Some remarks on the Rationality of Induction

This paper begins with a rigorous critique of David Stove's recent bookThe Rationality of Induction. In it, Stove produced four different proofs to refute Hume's sceptical thesis about induction. I show that Stove's attempts to vindicate induction are unsuccessful. Three of his proofs refute theses that are not the sceptical thesis about induction at all. Stove's fourth proof, which uses the sampling principle to justify one particular inductive inference, makes crucial use of an unstated assumption regarding randomness. Once this assumption is made explicit, Hume's thesis once more survives.The refutation of Stove's fourth proof leads to some observations which relate Goodman's grue paradox with randomness of a sample. I formulate a generalized version of Goodman's grue paradox, and argue that whenever a sample, no matter how large, is drawn from a predetermined smaller interval of a population that is distributed over a larger interval, any conclusion drawn about the characteristics of the population based on the observed characteristics of the sample is fatally vulnerable to the generalized grue paradox.Finally, I argue that the problem of justification of induction can be addressed successfully only from a cognitive point of view, but not from a metaphysical one. That is, we may ask whether an inductive inference is justified or not within the theories or cognitive structures of a subject, but not outside them. With this realization, induction is seen as a cognitive process, not unlike vision, that is useful at times, and yet has its own illusions that may make it a serious obstacle to cognition at other times.     

PREDICTION AND PREDICATION

Nelson Goodman's own solution to his new riddle of induction turns on the degree to which predicates are entrenched in our use of language. However, this solution requires that judgements concerning the degree to which a predicate is entrenched can be made independently of any canon of perceptible similarity. I argue that this requirement cannot be met. The riddle itself depends upon the claim that since ‘green’ can be defined positionally in terms of ‘grue’ and ‘bleen’, ‘grue’ and ‘bleen’ are symmetrically related to ‘green’ and ‘blue’, and positionality is therefore ‘an entirely relative matter’. However, in order to establish that positionality is an entirely relative matter, we would need to provide a non-positional definition of ‘grue’, rather than a positional definition of ‘green’. We therefore need to decide whether ‘grue’ can be defined ostensively by means of a single sample, as ‘green’ can. I argue that any attempt to provide such a definition is bound to fail, and hence the new riddle of induction does not reveal that ‘the roots of inductive validity are to be found in our use of language’.     

Testability and Ockham’s Razor: How Formal and Statistical Learning Theory Converge in the New Riddle of Induction

Nelson Goodman’s new riddle of induction forcefully illustrates a challenge that must be confronted by any adequate theory of inductive inference: provide some basis for choosing among alternative hypotheses that fit past data but make divergent predictions. One response to this challenge is to distinguish among alternatives by means of some epistemically significant characteristic beyond fit with the data. Statistical learning theory takes this approach by showing how a concept similar to Popper’s notion of degrees of testability is linked to minimizing expected predictive error. In contrast, formal learning theory appeals to Ockham’s razor, which it justifies by reference to the goal of enhancing efficient convergence to the truth. In this essay, I show that, despite their differences, statistical and formal learning theory yield precisely the same result for a class of inductive problems that I call strongly VC ordered, of which Goodman’s riddle is just one example.     

ROBUSTNESS AND THE NEW RIDDLE REVIVED

The problem of induction is perennially important in epistemology and the philosophy of science. In response to Goodman's ‘New Riddle of Induction’, Frank Jackson made a compelling case for there being no new riddle, by arguing that there are no nonprojectible properties. Although Jackson's denial of nonprojectible properties is correct, I argue here that he is mistaken in thinking that he thereby shows that there is no new riddle of induction, and demonstrate that his solution to the grue paradox fails to rule out the possibility of equally justified contradictory inductions. More importantly, in illuminating where Jackson's argument fails, the paper casts a new light on the problem of induction, locating the problem not in the nature of the next (unexamined) x, but in the counterfactual robustness of properties of already examined x's.     

