Consider the following. The first is a one-premise argument; the second has two premises. The question sign marks the conclusions as such. -/- Matthew, Mark, Luke, and John wrote Greek. ? Every evangelist wrote Greek. -/- Matthew, Mark, Luke, and John wrote Greek. Every evangelist is Matthew, Mark, Luke, or John. ? Every evangelist wrote Greek. -/- The above pair of premise-conclusion arguments is of a sort familiar to logicians and philosophers of science. In each case the first premise (...) is logically equivalent to the set of four atomic propositions: “Matthew wrote Greek”, “Mark wrote Greek”, “Luke wrote Greek”, and “John wrote Greek”. The universe of discourse is the set of evangelists. We presuppose standard first-order logic. -/- As many logic texts teach, the first of these two premise-conclusion arguments—sometimes called a complete enumerative induction— is invalid in the sense that its conclusion does not follow from its premises. To get a counterargument, replace ‘Matthew’, ‘Mark’, ‘Luke’, and ‘John’ by ‘two’,’four’, ‘six’ and ‘eight’; replace ‘wrote Greek’ by ‘are even’; and replace ‘evangelist’ by ‘number’. This replacement converts the first argument into one having true premises and false conclusion. -/- But the same replacement performed on the second argument does no such thing: it converts the second premise into the falsehood “Every number is two, four, six, or eight”. As many logic texts teach, there is no replacement that converts the second argument into one with all true premises and false conclusion. The second is valid; its conclusion is deducible from its two premises using an instructive natural deduction. -/- This paper “does the math” behind the above examples. The theorem could be stated informally: the above examples are typical. (shrink)

This paper addresses Harman’s approach to enumerative induction as a case of inference to the best explanation. Αfter taking under brief consideration Hume’s critique to induction, the paper argues that Harman’s proposal does not improve the situation since the same characteristics of induction and the kind of skepticism associated with it reappear in case of inference to the best explanation. Then the paper questions Armstrong’s attempt to upgrade Harman’s suggestion by regarding a necessitation relation among two universals (a natural (...) law) as the best explanans of the n observed cases of an enumerative induction procedure. It holds that Armstrong’s modification of Harman’s approach faces some difficulties which arise in the context of his views about instantiated universals. Instead, the paper investigates the possibility to assign the role of the explanans to a metaphysically stronger condition. It suggests that accounts of natural laws as metaphysically necessary facts concerning natural kinds can contribute in an attempt for a more fertile application of Harman’s approach. (shrink)

This paper argues that we should replace the common classification of theories of welfare into the categories of hedonism, desire theories, and objective list theories. The tripartite classification is objectionable because it is unduly narrow and it is confusing: it excludes theories of welfare that are worthy of discussion, and it obscures important distinctions. In its place, the paper proposes two independent classifications corresponding to a distinction emphasised by Roger Crisp: a four-category classification of enumerative theories (about which items (...) constitute welfare), and a four-category classification of explanatory theories (about why these items constitute welfare). (shrink)

Two kinds of theoretical framework for syntax are encountered in current linguistics. One emerged from the mathematization of proof theory, and is referred to here as generative-enumerative syntax (GES). A less explored alternative stems from the semantic side of logic, and is here called model-theoretic syntax (MTS). I sketch the outlines of each, and give a capsule summary of some mathematical results pertaining to the latter. I then briefly survey some diverse types of evidence suggesting that in some ways (...) MTS seems better suited to theorizing about the relevant linguistic phenomena. (shrink)

