Social epistemology may be defined as the conceptual and normative study of the social dimensions of knowledge. It studies the bearing of social relations, interests, roles, and institutions – what I will term “social conditions” – on the conceptual and normative conditions of knowledge. It differs from the sociology of knowledge in being a conceptual and normative, and not primarily empirical, study, and in limning the necessary and not merely the contingent social conditions of knowledge. The central question of social (...) epistemology is whether, and to what extent, the conditions of knowledge include social conditions. Is knowledge a property of knowers in isolation from their social setting (and in what sense of “isolation”), or does it involve a relation between knowers and their social circumstances? This question can take various forms and admit diverse answers, depending on the kinds of knowers, knowledge, and social relations we ask about. (shrink)

We locate winning strategies for various ${\mathrm{\Sigma }}_{3}^{0}$ -games in the L-hierarchy in order to prove the following: Theorem 1. KP+Σ₂-Comprehension $\vdash \exists \alpha L_{\alpha}\ models"\Sigma _{2}-{\bf KP}+\Sigma _{3}^{0}-\text{Determinacy}."$ Alternatively: ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}-{\mathrm{C}\mathrm{A}}_{0}\phantom{\rule{0ex}{0ex}}$ "there is a β-model of ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17 em}}{\mathrm{\Sigma }}_{3}^{0}$ -Determinacy." The implication is not reversible. (The antecedent here may be replaced with ${\mathrm{\Pi }}_{3}^{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({\mathrm{\Pi }}_{3}^{1}\right)-{\mathrm{C}\mathrm{A}}_{0}:\text{\hspace{0.17em}}{\mathrm{\Pi }}_{3}^{1}$ instances of Comprehension with only ${\mathrm{\Pi }}_{3}^{1}$ -lightface definable parameters—or even weaker theories.) Theorem 2. KP +Δ₂-Comprehension +Σ₂-Replacement + ${\mathrm{\Sigma }}_{3}^{0}\phantom{\rule{0ex}{0ex}}$ -Determinacy. (Here AQI (...) is the assertion that every arithmetical quasi-inductive definition converges.) Alternatively: $\Delta _{3}^{1}{\rm CA}_{0}+{\rm AQI}\nvdash \Sigma _{3}^{0}$ -Determinacy. Hence the theories: ${\mathrm{\Pi }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0},\text{\hspace{0.17em}}{\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}+\text{\hspace{0.17em}}{\mathrm{\Sigma }}_{3}^{0}-\mathrm{D}\mathrm{e}\mathrm{t}\phantom{\rule{0ex}{0ex}}$ -Det, ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}+\mathrm{A}\mathrm{Q}\mathrm{I}$ , and ${\mathrm{\Delta }}_{3}^{1}-{\mathrm{C}\mathrm{A}}_{0}\phantom{\rule{0ex}{0ex}}$ are in strictly descending order of strength. (shrink)

Goncharov and Peretyat'kin independently gave necessary and sufficient conditions for when a set of types of a complete theory is the type spectrum of some homogeneous model of. Their result can be stated as a principle of second order arithmetic, which is called the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which states (...) that every complete atomic theory has an atomic model. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and do the same for an analogous result of Peretyat'kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model. (shrink)

In this paper we study local induction w.r.t. Σ1‐formulas over the weak arithmetic. The local induction scheme, which was introduced in, says roughly this: for any virtual class that is progressive, i.e., is closed under zero and successor, and for any non‐empty virtual class that is definable by a Σ1‐formula without parameters, the intersection of and is non‐empty. In other words, we have, for all Σ1‐sentences S, that S implies, whenever is progressive. Since, in the weak context, we (...) have (at least) two definitions of Σ1, we obtain two minimal theories of local induction w.r.t. Σ1‐formulas, which we call Peano Corto and Peano Basso.In the paper we give careful definitions of Peano Corto and Peano Basso. We establish their naturalness both by giving a model theoretic characterization and by providing an equivalent formulation in terms of a sentential reflection scheme.The theories Peano Corto and Peano Basso occupy a salient place among the sequential theories on the boundary between weak and strong theories. They bring together a powerful collection of principles that is locally interpretable in. Moreover, they have an important role as examples of various phenomena in the metamathematics of arithmetical (and, more generally, sequential) theories. We illustrate this by studying their behavior w.r.t. interpretability, model interpretability and local interpretability. In many ways the theories are more like Peano arithmetic or Zermelo Fraenkel set theory, than like finitely axiomatized theories as Elementary Arithmetic, and. On the one hand, Peano Corto and Peano Basso are very weak: they are locally cut‐interpretable in. On the other hand, they behave as if they were strong: they are not contained in any consistent finitely axiomatized arithmetical theory, however strong. Moreover, they extend, the theory of parameter‐free Π1‐induction. (shrink)

