Inductive Reasoning

We information-theoretically reformulate two measures of capacity from statistical learning theory: empirical VC-entropy and empirical Rademacher complexity. We show these capacity measures count the number of hypotheses about a dataset that a learning algorithm falsifies when it finds the classifier in its repertoire minimizing empirical risk. It then follows from that the future performance of predictors on unseen data is controlled in part by how many hypotheses the learner falsifies. As a corollary we show that empirical VC-entropy quantifies the message (...) length of the true hypothesis in the optimal code of a particular probability distribution, the so-called actual repertoire. (shrink)

The paper demonstrates that falsifiability is fundamental to learning. We prove the following theorem for statistical learning and sequential prediction: If a theory is falsifiable then it is learnable -- i.e. admits a strategy that predicts optimally. An analogous result is shown for universal induction.

Inductive reasoning occurs when we examine a set of cases (a sample) and generalize on the basis of them. Hume's problem of inductive reasoning is legendary: inductive reasoning depends on an assumption of a regular world (future cases are like past cases etc.), the assumption is not self-evident and cannot be established by deduction nor (on pain of circularity) by inductive reasoning itself. (Note: Hume's problem is NOT best specified as: induction does not yield absolutely certain knowledge.) In the 20th (...) century, Nelson Goodman added his new problem of inductive reasoning. Unwisely perhaps, I want to join the club. This is my moral problem of induction. A person is brought up in a very strict moral household and he learns to accept in his childhood numerous moral rules, 400 let us say. One day as an adult he lists the 400. He examines number 400 and finds that he rejects it. The next day he examines number 399 and finds that he rejects it. The third day he examines number 398 and finds that he rejects it. When he gets to about number 50, he infers, "Probably all moral rules are wrong." He decides to become an amoralist. If you are a really moral person, I presume you think this: "You should not be able to exit morality like this. You have to keep going, in case one of the remaining moral rules turns out to withstand examination." But how can you argue for this perspective, if you accept the legitimacy of inductive reasoning? (Can you believe they rejected this from Analysis? What goes on in the evaluator's mind when he reads this?). (shrink)

This paper presents a paradox of inductive reasoning which occurred to me. You find that people in a certain profession, call it profession 1, do not do things in the way that you would expect. Likewise profession 2, profession 3, profession 4 ,and more. For some of these professions, you figure out why they might well depart from what you regard as the obvious or natural approach: why this actress intent on fame has only one memorable quote, say? Then you (...) suddenly find that people in a certain other profession, or one person at least, does things as you would expect. Given all your past experiences, this very doing of things as you would expect is paradoxical. Is this really what is going on? This football manager simply uses big strong mature reliable expensive players of some skill? This dramatist has produced a moving and superbly crafted tragedy featuring a legendary character? (shrink)

For years, the European world saw millions of swans, and all of them without exception were white. If inductive reasoning is valid, one may conclude that all swans are white. However, this would be incorrect: in 1667 Dutch explorer Willem de Vlamingh observed black swans in Australia, falsifying the hypothesis that all swans are white. While often used as a cautionary tale for the use of induction, such as with Popper’s falsification principle, I want to explore a slightly different idea: (...) does the existence of fabricated or otherwise failed accounts of Black swans give me a reason to ignore future arguments or doubt their existence? I argue that while faulty or fabricated reports do not give us a logical reason to reject a claim, they can give us a practical one for refusing to consider future arguments. This practical vs logical distinction gives an inductive reason for believing that future arguments will be invalid, but keeps one from being able to necessarily deny the claim in question without further argumentation. (shrink)

Inductivism is the conjoint thesis that there are genuine inductive inferences and that successful inductive inferences provide inductive knowledge. This paper presents two challenges to inductivism. The first is explaining how many observations are required for inductive inferences in a precise and non-arbitrary way. This is the threshold problem. The second is providing an account of how inductive knowledge can be a qualitative change from a mere quantity of observations. This is the qualitative leap problem. I anticipate and criticize some (...) attempts to solve these problems. One strategy is based on the idea that inferences are justification attempts and that different inference types require different theories of justification. Thus successful inductions can receive an externalist interpretation as reliable processes in which the evidential threshold isn't reflectively accessible. I object that this solution generates new versions of the threshold problem and doesn't address the qualitative leap problem. By relying on Galen's argument against induction, I also argue that any attempt to solve the two problems will be unsuccessful, since an evidential threshold for inductive inferences is implausible in principle. The conclusion is that inductive skepticism is justified. (shrink)

This paper provides an inductive interpretation of the sorites paradox. It is argued that the sorites paradox is an inductive meta-meta-inference that future modus ponens applications in a soritical series will be sound. The root of the paradox lies in the fact that inductive inferences rely on evidence and epistemic norms that are insensitive to incremental differences, since they are low probability events. This prevents an epistemicist solution given that cut-off points are incremental differences. This diagnosis is also adapted to (...) epistemic paradoxes such as the lottery paradox, the prediction paradox and the preface paradox. The other main idea of this paper is that the soritical nature of inductive inferences in addition to epistemicism motivates some arguments for inductive skepticism, such as that the evidentiary threshold of inductive inferences is unknowable, which can generate another soritical series that requires another unknowable cut-off point, and so on. The conclusion is that inductive inferences are unjustified. (shrink)

