The concept of induction (the kind of reasoning that leads us to draw general conclusions or make predictions concerning unobserved cases based on cases we have observed) has a long tradition in philosophy, which is worth revisiting.
Aristotle was the first to concern himself systematically with induction and he gave it a foundation based on the whole of his metaphysics, that is, what he thought concerning the nature of reality and knowledge. For Aristotle, scientific knowledge was essentially a system of classification. In a world of beings which organise themselves and rank themselves according to unchangeable forms or essences, a scientific statement confirms that an individual belongs to a certain species or that a certain species belongs to a genus. The individual is the specific case, the particular; stating that he belongs to a species defines his essence and shows what is universal in him.
According to Aristotle, the universal elements, the essences, exist in things, in particulars. Induction (in-ducere, to lead inward) consists precisely in this recognition of the concept (the universal) within the sensible (the particular). As we observe the behaviour of a phenomenon in various particular cases and recognize a regular pattern, we are naturally led to infer that this regular behaviour is a sign of the essence of the phenomenon and to forecast that the same behaviour will be shown in cases which will be observed in the future. This is basically the way in which Aristotle understands and justifies induction.
After 2000 years of almost complete dominance, Aristotle’s metaphysical ideas were challenged by modern philosophy. Hume, in particular, rejected Aristotelian essentialism, thus undermining the basis of induction. The link between particular cases and general law no longer depends on the presence of the universal within the thing and comes to be seen as simply the result of subjective expectation based on habit. This does not imply that Hume rejects or undervalues induction as a resource to be used in daily life or in empirical science: he simply removes the claim that knowledge obtained through it has an absolute, unquestionable metaphysical truth.
Hume’s criticism of Aristotle’s theory of induction did not prevent logical empiricism, the dominant concept in the philosophy of science until the 1950s, from defending an inductivist view of scientific method. It was felt that the general laws of empirical sciences were obtained through induction from the observation of specific cases, thus consisting in a simple summary or ‘condensation’ of concrete experience. As well as being acquired by induction, for the logical empiricists general laws could also be proved inductively. The greater the number of positive examples (specific cases conforming to the law) that could be observed, the greater would be the level of confirmation of the law or hypothesis. Actually, laws were only hypotheses with a sufficiently high degree of confirmation. The idea of the degree of confirmation led to attempts to apply calculations of probability to this discussion, but without any great success.
The discrediting of the inductive conception of scientific method was to a great extent the work of Popper. Proper claimed to have solved the problem of induction in a new and radical way: simply showing that the problem of induction does not exist in empirical science for the good reason that empirical science is not inductive. Scientific hypotheses are not obtained by inductive generalisation nor are they proved by the repetition of positive cases. Science moves forward by conjecture (bold generalisations with no logical support from experience) and refutations. What strengthens our hypotheses is their resistance to the clever and honest attempts to refute them to which they are subjected and that they manage to survive. Popper calls this process corroboration.
The whole of Popper’s theory on scientific method rests in the final analysis of an apparently curious logical property of universal statements. Scientific hypotheses and laws are usually expressed as universal statements: “Whenever there is the prospect of a price freeze, prices go up”, for example or, more simply, “All crows are black”. We may express the most basic logical structure of these laws thus: “Whenever A, B” or “A->B (A implies B)”. So the hundreds or thousands of cases where A is accompanied or followed by B do not eliminate the logical possibility that A might occur without B. However, a single case of A being observed without B overturns the general law that states universally the dependence of B on A.
This logical imbalance led Popper to claim that while universal statements cannot be confirmed, they can be refuted. Logically, “All crows are black” is the equivalent of saying “If something is not black, it is not a crow”. A single observation of a non-black crow overturns a general statement that agrees with thousands of observations of black crows. We must be careful here to avoid two logical mistakes that are very easily made, so much so that they have been given special names as fallacies: the fallacy of affirmation of the consequent, and the fallacy of negation of the antecedent. An example will help us to understand these fallacies.