Mainstream epistemologists have recently made a few isolated attempts to demonstrate the particular ways, in which specific types of knowledge are partly social. Two promising cases in point are Lackey’s dualism in the epistemology of testimony and Goldberg’s process reliabilist treatment of testimonial and coverage-support justification. What seems to be missing from the literature, however, is a general approach to knowledge that could reveal the partly social nature of the latter anytime this may be the case. Indicatively, even though Lackey (...) :345–361, 2007) has recently launched an attack against the Credit Account of Knowledge on the basis of testimony, she has not classified her view of testimonial knowledge into any of the alternative, general approaches to knowledge. Similarly, even if Goldberg’s attempt to provide a process reliabilist explanation of the social nature of testimonial knowledge is deemed satisfactory, his attempt to do the same in the case of coverage-support justification does not deliver the requisite result. This paper demonstrates that CAK can in fact provide, pace Lackey’s renunciation of the view, a promising account of the social nature of both testimonial and coverage-supported knowledge. Additionally, however, it can display further explanatory power by also revealing the social nature of knowledge produced on the basis of epistemic artifacts. Despite their disparities, all these types of knowledge count as partly social in nature, because in all these cases, according to CAK, the epistemic credit for the individual agent’s true belief must spread between the individual agent and certain parts of her epistemic community. Accordingly, CAK is a promising candidate for providing a unified approach to several and, perhaps all possible, instances of what we may call ‘weak epistemic anti-individualism’ within mainstream epistemology: i.e., the claim that the nature of knowledge can occasionally be both social and individual at the same time. (shrink)

The thesis of methodological individualism in social science is commonly divided into two different claims—explanatory individualism and ontological individualism. Ontological individualism is the thesis that facts about individuals exhaustively determine social facts. Initially taken to be a claim about the identity of groups with sets of individuals or their properties, ontological individualism has more recently been understood as a global supervenience claim. While explanatory individualism has remained controversial, ontological individualism thus understood is almost (...) universally accepted. In this paper I argue that ontological individualism is false. Only if the thesis is weakened to the point that it is equivalent to physicalism can it be true, but then it fails to be a thesis about the determination of social facts by facts about individual persons. Even when individualistic facts are expanded to include people’s local environments and practices, I shall argue, those still underdetermine the social facts that obtain. If true, this has implications for explanation as well as ontology. I first consider arguments against the local supervenience of social facts on facts about individuals, correcting some flaws in existing arguments and affirming that local supervenience fails for a broad set of social properties. I subsequently apply a similar approach to defeat a particularly weak form of global supervenience, and consider potential responses. Finally, I explore why it is that people have taken ontological individualism to be true. (shrink)

In Part I, the author argued for nonreductive individualism (NRI), an account of the individual-collective relation that is ontologically individualist yet rejects methodological individualism. However, because NRI is ontologically individualist, social entities and properties would seem to be only analytic constructs, and if so, they would seem to be epiphenomenal, since only real things can have causal power. In general, a nonreductionist account is a relatively weak defense of sociological explanation if it cannot provide an account of (...) how social properties can participate in causal relations. In this article, the author extends NRI to address this weakness and provides an account of social causation that he refers to as supervenient causation. Key Words: individualism • collectivism • social realism • social causation. (shrink)

Inductive generalization, where people go beyond the data provided, is a basic cognitive capability, and it underpins theoretical accounts of learning, categorization, and decision making. To complete the inductive leap needed for generalization, people must make a key ‘‘sampling’’ assumption about how the available data were generated. Previous models have considered two extreme possibilities, known as strong and weak sampling. In strong sampling, data are assumed to have been deliberately generated as positive examples of a concept, whereas (...) in weak sampling, data are assumed to have been generated without any restrictions. We develop a more general account of sampling that allows for an intermediate mixture of these two extremes, and we test its usefulness. In two experiments, we show that most people complete simple one-dimensional generalization tasks in a way that is consistent with their believing in some mixture of strong and weak sampling, but that there are large individual differences in the relative emphasis different people give to each type of sampling. We also show experimentally that the relative emphasis of the mixture is influenced by the structure of the available information. We discuss the psychological meaning of mixing strong and weak sampling, and possible extensions of our modeling approach to richer problems of inductive generalization. (shrink)