This paper introduces Crosshairs Audit Lab, a suite of nine interactive browser-based tools designed to help theologically inclined inquirers move from inherited confidence toward clearer, thinner, and more publicly defensible belief positions. The paper argues that the central problem in much apologetic reasoning is not simply false belief, but the collapse of several distinctions that responsible inquiry requires: between confidence and substantiation, private meaning and public evidence, thin conclusions and the thicker claims later built upon them, and sincerity and correction-readiness. (...) Framed as a design-and-pedagogy paper rather than an outcome study, the essay situates the suite within research on epistemic cognition, motivated reasoning, disconfirmation in religious belief, and scaffolded conceptual change. It argues that Crosshairs Audit Lab contributes a staged interactive method for teaching seekers how to audit their own claims without requiring immediate worldview abandonment. The suite's pedagogical aim is not to coerce unbelief, but to make belief answerable to explicit standards, explicit bridge premises, and explicit conditions of revision. (shrink)

The grue paradox, also called the new riddle of induction, posed a great challenge to the common understanding about induction. This paper shows that there is a close relation between the grue paradox and the problem of conditionals. This paper presents a general form of the grue predicate. Based on the general form, this paper argues that this kind of predicates can not be used for induction and prediction.

There are (at least) three approaches to quantifying information. The first, algorithmic information or Kolmogorov complexity, takes events as strings and, given a universal Turing machine, quantifies the information content of a string as the length of the shortest program producing it [1]. The second, Shannon information, takes events as belonging to ensembles and quantifies the information resulting from observing the given event in terms of the number of alternate events that have been ruled out [2]. The third, statistical learning (...) theory, has introduced measures of capacity that control (in part) the expected risk of classifiers [3]. These capacities quantify the expectations regarding future data that learning algorithms embed into classifiers. Solomonoff and Hutter have applied algorithmic information to prove remarkable results on universal induction. Shannon information provides the mathematical foundation for communication and coding theory. However, both approaches have shortcomings. Algorithmic information is not computable, severely limiting its practical usefulness. Shannon information refers to ensembles rather than actual events: it makes no sense to compute the Shannon information of a single string – or rather, there are many answers to this question depending on how a related ensemble is constructed. Although there are asymptotic results linking algorithmic and Shannon information, it is unsatisfying that there is such a large gap – a difference in kind – between the two measures. This note describes a new method of quantifying information, effective information, that links algorithmic information to Shannon information, and also links both to capacities arising in statistical learning theory [4, 5]. After introducing the measure, we show that it provides a non-universal analog of Kolmogorov complexity. We then apply it to derive basic capacities in statistical learning theory: empirical VC-entropy and empirical Rademacher complexity. A nice byproduct of our approach is an interpretation of the explanatory power of a learning algorithm in terms of the number of hypotheses it falsifies [6], counted in two different ways for the two capacities. We also discuss how effective information relates to information gain, Shannon and mutual information. (shrink)

On a popular and influential view, normative reasons are premises in good reasoning, where good reasoning manifests a pattern of reasoning that normally preserves fittingness. I develop two counterexamples to this view: isolated *pro tanto* reasons that are either weak or normally overridden.

In times of crisis, science tends to move fast. However, speeding up science often involves deliberate compromises to the scientific norms that are meant to guarantee the reliability of scientific results in normal circumstances. Call this hurried science. Some of the compromises involved in hurried science can be kept track of and later revisited—say, by conducting a new study with larger sample sizes and proper randomization. However, I argue that hurried science may also involve untracked compromises to reliability-enhancing scientific norms, (...) in which the negative epistemic effects of speeding up scientific research are not properly noticed or understood. I further argue that these are considerably more worrying for the overall reliability of scientific research, insofar as they are liable to lead both scientists and the general public to overestimate the trustworthiness of scientific results. (shrink)

The standard Bayesian solution to the paradox of the ravens maintains that the degree of confirmation provided by seeing a non-black non-raven is positive but negligible compared to that provided by seeing a black raven. I show that, unless we impose severe and unmotivated restrictions on the subject’s priors, this has the consequence that the cumulative confirmation provided by all the non-black non-ravens the subject expects to see is non-negligible compared to the cumulative confirmation provided by all the black ravens (...) the subject expects to see. If this is so, however, then the paradox retains its full force. (shrink)

The problem of induction continues to vex and beguile. How can we reliably draw universal conclusions from limited observations? To say we never can would itself be to make a universal statement. Maybe we need to step back and rethink long-held assumptions. • • • This is the first book to do that historically. It takes us from Socrates and Aristotle through to twentieth-century framers of the current conundrums. This comprehensive account does not look at how people of the past (...) answered the questions we ask but at how they understood the very meaning of the words epagōgē in Greek, inductio in Latin, istiqrāʾ in Arabic, Induktion in German, induction in English. The careful treatment of texts in their context dispels many long-standing myths. • • • Importantly, the account introduces us to a now unfamiliar way to think about what induction is—a way in which there simply is no “problem of induction.” It turns out the problem was one of our own making. An accurate history may help us recover old ways—and thereby introduce new ways—to think about the whole idea. (shrink)

