Kevin Kelly

The problem of induction reminds us that science cannot wait for empirical hypotheses to be verified and Duhem’s problem reminds us that we cannot expect full refutations either. We must settle for something less. The shape of this something less depends on which features of full verification and refutation we choose to emphasize. If we conceive of verification and refutation as arguments in which evidence entails the hypothesis or its negation, then the central problem of the philosophy of science is to explicate a relation of confirmation or support that is weaker than full entailment but which serves, nonetheless, to justify empirical conclusions.

Many distinct theories are compatible with current experience. Scientific realists recommend that we choose the simplest. Anti-realists object that such appeals to “Ockham’s razor” cannot be truth-conducive, since they lead us astray in complex worlds. I argue, on behalf of the realist, that always preferring the simplest theory compatible with experience is necessary for efficient convergence to the truth in the long run, even though it may point in the wrong direction in the short run. Efficiency is a matter of minimizing errors or retractions prior to convergence to the truth.

One construal of convergent realism is that for each clear question, scientific inquiry eventually answers it. In this paper we adapt the techniques of formal learning theory to determine in a precise manner the circumstances under which this ideal is achievable. In particular, we define two criteria of convergence to the truth on the basis of evidence. The first, which we call EA convergence, demands that the theorist converge to the complete truth "all at once". The second, which we call AE convergence, demands only that for every sentence in the theorist's language, there is a time at which the theorist settles the status of the sentence. The relative difficulties of these criteria are compared for effective and ineffective agents. We then examine in detail how the enrichment of an agent's hypothesis language makes the task of converging to the truth more difficult. In particular, we parametrize first-order languages by predicate and function symbol arity, presence or absence of identity, and quantifier prefix complexity. For nearly each choice of values of these parameters, we determine the senses in which effective and ineffective agents can converge to the complete truth on an arbitrary structure for the language. Finally, we sketch directions in which our learning theoretic setting can be generalized or made more realistic

Synchronic norms of theory choice, a traditional concern in scientific methodology, restrict the theories one can choose in light of given information. Diachronic norms of theory change, as studied in belief revision, restrict how one should change one’s current beliefs in light of new information. Learning norms concern how best to arrive at true beliefs. In this paper, we undertake to forge some rigorous logical relations between the three topics. Concerning, we explicate inductive truth conduciveness in terms of optimally direct convergence to the truth, where optimal directness is explicated in terms of reversals and cycles of opinion prior to convergence. Concerning, we explicate Ockham’s razor and related principles of choice in terms of the information topology of the empirical problem context and show that the principles are necessary for reversal or cycle optimal convergence to the truth. Concerning, we weaken the standard principles of agm belief revision theory in intuitive ways that are also necessary for reversal or cycle optimal convergence. Then we show that some of our weakened principles of change entail corresponding principles of choice, completing the triangle of relations between,, and.

Synchronic norms of theory choice, a traditional concern in scientific methodology, restrict the theories one can choose in light of given information. Diachronic norms of theory change, as studied in belief revision, restrict how one should change one’s current beliefs in light of new information. Learning norms concern how best to arrive at true beliefs. In this paper, we undertake to forge some rigorous logical relations between the three topics. Concerning, we explicate inductive truth conduciveness in terms of optimally direct convergence to the truth, where optimal directness is explicated in terms of reversals and cycles of opinion prior to convergence. Concerning, we explicate Ockham’s razor and related principles of choice in terms of the information topology of the empirical problem context and show that the principles are necessary for reversal or cycle optimal convergence to the truth. Concerning, we weaken the standard principles of agm belief revision theory in intuitive ways that are also necessary for reversal or cycle optimal convergence. Then we show that some of our weakened principles of change entail corresponding principles of choice, completing the triangle of relations between,, and.