We introduce a new axiom called inductive dichotomy, a weak variant of the axiom of inductive definition, and analyze the relationships with other variants of inductive definition and with related axioms, in the general second order framework, including second order arithmetic, second order set theory and higher order arithmetic. By applying these results to the investigations on the determinacy axioms, we show the following. (i) Clopen determinacy is consistency-wise strictly weaker than open determinacy in these frameworks, (...) except second order arithmetic; this is an enhancement of Schweber–Hachtman separation of open and clopen determinacy into the consistency-wise separation. (ii) Hausdorff–Kuratowski hierarchy of differences of opens is faithfully reflected by the hierarchy of consistency strengths of corresponding parameter-free determinacies in the aforementioned frameworks; this result is valid also in second order arithmetic only except clopen determinacy. (shrink)

The last couple of decades have witnessed a renewed interest in the notion of inductive risk among philosophers of science. However, while it is possible to find a number of suggestions about the mitigation of inductive risk in the literature, so far these suggestions have been mostly relegated to vague marginal remarks. This paper aims to lay the groundwork for a more systematic discussion of the mitigation of inductive risk. In particular, I consider two approaches to the (...) mitigation of inductive risk—the individualistic approach, which maintains that individual scientists are primarily responsible for the mitigation of inductive risk, and the socialized approach, according to which the responsibility for the mitigation of inductive risk should be more broadly distributed across the scientific community or, even more broadly, across society. I review some of the argument for and against the two approaches and introduce two new problems for the individualistic approach, which I call the problem of precautionary cascades and the problem of exogenous inductive risk, and I argue that a socialized approach might alleviate each of these problems. (shrink)

Jullien's indecomposability theorem (INDEC) states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis, and then, in the base system RCA₀ plus ${\mathrm{\Sigma }}_{1}^{1}\text{\hspace{0.17em}}$ induction, it was shown by Neeman to have strength strictly between weak ${\mathrm{\Sigma }}_{1}^{1}$ choice and ${\mathrm{\Delta }}_{1}^{1}$ comprehension. We prove in this paper that ${\mathrm{\Sigma }}_{1}^{1}$ induction is needed (...) for the reversal of INDEC, that is for the proof that INDEC implies weak ${\mathrm{\Sigma }}_{1}^{1}$ choice. This is in contrast with the typical situation in reverse mathematics, where reversals can usually be refined to use only ${\mathrm{\Sigma }}_{1}^{0}$ induction. (shrink)

By a well-known result of Kotlarski et al., first-order Peano arithmetic \ can be conservatively extended to the theory \ of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and commutes with all the propositional connectives and quantifiers. This result motivates the general question of determining natural axioms concerning the truth predicate that can be added to \ while maintaining conservativity over \. Our main result shows that conservativity fails even (...) for the extension of \ obtained by the seemingly weak axiom of disjunctive correctness \ that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, \ implies \\). Our main result states that the theory \ coincides with the theory \ obtained by adding \-induction in the language with the truth predicate. This result strengthens earlier work by Kotlarski and Cieśliński. For our proof we develop a new general form of Visser’s theorem on non-existence of infinite descending chains of truth definitions and prove it by reduction to Gödel’s second incompleteness theorem, rather than by using the Visser–Yablo paradox, as in Visser’s original proof. (shrink)

The account of meta-induction (G. Schurz, Hume’s problem solved: the optimality of meta-induction, MIT Press, Cambridge, 2019) proposes a two-step solution to the problem of induction. Step 1 consists in a mathematical a priori justification of the predictive optimality of meta-induction, upon which step 2 builds a meta-inductive a posteriori justification of object-induction based on its superior track record (Sect. 1). Sterkenburg (Br J Philos Sci, forthcoming. 10.1086/717068/) challenged this account by arguing that meta-induction can only provide a (non-circular) (...) justification of inductive predictions for now and for the next future, but not a justification of inductive generalizations (Sect. 2). This paper develops a meta-inductive method that does provide an a posteriori justification of inductive generalizations, in the form of exchangeability conditions (Sect. 3). In Sect. 4, a limitation of the proposed method is worked out: while the method can justify weakly lawlike generalizations, the justification of strongly lawlike generalizations (claimed to hold for all eternity) requires epistemic principles going beyond meta-induction based on predictive success. (shrink)