Calibration inductive logics are based on accepting estimates of relative frequencies, which are used to generate imprecise probabilities. In turn, these imprecise probabilities are intended to guide beliefs and decisions — a process called “calibration”. Two prominent examples are Henry E. Kyburg's system of Evidential Probability and Jon Williamson's version of Objective Bayesianism. There are many unexplored questions about these logics. How well do they perform in the short-run? Under what circumstances do they do better or worse? What is their (...) performance relative to traditional Bayesianism? -/- In this article, we develop an agent-based model of a classic binomial decision problem, including players based on variations of Evidential Probability and Objective Bayesianism. We compare the performances of these players, including against a benchmark player who uses standard Bayesian inductive logic. We find that the calibrated players can match the performance of the Bayesian player, but only with particular acceptance thresholds and decision rules. Among other points, our discussion raises some challenges for characterising “cautious” reasoning using imprecise probabilities. Thus, we demonstrate a new way of systematically comparing imprecise probability systems, and we conclude that calibration inductive logics are surprisingly promising for making decisions. (shrink)

We argue that a particular approach to satisfying the broad predictive ambitions of the sciences demands law confirmation. On this approach we confirm non-nomic generalizations by confirming there are no actually realized ways of causing disconfirming cases. This gives causal generalizations a crucial role in prediction. We then show how rational judgements of relevant causal similarity can be used to confirm that causal generalizations themselves have no actual disconfirmers, providing a distinctive and clearly viable methodology for inductively confirming them. Finally, (...) we argue that for agents in our epistemic position using this methodology to confirm causal generalizations of adequate breadth will commonly demand law confirmation. We make a prima facie assessment of the methodology’s fit with scientific practice and briefly consider the prospects for an associated analysis of laws. (shrink)

In this paper, we offer a novel normative analysis of the inclusion fallacy. This refers to the robust tendency of experimental participants to reason about categories in a way that violates the laws of probability. In contrast to the received view, we argue that participants’ preferences in this kind of task might be rationally defended. To this purpose, we employ the philosophical notion of truthlikeness as a normative benchmark for category-based induction. Our analysis complements the main empirical findings and theoretical (...) predictions of the psychological literature and it explains the difficulties that probabilistic models have in treating cases like the inclusion fallacy. (shrink)

This paper argues that Newton employed a non-hypothetical method of evidential reasoning called demonstrative induction in his chronological studies. In demonstrative induction, inductive risk is confined to the premises, and the secureness of the conclusion depends on the secureness of those premises. I show that Newton's approach to chronology exemplifies two key restrictions on demonstrative induction: (i) premises must themselves be supported by demonstrative induction or by inductive generalizations, and (ii) inductive risk should be confined to premises drawn from domains (...) of inquiry whose claims are better established, thereby “exporting” inductive risk to more certain domains. Drawing on Newton's use of sources from history, biology, social science, and astronomy, I analyze how he transformed data from these fields into evidence for his chronological scheme. The strategy of exporting inductive risk, I argue, reveals a common evidential pattern across his natural-philosophical, historical, and theological work, and sheds light on the deeper methodological unity that underlies his otherwise disparate intellectual pursuits. (shrink)

Some learning strategies that work well when computational considerations are abstracted away from become severely limiting when such considerations are taken into account. We illustrate this phenomenon for agents who attempt to extrapolate patterns in binary data streams chosen from among a countable family of possibilities. If computational constraints are ignored, then two strategies that will always work are learning by enumeration (enumerate the possibilities—in order of simplicity, say—then search for the one earliest in the ordering that agrees with your (...) data and use it to predict the next data point) and Bayesian learning. But there are many families of computable data streams that, although they can be successfully extrapolated by computable agents, cannot be handled by any computable learner by enumeration. And while there is a sense in which Bayesian learning is a fully general strategy for computable learners, the ability to mimic powerful learners comes at a price for Bayesians: they cannot, in general, become highly confident of their predictions in the limit of large data sets and they cannot, in general, use priors that incorporate all relevant background knowledge. (shrink)

Reasoning from inconclusive evidence, or 'induction', is central to science and any applications we make of it. For that reason alone it demands the attention of philosophers of science. This element explores the prospects of using probability theory to provide an inductive logic: a framework for representing evidential support. Constraints on the ideal evaluation of hypotheses suggest that the overall standing of a hypothesis is represented by its probability in light of the total evidence, and incremental support, or confirmation, indicated (...) by the hypothesis having a higher probability conditional on some evidence than it does unconditionally. This proposal is shown to have the capacity to reconstruct many canons of the scientific method and inductive inference. Along the way, significant objections are discussed, such as the challenge of inductive scepticism, and the objection that the probabilistic approach makes evidential support arbitrary. (shrink)

In this paper I show that James Africanus Beale Horton launched an internal critique of race science as it developed in the hands of Robert Knox, Carl Vogt, and James Hunt. The latter three held an inductivist Baconian conception of science. Horton shows that their practices as scientists and natural philosophers contradict their own conception of what one must do in order to do good science. Horton’s critique of race science has important implications for philosophical anthropology as it took shape (...) over the course of the nineteenth century and for our understanding of the history of African philosophy in its global context. (shrink)

