Philosophy and Phenomenological Research, EarlyView.

Philosophy and Phenomenological Research, EarlyView.

This article argues that a successful answer to Hume's problem of induction can be developed from a sub-genre of philosophy of science known as formal learning theory. One of the central concepts of formal learning theory is logical reliability: roughly, a method is logically reliable when it is assured of eventually settling on the truth for every sequence of data that is possible given what we know. I show that the principle of induction (PI) is necessary and sufficient for logical (...) reliability in what I call simple enumerative induction. This answer to Hume's problem rests on interpreting PI as a normative claim justified by a non-empirical epistemic means-ends argument. In such an argument, a rule of inference is shown by mathematical or logical proof to promote a specified epistemic end. Since the proof concerning PI and logical reliability is not based on inductive reasoning, this argument avoids the circularity that Hume argued was inherent in any attempt to justify PI. (shrink)

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. (shrink)

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. (shrink)

This paper describes the corner-stones of a means-ends approach to the philosophy of inductive inference. I begin with a fallibilist ideal of convergence to the truth in the long run, or in the 'limit of inquiry'. I determine which methods are optimal for attaining additional epistemic aims (notably fast and steady convergence to the truth). Means-ends vindications of (a version of) Occam's Razor and the natural generalizations in a Goodmanian Riddle of Induction illustrate the power of this approach. The paper (...) establishes a hierarchy of means-ends notions of empirical success, and discusses a number of issues, results and applications of means-ends epistemology. (shrink)

This paper develops a means–end analysis of an inductive problem that arises in particle physics: how to infer from observed reactions conservation principles that govern all reactions among elementary particles. I show that there is a reliable inference procedure that is guaranteed to arrive at an empirically adequate set of conservation principles as more and more evidence is obtained. An interesting feature of reliable procedures for finding conservation principles is that in certain precisely defined circumstances they must introduce hidden particles. (...) Among the reliable inductive methods there is a unique procedure that minimizes convergence time as well as the number of times that the method revises its conservation principles. Thus the aims of reliable, fast and steady convergence to an empirically adequate theory single out a unique optimal inference for a given set of observed reactions—including prescriptions for when exactly to introduce hidden particles. (shrink)

In his commentary on my paper, “Means-Ends Epistemology”, David Chart constructs a Riddle of Induction with the following feature: Means-ends analysis, as I formulated it in the paper, selects “all emeralds are grue” as the optimal conjecture after observing a sample of all green emeralds. Chart’s construction is rigorous and correct. If we disagree, it is in the philosophical morals to be drawn from his example. Such morals are best discussed by elucidating some of the larger epistemological issues involved. “Means-ends (...) Epistemology” sought a normative theory of hypothesis selection. I defined what it means for an inductive method to reliably and efficiently find a correct hypothesis from a set of alternative hypotheses. (In fact, I investigated a number of standards of empirical success for inductive methods.) Call such methods optimal. We may take optimal inferences to be those made by optimal methods. This defines a relation Optimal-Inference(h,e,H): “given the set of alternative hypotheses H, and evidence e, hypothesis h is an optimal inference”. One fundamental difference between the means-ends approach and traditional confirmation theory is that the latter has sought a two-place relation between theory and evidence alone, something like “hypothesis h is highly confirmed given evidence e”. From my point of view, posing the problem of induction as discerning the right relation between theory and evidence is elliptical because it leaves unspecified the set of alternative hypotheses under investigation (as well as other relevant factors, such as the scientist’s background knowledge, observational means, cognitive capacities and epistemic values). Chart’s Riddle highlights the fact that depending on the space of alternative hypotheses, means-ends analysis may select a different hypothesis on the same evidence: “all emeralds are green” in my Goodmanian Riddle, and “all emeralds are grue” in his. To my mind, his example.. (shrink)

Severity of Test (SoT) is an alternative to Popper's logical falsification that solves a number of problems of the logical view. It was presented by Popper himself in 1963. SoT is a less sophisticated probabilistic model of hypothesis testing than Oaksford & Chater's (O&C's) information gain model, but it has a number of striking similarities. Moreover, it captures the intuition of everyday hypothesis testing.

Human cognition requires coping with a complex and uncertain world. This suggests that dealing with uncertainty may be the central challenge for human reasoning. In Bayesian Rationality we argue that probability theory, the calculus of uncertainty, is the right framework in which to understand everyday reasoning. We also argue that probability theory explains behavior, even on experimental tasks that have been designed to probe people's logical reasoning abilities. Most commentators agree on the centrality of uncertainty; some suggest that there is (...) a residual role for logic in understanding reasoning; and others put forward alternative formalisms for uncertain reasoning, or raise specific technical, methodological, or empirical challenges. In responding to these points, we aim to clarify the scope and limits of probability and logic in cognitive science; explore the meaning of the explanation of cognition; and re-evaluate the empirical case for Bayesian rationality. (shrink)

A standard way to challenge convergence-based accounts of inductive success is to claim that they are too weak to constrain inductive inferences in the short run. We respond to such a challenge by answering some questions raised by Juhl (1994). When it comes to predicting limiting relative frequencies in the framework of Reichenbach, we show that speed-optimal convergence—a long-run success condition—induces dynamic coherence in the short run.