Over the past two decades, several consistent procedures have been designed to infer causal conclusions from observational data. We prove that if the true causal network might be an arbitrary, linear Gaussian network or a discrete Bayes network, then every unambiguous causal conclusion produced by a consistent method from non-experimental data is subject to reversal as the sample size increases any finite number of times. That result, called the causal flipping theorem, extends prior results to the effect that causal discovery cannot be reliable on a given sample size. We argue that since repeated flipping of causal conclusions is unavoidable in principle for consistent methods, the best possible discovery methods are consistent methods that retract their earlier conclusions no more than necessary. A series of simulations of various methods across a wide range of sample sizes illustrates concretely both the theorem and the principle of comparing methods in terms of retractions

then essentially characterized the hypotheses that mechanical scientists can successfully decide in the limit in terms of arithmetic complexity. These ideas were developed still further by Peter Kugel [4]. In this paper, I extend this approach to obtain characterizations of identification in the limit, identification with bounded mind-changes, and identification in the short run, both for computers and for ideal agents with unbounded computational abilities. The characterization of identification with n mind-changes entails, as a corollary, an exact arithmetic characterization of Putnam's n-trial predicates, which closes a gap of a factor of two in Putnam's original characterization [12].

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

Belief revision theory aims to describe how one should change one’s beliefs when they are contradicted by newly input information. The guiding principle of belief revision theory is to change one’s prior beliefs as little as possible in order to maintain consistency with the new information. Learning theory focuses, instead, on learning power: the ability to arrive at true beliefs in a wide range of possible environments. The goal of this paper is to bridge the two approaches by providing a learning theoretic analysis of the learning power of belief revision methods proposed by Spohn, Boutilier, Darwiche and Pearl, and others. The results indicate that learning power depends sharply on details of the methods. Hence, learning power can provide a well-motivated constraint on the design and implementation of concrete belief revision methods.

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].

I have applied a fairly general, learning theoretic perspective to some questions raised by Reichenbach's positions on induction and discovery. This is appropriate in an examination of the significance of Reichenbach's work, since the learning-theoretic perspective is to some degree part of Reichenbach's reliabilist legacy. I have argued that Reichenbach's positivism and his infatuation with probabilities are both irrelevant to his views on induction, which are principally grounded in the notion of limiting reliability. I have suggested that limiting reliability is still a formidable basis for the formulation of methodological norms, particularly when reliability cannot possibly be had in the short run, so that refined judgments about evidential support must depend upon measure-theoretic choices having nothing to do in the short run with the truth of the hypothesis under investigation. To illustrate the generality of Reichenbach's program, I showed how it can be applied to methods that aim to solve arbitrary assessment and discovery problems in various senses. In this generalized Reichenbachian setting, we can characterize the intrinsic complexity of reliable inductive inference in terms of topological complexity. Finally, I let Reichenbach's theory of induction have the last say about hypothetico-deductive method.

The approach to scientific methodology developed in my recent book The Logic of Reliable Inquiry (LRI) shares many general features with that summarized in Larry Laudan’s concurrently published collection of papers Beyond Positivism and Relativism (BPR). Nonetheless, this fact might not be apparent, as my own work emphasizes mathematical theorems, whereas Laudan’s draws primarily upon historiography. It is, therefore, of some interest to discuss the extent of the agreement and the significance of the differences. More generally, the discussion will (I) provide a logical analysis of the instrumental significance of empirical meta-methodology and (II) redefine the role of logic in a post-positivistic, naturalized approach to epistemology and scientific method.

This book is a tribute to Kevin Kelly, who has been one of the most influential British theologians for a number of decades. On its own merits, however, it is groundbreaking collection of essays on key themes, issues and concepts in contemporary moral theology and Christian ethics. The focus is on perspectives to inform moral debate and discernment in the future. The main themes covered are shown in the list of contents below. Several of the of the contributors are from the United States, three others live and work in Continental Europe and the rest are from various parts of the British Isles. Many of the authors are among the best known in their fields on both sides of the Atlantic.

Simplicity has long been recognized as an apparent mark of truth in science, but it is difficult to explain why simplicity should be accorded such weight. This chapter examines some standard, statistical explanations of the role of simplicity in scientific method and argues that none of them explains, without circularity, how a reliance on simplicity could be conducive to finding true models or theories. The discussion then turns to a less familiar approach that does explain, in a sense, the elusive connection between simplicity and truth. The idea is that simplicity does not point at or reliably indicate the truth but, rather, keeps inquiry on the cognitively most direct path to the truth