Online research in philosophy

This chapter1 concerns the relation between statistics and inductive logic. I start by describing induction in formal terms, and I introduce a general notion of probabilistic inductive inference. This provides a setting in which statistical procedures and inductive logics can be cap- tured. Speciacally, I discuss three statistical procedures (hypotheses testing, parameter estimation, and Bayesian statistics) and I show to what extend they can be captured by certain inductive logics. I end with some suggestions on how inductive.

This paper aims to be a friendly introduction to formal learning theory. I introduce key concepts at a slow pace, comparing and contrasting with other approaches to inductive inference such as con…rmation theory. A number of examples are discussed, some in detail, such as Goodman’s Riddle of Induction. I outline some important results of formal learning theory that are of philosophical interest. Finally, I discuss recent developments in this approach to inductive inference.

In this article I take a loose, functional approach to defining induction: Inductive forms of reasoning include those prima facie reasonable inference patterns that one finds in science and elsewhere that are not clearly deductive. Inductive inference is often taken to be reasoning from the observed to the unobserved. But that is incorrect, since the premises of inductive inferences may themselves be the results of prior inductions. A broader conception of inductive inference regards any ampliative inference as inductive, where an ampliative inference is one where the conclusion ‘goes beyond’ the premises. ‘Goes beyond’ may mean (i) ‘not deducible from’ or (ii) ‘not entailed by’. Both of these are problematic. Regarding (i), some forms of reasoning might have a claim to be called ‘inductive’ because of their role in science, yet turn out to be deductive after all—for example eliminative induction (see below) or Aristotle’s ‘perfect induction’ which is an inference to a generalization from knowledge of every one of its instances. Interpretation (ii) requires that the conclusions of scientific reasoning are always contingent propositions, since necessary propositions are entailed by any premises. But there are good reasons from metaphysics for thinking that many general propositions of scientific interest and known by inductive inference (e.g. “all water is H2O”) are necessarily true. Finally, both (i) and (ii) fail to take account of the fact that there are many ampliative forms of inference one would not want to call inductive, such as counter-induction (exemplified by the ‘gambler’s fallacy’ that the longer a roulette wheel has come up red the more likely it is to come up black on the next roll). Brian Skyrms (1999) provides a useful survey of the issues involved in defining what is meant by ‘inductive argument’. Inductive knowledge will be the outcome of a successful inductive inference. But much discussion of induction concerns the theory of confirmation, which seeks to answer the question, “when and to what degree does evidence support an hypothesis?” Usually, this is understood in an incremental sense and in a way that relates to the rational credibility of a hypothesis: “when and by how much does e add to the credibility of h?”, although ‘confirms’ is sometimes used in an absolute sense to indicate total support that exceeds some suitably high threshold..

Since the mid-1970s, scholars have recognized that the skeptical interpretation of Hume’s central argument about induction is problematic. The science of human nature presupposes that inductive inference is justified and there are endorsements of induction throughout Treatise Book I. The recent suggestion that I.iii.6 is confined to the psychology of inductive inference cannot account for the epistemic flavor of its claims that neither a genuine demonstration nor a non-question-begging inductive argument can establish the uniformity principle. For Hume, that inductive inference is justified is part of the data to be explained. Bad argument is therefore excluded as the cause of inductive inference; and there is no good argument to cause it. Does this reinstate the problem of induction, undermining Hume’s own assumption that induction is justified? It does so only if justification must derive from “reason”, from the availability of a cogent argument. Hume rejects this internalist thesis; induction’s favorable epistemic status derives from features of custom, the mechanism that generates inductive beliefs. Hume is attracted to this externalist posture because it provides a direct explanation of the epistemic achievements of children and non-human animals—creatures that must rely on custom unsupplemented by argument.

My purpose in this chapter is to survey some of the principal approaches to inductive inference in the philosophy of science literature. My first concern will be the general principles that underlie the many accounts of induction in this literature. When these accounts are considered in isolation, as is more commonly the case, it is easy to overlook that virtually all accounts depend on one of very few basic principles and that the proliferation of accounts can be understood as efforts to ameliorate the weaknesses of those few principles. In the earlier sections, I will lay out three inductive principles and the families of accounts of induction they engender. In later sections I will review standard problems in the philosophical literature that have supported some pessimism about induction and suggest that their import has been greatly overrated. In the final sections I will return to the proliferation of accounts of induction that frustrates efforts at a final codification. I will suggest that this proliferation appears troublesome only as long as we expect inductive inference to be subsumed under a single formal theory. If we adopt a material theory of induction in which individual inductions are licensed by particular facts that prevail only in local domains, then the proliferation is expected and not problematic.

It has been common wisdom for centuries that scientific inference cannot be deductive; if it is inference at all, it must be a distinctive kind of inductive inference. According to demonstrative theories of induction, however, important scientific inferences are not inductive in the sense of requiring ampliative inference rules at all. Rather, they are deductive inferences with sufficiently strong premises. General considerations about inferences suffice to show that there is no difference in justification between an inference construed demonstratively or ampliatively. The inductive risk may be shouldered by premises or rules, but it cannot be shirked. Demonstrative theories of induction might, nevertheless, better describe scientific practice. And there may be good methodological reasons for constructing our inferences one way rather than the other. By exploring the limits of these possible advantages, I argue that scientific inference is neither of essence deductive nor of essence inductive.

The use of the material theory of induction to vindicate a scientist’s claims of evidential warrant is illustrated with the cases of Einstein’s thermodynamic argument for light quanta of 1905 and his recovery of the anomalous motion of Mercury from general relativity in 1915. In a survey of other accounts of inductive inference applied to these examples, I show that, if it is to succeed, each account must presume the same material facts as the material theory and, in addition, some general principle of inductive inference not invoked by the material theory. Hence these principles are superfluous and the material theory superior in being more parsimonious.

In a material theory of induction, inductive inferences are warranted by facts that prevail locally. This approach, it is urged, is preferable to formal theories of induction in which the good inductive inferences are delineated as those conforming to some universal schema. An inductive inference problem concerning indeterministic, non-probabilistic systems in physics is posed and it is argued that Bayesians cannot responsibly analyze it, thereby demonstrating that the probability calculus is not the universal logic of induction.

For several years, through the “material theory of induction,” I have urged that inductive inferences are not licensed by universal schemas, but by material facts that hold only locally (Norton, 2003, 2005). My goal has been to defend inductive inference against inductive skeptics by demonstrating when and how inductive inferences are properly made. Since I have always admired Peter Achinstein as a staunch defender of induction, it was a surprise when Peter..

In a formal theory of induction, inductive inferences are licensed by universal schemas. In a material theory of induction, inductive inferences are licensed by facts. With this change in the conception of the nature of induction, I argue that Hume’s celebrated “problem of induction” can no longer be set up and is thereby dissolved.