The present research aims to examine the different accounts of induction given by Aristotle, Leibniz, Hume, Carnap and De Finetti, trying to support that probability calculus offers a sufficient grounding of inductive logic. The term induction had been contrasted to deduction, by Aristotle. The Neoplatonic philosopher Alcinous suggested that dialectic firstly investigates the substances and then the accidents. There are five kinds of dialectic reasoning: division, definition, analysis, induction and syllogistic. The first three concern with substances, the last two with (...) accidents. Although Leibniz regarded probability theory as a basis of inductive logic, Hume’s skepticism was seminal for the reappraisal of the role of induction in modern philosophy. Enhancing Hume’s criticism, Popper and Wittgenstein completely denied that scientists use induction. Hans Reichenbach, however, attempted to build a theory of justification for the use of induction, based on a factual basis of other successful predictive methods that make induction feasible (Earman & Salmon 1999). Moreover, Buchdall (1969) stressed that we must distinguish the inductive process of the scientist from the inductive conclusion, which comes after the completion of observation and experimentation. (shrink)

Scientists and philosophers of science often draw methodological lessons from successful theories to justify methods of theory construction and to guide research programs. This paper proposes an epistemic framework for this practice, articulated in terms of the notion of meta-induction. By analogy to Goodman's `New Riddle of Induction', it introduces the concept of projectability of meta-inductive arguments, and demonstrates its significance in any account of meta-inductive reasoning. Likewise to scientific induction, meta-induction is shown to be constrained by naturalist epistemology in (...) determining which classifications of theoretical methods are `projectable'. Projectability judgments in meta-inductive arguments help ensure that when taking lessons from past theorizing (either as a part of scientific discussions or in a philosophical reflections about science), they would be based on the way evidence was exploited in theory construction or on theoretically plausible hypothesis about the natural world, rather than on human-made concepts. This framework emphasizes the primacy of empirically guided patterns of conjecture over purely formal considerations, enabling epistemic evaluation of research projects, even when theorizing is primarily grounded on abstract theoretical methods and at stages when direct evidence is unavailable. The paper discusses possible implications on philosophical debates concerning theory assessment in fundamental physics, particularly regarding meta-empirical justification, as well as discussions on model transfer in the special sciences. (shrink)

I attempt to refute Hume’s problem of induction. I raise the problem of counter-induction. I develop a template for generating philosophical problems. I develop and defend stage theory of scientific development for scientific realism. I develop and defend two new metaethical theories: cultural emotivism and moral functionalism. I argue that to compare morality to mathematics is to lose credibility. I develop the problem of moral induction against moral realism.

Douven observes that Schurz’s meta-inductive justification of induction cannot explain the great empirical success of induction, and offers an explanation based on computer simulations of the social and evolutionary development of our inductive practices. In this article, I argue that Douven’s account does not address the explanatory question that Schurz’s argument leaves open, and that the assumption of the environment’s induction-friendliness that is inherent to Douven’s simulations is not justified by Schurz’s argument.

According to the sensitivity account of knowledge, S knows that p only if S’s belief in p is sensitive in the sense that S would not believe that p if p were false. The sensitivity condition is usually relativized to belief-formation methods to avoid putative counterexamples. A remaining issue for the account is where methods should be mentioned in sensitivity subjunctives. In this paper, I argue that if methods are mentioned in the antecedent, then the account is too strong to (...) accommodate inductive knowledge; if methods are mentioned in the consequent, then the account is too weak to eliminate some luckily true beliefs from the realm of knowledge. Therefore, the strategy to relativize the sensitivity condition is undermined by inductive knowledge and some luckily true beliefs. (shrink)

In the last quarter of 2024 four articles on survival after death were published online ahead of print in the International Review of Psychiatry. Two of these included gratuitous attempts to discredit the work of a well-known survival critic. After correcting these misrepresentations, in this response Keith Augustine lays out how those who appeal to survival research to support their beliefs bungle their purportedly scientific assessments of the evidence concerning whether or not individual human minds survive death in a discarnate (...) state (without a normal physical body). The response also lays out why the still-common arguments that the late eminent thanatologist Robert Kastenbaum offered in favor of a discarnate afterlife are fallacious. It canvasses erroneous tropes about discovering white crows and the supposed impossibility of proving negatives, how scientific facts are established as highly probable, how psychical research is grounded on fallacious arguments from ignorance, and when all else fails, how survival researchers resort to psychologizing their critics rather than address their critics' arguments. Their collective failure to engage their critics' actual arguments raises the issue of whether survival proponents are able to adequately answer them—and if not, why not. Augustine submits that they are unable to answer their critics because the scientific evidence against their afterlife beliefs is incredibly strong. 1. Introduction -- 2. Gregory Shushan -- 3. Andreas Sommer -- 3.1 The Irrelevance of My Work to Sommer’s Thesis -- 3.2 How My Use is Comparable to What Robert Kastenbaum Intended -- 3.3 Sommer’s Characterization of The Myth of an Afterlife -- 3.4 The Context of What Kastenbaum was Trying to Do -- 3.5 Where Kastenbaum Fell on the Survival Question—and Why -- 3.5.1 Fallacious White Crows -- 3.5.2 Proof, Certainty, and Proving Negatives -- 3.5.3 Causal Ignorance is not Causal Evidence -- 3.5.4 Psychologizing Opponents -- 4. Conclusion: A Belief in Search of Evidence, or Evidence-Driven Belief? (shrink)

