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 discusses Kevin Kelly’s recent attempt to justify Ockham’s Razor in terms of truth-finding efficiency. It is argued that Kelly’s justification fails to warrant confidence in the empirical content of theories recommended by Ockham’s Razor. This is a significant problem if, as Kelly and many others believe, considerations of simplicity play a pervasive role in scientific reasoning, underlying even our best tested theories, for the proposal will fail to warrant the use of these theories in practical prediction.

There is a tradition, tracing back to Kant, of recasting metaphysical questions as questions about the utility of a conceptual scheme, linguistic framework, or methodological rule for achieving some particular end. Following in this tradition, I propose a ‘means-ends metaphysics ’, in which one rigorously demonstrates the suitability of some conceptual framework for achieving a specified goal. I illustrate this approach using a debate about the nature of events. Specifically, the question is whether the time at which an event occurs (...) is an essential property of that event. I argue that this question is naturally transformed into a question about the methodology of causal modeling. In this new framework, the question concerns what kind of variables to use to represent the effects of potential interventions on a system. This question has a demonstrably correct answer, which sheds new light on the original question. (shrink)

In a recent article in this journal, Daniel Steel charges me with committing a fallacy in my discussion of inductive rules. I show that the charge is false, and that Steel's own attempt to validate enumerative induction in terms of formal learning theory is itself fallacious. I go on to argue that, contra Steel, formal learning theory is in principle incapable of answering Hume's famous claim that any attempt to justify induction will beg the question.

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)

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)

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)

Explaining the connection, if any, between simplicity and truth is among the deepest problems facing the philosophy of science, statistics, and machine learning. Say that an efficient truth finding method minimizes worst case costs en route to converging to the true answer to a theory choice problem. Let the costs considered include the number of times a false answer is selected, the number of times opinion is reversed, and the times at which the reversals occur. It is demonstrated that (1) (...) always choosing the simplest theory compatible with experience, and (2) hanging onto it while it remains simplest, is both necessary and sufficient for efficiency. †To contact the author, please write to: Department of Philosophy, Carnegie Mellon University, Baker Hall 135, Pittsburgh, PA 15213-3890; e-mail: [email protected]. (shrink)

Falsificationism has dominated 20th century philosophy of science. It seemed to have eclipsed all forms of inductivism. Yet recent debates have revived a specific form of eliminative inductivism, the basic ideas of which go back to F. Bacon and J.S. Mill. These modern endorsements of eliminative inductivism claim to show that progressive problem solving is possible using induction, rather than falsification as a method of justification. But this common ground between falsificationism and eliminative inductivism has not led to a detailed (...) investigation into the relationship, if any, which may exist between these two methodologies. This paper reviews several versions of eliminative inductivism, establishes a natural relation between eliminative inductivism and falsificationism, which derives from the distinction between models and theories, and carries out this investigation against a case study of the construction of atom models. The result of the investigation is that falsificationism is a form of eliminative inductivism in the limit of certain constraints. (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)

Explainable artificial intelligence (XAI) seeks to produce explanations for those machine learning methods which are deemed opaque. However, there is considerable disagreement about what this means and how to achieve it. Authors disagree on what should be explained (topic), to whom something should be explained (stakeholder), how something should be explained (instrument), and why something should be explained (goal). In this paper, I employ insights from means-end epistemology to structure the field. According to means-end epistemology, different means ought to be (...) rationally adopted to achieve different epistemic ends. Applied to XAI, different topics, stakeholders, and goals thus require different instruments. I call this the means-end account of XAI. The means-end account has a descriptive and a normative component: on the one hand, I show how the specific means-end relations give rise to a taxonomy of existing contributions to the field of XAI; on the other hand, I argue that the suitability of XAI methods can be assessed by analyzing whether they are prescribed by a given topic, stakeholder, and goal. (shrink)

Philosophy and Phenomenological Research, EarlyView.

