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 overall support for a hypothesis is represented by its probability in light of the total evidence, and incremental support, or (...) confirmation, indicated by an increased probability given the evidence than otherwise. 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 from inductive scepticism and the objection that the probabilistic approach makes evidential support arbitrary. (shrink)

Laurence BonJour has recently proposed a novel and interesting approach to the problem of induction. He grants that it is contingent, and so not a priori, that our patterns of inductive inference are reliable. Nevertheless, he claims, it is necessary and a priori that those patterns are highly likely to be reliable, and that is enough to ground an a priori justification induction. This paper examines an important defect in BonJour's proposal. Once we make sense of the claim (...) that inductive inference is "necessarily highly likely" to be reliable, we find that it is not knowable a priori after all. (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)

John Maynard Keynes’s A Treatise on Probability is the seminal text for the logical interpretation of probability. According to his analysis, probabilities are evidential relations between a hypothesis and some evidence, just like the relations of deductive logic. While some philosophers had suggested similar ideas prior to Keynes, it was not until his Treatise that the logical interpretation of probability was advocated in a clear, systematic and rigorous way. I trace Keynes’s influence in the philosophy of (...) class='Hi'>probability through a heterogeneous sample of thinkers who adopted his interpretation. This sample consists of Frederick C. Benenson, Roy Harrod, Donald C. Williams, Henry E. Kyburg and David Stove. The ideas of Keynes prove to be adaptable to their diverse theories of probability. My discussion indicates both the robustness of Keynes’s probability theory and the importance of its influence on the philosophers whom I describe. I also discuss the Problem of the Priors. I argue that none of those I discuss have obviously improved on Keynes’s theory with respect to this issue. (shrink)

Many theorists have proposed that we can use the principle of indifference to defeat the inductive sceptic. But any such theorist must confront the objection that different ways of applying the principle of indifference lead to incompatible probability assignments. Huemer offers the explanatory priority proviso as a strategy for overcoming this objection. With this proposal, Huemer claims that we can defend induction in a way that is not question-begging against the sceptic. But in this article, I argue that (...) the opposite is true: if anything, Huemer’s use of the principle of indifference supports the rationality of inductive scepticism. (shrink)

It was Hume's concept of knowledge that marked Masaryk's philosophy more than influential Kant. Masaryk devoted Hume his inaugural lecture The Probability Calculus and Hume's Scepticism at Prague University in 1882. He tried to face his scepticism concerning causation and induction by a formula P = n: or P = : where P means the increasing probability that an event that had happened n times in the past will happen again. Hume stresses that there is an essential (...) difference between probability and certainty, and we have no logical reason to presuppose that the past experience is legitimate for expectation that the future will resemble to the past. Only our belief, not knowledge, leads us to that. Although Masaryk did not accept Hume's scepticism, he had deep understanding for his way of thinking. Reflection of Hume's influence on Masaryk in Czech philosophy is unsatisfactory, which is stated in this article. (shrink)

Medical diagnosis has been traditionally recognized as a privileged field of application for so called probabilistic induction. Consequently, the Bayesian theorem, which mathematically formalizes this form of inference, has been seen as the most adequate tool for quantifying the uncertainty surrounding the diagnosis by providing probabilities of different diagnostic hypotheses, given symptomatic or laboratory data. On the other side, it has also been remarked that differential diagnosis rather works by exclusion, e.g. by modus tollens, i.e. deductively. By drawing on (...) a case history, this paper aims at clarifying some points on the issue. Namely: 1) Medical diagnosis does not represent, strictly speaking, a form of induction, but a type, of what in Peircean terms should be called ‘abduction’ (identifying a case as the token of a specific type); 2) in performing the single diagnostic steps, however, different inferential methods are used for both inductive and deductive nature: modus tollens, hypothetical-deductive method, abduction; 3) Bayes’ theorem is a probabilized form of abduction which uses mathematics in order to justify the degree of confidence which can be entertained on a hypothesis given the available evidence; 4) although theoretically irreconcilable, in practice, both the hypothetical- deductive method and the Bayesian one, are used in the same diagnosis with no serious compromise for its correctness; 5) Medical diagnosis, especially differential diagnosis, also uses a kind of “probabilistic modus tollens”, in that, signs (symptoms or laboratory data) are taken as strong evidence for a given hypothesis not to be true: the focus is not on hypothesis confirmation, but instead on its refutation [Pr (¬ H/E1, E2, …, En)]. Especially at the beginning of a complicated case, odds are between the hypothesis that is potentially being excluded and a vague “other”. This procedure has the advantage of providing a clue of what evidence to look for and to eventually reduce the set of candidate hypotheses if conclusive negative evidence is found. 6) Bayes’ theorem in the hypothesis-confirmation form can more faithfully, although idealistically, represent the medical diagnosis when the diagnostic itinerary has come to a reduced set of plausible hypotheses after a process of progressive elimination of candidate hypotheses; 7) Bayes’ theorem is however indispensable in the case of litigation in order to assess doctor’s responsibility for medical error by taking into account the weight of the evidence at his disposal. (shrink)