Smith is generally regarded as an individualist without qualification. This paper argues that his predominantly individualist policy prescription is rooted in a more complex philosophy. He sees nature, including human nature, as a vast machine supervised by God and designed to maximise human happiness. Human weaknesses, as well as strengths, display the wisdom of God and play their part in this scheme. While Smith pays lip service to justice, it is really social order that pre-occupies him, and within that, the (...) defence of property. Individuals are valued as bearers of property. As persons, individuals are deceived by nature into acting in a socially beneficial way. In different ways Smith systematically denies the autonomy of the individual with respect to the whole of which he is part. For Smith, individual liberty is, not the end, but the means, of sustaining social order and property. (shrink)

The elementary arithmetic operations +,·,≤\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${+,\cdot,\le}$$\end{document} on integers are well-known to be computable in the weak complexity class TC0, and it is a basic question what properties of these operations can be proved using only TC0-computable objects, i.e., in a theory of bounded arithmetic corresponding to TC0. We will show that the theory VTC0 extended with an axiom postulating the totality of iterated multiplication proves induction for quantifier-free formulas in the language (...) ⟨+,·,≤⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle{+,\cdot,\le}\rangle}$$\end{document}, and more generally, minimization for Σ0b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma_{0}^{b}}$$\end{document} formulas in the language of Buss’s S2. (shrink)

We develop measure theory in the context of subsystems of second order arithmetic with restricted induction. We introduce a combinatorial principleWWKL (weak-weak König's lemma) and prove that it is strictly weaker thanWKL (weak König's lemma). We show thatWWKL is equivalent to a formal version of the statement that Lebesgue measure is countably additive on open sets. We also show thatWWKL is equivalent to a formal version of the statement that any Borel measure on a compact metric space (...) is countably additive on open sets. (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)

This chapter contains sections titled: Liberals versus Communitarians? Individualism Community From Community to Republic Acknowledgments Notes References.

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)

In [10], we have shown that the statement that all ∑ 1 1 partitions are Ramsey is deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition,but the reversal needs П 1 1 - CA 0 rather than ATR 0 . By contrast, we show in this paper that the statement that all ∑ 0 2 games are determinate is also deducible over ATR 0 from the axiom of ∑ 1 1 monotone inductive definition, (...) but the reversal is provable even in ACA 0 . These results illuminate the substantial differences among lightface theorems which can not be observed in boldface. (shrink)

We consider two formalizations of the notion of irrelevance as a rationality principle within the framework of (Carnapian) Inductive Logic: Johnson's Sufficientness Principle, JSP, which is classically important because it leads to Carnap's influential Continuum of Inductive Methods and the recently proposed Weak Irrelevance Principle, WIP. We give a complete characterization of the language invariant probability functions satisfying WIP which generalizes the Nix-Paris Continuum. We argue that the derivation of two very disparate families of inductive methods (...) from alternative perceptions of 'irrelevance' is an indication that this notion is imperfectly understood at present. (shrink)

Girard’s maxim, that Peano Arithmetic is a theory of one inductive definition, is re-examined in the light of a weak theory EA formalising basic principles of Nelson’s predicative Arithmetic.

In this paper, I intend to clarify Alfred Schutz’s complex relationship to methodological individualism (MI), showing that he defends a “hermeneutic,” “weak” and “partial” variant of this approach. I will do so by focusing mainly on his reception of Max Weber, given the latter’s centrality in the MI paradigm.. In order to achieve my objective, I will proceed in three steps. First (1), to avoid misunderstandings, I will provide an updated definition of MI, distinguishing it from ontological and (...) normative individualism and giving an overview of its different variants. Second (2), I will systematically reconstruct Schutz’s convergences with Weber’s “hermeneutic” MI. Finally (3), I will develop the thesis of the “partiality” and “weakness” of Schutz’s MI with the help of recent literature on his work. (shrink)

