadly fallacy. Let us explore which ones these necessary guarantees might be.
It was explained in chapter nine that universal proposition is a sort of conjunctive molecular one, except for the fact that the number of components included in the conjunction remains indefinite. Knowing the number of individuals that constitute our universe of discourse, however, makes the number of components in the conjunction determinate; namely, it becomes equal to the number of individuals. For instance, a proposition relative to all members of the U. S. Senate could be expressed as a conjunction with 101 different constituents. “All senators were elected by the people” would translate into “Senator A was elected by the people, Senator B was elected by the people, …” until reaching the singular proposition referring to the current Vice President of the United States of America.
Let us assume that we are conducting a research on the U. S. Senate. We want to know whether all its members are susceptible to being influenced by lobbying; that is to say, we wish to prove the following conclusion, referred to the universe of members of the U. S. Senate: “All are susceptible to being lobbied.” The most economical method for organizing this research would be to select at random a sample of some members and hold interviews with their staff about his or her conduct. If results were totally favorable to the generalization, we will jump to the conclusion “all are susceptible...” referring to the 101 members of the U. S. Senate. This method, however, is not trustworthy; we could have selected by sheer chance for our sample only susceptible members!
A far safer method to arrive at the same conclusion would involve much more work on our part, namely, to interview all aids of every one of the members of the U. S. Senate. If the result were favorable in all cases, we would have proved conclusively our hypothesis. We call this procedure complete induction and it is actually a case of deduction, since the truth of the conclusion can be asserted with the same certainty as the truth of the premises. However, it is not very useful in practice: it does not allow us to predict what we have not known already by direct experience. In contrast, it is extremely useful theoretically, inasmuch as it can help us to distinguish the correct application of inductive leap, from the imprudent one associated with illegitimate generalization. The inductive leap will be justified if there is some substantial reason to believe that induction by sample will give us the same results as complete induction. As a good statistician will tell you, this will be the case if the universe investigated is rather large and the sample is selected is done with extreme care in the avoidance of bias. If number, on the other hand, ware not large, as in our example, then we should definitely opt for theoretical considerations relevant for the case. For instance, if what we want to know is whether a specific law will be approved by Congress, we should count upon party discipline, interview leader of each political parties, take into consideration the ideological inclination of each, their campaign promises, past record, etc. On the other hand, this consideration may no help much if what we are interested in determining is susceptibility to being lobbied.
Let us imagine a group of scientists arriving at a beach and seeing hundreds of animals nesting in the sand. Let us assume those scientists know that there are two similar species in the area, which never nest together in the same section of the beach. Scientists will be inclined to writing down in their observation protocols something like: “We saw hundreds of species A specimens nesting at the beach,” or “we saw hundreds of species B specimens nesting at the beach.” They must decide which of these two propositions is true. To do so, they have the inductive method at hand: they may capture a sample of the animals and, through careful observation, determine which of the species is currently depositing eggs there. The sample may be small, although not too small, to reduce the danger of errors in appreciation, or even to compensate for the possibility that some specimens from the not present species might have mixed by accident with the present one. But clearly, a good observation of a few specimens will be as good as the observation (impracticable for the rest) of all specimens present.
We have here a clear case where induction by sampling is as methodologically effective as complete induction. Why is that so? The answer lies in the additional knowledge that the scientists have about the fact of the two species never nesting together; i.e. their certain knowledge of a regularity in the phenomenon in question. “If some cases are A, all cases are so,” such is the implicit premise that transforms the inductive leap into an inference procedure with as much certainty as the deductive method itself. The same is true for the members-of-the-legislature example examined before: the tacit premise that party discipline exists can transform the induction by sampling into a perfectly valid deductive argument.
We may extend the finding we have made in relation with these simple examples to the whole field of scientific research. Induction is applicable as a serious method, different from illegitimate instantiation or generalization fallacies, provided conditions are such that allow trust on the existence of regularities in the field studied. These regularities are often called natural laws; we are never sure of their existence; but we can assert that, if they exist, inductive method will surely lead to their discovery. We all have examined, in our spare time, the design in a wall paper: a small flower here, a scribble there, then a star, and again the small flower. We experience a feeling of contentment when we discover the repeating model, and we stop examining further. The same happens in the study of nature: If we trust that nature is made, at least metaphorically speaking, according to a plan, we will not have to resort to the (normally impossible) procedure of complete induction; induction by sampling will do as well to exhibit for us the mold used in the construction of the universe.
It seems, thus, that we may be optimistic: knowledge of nature is attainable. Nevertheless, this optimism is justified only in practice, and for certain fields of knowledge. In theory, we have no guarantee that this “plan of the universe” really exists. Furthermore, in many fields, especially from the social sciences, we cannot trust in fact on the regularity of our universe. Party discipline, for instance, is a frail regularity to use while doing an induction about Congress: its human character makes it liable to many and varied exceptions.
The principle of the regularity of nature is commonly formulated saying that “every effect has its cause.” This principle seems an obvious truth, for by definition “cause” and “effect” implicate each other. But what is meant with this phrase is that the universe is such that things do not happen just by chance; rather, each event is explainable in terms of specific regularities of a permanent character. "Principle of causality” is another name often used to designate the assertion of the regularity of nature. By whatever name, this principle is then, in a certain sense, the very foundation of the scientific method.