It is argued that two observers with the same information may rightly disagree about the probability of an event that they are both observing. This is a correct way of describing the view of a lottery outcome from the perspective of a winner and from the perspective of an observer not connected with the winner - the outcome is improbable for the winner and not improbable for the unconnected observer. This claim is both argued for and extended by developing (...) a case in which a probabilistic inference is supported for one observer and not for another, though they relevantly differ only in perspective, not in any information that they have. It is pointed out, finally, that all probabilities are in this way dependent on perspective. (shrink)

Reasoning over our knowledge bases and theories often requires non-deductive inferences, especially – but by no means only – when commonsense reasoning is the case, i.e. when practical agency is called for. This kind of reasoning can be adequately formalized via the notion of supraclassical consequence, a non-deductive consequence tightly associated with default and non-monotonic reasoning and featuring centrally in abductive, inductive, and probabilistic logical systems. In this paper, we analyze core concepts and problems of these systems in the light (...) of supraclassical consequence. (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)

This article explores some open questions related to the problem of verification of theories in the context of empirical sciences by contrasting three epistemological frameworks. Each of these epistemological frameworks is based on a corresponding central metaphor, namely: (a) Neo-empiricism and the gambling metaphor; (b) Popperian falsificationism and the scientific tribunal metaphor; (c) Cognitive constructivism and the object as eigen-solution metaphor. Each of one of these epistemological frameworks has also historically co-evolved with a certain statistical theory and method for testing (...) scientific hypotheses, respectively: (a) Decision theoretic Bayesian statistics and Bayes factors; (b) Frequentist statistics and p-values; (c) Constructive Bayesian statistics and e-values. This article examines with special care the Zero Probability Paradox (ZPP), related to the verification of sharp or precise hypotheses. Finally, this article makes some remarks on Lakatos’ view of mathematics as a quasi-empirical science. (shrink)

Some patients, following brain injury, do not outwardly respond to spoken commands, yet show patterns of brain activity that indicate responsiveness. This is “cognitive-motor dissociation” (CMD). Recent research has used machine learning to diagnose CMD from electroencephalogram (EEG) recordings. These techniques have high false discovery rates, raising a serious problem of inductive risk. It is no solution to communicate the false discovery rates directly to the patient’s family, because this information may confuse, alarm and mislead. Instead, we need a procedure (...) for generating case-specific probabilistic assessments that can be communicated clearly. This article constructs a possible procedure with three key elements: (1) a shift from categorical “responding or not” assessments to degrees of evidence; (2) the use of patient-centred priors to convert degrees of evidence to probabilistic assessments; and (3) the use of standardized probability yardsticks to convey those assessments as clearly as possible. (shrink)

I present a solution to the epistemological or characterisation problem of induction. In part I, Bayesian Confirmation Theory (BCT) is discussed as a good contender for such a solution but with a fundamental explanatory gap (along with other well discussed problems); useful assigned probabilities like priors require substantive degrees of belief about the world. I assert that one does not have such substantive information about the world. Consequently, an explanation is needed for how one can be licensed to act (...) as if one has substantive information about the world when one does not. I sketch the outlines of a solution in part I, showing how it differs from others, with full details to follow in subsequent parts. The solution is pragmatic in sentiment (though differs in specifics to arguments from, for example, William James); the conceptions we use to guide our actions are and should be at least partly determined by preferences. This is cashed out in a reformulation of decision theory motivated by a non-reductive formulation of hypotheses and logic. A distinction emerges between initial assumptions--that can be non-dogmatic--and effective assumptions that can simultaneously be substantive. An explanation is provided for the plausibility arguments used to explain assigned probabilities in BCT. -/- In subsequent parts, logic is constructed from principles independent of language and mind. In particular, propositions are defined to not have form. Probabilities are logical and uniquely determined by assumptions. The problems considered fatal to logical probabilities--Goodman's `grue' problem and the uniqueness of priors problem are dissolved due to the particular formulation of logic used. Other problems such as the zero-prior problem are also solved. -/- A universal theory of (non-linguistic) meaning is developed. Problems with counterfactual conditionals are solved by developing concepts of abstractions and corresponding pictures that make up hypotheses. Spaces of hypotheses and the version of Bayes' theorem that utilises them emerge from first principles. -/- Theoretical virtues for hypotheses emerge from the theory. Explanatory force is explicated. The significance of effective assumptions is partly determined by combinatoric factors relating to the structure of hypotheses. I conjecture that this is the origin of simplicity. (shrink)