The question of how, according to Aristotle, the principles of science are acquired remains contested among scholars. An aspect of this broader topic concerns the role of induction, and whether it is able to provide us with knowledge of natural necessity without the assistance of intuition. In a recent publication in this journal, David Botting argues in favour of the enumerative/empiricist interpretation of induction and criticizes the intuitive/rationalist interpretation of it, a version of which was defended in one (...) of my publications. He thinks that Aristotle is like Hume: both understand the cognitive process of induction similarly; and, both are equally skeptical about acquiring knowledge of natural necessity through induction. My reply argues that reading Aristotle’s induction in Humean terms is problematic in several respects. I argue, in addition, that natural necessity can be known through induction if nous is involved. My explanation of how this is possible relies on thinking of the act of noēsis in terms of an act of recognition. Botting claims, furthermore, that Aristotle only differs from Hume in that the former does have a non-inductive and non-intuitive method by which natural necessity may become known, and which Botting calls “the constructive proof of necessity”. My reply examines this method, showing how certain steps in it rely on cognitive acts that are really acts of intuition merely expressed in Humean terms. Despite the criticisms, I end with suggestions for how Botting’s account might offer original paths of research to Aristotle scholars seeking to answer the question of the acquisition of principles of science, particularly in the early stages of this process. (shrink)

This book addresses controversies concerning the epistemological foundations of data science: Is it a genuine science? Or is data science merely some inferior practice that can at best contribute to the scientific enterprise, but cannot stand on its own? The author proposes a coherent conceptual framework with which these questions can be rigorously addressed. Readers will discover a defense of inductivism and consideration of the arguments against it: an epistemology of data science more or less by definition has to be (...) inductivist, given that data science starts with the data. As an alternative to enumerative approaches, the author endorses Federica Russo’s recent call for a variational rationale in inductive methodology. Chapters then address some of the key concepts of an inductivist methodology including causation, probability and analogy, before outlining an inductivist framework. The inductivist framework is shown to be adequate and useful for an analysis of the epistemological foundations of data science. The author points out that many aspects of the variational rationale are present in algorithms commonly used in data science. Introductions to algorithms and brief case studies of successful data science such as machine translation are included. Data science is located with reference to several crucial distinctions regarding different kinds of scientific practices, including between exploratory and theory-driven experimentation, and between phenomenological and theoretical science. Computer scientists, philosophers and data scientists of various disciplines will find this philosophical perspective and conceptual framework of great interest, especially as a starting point for further in-depth analysis of algorithms used in data science. (shrink)

A major difficulty for currently existing theories of inductive inference involves the question of what to do when novel, unknown, or previously unsuspected phenomena occur. In this paper one particular instance of this difficulty is considered, the so-called sampling of species problem.The classical probabilistic theories of inductive inference due to Laplace, Johnson, de Finetti, and Carnap adopt a model of simple enumerative induction in which there are a prespecified number of types or species which may be observed. But, (...) realistically, this is often not the case. In 1838 the English mathematician Augustus De Morgan proposed a modification of the Laplacian model to accommodate situations where the possible types or species to be observed are not assumed to be known in advance; but he did not advance a justification for his solution. (shrink)

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

This article grew out of a profound disinclination to make the kind of comment that I was invited to make on Angus Fletcher's article in a previous issue [June 1975]. I felt that such a writer as Mr. Fletcher, who clearly understands me, and, more important, himself, ought to be allowed the last word on both subjects. Besides that, I have a rooted dislike of the "position paper" genre. In all arts, adhering to a school and issuing group manifestoes and (...) statements of common aims is a sign of youthfulness, and to some degree of immaturity; as a painter or writer or other creative person grows older and acquires more authority, he tends to withdraw from all such organizations and become simply himself. Others in the same field become friends or colleagues rather than allies. I see no reason why that should not be the normal tendency in criticism and scholarship also. About twenty years ago I was asked, in a hotel lobby during an MLA conference, "What is your position relative to Kenneth Burke?" I forgot what I mumbled, but my real answer was, first, that I hadn't the least idea and, second, that anyone who could really answer such a question would have to be a third person, neither Burke nor Frye. Northrop Frye's contribution to contemporary thought has been discussed by prominent critics in Northup Frye in Modern Criticism: Selected Papers from the English Institute . The most comprehensive bibliography of his publications and of commentaries on them has been compiled by Robert D. Denham in Northup Frye: An Enumerative Bibliography . Angus Fletcher's interpretation of Professor Frye's works, "The Critical Passion," appeared in the June 1975 issue of Critical Inquiry. (shrink)