Let us clarify the concept of cause, so that what was said about the principle of causality can be properly understood. Every human mind, small children and primitive people included, ask themselves the why of things; the immediate and obvious answer being that the thing exists, or is determined in some fashion, because another thing –which we call cause– made it happen or be the way it is. From these essentials of human knowledge it is easy to pass to the belief that things are endowed with powers, not perceptible through the senses, but which make them produce things or transformations on things. From such primordial interpretation, thus, the principle of causality is perceived as the constant influx of such powers, an invisible action in itself, as invisible as are such hidden powers of things. We must take heed that this idea of causality has little to do with the methodology or content of modern science. It is rather a magical or religious idea, perhaps also poetic, about which science has absolutely nothing to say. Science does not study hidden or invisible powers, just the visible and the manifest, perceivable through the senses, with the help or not of instruments; we can even say that science does not actually study things, what they are in themselves, only their appearances, what is shown of them, i.e. –using a philosophical term- its phenomena. Science studies phenomena and their relations, and only that.
Thus, causality as used in science is simply regularity of phenomena. We say that two phenomena are causally connected if there is a constant regularity between the two. In this context, it is not even necessary to speak of causes: we may as well do away with this ambiguous term and speak instead simply of conditions. Not to say that a phenomenon is cause of another, but rather only that the phenomenon is the condition, necessary or sufficient, to make another phenomenon happen.
126. Different Kinds of Conditions
Regularity of nature may be expressed based on propositions of condition –conditionals– indicating factual connections between phenomena. We already know the meaning of the condition connective, and the way to analyze the propositions where it occurs (see section 48). Conditional propositions may have the form “if… then…,” “only if… then…,” or “if and only if… then….” In the first case we deal with the idea of sufficient condition; in the second, with necessary condition; and in the third, with sufficient and necessary condition.
We say that A is a sufficient condition for B when the proposition “if A then B” is true; in this case, it is enough that A occurs for B to occur. We say that A is necessary condition for B when the converse proposition “if B then A” occurs, or equivalently “only if A then B;” in this case, if A does not occur, neither will B. We say that A is necessary and sufficient condition of B when both aforementioned propositions are true, that is “if A then B” and “if B then A;” in this case, if one of the constituents does not occur, the other will not occur, and if one of them occurs, the other will necessarily also occur. Applying this to science, “A” and “B” represent phenomena between which there exists a constant connection. The ideal goal of science is to establish necessary and sufficient conditions for every phenomenon; nevertheless, sometimes one has to settle for less and conform oneself with one of the two weaker conditions: only the sufficient or only the necessary one.
We have seen that correct induction must be founded on the existence of regularities between phenomena. In addition, that those regularities can be expressed as sufficient or necessary conditions. Rules of correct induction are, then, those that allow us to discover and verify the existence of those conditions. Among the attempts to formulate such rules, or canons of scientific inquiry, one of the most influential is due to John Stuart Mill in his work System of Logic, written in the middle of the XIX century. Mill proposes several, the first two –agreement and difference– being the most important of them. Here his own words:
128. Appreciation of Mill’s Methods
Two interesting consequences follow from the above text: the method of agreement allows us to conclude that what may be eliminated without making the phenomenon disappear is not a necessary condition of it; if we add as an additional premise that every phenomenon has at least one sufficient condition, that which has not been eliminated will contain such condition. The method of agreement allows us, then, to identify the sufficient condition of the phenomenon. On the other hand, the method of difference allows us to conclude that what cannot be eliminated without making the phenomenon disappear is a necessary condition of it. This method, then, allows us to identify this type of condition.
Let us see application instances of both methods within the common experience of every student. Several members of a class have in common the fact that they all pass the course with good grades; we check the work of all of them during the period and we find that some studied with the text book but not all did; some in contrast studied with notes taken in class; in the group, some students have an appropriate behavior, others do not behave so well; there are some who attended all the time and others that were absent some times; they all agree in having studied the subject all semester long, following the teacher instructions. Applying the method of agreement, we may conclude that to study all semester long is the cause (or sufficient condition) of passing the course. One semester you attend the course punctually, study with the text book and with notes taken in class, every day a little; you pass the course. Another semester you attend the course punctually, study with text and notes, but leave all the material for the last days of the semester; you flunk the course. By the method of difference, we may conclude that to study all semester long is the cause (necessary condition) to pass the course.
Notwithstanding their undeniable usefulness in research, Mill’s canons have the limitation that they suppose an exhaustive and correct analysis of the concrete circumstances of the case. If such analysis is not thoroughly performed, we may incur in the fallacy of the scientific drinker: wanting to know the cause of his drunkenness, he resorted to Mill’s methods. One day he got drunk with rum with coke; another day he got drunken drinking tequila with coke; another, drinking gin with coke. He arrived at the conclusion that he should never again drink coke. The mistake he committed was not having sufficiently analyzed the constituents of the problem; if he had, he would have discovered that coke was not the only common constituent to the three beverages: alcohol was also present in all of them.
As you may conclude from the previous fallacy, the analysis of the circumstances of the problem is essential for the validity of Mill’s methods. Hence, we arrive to the conclusion that these rules are not in themselves enough to base science. Canons must be applied conjoined with hypotheses about what a correct analysis of the situation would be and within a theoretical context which make probable that those hypotheses adequately describe reality. We will devote the following and last chapter of this work to study the nature of such hypotheses and theories.