The absolute pessimistic induction states that earlier theories, although successful, were abandoned, so current theories, although successful, will also be abandoned. By contrast, the relative pessimistic induction states that earlier theories, although superior to their predecessors, were discarded, so current theories, although superior to earlier theories, will also be discarded. Some pessimists would have us believe that the relative pessimistic induction avoids empirical progressivism. I argue, however, that it has the same problem as the absolute pessimistic (...) class='Hi'>induction, viz., either its premise is implausible or its conclusion does not probably follow from its premise. (shrink)

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability (...) of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic. (shrink)

In this chapter we will deal with “mechanizing” induction, i.e. with ways in which theoretical computer science approaches inductive generalization. In the field of Machine Learning, algorithms for induction are developed. Depending on the form of the available data, the nature of these algorithms may be very different. Some of them combine geometric and statistical ideas, while others use classical reasoning based on logical formalism. However, we are not so much interested in the algorithms themselves, but more on (...) the philosophical and theoretical foundations they share. Thus in the first of two parts, we will examine different approaches and inductive assumptions in two particular learning settings. While many machine learning algorithms work well on a lot of tasks, the interpretation of the learned hypothesis is often difficult. Thus, while e.g. an algorithm surprisingly is able to determine the gender of the author of a given text with about 80 percent accuracy [Argamon and Shimoni, 2003], for a human it takes some extra effort to understand on the basis of which criteria the algorithm is able to do so. With that respect the advantage of approaches using logic are obvious: If the output hypothesis is a formula of predicate logic, it is easy to interpret. However, if decision trees or algorithms from the area of inductive logic programming are based purely on classical logic, they suffer from the fact that most universal statements do not hold for exceptional cases, and classical logic does not offer any convenient way of representing statements which are meant to hold in the “normal case”. Thus, in the second part we will focus on approaches for Nonmonotonic Reasoning that try to handle this problem. Both Machine Learning and Nonmonotonic Reasoning have been anticipated partially by work in philosophy of science and philosophical logic. At the same time, recent developments in theoretical computer science are expected to trig- ger further progress in philosophical theories of inference, confirmation, theory revision, learning, and the semantic and pragmatics of conditionals. We hope this survey will contribute to this kind of progress by building bridges between computational, logical, and philosophical accounts of induction. (shrink)

Karl Popper, in "The Logic of Scientific Discovery" Section *vii, argues that if you find that some objecta a,b, c ... have a specific property P, then this discovery by itself does not increase the probability that some other object also has P. He concludes that there can be no effective principle of induction. My paper disproves Popper's claim, using very elementary considerations..

How were reliable predictions made before Pascal and Fermat's discovery of the mathematics of probability in 1654? What methods in law, science, commerce, philosophy, and logic helped us to get at the truth in cases where certainty was not attainable? The book examines how judges, witch inquisitors, and juries evaluated evidence; how scientists weighed reasons for and against scientific theories; and how merchants counted shipwrecks to determine insurance rates. Also included are the problem of induction before Hume, design (...) arguments for the existence of God, and theories on how to evaluate scientific and historical hypotheses. It is explained how Pascal and Fermat's work on chance arose out of legal thought on aleatory contracts. The book interprets pre-Pascalian unquantified probability in a generally objective Bayesian or logical probabilist sense. (shrink)

We extend the framework of Inductive Logic to Second Order languages and introduce Wilmers' Principle, a rational principle for probability functions on Second Order languages. We derive a representation theorem for functions satisfying this principle and investigate its relationship to the first order principles of Regularity and Super Regularity.