Let “Whenever there is the prospect of a price freeze, prices go up” be a hypothesis or general law, which we can represent by “if A, then B” or “A -> B”. We may be tempted to think that it would be justifiable to conclude, based on this law, that “if there is no prospect of a freeze, prices will not go up” (not-B -> not-A), or “if the prices went up, then there was the prospect of a freeze” (B -> A). It happens, though, that none of these two statements is the result of our initial hypothesis and both could well be false at the same time as the former is true. It could perfectly well be true that when there is no prospect of a freeze, prices will go up for other reasons (a bad harvest, for example); this shows that it is possible for prices to rise even when there is no prospect of a freeze. These two last statements are logical equivalents to each other, but not to the first.
This is the point at which Hempel comes in with his famous paradox of the crows. Hempel also starts from the basis of the logical equivalence between “A -> B” and “not-B -> not-A”. His aim is to criticise the idea of confirming a general statement by its positive examples. It is reasonable to suppose that everything that proves a statement also proves those statements that are logically equivalent to it. But in this case each proof of the statement “not-B -> not-A” is also proof of the statement “A -> B”. The most disparate observations ‘confirm’ the statement “If something is not black, then it is not a crow”. Every non-black object we see which is not a crow confirms this: this wall, my shoe or the Taj Mahal are not black and are not crows. But our intuition tells us that this has nothing to do with the logically equivalent statement: “All crows are black”.
What Hempel’s paradox demonstrates is that proof is not a duly logical operation; something which should not be too surprising. When dealing with empirical science not all problems can be solved simply through logic! Hempel does not question the legitimacy of induction as a methodological principle in empirical science but he does make us question the philosophical theory that claims that the general statements in empirical sciences are logically proved by the observation of positive cases.
If we add together the criticisms made by Popper and Hempel we find that neither is induction the path to formulating scientific laws, nor does the inductive confrontation of those laws with experience ensure their confirmation.
Other logical paradoxes are associated with confirmation, such as Nelson Goodman’s ‘grue’ (green-blue) emeralds. The idea is too complicated to be explained here, but he shows that the same empirical base - the same observed facts - may provide different (and incompatible) inductive projections, which provides a healthy dose of scepticism to counteract the obligation to accept inductive conclusions, whatever they may be.
Paul Feyerabend’s counter-inductive ideas go further than this. For Popper, the most vital moment in scientific method (the point at which bold ideas and conjectures must confront the court of experience) is the search for data that run counter to the theory. In Feyerabend’s view we must not only look for data that contradict our theories, but we also need to look for theories that contradict our data (and our old theories). Taking his inspiration from John Stuart Mill and basing himself on humanist rather than logical reasons, Feyerabend proposes thereby to enrich scientific methodology. Contradiction would work as a “principle of proliferation” that would be critical, creative and pluralist, working to achieve flexibility in scientific categories and enable us to think, feel, see and experience the world in alternative ways.
And as well as looking for theories that agree with our facts, Feyerabend feels that empirical science should also work with hypotheses that are inconsistent with theories or with well-established data. While apparently absurd, this ‘anarchic’ suggestion boasts a solid historical foundation. In making their claims for the movement of the Earth, the evolution of species or infantile sexuality, were not Copernicus, Darwin and Freud possibly contradicting well-established and accepted theories and ‘facts’?
We may conclude this summary of the topic by stating that empirical theories, especially the more advanced ones, should not be understood simply as inductive generalisations based on observations gathered spontaneously or in a more or less systematic way. They are highly complex constructions which can involve drastic simplifications as their starting point (such as when Physics claims to be dealing with ‘isolated systems’ or Economics claims to talk of ‘perfect competition’); assumptions concerning non-observable processes and bodies; sophisticated mathematical apparatuses; metaphysical suppositions that are often not explained, and even views of reality that are often loaded with ideological preconceptions, among so many other elements!
Trying to understand how all of this works is a fascinating challenge and understanding the relevance of this question is vital if we are to achieve a dynamic and not a dogmatic view of empirical science. Induction, in science as in daily life, is methodologically useful and indispensable in practical terms, but it cannot guarantee that its results show in any way an absolute and unchanging truth. On the contrary, it gives them that porosity and openness that are crucial for scientific progress as well as for the freedom of creativity of human thought.
Rejane Xavier
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