This study had two purposes: to explore the components of the pedagogical content knowledge (PCK) of an experienced elementary science teacher and to reveal the presumed interactions among these components. A naturalistic inquiry was conducted as a single case study in which in-depth qualitative data were gathered through semi-structured interview questions. After the theory-based and data-driven analysis of the qualitative data, the verbal communication was quantitated into numerical data for the enumerative analysis. The results revealed that the teacher’s knowledge (...) of students’ understanding and difficulties, knowledge of the elementary science curriculum and knowledge of instructional strategies and representations were found to be mostly intersected PCK components, whereas orientations towards teaching science and knowledge of assessment in science had limited connections to the other components. Further discussions and context-based suggestions are given regarding professional development programmes. (shrink)

In this paper, I suggest an approach to the alleged problem with the Tarskian formal definition of truth: its enumerative character seems to make it unable to capture our pretheoretic general understanding of truth. For this purpose, after spelling out two requirements for extending an enumerative definition to new cases, I examine to what extent Tarski's Convention T provides what are needed for extending the Tarski's enumerative definition. I conclude that, though not explicitly providing what are needed, (...) Convention T does implicitly suggest something quite promising and provide necessary conceptual resources for its further modifications. Then I suggest a Tarskian explicitly general definition of what it is to be a truth-definition on the basis of some theoretical and conceptual resources within Tarski's semantic theory, and I explain how it would make the Tarskian seemingly enumerative formal definition have a general character in accord with our pretheoretic understanding of truth. Finally, I examine the current proposal in comparison with Davidson's approach and in the light of Hintikka's new perspective. (shrink)

RÉSUMÉDans l’esprit du scepticisme modal, Peter Hawke propose une épistémologie modale, la théorie de l’explication sûre (SET), qui adopte la forme de l’empirisme modal. Avec la SET, il essaie de défendre l’induction énumérative (EI) : on croit raisonnablement que tout X est F sur la base d’un échantillon suffisamment grand dans lequel tout X est F. Dans cet article, je soutiens que l’argmentation de Hawke échoue. De plus, je signale un problème avec la SET, lequel cause cet échec : la (...) SET est trop stricte pour expliquer certaines affirmations possibles que nous avons le droit de croire. (shrink)

Number theory abounds with conjectures asserting that every natural number has some arithmetic property. An example is Goldbach’s Conjecture, which states that every even number greater than 2 is the sum of two primes. Enumerative inductive evidence for such conjectures usually consists of small cases. In the absence of supporting reasons, mathematicians mistrust such evidence for arithmetical generalisations, more so than most other forms of non-deductive evidence. Some philosophers have also expressed scepticism about the value of enumerative inductive (...) evidence in arithmetic. But why? Perhaps the best argument is that known instances of an arithmetical conjecture are almost always small: they appear at the start of the natural number sequence. Evidence of this kind consequently suffers from size bias. My essay shows that this sort of scepticism comes in many different flavours, raises some challenges for them all, and explores their respective responses. (shrink)

The paper is based on ranking theory, a theory of degrees of disbelief (and hence belief). On this basis, it explains enumerative induction, the confirmation of a law by its positive instances, which may indeed take various schemes. It gives a ranking theoretic explication of a possible law or a nomological hypothesis. It proves, then, that such schemes of enumerative induction uniquely correspond to mixtures of such nomological hypotheses. Thus, it shows that de Finetti's probabilistic representation theorems may (...) be transformed into an account of confirmation of possible laws and that enumerative induction is equivalent to such an account. The paper concludes with some remarks about the apriority of lawfulness or the uniformity of nature. (shrink)