Scientific realism says of our best scientific theories that (1) most of their important posits exist and (2) most of their central claims are approximately true. Antirealists sometimes offer the pessimistic induction in reply: since (1) and (2) are false about past successful theories, they are probably false about our own best theories too. The contemporary debate about this argument has turned (and become stuck) on the question, Do the central terms of successful scientific theories refer? For example, Larry (...) Laudan offers a list of successful theories that employed central terms that failed to refer, and Philip Kitcher replies with a view about reference in which the central terms of such theories did sometimes refer. This article attempts to break this stalemate by proposing a direct version of the pessimistic induction, one that makes no explicit appeal to a substantive notion or theory of reference. While it is premature to say that this argument succeeds in showing that realism is probably false, the direct pessimistic induction is not subject to any kind of reference-based objection that might cripple a weaker, indirect version of the argument. Any attempt to trounce the direct pessimistic induction with a theory of reference fails. (shrink)

: In this paper I argue that Hume's famous discussion of probability and induction, as originally presented in the Treatise, is significantly motivated by irreligious objectives. A particular target of Hume's arguments is Joseph Butler's Analogy of Religion. In the Analogy Butler intends to persuade his readers of both the credibility and practical importance of the doctrine of a future state of rewards and punishments. The argument that he advances relies on probable reasoning and proceeds on the assumption (...) that our past experience in this life serves as a reliable and effective guide for our expectations concerning a future state. In the relevant sections of the Treatise Hume aims to discredit this religious argument and the practical objectives associated with it. (shrink)

In this paper I raise concerns that I have with what I perceive to be as a complete lack of warrant for Hume’s assertion that there are only two types of reasoning – demonstrative and probable. (Hume’s fork) What I leave for others to decide is whether this lack of warrant therefore successfully undercuts Hume’s argument for the Problem of Induction.

Although we often see references to Carnap’s inductive logic even in modern literatures, seemingly its confusing style has long obstructed its correct understanding. So instead of Carnap, in this paper, I devote myself to its necessary and sufficient commentary. In the beginning part (Sections 2-5), I explain why Carnap began the study of inductive logic and how he related it with our thought on probability (Sections 2-4). Therein, I trace Carnap’s thought back to Wittgenstein’s Tractatus as well (Section 5). (...) In the succeeding sections, I attempt the simplest exhibition of Carnap’s earlier system, where his original thought was thoroughly provided. For this purpose, minor concepts to which researchers have not paid attention are highlighted, for example, m-function (Section 8), in-correlation (Section 10), C-correlate (Section 10), statistical distribution (Section 12), and fitting sequence (Section 17). The climax of this paper is the proof of theorem (56). Through this theorem, we will be able to overview Carnap’s whole system. (shrink)

This paper discusses ethical issues surrounding Integrated Assessment Models (IAMs) of the economic effects of climate change, and how climate economists acting as policy advisors ought to represent the uncertain possibility of catastrophe. Some climate economists, especially Martin Weitzman, have argued for a precautionary approach where avoiding catastrophe should structure climate economists’ welfare analysis. This paper details ethical arguments that justify this approach, showing how Weitzman’s “fat tail” probabilities of climate catastrophe pose ethical problems for widely used IAMs. The main (...) claim is that economists who ignore or downplay catastrophic risks in their representations of uncertainty likely fall afoul of ethical constraints on scientists acting as policy advisors. Such scientists have duties to honestly articulate uncertainties and manage (some) inductive risks, or the risks of being wrong in different ways. (shrink)

Whereas many philosophers accept the validity of 'probability' and confine themselves to interpreting it, this paper challenges its conceptual coherence by critically examining its use in the empirical world. While measure theory provides a rigorous mathematical framework for manipulating probability functions, we argue that applying precise probability measures to empirically uncertain outcomes introduces a fundamental contradiction. Probability measures claim to quantify uncertainty while simultaneously implying a degree of understanding about events that we do not fully possess. (...) This inconsistency undermines the idea that probability offers objective or reliable insights into reality. Moreover, we argue that it is impossible to assign a correct probability in the empirical world—neither deduction nor induction provides a justifiable basis for doing so. Therefore, despite probability appearing to work as a practical tool for managing uncertainty, its theoretical foundation collapses under scrutiny in empirical applications. (shrink)