Mitchell & Gronenborn ( 2017 ) propose that we account for the presence of multiple models of protein structure, each produced in different contexts, through the framework of integrative pluralism. I argue that two interpretations of this framework are available, neither of which captures the relationship between a model and the protein structure it represents or between multiple models of protein structure. Further, it inclines us toward concluding prematurely that models of protein structure are right in their contexts and makes (...) extrapolation of findings from one context to another seem unwarranted. Instead, protein structure determination ought to be understood as _modestly monistic_. There is one model for every protein in each physicochemical context, and models of the same protein produced in different contexts are compatible with one another. ‘Integrating’ multiple models amounts to extrapolating from one context to another; this is possible because the effect of context on protein folding is relatively weak and predictable. Modest monism better describes the practice of protein structure determination than integrative pluralism and enables greater attention to how context affects protein folding. (shrink)

Most social policies cannot be defended without making inductive inferences. For example, consider certain arguments for racial profiling and affirmative action, respectively. They begin with statistics about crime or socioeconomic indicators. Next, there is an inductive step in which the statistic is projected from the past to the future. Finally, there is a normative step in which a policy is proposed as a response in the service of some goal—for example, to reduce crime or to correct socioeconomic imbalances. In comparison (...) to the normative step, the inductive step of a policy defense may seem trivial. We argue that this is not so. Satisfying the demands of the inductive step is difficult, and doing so has important but underappreciated implications for the normative step. In this paper, we provide an account of induction in social contexts and explore its implications for policy. Our account helps to explain which normative principles we ought to accept, and as a result it can explain why it is acceptable to make inferences involving race in some contexts (e.g., in defense of affirmative action) but not in others (e.g., in defense of racial profiling). (shrink)

Error Management Theory (EMT) suggests that cognitive adaptations evolve to minimize the cost of false negative and false positive errors in detections of consequential environmental conditions. These adaptations manifest as biases tailored to specific environmental conditions. This paper proposes that the same selection pressure fostered the evolution of a self-biasing ability, allowing us to minimize such costs based on experience and culturally transmitted information. The research indicates that this ability specifically applies to productions of belief or doubt about the existence (...) of an environmental condition that is not adequately perceptible to the senses, e.g., doubt that the man is being honest. A model of this self-biasing process, rooted in signal detection theory, is developed and its explanatory reach is demonstrated through many diverse examples from epistemological and social psychological literatures. These examples feature an evidentiary standard for belief that appears too high or low, or seems to have modulated. This study aims to correct and to significantly enhance EMT by introducing this evolved self-biasing ability. (shrink)

When an empirical prediction E of hypothesis H is observed to be true, such observation is said to confirm, i.e., support (although not prove) the truth of the hypothesis. But why? What justifies the claim that such evidence supports the hypothesis? The widely accepted answer is that it is justified by induction. More specifically, it is commonly held that the following argument, (1) If H then E; (2) E; (3) Therefore, (probably) H (here referred to as ‘hypothetico-deductive confirmation argument’), is (...) inductively strong. Yet this argument looks nothing like an inductive generalization, i.e., it does not seem inductive in the term’s traditional, enumerative sense. If anything, it has the form of the fallacy of affirming the consequent. This paper aims to solve this puzzle. True, in recent decades, ‘induction’ has been sometimes used more broadly to encompass any non-deductive, i.e., ampliative, argument. Applying Bayesian confirmation theory has famously demonstrated that hypothetico-deductive confirmation is indeed inductive in this broader, ampliative sense. Nonetheless, it will be argued here that, despite appearance, hypothetico-deductive confirmation can also be recast as enumerative induction. Hence, by being enumeratively inductive, the scientific method of hypothetico-deductive confirmation is justified through this traditional, more restrictive type of induction rather than merely by ampliative induction. (shrink)

In a world filled with information overload and complex problems, the ability to think logically is a superpower. "A Little More Logical" is your guide to mastering this essential skill. This engaging and accessible open educational resource is perfect for students, teachers, and lifelong learners who want to improve their critical thinking abilities and make better decisions in all aspects of life. -/- Through a series of fun and interactive chapters, "A Little More Logical" covers a wide range of topics, (...) from the basics of constructing sound arguments to the application of logic in fields like ethics, science, and computing. You'll learn how to spot and avoid common fallacies, analyze real-world arguments, and use logical reasoning to solve problems and make informed choices. -/- But this book isn't just about abstract concepts – it's about the human story behind logic. You'll discover the fascinating lives and ideas of diverse thinkers, from ancient philosophers to modern-day pioneers, who have shaped the development of logical reasoning throughout history. With thought-provoking examples, engaging exercises, and plenty of humor along the way, "A Little More Logical" makes the study of logic both rigorous and entertaining. -/- Whether you're a student looking to ace your philosophy class, a teacher searching for high-quality educational resources, or a curious reader who wants to sharpen your mind, "A Little More Logical" is the perfect companion on your journey to becoming a more effective thinker. Unlock the power of logical thinking today with this one-of-a-kind open educational resource! (shrink)