This paper starts from the familiar premise that psychological anti-individualism is incompatible with materialism. It attempts to state more clearly what this incompatibility consists in, and — rather than arguing in detail for any particular resolution — to inquire whether this incompatibility admits any resolution. However, the paper does offer a conditional argument concerning the possibility that the incompatibility is genuine and cannot be resolved. Provided that anti-individualism and materialism cannot be squared, and anti-individualism is correct, it (...) follows that materialism has to be abandoned. If so, the situation is not as disastrous as it might at first seem. We need not, in consequence of our inability to construe a coherent metaphysics of mind, give up on intentional vocabulary any more than we must stop, in consequence of our inability to make sense of induction, anticipating the future. (shrink)

A standard way to challenge convergence-based accounts of inductive success is to claim that they are too weak to constrain inductive inferences in the short run. We respond to such a challenge by answering some questions raised by Juhl (1994). When it comes to predicting limiting relative frequencies in the framework of Reichenbach, we show that speed-optimal convergence—a long-run success condition—induces dynamic coherence in the short run.

In his previous work, the author has introduced the axiom schema of inductive dichotomy, a weak variant of the axiom schema of inductive definition, and used this schema for elementary ) positive operators to separate open and clopen determinacies for those games in which two players make choices from infinitely many alternatives in various circumstances. Among the studies on variants of inductive definitions for bounded ) positive operators, the present article investigates inductive dichotomy for these (...) operators, and applies it to constructive investigations of variants of determinacy statements for those games in which the players make choices from only finitely many alternatives. As a result, three formulations of open determinacy, that are all classically equivalent with each other, are equivalent to three different semi-classical principles, namely Markov’s Principle, Lesser Limited Principle of Omniscience and Limited Principle of Omniscience, over a suitable constructive base theory that proves clopen determinacy. Open and clopen determinacies for these games are thus separated. Some basic results on variants of inductive definitions for \ positive operators will also be given. (shrink)

Inductive generalization asserts that what obtains in known instances can be generalized to all. Its original form is enumerative induction, the earliest form of inductive inference, and it has been elaborated in various ways, largely with the goal of extending its reach. Its principal problem is that it supplies no intrinsic notion of strength of support so that one cannot tell if the generalization has weak or strong support.

The notion of countable well order admits an alternative definition in terms of embeddings between initial segments. We use the framework of reverse mathematics to investigate the logical strength of this definition and its connection with Fraïssé’s conjecture, which has been proved by Laver. We also fill a small gap in Shore’s proof that Fraïssé’s conjecture implies arithmetic transfinite recursion over $\mathbf {RCA}_0$, by giving a new proof of $\Sigma ^0_2$ -induction.

This paper argues that the notion of weak intentionalism in Mark Bevir's The Logic of the History of Ideas is incoherent. Bevir's proposal for weak intentionalism as procedural individualism relies on the argument that the object of study for historians of ideas is given by the beliefs that are expressed by individuals since these beliefs constitute the historical meaning of the work for those individuals as historical figures. Historical meanings are thus hermeneutic meanings. In the case of (...) insincere, unconscious, and irrational beliefs, however, the beliefs expressed by individuals are not in fact their actual beliefs, and their actual beliefs are now taken to be those expressed by the works. It thus turns out that it is not the beliefs expressed by individuals that are the object of study for historians but the works themselves, since the overriding requirement for historians of ideas is to "make sense of their material" and it is this requirement that determines whether or not the beliefs are to be construed as expressed by individuals or by the works. But once it is accepted that the beliefs that are the object of study for historians are expressed by the works and not by individuals, the original argument that such beliefs are historical hermeneutic meanings for historical figures no longer applies. The argument for weak intentionalism thus turns out to be incoherent. Bevir's argument fails to establish that the object of study for the history of ideas is external to the works, and the attempted distinction between interpreting a work and reading a text also fails. (shrink)

