A term encompassing a variety of forms of inference, commonly, but not always, in contrast to
deduction. If 'proposition' is defined as a thought expressible by a grammatical sentence having either prescriptive or descriptive force, 'inference' may be defined as a transition in thought between one or more propositions (premisses) and a further proposition (conclusion), where the premisses purport to be reasons for the conclusion. The conclusions of deductive inferences cannot be rejected without contradicting the thoughts contained in the premisses, and in this sense are already contained in the premisses. Deductive inferences were consequently classified by
C. S. Peirce as 'explicative', while inferences whose conclusions were not already implicit in their premisses were called 'ampliative'. 'Induction' is sometimes used in the sense of 'ampliative inference'. Classically, however, following
Aristotle's use of '
epagōgē' (from the Latin translation of which we have 'induction'), the term applies to a subclass of ampliative inferences, namely those in which the conclusion is more general (applies to a wider range of instances) than the premisses.
The root idea of a movement in thought from particular to general has given rise to the practice of applying the term 'induction' to two forms of inference which are in fact deductive. The first of these is complete induction (in Aristotle, 'deduction from induction,
ex epagōgēs sullogismos'), where the premisses are all less general than the conclusion, but collectively exhaust the instances covered by the conclusion. If chimpanzees, gorillas, humans, etc. are found species by species to react in a certain way to a certain virus, and these species exhaust the class of primates, one may conclude that all primates react in this way to the given virus. The second is the large family of proof procedures used by mathematicians on a variety of order structures, the simplest example of which is numerical induction: if (1)
F is a property of the number one and (2) if
F is a property of the number
n, then it is a property of
n+1, then (1) and (2) together
entail that
F is a property of all (natural) numbers.
The narrower classical idea of induction as inference from particular to general excludes certain ampliative inferences involving probability, e.g. inferring that
x is
F from the high probability that a member of a class,
C, to which
x belongs, will be
F. It also excludes ampliative inference from one or more descriptions of an individual case to some further descriptions of that case. (Where the further description stands as the best explanation of why the first descriptions apply to that case, Peirce distinguished what he regarded as a scientifically vital form of inference, and which he called 'hypothesis' or 'abduction'.)
It is clear from this account that ampliative inferences are by definition not deductions and hence not deductively valid. Where, in other words, an induction is not complete — that is, the cases covered in the premisses do not exhaust those referred to in the conclusion (e.g. when concluding that all chimpanzees react in a certain way to a certain virus on the basis of having examined any number,
n, of chimpanzees) — the premisses may all be true and the conclusion false, and the inference may be reasonable but unfortunately misleading. This gives rise to the so-called problem of induction, but from a purely logical standpoint it can appear to be a matter of lamenting the fact that not all of the inferences which we make have the rigour and compulsion of deductions, coupled, perhaps, with the insinuation that only deductive inferences are rationally grounded. But, against the insinuation, it is far from obvious why good reasons for a conclusion must preclude its negation on pain of contradiction. It is clearly possible to distinguish good from bad reasoning which is not in this way absolutely compelling.
From the standpoint of certain epistemological presuppositions, however, the problem is acute. If we assume, as is done in traditional empiricism, that all our knowledge is founded on (sensory) observations of individual instances, inductive inference presents itself as the only, however doubtful, means at our disposal for building on this modest foundation the vast edifice of our beliefs about the natural world. Unless there are general statements whose truth can be known
a priori, all premisses of our deductions must be reached by induction; and every attempt to infer what holds for some unobserved or unobservable case on the basis of what has been observed must explicitly or implicitly appeal to a general proposition, which can only rest on induction from observed cases.
But the most, it is held, we are able to observe in individual cases are certain similarities among them, and it seems reckless in the extreme to expect such similarities to appear elsewhere unless we can identify some
cause or constraint which ensures that a pattern we have observed will occur elsewhere. However, in seeking such a causal constraint in what we observe, all we will find are further patterns of similarities in the features of what we observe which are constantly conjoined. We can find, in other words, no basis for a causal constraint which is not itself in need of the very justification that we are seeking to provide.