In an enumerative induction we project a property found in all the examined members of a class to a hitherto unexamined member of that class. I consider an unresolved disagreement between Stephen Thomas and John Nolt about how likely the conclusion of one example of such reasoning is, given the premisses. Reflection on their controversy, and on the example, indicates that many textbook commonplaces about enumerative inductions are false. In particular, uniformity of the examined members of a class (...) does not necessarily make it highly likely that the next member will have the target property; this proposition may still be unlikely, or on the other hand be quite definite, depending on our background knowledge. Also, enumerative induction need not rely on any general assumption about the uniformity of nature or the resemblance of the future to the past. (shrink)

I propose a new definition of identification in the limit, also called convergence to the truth, as a new success criterion that is meant to complement, but not replace, the classic definition due to Putnam (1963) and Gold (1967). The new definition is designed to explain how it is possible to have successful learning in a kind of scenario that the classic account ignores—the kind of scenario in which the entire infinite data stream to be presented incrementally to the learner (...) is not presupposed to completely determine the correct learning target. For example, suppose that a scientists is interested in whether all ravens are black, and that she will never observe a counterexample in her entire life. This still leaves open whether all ravens (in the universe) are black. From a purely mathematical point of view, the proposed definition of convergence to the truth employs a convergence concept that generalizes net convergence and sits in between pointwise convergence and uniform convergence. Two results are proved to suggest that the proposed definition provides a success criterion that is by no means weak: (i) Between the proposed identification in the limit and the classic one, neither implies the other. (ii) If a learning method identifies the correct target in the limit in the proposed sense, any U-shaped learning involved therein has to be essentially redundant. I conclude that we should have (at least) two success criteria that correspond to two senses of identification in the limit: the classic one and the one proposed here. They are complementary: meeting any one of the two is good; meeting both at the same time, if possible, is even better. (shrink)

This paper represents one of the first critical social science interrogations of an agrifood just-in-case transition. The just-in-case transition speaks to a philosophy that values building buffers and flexibility into longer value chains to make them more resilient to shocks, which stands in contrast to the just-in-time philosophy with its emphasis on long, specialized, and often inflexible networks. Influenced by COVID-related disruptions and climate change induced uncertainties, the just-in-case transition examined here centers on the heightened interest in vertical farm-anchored supply (...) chains. Interviewing actors responsible for promoting vertical farm-anchored local supply chains in the US and Canada, I attempt to sketch out how these spaces, infrastructures, and practices care. Put differently, as understood through a feminist ethics of care, _whom_ and _what_ are cared for and _how_ is care practiced in these just-in-case transitions and _why_? Enumerative politics was observed in the data—the idea that we can make care count. Practices and discourses linked to infrastructural/supply chain transitions are highlighted that result in care being narrowly conceived as a technical or transactional matter. The paper concludes reflecting on what it means to afford just-in-case agrifood transitions animated by matters of care that hold greater emancipatory potentials. (shrink)

One recurring criticism of the best interests standard concerns its vagueness, and thus the inadequate guidance it offers to care providers. The lack of an agreed definition of ‘best interests’, together with the fact that several suggested considerations adopted in legislation or professional guidelines for doctors do not obviously apply across different groups of persons, result in decisions being made in murky waters. In response, bioethicists have attempted to specify the best interests standard, to reduce the indeterminacy surrounding medical decisions. (...) In this paper, we discuss the bioethicists’ response in relation to the state's possible role in clarifying the best interests standard. We identify and characterise two clarificatory strategies employed by bioethicists —elaborative and enumerative—and argue that the state should adopt the latter. Beyond the practical difficulties of the former strategy, a state adoption of it would inevitably be prejudicial in a pluralistic society. Given the gravity of best interests decisions, and the delicate task of respecting citizens with different understandings of best interests, only the enumerative strategy is viable. We argue that this does not commit the state to silence in providing guidance to and supporting healthcare providers, nor does it facilitate the abuse of the vulnerable. Finally, we address two methodological worries about adopting this approach at the state level. The adoption of the enumerative strategy is not defeatist in attitude, nor does it eventually collapse into (a form of) the elaborative strategy. (shrink)

What happens now is that we (by which I mean the West) eradicate state sponsored terrorism. And we can achieve that only by replacing all political systems that perpetrate or collaborate with terrorism, by systems that respect human rights both domestically and internationally.