Ockham’s razor is the principle that, all other things being equal, scientists ought to prefer simpler theories. In recent years, philosophers have argued that simpler theories make better predictions, possess theoretical virtues like explanatory power, and have other pragmatic virtues like computational tractability. However, such arguments fail to explain how and why a preference for simplicity can help one find true theories in scientific inquiry, unless one already assumes that the truth is simple. One new solution to that problem is (...) the Ockham efficiency theorem, which states that scientists who heed Ockham’s razor retract their opinions less often and sooner than do their non-Ockham competitors. The theorem neglects, however, to consider competitors following random strategies and in many applications random strategies are known to achieve better worst-case loss than deterministic strategies. In this paper, we describe two ways to extend the result to a very general class of random, empirical strategies. The first extension concerns expected retractions, retraction times, and errors and the second extension concerns retractions in chance, times of retractions in chance, and chances of errors. (shrink)

Both early analytic philosophy and the branch of mathematics now known as topology were gestated and born in the early part of the 20th century. It is not well recognized that there was early interaction between the communities practicing and developing these fields. We trace the history of how topological ideas entered into analytic philosophy through two migrations, an earlier one conceiving of topology geometrically and a later one conceiving of topology algebraically. This allows us to reassess the influence and (...) significance of topological methods for philosophy, including the possible fruitfulness of a third conception of topology as a structure determining similarity. (shrink)

Bayesian models of human learning are becoming increasingly popular in cognitive science. We argue that their purported confirmation largely relies on a methodology that depends on premises that are inconsistent with the claim that people are Bayesian about learning and inference. Bayesian models in cognitive science derive their appeal from their normative claim that the modeled inference is in some sense rational. Standard accounts of the rationality of Bayesian inference imply predictions that an agent selects the option that maximizes the (...) posterior expected utility. Experimental confirmation of the models, however, has been claimed because of groups of agents that probability match the posterior. Probability matching only constitutes support for the Bayesian claim if additional unobvious and untested (but testable) assumptions are invoked. The alternative strategy of weakening the underlying notion of rationality no longer distinguishes the Bayesian model uniquely. A new account of rationality—either for inference or for decision-making—is required to successfully confirm Bayesian models in cognitive science. (shrink)

Oaksford & Chater (O&C) aim to provide teleological explanations of behavior by giving an appropriate normative standard: Bayesian inference. We argue that there is no uncontroversial independent justification for the normativity of Bayesian inference, and that O&C fail to satisfy a necessary condition for teleological explanations: demonstration that the normative prescription played a causal role in the behavior's existence.

Some think that issues to do with scientific method are last century's stale debate; Popper was an advocate of methodology, but Kuhn, Feyerabend, and others are alleged to have brought the debate about its status to an end. The papers in this volume show that issues in methodology are still very much alive. Some of the papers reinvestigate issues in the debate over methodology, while others set out new ways in which the debate has developed in the last decade. The (...) book will be of interest to philosophers and scientists alike in the reassessment it provides of earlier debates about method and current directions of research. (shrink)

Showcasing original arguments for well-defined positions, as well as clear and concise statements of sophisticated philosophical views, this volume is an ...

This essay demonstrates a previously unnoticed connection between formal and statistical learning theory with regard to Nelson Goodman’s new riddle of induction. Discussions of Goodman’s riddle in formal learning theory explain how conjecturing “all green” before “all grue” can enhance efficient convergence to the truth, where efficiency is understood in terms of minimizing the maximum number of retractions or “mind changes.” Vapnik-Chervonenkis (VC) dimension is a central concept in statistical learning theory and is similar to Popper’s notion of degrees of (...) testability. I show that for a class inductive problems of which Goodman’s riddle is one example, a reliable inductive method minimizes the maximum number of mind changes exactly if it always conjectures the hypothesis from the set with lowest VC dimension consistent with the data. I also discuss the relevance of these results to language invariance and curve fitting. (shrink)

This essay defends the view that inductive reasoning involves following inductive rules against objections that inductive rules are undesirable because they ignore background knowledge and unnecessary because Bayesianism is not an inductive rule. I propose that inductive rules be understood as sets of functions from data to hypotheses that are intended as solutions to inductive problems. According to this proposal, background knowledge is important in the application of inductive rules and Bayesianism qualifies as an inductive rule. Finally, I consider a (...) Bayesian formulation of inductive skepticism suggested by Lange. I argue that while there is no good Bayesian reason for judging this inductive skeptic irrational, the approach I advocate indicates a straightforward reason not to be an inductive skeptic. (shrink)

A finite data set is consistent with infinitely many alternative theories. Scientific realists recommend that we prefer the simplest one. Anti-realists ask how a fixed simplicity bias could track the truth when the truth might be complex. It is no solution to impose a prior probability distribution biased toward simplicity, for such a distribution merely embodies the bias at issue without explaining its efficacy. In this note, I argue, on the basis of computational learning theory, that a fixed simplicity bias (...) is necessary if inquiry is to converge to the right answer efficiently, whatever the right answer might be. Efficiency is understood in the sense of minimizing the least fixed bound on retractions or errors prior to convergence. (shrink)

I develop a critique of Hume’s infamous problem of induction based upon the idea that the principle of induction (PI) is a normative rather than descriptive claim. I argue that Hume’s problem is a false dilemma, since the PI might be neither a “relation of ideas” nor a “matter of fact” but rather what I call a contingent normative statement. In this case, the PI could be justified by a means-ends argument in which the link between means and end is (...) established solely by deductive reasoning. The means-ends argument is an elementary result from formal learning theory that you must be willing to make inductive generalizations if you want to be logically reliable in the types of examples Hume described. This justification of the PI avoids both horns of Hume’s dilemma. Since no contradiction ensues from rejecting logical reliability as an aim, the PI is contingent. Yet since the proof concerning the PI and logical reliability is not based on inductive reasoning, there is no threat of circularity. (shrink)