Inductivism and probabilism

Synopsis I I set out my view that all inference is essentially deductive and pinpoint what I take to be the major shortcomings of the induction rule.II The import of data depends on the probability model of the experiment, a dependence ignored by the induction rule. Inductivists admit background knowledge must be taken into account but never spell out how this is to be done. As I see it, that is the problem of induction.III The induction rule, far from providing a method of discovery, does not even serve to detect pattern. Knowing that there is uniformity in the universe is no help to discovering laws. A critique of Reichenbach's justification of the straight rule is constructed along these lines.IV The induction rule, by itself, cannot account for the varying rates at which confidence in an hypothesis mounts with data. The mathematical analysis of this salient feature of inductive reasoning requires prior probabilities. We also argue, against orthodox statisticians, that prior probabilities make a substantive contribution to the objectivity of inductive methods, viz. to the design of experiments and the selection of decision rules.V Carnap's general criticisms of various estimation rules, like the straight rule and the impervious rule, are seen to be misguided when the prior densities to which they correspond are taken into account.VI Analysis of Hempel's definition of confirmation qua formalization of the enumerative (naive) conception of instancehood. We show that from the standpoint of the quantitative measure P(H/E):P(H) for the degree to which E confirms H, Hempel's classificatory concept yields correct results only for sampling at large from a finite population with a two-way classification all of whose compositions are equally probable. We extend the analysis to Goodman's paradox, finding cases in which grue-like hypotheses do receive as much confirmation as their opposite numbers. We argue, moreover, the irrelevancy of entrenchment, and maintain that Goodman's paradox is no more than a straightforward counter-example to the enumerative conception of instancehood embodied in Hempel's definition.VII We rebutt the objection that prior probabilities, qua inputs of Bayesian analysis, can only be obtained by enumerative induction (insofar as they are objective). The divergence in the prior densities of two rational agents is less a function of subjectivity, we maintain, than of vagueness.VIII Our concluding remarks stress that, for Bayesians, there is no problem of induction in the usual sense.     

On probabilism and induction

R. C. Jeffrey has proposed probabilism as a solution to Hume's problem of justifying induction. This paper shows that the assumptions of his Estimation Theorem, used to justify induction, can be weakened to provide a more satisfactory interpretation. It is also questioned whether the use of probabilism adds significantly to our understanding (or even Hume's understanding) of the problem of induction.     

On Not Changing the Problem: A Reply to Howson

Howson's critique of my essay on Hume's problem of induction levels two main charges. First, Howson claims that I have attributed to him an error that he never made, and in fact which he warned against in the very text that I cite. Secondly, Howson argues that my proposed solution to Hume's problem is flawed on technical and philosophical grounds. In response to the first charge, I explain how Howson's text justifies attributing to him the claim that the principle of induction is shown to be inconsistent by Goodman's riddle. In regards to the second, I show that Howson's objections rest on misunderstandings of formal learning theory and on conflating the problem of induction with the problem of unconceived alternatives.     

A material dissolution of the problem of induction

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 the celebrated “problem of induction” can no longer be set up and is thereby dissolved. Attempts to recreate the problem in the material theory of induction fail. They require relations of inductive support to conform to an unsustainable, hierarchical empiricism.     

CAN RATIONALIST ABDUCTIVISM SOLVE THE PROBLEM OF INDUCTION?

Abstract: According to Laurence BonJour, the problem of induction can be solved by recognizing the a priori necessity that inductive conclusions constitute the best explanations of inductive premises. I defend an interpretation of the key probability claims BonJour makes about inductive premises and show that they are not susceptible to many of the objections that have been lodged against them. I then argue that these purportedly necessary probability claims nevertheless remain deeply problematic and that, as a result, BonJour's proposal fails to provide a satisfactory resolution of the problem of induction.     

Humean Naturalism and the Problem of Induction

Naturalised epistemology has shunned rationality, a hallmark of humanity since ancient Greece. One of Quine's explicit motivations is that Hume's problem of induction cannot be solved. However, Hume himself suggests a solution and the narrow focus of the paper is to present a 'Humean Solution' which is an elaboration and defence of Hume's suggestion. What emerges will be argued to be a naturalised conception of rationality which makes naturalised epistemology more continuous with traditional epistemology's focus on rationality.     

Causation as a Philosophical Relation in Hume

By giving the proper emphasis to both radical skepticism and naturalism as two independent standpoints in Hume, I wish to propose a more satisfactory account of some of the more puzzling Humean claims on causation. I place these claims alternatively in either the philosophical standpoint of the radical skeptic or in the standpoint of everyday and scientific beliefs. I characterize Hume's radical skeptical standpoint in relation to Hume's perceptual model of the traditional theory of ideas, and I argue that Hume's radical skeptical argument concerning our causal inferences is inextricably linked to his skeptical argument concerning our idea of a necessary connection between cause and effect. I discuss Hume's naturalistic account of the origin of our idea of necessity and offer a new reading of Hume's two "definitions" of cause. I argue along the way against central aspects of two opposing styles of interpretation—Norman Kemp Smith's and Annette Baier's, on the one hand, and Robert Fogelin's, on the other—that in my view do not appreciate the mutual autonomy of radical skepticism and naturalism in Hume.     

Psychology, epistemology, and skepticism in Hume’s argument about induction

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.     