Evaluative studies of inductive inferences have been pursued extensively with mathematical rigor in many disciplines, such as statistics, econometrics, computer science, and formal epistemology. Attempts have been made in those disciplines to justify many different kinds of inductive inferences, to varying extents. But somehow those disciplines have said almost nothing to justify a most familiar kind of induction, an example of which is this: “We’ve seen this many ravens and they all are black, so all ravens are black.” This is (...) enumerative induction in itsfullstrength. For it does not settle with a weaker conclusion (such as “the ravens observed in the future will all be black”); nor does it proceed with any additional premise (such as the statistical IID assumption). The goal of this paper is to take some initial steps toward a justification for the full version of enumerative induction, against counterinduction, and against the skeptical policy. The idea is to explore various epistemic ideals, mathematically defined as different modes of convergence to the truth, and look for one that is weak enough to be achievable and strong enough to justify a norm that governs both the long run and the short run. So the proposal is learning-theoretic in essence, but a Bayesian version is developed as well. (shrink)

A longstanding philosophical premise perceives simplicity as a desirable attribute of scientific theories. One of several raised justifications for this notion is that simple theories are more likely to indicate the true makeup of natural systems. Qualitatively parsimonious hypotheses and theories keep to a minimum the number of different postulated entities within a system. Formulation of such ontologically simple working hypotheses proved to be useful in the experimental probing of narrowly defined bio systems. It is less certain, however, whether qualitatively (...) parsimonious theories are effective indicators of the true nature of complex biological systems. This paper assesses the success of ontologically simple theories in envisaging the makeup of three complex systems in bacteriology, immunology, and molecular biology. Evidence shows that parsimonious theories completely misconstrued the actual ontologically complex constitutions of the three examined systems. Since evolution and selective pressures typically produce ontologically intricate rather than simple bio systems, qualitatively parsimonious theories are mostly inapt indicators of the true nature of complex biological systems. (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.

Occam’s razor directs us to adopt the simplest hypothesis consistent with the evidence. Learning theory provides a precise definition of the inductive simplicity of a hypothesis for a given learning problem. This definition specifies a learning method that implements an inductive version of Occam’s razor. As a case study, we apply Occam’s inductive razor to causal learning. We consider two causal learning problems: learning a causal graph structure that presents global causal connections among a set of domain variables, and learning (...) context-sensitive causal relationships that hold not globally, but only relative to a context. For causal graph learning, Occam’s inductive razor directs us to adopt the model that explains the observed correlations with a minimum number of direct causal connections. For expanding a causal graph structure to include context-sensitive relationships, Occam’s inductive razor directs us to adopt the expansion that explains the observed correlations with a minimum number of free parameters. This is equivalent to explaining the correlations with a minimum number of probabilistic logical rules. The paper provides a gentle introduction to the learning-theoretic definition of inductive simplicity and the application of Occam’s razor for causal learning. (shrink)

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)

This paper pursues a thorough-going instrumentalist, or means-ends, approach to the theory of inductive inference. I consider three epistemic aims: convergence to a correct theory, fast convergence to a correct theory and steady convergence to a correct theory (avoiding retractions). For each of these, two questions arise: (1) What is the structure of inductive problems in which these aims are feasible? (2) When feasible, what are the inference methods that attain them? Formal learning theory provides the tools for a complete (...) set of answers to these questions. As an illustration of the results, I apply means-ends analysis to various versions of Goodman's Riddle of Induction. (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)

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)

Formal learning theory is the mathematical embodiment of a normative epistemology. It deals with the question of how an agent should use observations about her environment to arrive at correct and informative conclusions. Philosophers such as Putnam, Glymour and Kelly have developed learning theory as a normative framework for scientific reasoning and inductive inference.

Statistical learning theory is often associated with the principle of Occam’s razor, which recommends a simplicity preference in inductive inference. This paper distills the core argument for simplicity obtainable from statistical learning theory, built on the theory’s central learning guarantee for the method of empirical risk minimization. This core “means-ends” argument is that a simpler hypothesis class or inductive model is better because it has better learning guarantees; however, these guarantees are model-relative and so the theoretical push towards simplicity is (...) checked by our prior knowledge. (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)

I propose that empirical procedures, like computational procedures, are justified in terms of truth-finding efficiency. I contrast the idea with more standard philosophies of science and illustrate it by deriving Ockham's razor from the aim of minimizing dramatic changes of opinion en route to the truth.