Consider a set of shuffled observations drawn from a fixed probability distribution over some instance domain. What enables learning of inductive generalizations which proceed from such a set of observations? The scenario is worthwhile because it epistemically characterizes most of machine learning. This kind of learning from observations is also inverse and ill-posed. What reduces the non-uniqueness of its result and, thus, its problematic epistemic justification, which stems from a one-to-many relation between the observations and many learnable generalizations? The paper (...) argues that this role belongs to any complexity regularization which satisfies Norton’s Material Theory of Induction (MTI) by localizing the inductive risk to facts in the given domain. A prime example of the localization is the Lottery Ticket Hypothesis (LTH) about overparameterized neural networks. The explanation of MTI’s role in complexity regularization of neural networks is provided by analyzing the stability of Empirical Risk Minimization (ERM), an inductive rule that controls the learning process and leads to an inductive generalization on the given set of observations. In cases where ERM might become asymptotically unstable, making the justification of the generalization by uniform convergence unavailable, LTH and MTI can be used to define a local stability. A priori, overparameterized neural networks are such cases and the combination of LTH and MTI can block ERM’s trivialization caused by equalizing the strengths of its inductive support for risk minimization. We bring closer the investigation of generalization in artificial neural networks and the study of inductive inference and show the division of labor between MTI and the optimality justifications (developed by Gerhard Schurz) in machine learning. (shrink)

In 2021, the Bigelow Institute for Consciousness Studies (hereafter, BICS) sponsored an essay competition (hereafter, the Contest) designed to solicit the best evidence for the hypothesis that human consciousness survives bodily death, and more specifically, evidence that would prove this hypothesis beyond a reasonable doubt. The summer 2022 issue of the Journal of Scientific Exploration featured a special subsection on the BICS contest and its winning essays. Robert Bigelow and Colm Kelleher outlined the motivation, design, and judging criteria for the (...) competition. Keith Augustine provided an extensive critical commentary on the contest design and eight of its prominent winning essays. Stephen Braude and several coauthors responded to Augustine’s criticisms, and Augustine provided a reply to Braude and his collaborators. Finally, the subsection concluded with a collaborative paper in which Etienne LeBel, Adam Rock, and Augustine proposed a more rigorous experimental design for testing the survival hypothesis. Augustine presented important criticisms of the Contest and several of its prominent winning essays. Moreover, many of his criticisms apply to survival literature in general and so are instructive for the wider survival debate. And Braude, one of the most important contributors to the survival debate since C.D. Broad and C.J. Ducasse, was at the helm of the reply to Augustine. Although the response to Augustine raised important concerns about skeptical assessments of the evidence for survival, it was hamstrung with several defects. 1. Preliminaries 2. Problematic Aspects of the Reply to Augustine 3. What is the Question? Honing in on the Actual Disagreement 4. Inference to Best Explanation 5. Confirmation Approaches to Evidential Support -- 5.1 Evidential Justification -- 5.2 When Does an Observation Favor One Hypothesis Over Another Hypothesis? The Law of Likelihood -- 5.3 When Should We Up Our Confidence in a Hypothesis? Incremental Confirmation -- 5.4 How Can We Determine the Net Plausibility of a Hypothesis? Bayes’ Theorem and Posterior Probabilities -- 5.5 Which of Two Hypotheses is More Probable than the Other? Principle of Contrastive Posterior Probabilities -- 5.6 When Does Evidence Justify Believing a Hypothesis? Principle of Absolute Confirmation -- 5.7 Survivalist Confusions about Bayesian Analyses -- 5.8 The Washing Out of Priors -- 5.9 The Subjectivism Objection -- 5.10 Summing Up Insights from Confirmation Theory -- 6. Back to Braude 7. Varieties of Skepticism 8. Neurophysiological Evidence and the Survival Hypothesis 9. The Data from Mediumship 10. The Alleged Improbability of Fraud and Other Counterexplanations 11. Mediumship and the Logic of Confirmation 12. The Significance of Failed Tests 13. Survivalist Rescues and Contrastive Confirmation 14. Evidential Support without Predictions -- 15. The Testability Problem -- Concluding Remarks. (shrink)

In this paper, we analyse conceptions of induction and certainty in Wolff and Crusius, highlighting their competing conceptions of physics. We discuss (i) the perspective of Wolff, who assigned induction an important role in physics, but argued that physics should be an axiomatic science containing certain statements, and (ii) the perspective of Crusius, who adopted parts of the ideal of axiomatic physics but criticized the scope of Wolff’s ideal of certain science. Against interpretations that take Wolff’s proofs in physics to (...) be based on empirical statements that are probable, we show how inductively established statements, coupled with assumptions concerning the uniformity of nature, can be certain according to Wolff. We examine Crusius’ little known work on physics and show that he attacked Wolff, arguing that many statements of physics are probable. We provide a discussion of the reception of methodological rules similar to Newton’s Regulae Philosophandi, discussing different interpretations of such rules. We conclude by briefly hinting at how the debate between Wolff and Crusius provides historical context for understanding Kant’s philosophy of natural science. (shrink)