意见收敛定理与休谟问题

意见收敛定理是主观主义概率论的一条重要定理,它表明随着证据的增加,验前概率的主观性将被验后概率的客观性所代替。意见收敛定理被看作主观概率的动态合理性原则,因而被用来解决休谟问题,即归纳合理性问题。然而,哈金有说服力地表明,意见收敛定理证明的是条件概率Pr（h／e）的收敛,而不是验后概率Pre（h）的收敛。主观主义概率论暗中接受的一个等式是：Pre（h）=Pr（h／e）,通常称之为“条件化规则”。这样,归纳法的合理性问题变成条件化规则的合理性问题。为此,本文提出一个新的合理性原则,即“最少初始概率原则”,将它同“局部合理性”观念结合起来便可为条件化规则的合理性加以辩护。     

"是-应该"问题及其解答

休谟提出如何从“是”推出“应该”,亦即如何从事实判断推出价值判断的问题。本文以该问题为线索,评述和澄清休谟道德哲学的主要观点,并对与此问题密切相关的“囚徒困境”作了探讨。在此基础上,提出一种新的解答,即：“是一应该”问题是不能彻底解决的,在这个意义上,休谟的“是一应该”论题是正确的。但是,在另一种意义上,“是一应该”问题是可以解决的,即对于那些已经把长远利益作为追求目标的人而言是如此。因为对他们来说,追求长远利益已经是一个事实,因而必须考虑与他人或社会的合作,这使他们原有的同情心得到加强,从而产生名誉感和自尊心,进而达到正义和仁慈的德行。这样便从追求长远利益的“是”得出正义和仁慈的“应该”。     

VII—A Compleat Chain of Reasoning: Hume's Project inA Treatise of Human Nature, Books One and Two

In this paper I consider the context and significance of the first instalment of Hume's A Treatise of Human Nature , Books One and Two, on the understanding and on the passions, published in 1739 without Book Three. I argue that Books One and Two taken together should be read as addressing the question of the relation between reason and passion, and place Hume's discussion in the context of a large early modern philosophical literature on the topic. Hume's goal is to show that the passions do not require government by reason, and to illustrate various ways in which the passions of social beings regulate themselves. The underlying theme of the first Treatise is thus a new theory of sociability: sympathetic sociability.     

The world in itself: Neither uniform nor physical

Since Hume, philosophers of induction have debated the question of whether we have any reason for assuming that nature is uniform. This debate has always presumed that the uniformity hypothesis is itself coherent. In Part 1 of the following I argue that a proper appreciation of Nelson Goodman's so-called grue-green problem¹ should lead us to the conclusion that the uniformity hypothesis, under its usual interpretation as a strictly ontological thesis, is incoherent. In Part 2 I argue that further consideration of the grue-green problem leads to the conclusion that certain popular versions of the thesis of physical supervenience/the primacy of physics, under their usual interpretation as strictly ontological theses, are false. In Part 3 I argue that the notions of natural kinds and nature's joints should not be taken as ontologically objective notions but as interest relative. Together Parts 1, 2, and 3 provide support for the Nietzsche-Goodman thesis that philosophers are prone to mistakenly identify as absolute, mind and language independent, features of the world which are in fact only features of a particular discourse, or of the world relative to a particular discourse.     

Integrating Hume's Accounts of Belief and Justification

Hume's claim that a state is a belief is often intertwined—though without his remarking on this fact—with epistemic approval of the state. This requires explanation. Beliefs, in Hume's view, are steady dispositions (not lively ideas), nature's provision for a steady influence on the will and action. Hume's epistemic distinctions call attention to circumstances in which the presence of conflicting beliefs undermine a belief's influence and thereby its natural function. On one version of this interpretation, to say that a belief is justified, ceteris paribus , is to say that for all that has been shown the belief would be steady in its influence under suitable reflection. On a second version, it is to say that prima facie justification is an intrinsic property of the state, in virtue of its steadiness. These versions generate different understandings of the relationship between Parts iii and iv of Book I of the Treatise .     

Tracking Track Records: John WorrallRelying on Meta-induction?

From a reliabilist point of view, our inferential practices make us into instruments for determining the truth value of hypotheses where, like all instruments, reliability is a central virtue. I apply this perspective to second-order inductions, the inductive assessments of inductive practices. Such assessments are extremely common, for example whenever we test the reliability of our instruments or our informants. Nevertheless, the inductive assessment of induction has had a bad name ever since David Hume maintained that any attempt to justify induction by means of an inductive argument must beg the question. I will consider how the inductive justification of induction fares from the reliabilist point of view. I will also consider two other well-known arguments that can be construed as inductive assessments of induction. One is the miracle argument, according to which the truth of scientific theories should be inferred as the best explanation of their predictive success; the other is the disaster argument, according to which we should infer that all present and future theories are false on the grounds that all past theories have been found to be false.
'And here it is constantly supposed, that there is a connexion between the present fact and that which is inferred from it. Were there nothing to bind them together, the inference would be entirely precarious.' (David Hume, Enquiry IV)