David Hume, who is the classic source for this problem (although he did not formulate it using the word 'induction'), considered whether a global principle such as the uniformity of nature could underwrite our inferences from observed to unobserved cases. But such a global principle seems a non-starter: nature is uniform only in certain respects, and in other respects is highly variable. This is reflected in our practice of making inductive inferences. In some cases it is reasonable to generalize on the basis of very few instances; in others a very large number of observed instances is no basis at all for a generalization. Traditional empiricism tends to obscure this difference because it presents all inductive inferences as ultimately proceeding by 'simple enumerations', which have to be assessed in the absence of any background of established beliefs and experimental techniques. Popular and oversimplified views of Karl Popper's response to Hume, namely that science does not rely on induction but on finding exceptions to its generalizations, are likewise based on the notion that induction is simply a matter of projecting a similarity in our experience of part of a class of cases onto the whole of that class (e.g. of expecting what we have observed in some chimpanzees to hold of all chimpanzees which have yet to be observed). In fact we move no closer to real science by saying that the aim is rather to find exceptions to such attempts at superficial generalization.
Francis Bacon rejected the procedure of applying a global principle as 'childish', insisting that induction must proceed 'by proper rejections and exclusions'. The underlying principle of this genuine Baconian induction (known as 'eliminative induction') is the control we exercise over the circumstances of our observations. We must, in other words, move beyond simply projecting our observations to cases not yet observed and project in the form of experimental hypotheses, which ascribe a network of links between circumstances, which can be varied, and phenomena, which we can observe. We have observed a link between heavy smoking and lung cancer, but do not yet know how the circumstances may be varied so as to interfere with the link (and therefore cannot yet explain why many heavy smokers do not get lung cancer). In this spirit Peirce defined induction as 'the operation of testing a hypothesis by experiment'.
J. S. Mill's 'four methods of experimental inquiry' were designed to help identify the laws and causal factors governing phenomena. We need (i) to find what is common among the differences in the instances that we have observed ('method of agreement') and (ii) to compare the instances in which the phenomena occur with those in which they do not ('method of difference'). We can (iii) 'subduct' from the phenomena all portions which we can assign to known causes and the remainder will be the effects of causes still to be determined ('method of residues'). And we can (iv) look for functional relations between variations in phenomena ('method of concomitant variations'). The application of such methods and the testing of the hypotheses which they yield evidently involve a procedure of thought which goes well beyond the projection of superficial similarities (expecting future crows to be black because all observed crows have been black); it requires an account of the causal mechanism, which we can then test by creating circumstances which would very likely not occur in nature without our intervention.
Arguably, traditional empiricism generalized inadequately on the procedures by which all humans learn about their environment. Some things are learned by simple
habituation, developing a uniform response to similar stimuli, but a great deal of human learning involves interfering with the environment.
Such a response to traditional empiricism and the problem it has with induction will, however, appear to beg the question unless at the same time one calls into question the assumptions that observation consists in the passive reception of sensory qualities and that the concepts we apply to what we observe derive wholly from this source rather than, in a large and important part, from the control we are able to exercise over what we observe. Bacon, for one, saw that induction properly conducted (i.e. as experimental inquiry) needed to be used not only 'to discover axioms but also in the formation of notions'. If we allow that induction is a procedure through which we develop our concepts of what is or is not a (natural) possibility the traditional problem of induction appears in a quite different light.
(Published 1987)— J. E. Tiles
Bibliography- Bacon, F. (1620). The New Organon (1960 edn.), esp. Bk. I.
- Hume, D. (1739–40). A Treatise of Human Nature (1888 edn.), Bk. I, pt. iii.
- Mill, J. S. (1843). A System of Logic (1879 edn.), Bk. III.
- Peirce, C. S. (1955). Philosophical Writings of Peirce. Ed. J. Buchler, chs. 11–15.
- Swinburne, R. (ed.) (1974). The Justification of Induction.