I set up two axiomatic theories of inductive support within the framework of Kolmogorovian probability theory. I call these theories ‘Popperian theories of inductive support’ because I think that their specific axioms express the core meaning of the word ‘inductive support’ as used by Popper (and, presumably, by many others, including some inductivists). As is to be expected from Popperian theories of inductive support, the main theorem of each of them is an anti-induction theorem, the stronger one of (...) them saying, in fact, that the relation of inductive support is identical with the empty relation. It seems to me that an axiomatic treatment of the idea(s) of inductive support within orthodox probability theory could be worthwhile for at least three reasons. Firstly, an axiomatic treatment demands from the builder of a theory of inductive support to state clearly in the form of specific axioms what he means by ‘inductive support’. Perhaps the discussion of the new anti-induction proofs of Karl Popper and David Miller would have been more fruitful if they had given an explicit definition of what inductive support is or should be. Secondly, an axiomatic treatment of the idea(s) of inductive support within Kolmogorovian probability theory might be accommodating to those philosophers who do not completely trust Popperian probability theory for having theorems which orthodox Kolmogorovian probability theory lacks; a transparent derivation of anti-induction theorems within a Kolmogorovian frame might bring additional persuasive power to the original anti-induction proofs of Popper and Miller, developed within the framework of Popperian probability theory. Thirdly, one of the main advantages of the axiomatic method is that it facilitates criticism of its products: the axiomatic theories. On the one hand, it is much easier than usual to check whether those statements which have been distinguished as theorems really are theorems of the theory under examination. On the other hand, after we have convinced ourselves that these statements are indeed theorems, we can take a critical look at the axioms—especially if we have a negative attitude towards one of the theorems. Since anti-induction theorems are not popular at all, the adequacy of some of the axioms they are derived from will certainly be doubted. If doubt should lead to a search for alternative axioms, sheer negative attitudes might develop into constructive criticism and even lead to new discoveries. -/- I proceed as follows. In section 1, I start with a small but sufficiently strong axiomatic theory of deductive dependence, closely following Popper and Miller (1987). In section 2, I extend that starting theory to an elementary Kolmogorovian theory of unconditional probability, which I extend, in section 3, to an elementary Kolmogorovian theory of conditional probability, which in its turn gets extended, in section 4, to a standard theory of probabilistic dependence, which also gets extended, in section 5, to a standard theory of probabilistic support, the main theorem of which will be a theorem about the incompatibility of probabilistic support and deductive independence. In section 6, I extend the theory of probabilistic support to a weak Popperian theory of inductive support, which I extend, in section 7, to a strong Popperian theory of inductive support. In section 8, I reconsider Popper's anti-inductivist theses in the light of the anti-induction theorems. I conclude the paper with a short discussion of possible objections to our anti-induction theorems, paying special attention to the topic of deductive relevance, which has so far been neglected in the discussion of the anti-induction proofs of Popper and Miller. (shrink)

Inductive logic would be the logic of arguments that are not valid, but nevertheless justify belief in something like the way in which valid arguments would. Maybe we could describe it as the logic of “almost valid” arguments. There is a sort of transitivity to valid arguments. Valid arguments can be chained together to form arguments and such arguments are themselves valid. One wants to distinguish the “almost valid” arguments by noting that chains of “almost valid” arguments are weaker than (...) the links that form them. But it is not clear that this is so. I have an apparent counterexample the claim. Though: as is typical in these sorts of situations, it is hard to tell where the problem lies. (shrink)

In the replies to my critics that follow I offer a more detailed account of the specific papers that they discuss or examine. The papers that they are especially concerned with are: “The Material World and Natural Religion in Hume’s Treatise” (Ryan) [Essay 3], “Hume’s Skepticism and the Problem of Atheism” (Fosl) [Essay 12], and “Hume’s Philosophy of Irreligion and the Myth of British Empiricism (Gautier) [Essay 16].