I examine Du Châtelet’s methodology for physics and metaphysics through the lens of her engagement with Newton’s Rules for Reasoning in Natural Philosophy. I first show that her early manuscript writings discuss and endorse these Rules. Then, I argue that her famous published account of hypotheses continues to invoke close analogues of Rules 3 and 4, despite various developments in her position. Once relevant experimental evidence and some basic constraints are met, it is legitimate to inductively generalize from observations; general (...) hypotheses can thereafter be assumed as true until contrary experiments show otherwise. I conclude by arguing that this account of induction plays an essential role in her metaphysics, both in an argument for simple substances—which has an inductive premise—and in her attempt to distinguish acceptable and unacceptable metaphysical commitments. (shrink)

This paper argues that the linguistic approach to analyzing induction, according to which induction is a type of inference or argument composed of statements or propositions, is unsuitable to account for scientific reasoning. Consequently, a novel approach to induction in model-based science is suggested. First, in order to show their adherence to the linguistic treatment of induction, two strategies are reviewed: (i) Carnap and Reichenbach’s attempts to justify induction and (ii) Norton’s recent material theory of induction. Second, three reasons are (...) provided to support the claim that the linguistic treatment of induction is insufficient in accounting for model-based reasoning in science. Finally, a framework focused on models—rather than statements or propositions—is suggested to address induction in science. William Whewell’s theory of induction is briefly outlined as an example of a non-propositional treatment of induction that is consistent with model-based scientific practice. (shrink)

This chapter presents a typology of the different kinds of inductive inferences we can draw from our evidence, based on the explanatory relationship between evidence and conclusion. Drawing on the literature on graphical models of explanation, I divide inductive inferences into (a) downwards inferences, which proceed from cause to effect, (b) upwards inferences, which proceed from effect to cause, and (c) sideways inferences, which proceed first from effect to cause and then from that cause to an additional effect. I further (...) distinguish between direct and indirect forms of downwards and upwards inferences. I then show how we can subsume canonical forms of inductive inference mentioned in the literature, such as inference to the best explanation, enumerative induction, and analogical inference, under this typology. Along the way, I explore connections with probability and confirmation, epistemic defeat, the relation between abduction and enumerative induction, the compatibility of IBE and Bayesianism, and theories of epistemic justification. (shrink)

The framework of Solomonoff prediction assigns prior probability to hypotheses inversely proportional to their Kolmogorov complexity. There are two well-known problems. First, the Solomonoff prior is relative to a choice of Universal Turing machine. Second, the Solomonoff prior is not computable. However, there are responses to both problems. Different Solomonoff priors converge with more and more data. Further, there are computable approximations to the Solomonoff prior. I argue that there is a tension between these two responses. This is because computable (...) approximations to Solomonoff prediction do not always converge. (shrink)

This paper argues that the linguistic approach to analyzing induction, according to which induction is a type of inference or argument composed of statements or propositions, is unsuitable to account for scientific reasoning. Consequently, a novel approach to induction in model-based science is suggested. First, in order to show their adherence to the linguistic treatment of induction, two strategies are reviewed: (i) Carnap and Reichenbach’s attempts to justify induction and (ii) Norton’s recent material theory of induction. Second, three reasons are (...) provided to support the claim that the linguistic treatment of induction is insufficient in accounting for model-based reasoning in science. Finally, a framework focused on models—rather than statements or propositions—is suggested to address induction in science. William Whewell’s theory of induction is briefly outlined as an example of a non-propositional treatment of induction that is consistent with model-based scientific practice. (shrink)

We are often justified in acting on the basis of evidential confirmation. I argue that such evidence supports belief in non-quantificational generic generalizations, rather than universally quantified generalizations. I show how this account supports, rather than undermines, a Bayesian account of confirmation. Induction from confirming instances of a generalization to belief in the corresponding generic is part of a reasoning instinct that is typically (but not always) correct, and allows us to approximate the predictions that formal epistemology would make.

In this thesis I motivate and develop a Hybrid Theory of Induction (HTI), and I explore some of its virtues and implications. The HTI is a hybrid second-order model of inductive support. It is a "hybrid" model of inductive support because it holds that two ingredients play a necessary role in understanding inductive support: rules and facts. It is a "second-order" model of inductive support because it is a model within which first-order models of inductive support (i.e. logics of induction) (...) can fit. In chapter 1 I argue that we need both rules and facts to play a role in a successful account of inductive support. Rules of induction accurately describe relations of inductive support when they are warranted; facts do the warranting work. I call this type of warrant "factual warrant". The resulting account is both functional and accurate, it helps us make sense of how different rules of induction can coexist and it allows us to resolve some current debates in induction. For the purposes of chapter 1 I adopt an existing binary account of factual warrant. In chapter 2 I develop a Graded account of Factual Warrant (GFW), according to which factual warrant comes in degrees. I integrate the GFW in the HTI. I then show that the GFW illuminates the connection between factual warrant and inductive support, and it can successfully account for the role of idealisations and theory in our understanding of inductive support. In chapter 3 I argue that the HTI is also useful for agents, since it can provide methodological guidance to ensure strong inferences and conceptual guidance to assess the strength of our inferences. Finally, in chapter 4, I explore Bayesian inductive logics from the perspective of the HTI. This analysis brings to light the central role that probability models play in Bayesian inductive logics, offering a logical underpinning for some recent suggestions in Bayesian epistemology. Furthermore, throughout this thesis I analyse in detail three rules of induction from the perspective of the HTI: enumerative induction in chapter 2, causal inference in chapter 3 and Bayesian inductive logics in chapter 4. These analyses illustrate how the HTI can help us think more clearly about rules of induction, offering new tools to tackle existing challenges. (shrink)

A volume of papers inspired by the work of George E. Smith on confirmation and evidence in advanced science—from Newton's gravitation theory to the physics of molecules.