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)

I resolve an apparently unresolved dispute about how probable uniform experience makes an extrapolation from it, and draw some general lessons about such enumerative induction. Uniform experience does not necessarily confer a high probability on an extrapolation of or generalization from that experience. Rational extrapolation or generalization typically involves a lot of specific background information, though not necessarily a general assumption that nature is uniform or that the future will resemble the past. And new evidence which is highly likely (...) on one hypothesis but highly unlikely on any of its competitors does not necessarily make the former hypothesis highly probable. (shrink)

Responding rationally to the information that others disagree with one’s beliefs requires assessing the epistemic credentials of the opposing beliefs. Conciliatory accounts of disagreement flow in part from holding that these assessments must be independent from one’s own initial reasoning on the disputed matter. I argue that this claim, properly understood, does not have the untoward consequences some have worried about. Moreover, some of the difficulties it does engender must be faced by many less conciliatory accounts of disagreement.

Under the influence of Quine’s famous manifesto, many philosophers have thought that logical theories are scientific theories that can be ‘adopted’ and tested as scientific theories. Here we argue that this idea is untenable. We discuss it with special reference to Putnam’s proposal to ‘adopt’ a particular non-classical logic to solve the foundational problems of quantum mechanics in his famous paper ‘Is Logic Empirical?’ (1968), which we argue was not really coherent.

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

In this paper I consider whether traditional behaviors of animals, like traditions of humans, are transmitted by imitation learning. Review of the literature on problem solving by captive primates, and detailed consideration of two widely cited instances of purported learning by imitation and of culture in free-living primates (sweet-potato washing by Japanese macaques and termite fishing by chimpanzees), suggests that nonhuman primates do not learn to solve problems by imitation. It may, therefore, be misleading to treat animal traditions and human (...) culture as homologous (rather than analogous) and to refer to animal traditions as cultural. (shrink)

Attitude verbs fall in different categories depending on the kind of sentential complements which they can embed. In English, a verb like know takes both declarative and interrogative complements. By contrast, believe takes only declarative complements and wonder takes only interrogative complements. The present paper examines the hypothesis, originally put forward by Hintikka (1975), that the only verbs that can take both that -complements and whether -complements are the factive verbs. I argue that at least one half of the hypothesis (...) is empirically correct, namely that all veridical attitude verbs taking that -complements take whether -complements. I distinguish veridical verbs from factive verbs, and present one way of deriving the generalization. Counterexamples to both directions of the factivity hypothesis are discussed, in particular the case of emotive factive verbs like regret , and the case of non-veridical verbs that licence whether complements, in particular tell, guess, decide and agree . Alternative accounts are discussed along the way, in particular Zuber (1982), Ginzburg (1995) and Saebø (2007). (shrink)

White's dense article on narrative discusses the ways that different groups of 20th century historians, particularly historical theorists (see pp.8-9), have constructed and deconstructed narrative as a means of communicating history. White himself acknowledges that narrativity challenges the scientific of history, but suggests that narrativity is not only unavoidable, but also offers a form of literary or allegorical truth.\n\nWhite first discusses the critiques of narrative as a means of communication--it focuses too heavily on political players, it is "unscientific," it is (...) inherently limited because narrative itself cannot be objective (although it's authors sometimes claim it to be).\n\nHe then moves on to discuss other, heavily theoretical, takes on why narrative is still useful as a means of historical communication: that narrative works because it performs as a trope which readers can easily recognize from myth and fiction, or, that narrative works because it allows us to transform events from the present to a synthesized past.\n\nWhite's article is difficult to read, but important background for anyone who wants to see what people have said about narrative in the past 100 years--and why it is still important. (shrink)

Introduces the phenomenology of the Other, taking into account the work of Husserl, Merleau-Ponty, Levinas, Schutz, and Derrida, but mostly going back to things themselves.