Hope is an attitude with a distinctive epistemological dimension: it is incompatible with knowledge. This chapter examines hope as it relates to knowledge but also to probability and inductive considerations. Such epistemic constraints can make hope either impossible, or, when hope remains possible, they affect how one’s epistemic situation can make hope rational rather than irrational. Such issues are especially relevant to when hopefulness may permissibly figure in practical deliberation over a course of action. So I consider cases of (...) second-order inductive reflection on when one should, or should not, be hopeful for an outcome with which one has a long record of experience: in other words, what is the epistemology behind when one should, if ever, stop hoping for outcomes which have failed one many times in the past? (shrink)

In The Problems of Philosophy, Bertrand Russell presents a justification of induction based on a principle he refers to as “the principle of induction”. Owing to the ambiguity of the notion of probability, the principle of induction may be interpreted in two different ways. If interpreted in terms of the subjective interpretation of probability, the principle of induction may be known a priori to be true. But it is unclear how this should give us (...) any confidence in our use of induction, since induction is applied to the external world outside our minds. If the principle is interpreted in light of the objective interpretation of induction, it cannot be known to be true a priori, since it applies to frequencies that occur in the world outside the mind, and these cannot be known without recourse to experience. Russell’s principle of induction therefore fails to provide a satisfactory justification of induction. (shrink)

A collective understanding that traces a debate between 'what is science?’ and ‘what is a science about?’ has an extraction to the notion of scientific knowledge. The debate undertakes the pursuit of science that hardly extravagance the dogma of pseudo-science. Scientific conjectures invoke science as an intellectual activity poured by experiences and repetition of the objects that look independent of any idealist views (believes in the consensus of mind-dependence reality). The realistic machinery employs in an empiricist exposition of the objective (...) phenomenon by synchronizing the general method to make observational predictions that cover all the phenomena of the particular entity without any exception. The formation of science encloses several epistemological purviews and a succession of conjectures cum refutation that a newer theorem could reinstate. My attempt is to advocate a holistic plea of scientific conjectures that outruns the restricted regulation of experience or testable hypothesis to render the validity of a chain of logical reasoning (deductive or inductive) of basic scientific statements. The milieu of scientific intensification integrates speculation that loads efficiency towards a new experimental dimension where the reality is not itself objective or observers relative; in fact the observed phenomenon divulges in the constructive progression of preferred methods of falsifiability and uncertainty. (shrink)

The prospect of cognitive enhancement well beyond current human capacities raises worries that the fundamental equality in moral status of human beings could be undermined. Cognitive enhancement might create beings with moral status higher than persons. Yet, there is an expressibility problem of spelling out what the higher threshold in cognitive capacity would be like. Nicholas Agar has put forward the bold claim that we can show by means of inductive reasoning that indefinite cognitive enhancement will probably mark a difference (...) in moral status. The hope is that induction can determine the plausibility of post‐personhood existence in the absence of an account of what the higher status would be like. In this article, we argue that Agar's argument fails and, more generally, that inductive reasoning has little bearing on assessing the likelihood of post‐personhood in the absence of an account of higher status. We conclude that induction cannot bypass the expressibility problem about post‐persons. (shrink)

This paper has three interdependent aims. The first is to make Reichenbach’s views on induction and probabilities clearer, especially as they pertain to his pragmatic justification of induction. The second aim is to show how his view of pragmatic justification arises out of his commitment to extensional empiricism and moots the possibility of a non-pragmatic justification of induction. Finally, and most importantly, a formal decision-theoretic account of Reichenbach’s pragmatic justification is offered in terms both of the minimax (...) principle and the dominance principle. (shrink)

To justify inductive inference and vanquish classical skepticisms about human memory, external world realism, etc., Richard Fumerton proposes his “inferential internalism,” an epistemology whereby humans ‘see’ by Russellian acquaintance Keynesian probable relations (PRs) between propositions. PRs are a priori necessary relations of logical probability, akin to but not reducible to logical entailments, such that perceiving a PR between one’s evidence E and proposition P of unknown truth value justifies rational belief in P to an objective degree. A recent critic (...) of inferential internalism is Alan Rhoda, who questions its psychological plausibility. Rhoda argues that in order to see necessary relations between propositions E and P, one would need acquaintance with too many propositions at once, since our evidence E is often complex. In this paper, I criticize Rhoda’s implausibility objection as too quick. Referencing the causal status effect (CSE) from psychology, I argue that some of the complex features of evidence E contribute to our type-categorizing it as E-type, and thus we do not need to ‘see’ all of the complex features when we see the PR between E and P. My argument leaves unchanged Fumerton’s justificatory role for the PR, but enhances its psychological plausibility. (shrink)

I examine what the mathematical theory of random structures can teach us about the probability of Plenitude, a thesis closely related to David Lewis's modal realism. Given some natural assumptions, Plenitude is reasonably probable a priori, but in principle it can be (and plausibly it has been) empirically disconfirmed—not by any general qualitative evidence, but rather by our de re evidence.