The paper presents a comprehensive solution to the new riddle of induction. Gruesome induction is blocked because “grue” is not independent of our sampling and observation methods. Before presenting my theory, I critically survey previous versions of what I call the “independence strategy”, tracing the strategy to three different papers from the 1970s by (respectively) Wilkerson, Moreland, and Jackson. Next I critically examine recent approaches by Okasha, Godfrey-Smith, Schramm, and Freitag. All of these approaches have their virtues, but none of (...) them rules out every problem case. Nor do any of them alone explain exactly what goes wrong in all types of gruesome reasoning. My account aims to do both. Learning lessons from my criticisms of previous theories, I argue that there are several crucial types of independence. All of them are needed to rule out every problem case. I close by suggesting that the independence solution can also be adopted by proponents of other approaches—Bayesians, fans of natural kinds, and pragmatists. (shrink)

Michael Nahm's preceding commentary accuses me of seven misrepresentations. One of these is an acknowledged good-faith error about a peripheral detail, while the remaining six are demonstrably accurate descriptions of Nahm's statements. At the same time, Nahm verifiably misrepresents me frequently and intentionally over issues that he takes to be consequential, which is a much more serious offense. All authors should call out when an interlocutor get their points wrong, but only when they can definitively back up the charge. Where (...) Nahm weakly attempts to show that I misrepresented him, I will show that, if anything, his showcase consists of six verifiably accurate characterizations of his Bigelow Institute contest-winning essay's conclusions. His commentary exemplifies the truism that one can appeal to a million frivolous reasons to dismiss what an opponent has to say if one is absolutely determined not to hear him. Though committed survivalists will undoubtedly be satisfied that survival researchers have responded to me regardless of whether they have responded well, those that care about the underlying issues will hopefully find value in my reply. 1. Introduction -- 2. My (One) Accidental Misattribution -- 3. Nahm's (Repeated) Intentional Mischaracterizations -- 4. Nahm's Mischaracterizations to Shift the Burden of Proof -- 5. Nahm's Seven Wonders of the World -- 6. Conclusion: Assertions Should be Supported. (shrink)

The recent Bigelow Institute contest rewarding the "best" evidence for life after death epitomizes much of what's wrong with the current state of survival research, its participants constituting a who's who list of contemporary survival researchers. Cases that are regularly hyped as among the best evidence for an afterlife are all too often easily susceptible to normal explanations--if only survival researchers would give them a chance. The consistently negative results of 121 years of experimental survival research ought to have spurred (...) soul-searching questions for survival researchers by now. And if we treat discarnate personal survival as a scientific hypothesis, then researchers are rationally obliged to seriously consider biological facts that tell against it, too. Limiting one's inquiry to attempts to only collect data that might confirm survival is one of the chief hallmarks of pseudoscience, and it's sadly a feature, not a bug, of the survival literature. This systematic review reveals that survival researchers would better serve science by setting aside their feelings and heeding what the data are telling them, for the probabilities should drive our beliefs, not the other way around. Is discarnate personal survival likely to occur in light of the total available evidence? The overall evidence doesn't even make personal survival more probable than not. 1. Introduction 2. In Science the Quality of the Evidence is Paramount -- 2.1 Prospective Experimental Tests of Survival -- 2.2 Where Have All the Deceased Survival Researchers Gone? -- 3. What Does the Total Available Relevant Evidence Tell Us? -- 4. Ranking the Survival Evidence -- 5. The Mind-Body Problem, Botched -- 6. Conclusion: Not Much Better than Religious Faith. (shrink)

What is the relation or connection between formalizations of induction and the actual inductive inferences of scientists? Building from recent works in the philosophy of logic, this paper argues that these formalizations of induction are best viewed as models and not literal descriptions of inductive inferences in science. Three arguments are put forward to support this claim. First, I argue that inductive support is the kind of phenomenon that can be justifiably modeled. Second, I argue that these formalizations have the (...) features that define models—that they have representors, artefacts, and idealizations. Third, I argue that these formalizations being models explains their plurality and revisability. (shrink)

This article compares and contrasts the reception of Comte’s positivism in the works of William Whewell, John Stuart Mill and (to a lesser extent) Franz Brentano. It is argued that Whewell’s rejection of positivism derives from his endorsement of (what I shall call) a constructivist account of the inductive sciences, while Mill and Brentano’s sympathies for positivism are connected to their endorsement of an empiricist account. The mandate of the article is to spell out the chief differences between these two (...) rival accounts. In the last, conclusive section, Whewell’s anti-positivist argument is briefly assessed and rebutted. (shrink)