David Stove was a philosopher strong on argument and polemic. His work on the logical intepretation of probability led to a defence of induction in The Rationality of Induction (1986). It resulted too in his denunciation of Popper, Kuhn, Lakatos and Feyeraband as irrationalists because of their "deductivism" (the thesis that the only logic is deductive logic). Stove also defended controversial views on the intelligence of women and on Darwinism. The article surveys his life and work.

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)

On the basis of the distinction between logical and factual probability, epistemic justification is distinguished from logical justification of induction. It is argued that, contrary to the accepted interpretation of Hume, Hume believes that inductive inferences are epistemically legitimate and justifiable. Hence the beliefs arrived at via (correct) inductive inferences are rational beliefs. According to this interpretation, Hume is not a radical skeptic about induction.

Michael Tooley, in Plantinga & Tooley, "Knowledge of God" (Blackwell 2008) argues that, in the absence of strong evidence in favour of the existence of God, the logical probability of God's existence is extremely low. His argument focusses on rightmaking and wrongmaking properties of divine actions, and employs Carnap's inductive logic to reach his conclusion. I argue that Tooley's argument's conceptual foundations are problematic, and that his application of Carnap's inductive logic is flawed. I then provide an alternative analysis (...) of the relevant inductive issues. (shrink)

There is good reason to believe that scientific realism requires a commitment to the objective modal structure of the physical world. Causality, equilibrium, laws of nature, and probability all feature prominently in scientific theory and explanation, and each one is a modal notion. If we are committed to the content of our best scientific theories, we must accept the modal nature of the physical world. But what does the scientific realist’s commitment to physical modality require? We consider whether scientific (...) realism is compatible with Humeanism about the laws of nature, and we conclude that it is not. We specifically identify three major problems for the best-systems account of lawhood: its central concept of strength cannot be formulated non-circularly, it cannot offer a satisfactory account of the laws of the special sciences, and it can offer no explanation of the success of inductive inference. In addition, Humeanism fails to be naturalistically motivated. For these reasons, we conclude that the scientific realist must embrace natural necessity. (shrink)

The idea of the uniformity of nature, as a solution to the problem of induction, has at least two contemporary versions: natural kinds and natural necessity. Then there are at least three alternative ontological ideas addressing the problem of induction. In this paper, I articulate how these ideas are used to justify the practice of inductive inference, and compare them, in terms of their applicability, to see whether each of them is preferred in addressing the problem of (...) class='Hi'>induction. Given the variety of contexts in which inductive inferences are made, from natural science to social science and to everyday thinking, I suggest that no singular idea is absolutely preferred, and a proper strategy is probably to welcome the plurality of ideas helpful to induction, and to take pragmatic considerations into account, in order to judge in every single case. (shrink)

According to Karl Popper, science cannot verify its theories empirically, but it can falsify them, and that suffices to account for scientific progress. For Popper, a law or theory remains a pure conjecture, probability equal to zero, however massively corroborated empirically it may be. But it does just seem to be the case that science does verify empirically laws and theories. We trust our lives to such verifications when we fly in aeroplanes, cross bridges and take modern medicines. We (...) can do some justice to this apparent capacity of science to verify if we make a number of improvements to Popper’s philosophy of science. The key step is to recognize that physics, in accepting unified theories only, thereby makes a big metaphysical assumption about the nature of the universe. The outcome is a conception of scientific method which facilitates the criticism and improvement of metaphysical assumptions of physics. This view provides, not verification, but a perfect simulacrum of verification indistinguishable from the real thing. (shrink)

The usage of Randomised Evaluations(REs) in social inquiry has been recent and responses to them have been wide ranging.RE seek to make predictions on the impact of an intervention, when it is attempted in a new situation. REs work by first determining the impact of the intervention. Subsequently, for the new situation it is expected that the impact would be similar. -/- The problem of induction poses one of the most serious challenges to the epistemological status of RE claims (...) as the value of REs themselve stem from their predictive validity. If the problem of induction is not solved or evaded then belief in projections based on REs is undermined. -/- In this paper, I argue that the unique methodology of REs used in Social Sciences allows itself to evade the problem the successfully, warranting belief that future interventions would have similar impact. (shrink)

Recasting Hume and Early Modern Philosophy is a collection of essays that are all concerned with major figures and topics in the early modern philosophy. Most of the essays are concerned, more specifically, with the philosophy of David Hume (1711-1776). The sixteen essays included in this collection are divided into five parts. These parts are arranged under the headings of: (1) Metaphysics and Epistemology; (2) Free Will and Moral Luck; (3) Ethics, Virtue and Optimism; (4) Skepticism, Religion and Atheism; and (...) (5) Irreligion and the Unity of Hume’s Thought. A particularly important theme running through many of these essays is the subject of Hume’s irreligious aims and intentions. The fifth and final part of the collection is devoted to an articulation and defence of this specific understanding of Hume’s philosophical thought. (shrink)

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or (...) objective Bayesianism or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

Ambiguity occurs insofar as a reasoner lacks information about the relevant physical probabilities. There are objections to the application of standard Bayesian inductive logic and decision theory in contexts of significant ambiguity. A variety of alternative frameworks for reasoning under ambiguity have been proposed. Two of the most prominent are Imprecise Bayesianism and Dempster–Shafer theory. We compare these inductive logics with respect to the Ambiguity Dilemma, which is a problem that has been raised for Imprecise Bayesianism. We develop an agent-based (...) model comparison that isolates the difference between the two inductive logics in their updating methods. We find that Dempster–Shafer theory does not avoid the Ambiguity Dilemma. We discuss the implications of this result. (shrink)

After long arguments between positivism and falsificationism, the verification of universal hypotheses was replaced with the confirmation of uncertain major premises. Unfortunately, Hemple proposed the Raven Paradox. Then, Carnap used the increment of logical probability as the confirmation measure. So far, many confirmation measures have been proposed. Measure F proposed by Kemeny and Oppenheim among them possesses symmetries and asymmetries proposed by Elles and Fitelson, monotonicity proposed by Greco et al., and normalizing property suggested by many researchers. Based on (...) the semantic information theory, a measure b* similar to F is derived from the medical test. Like the likelihood ratio, measures b* and F can only indicate the quality of channels or the testing means instead of the quality of probability predictions. Furthermore, it is still not easy to use b*, F, or another measure to clarify the Raven Paradox. For this reason, measure c* similar to the correct rate is derived. Measure c* supports the Nicod Criterion and undermines the Equivalence Condition, and hence, can be used to eliminate the Raven Paradox. An example indicates that measures F and b* are helpful for diagnosing the infection of Novel Coronavirus, whereas most popular confirmation measures are not. Another example reveals that all popular confirmation measures cannot be used to explain that a black raven can confirm “Ravens are black” more strongly than a piece of chalk. Measures F, b*, and c* indicate that the existence of fewer counterexamples is more important than more positive examples’ existence, and hence, are compatible with Popper’s falsification thought. (shrink)

In this paper, we consider recent attempts to metaphysically explain agentive modality in terms of conditionals. We suggest that the best recent accounts face counterexamples, and more worryingly, they take some agentive modality for granted. In particular, the ability to perform basic actions features as a primitive in these theories. While it is perfectly acceptable for a semantics of agentive modal claims to take some modality for granted in getting the extension of action claims correct, a metaphysical explanation of agentive (...) modality cannot, at least not in the way that conditional approaches to agentive modality do. We argue that this problem was present even in the classical conditional analysis. By a pessimistic induction, we suggest that, probably, no conditional approach to agentive modality will succeed. (shrink)

Objective: In this essay, I will try to track some historical and modern stages of the discussion on the Gettier problem, and point out the interrelations of the questions that this problem raises for epistemologists, with sceptical arguments, and a so-called problem of relevance. Methods: historical analysis, induction, generalization, deduction, discourse, intuition results: Albeit the contextual theories of knowledge, the use of different definitions of knowledge, and the different ways of the uses of knowledge do not resolve all the (...) issues that the sceptic can put forward, but they can be productive in giving clarity to a concept of knowledge for us. On the other hand, our knowledge will always have an element of intuition and subjectivity, however not equating to epistemic luck and probability. Significance novelty: the approach to the context in general, not giving up being a Subject may give us a clarity about the sense of what it means to say – “I know”. (shrink)

We characterize those identities and independencies which hold for all probability functions on a unary language satisfying the Principle of Atom Exchangeability. We then show that if this is strengthen to the requirement that Johnson's Sufficientness Principle holds, thus giving Carnap's Continuum of inductive methods for languages with at least two predicates, then new and somewhat inexplicable identities and independencies emerge, the latter even in the case of Carnap's Continuum for the language with just a single predicate.