1. Introduction. Attempts to utilize the probability calculus to prove or disprove various inductive or inductive skeptical theses are, I believe, highly problematic. Inductivism and inductive skepticism are substantive philosophical positions that do not allow of merely formal proofs or disproofs. Often the problems with particular alleged formal proofs of inductive or inductive sceptical theses turn on subtle technical considerations. In the following I highlight such considerations in pointing out the flaws of two proofs, one (...) an alleged proof of an inductive sceptical conclusion due to Karl Popper, the other an alleged proof of an inductivist thesis originally due to Harold Jeffreys and later advocated by John Earman. Surprisingly, in examining Popper's argument it is shown that certain apparently weak premises, often embraced by both inductivists and deductivists, lend themselves to inductive conclusions. However it is argued that those premises are still philosophically substantive and not amenable to a purely formal demonstration. The lesson to be learnt here is twofold. First, we need to be very careful in determining which formal theses entail, and which are entailed by, inductive skepticism and inductivism. Second, we need to take great care in laying out and examining the assumptions presumed in formal arguments directed for and against such formal theses. In a follow up article I will consider various attempts by David Stove and Karl Popper and David Miller to identify the exact content of inductive skepticism and propose a new identification based on the theory of content developed in Gemes 1994. Finally I will compare this new version with that proposed recently in Alberto Mura 1990. (shrink)

In intuitionistic generalized predicative systems as constructive set theory, or constructive type theory, two categories have been proposed to play the role of the category of locales: the category FSp of formal spaces, and its full subcategory FSpi of inductively generated formal spaces. Considered in impredicative systems as the intuitionistic set theory IZF, FSp and FSpi are both equivalent to the category of locales. However, in the mentioned predicative systems, FSp fails to be closed under basic constructions such as that (...) of finite products, while FSpi is complete , but does not contain some naturally occurring classes of formal spaces. Moreover, completeness of FSpi only holds under the assumption of strong principles for the existence of inductively defined sets.In this paper, working in the context of constructive set theory, the category ImLoc of imaginary locales is introduced. ImLoc is complete already over weak formulations of constructive set theory, contains FSp and FSpi as full subcategories, and, as FSp and FSpi, is equivalent to the category of locales in IZF.Applications of the concept of imaginary locale to a proof of a point-free version of Tychonoff embedding theorem, and to set-representation theorems, are then presented. (shrink)

Depending on whether we are somewhat tolerant of nearby error-possibilities or not, the safety condition on knowledge is open to a strong reading and a weak reading. In this paper, it is argued that induction and conjunction introduction constitute two horns of a dilemma for the safety account of knowledge. If we opt for the strong reading, then the safety account fails to account for inductive knowledge. In contrast, if we opt for the weak reading, then the (...) safety account fails to accommodate knowledge obtained via the method of conjunction introduction. (shrink)

Jerry Fodor has defended the claim that psychological theories should appeal to narrow rather than wide intentional properties. One of his arguments relies upon the cross contexts test, a test that purports to determine whether two events have the same causally relevant properties. Critics have charged that this test is too weak, since it counts certain genuinely explanatory relational properties in science as being causally irrelevant. Further, it has been claimed, the test is insensitive to the fact that special (...) scientific laws allow for exceptions which do not undermine those laws. This paper refines the cross contexts test to meet these objections while still allowing it to play its role in Fodor. (shrink)

This paper starts from the familiar premise that psychological anti-individualism is incompatible with materialism. It attempts to state more clearly what this incompatibility consists in, and {\textemdash} rather than arguing in detail for any particular resolution {\textemdash} to inquire whether this incompatibility admits any resolution. However, the paper does offer a conditional argument concerning the possibility that the incompatibility is genuine and cannot be resolved. Provided that anti-individualism and materialism cannot be squared, and anti-individualism is correct, it (...) follows that materialism has to be abandoned. If so, the situation is not as disastrous as it might at first seem. We need not, in consequence of our inability to construe a coherent metaphysics of mind, give up on intentional vocabulary any more than we must stop, in consequence of our inability to make sense of induction, anticipating the future. (shrink)

This study aims to develop and defend an account of practical rationality and intention that explains how weakness of the will is conceptually possible. I first present two sceptical arguments against the possibility of weakness and then distinguish two different responses to scepticism that defends its possibility. Both sceptical arguments are motivated by what many have believed is an analogy between theoretical and practical reasoning. This analogy holds that the conclusion of practical reasoning is an intention just as the conclusion (...) of theoretical reasoning is a belief. The particular version held by what I call the Socratic theorists claims, in addition, that the value of desirability is transferred from premisses to conclusion in practical reasoning, just as probability is transferred from premisses to conclusion in theoretical reasoning. One example of an author who takes the analogy as a basic premise and seek to explain how weakness of the will still may be possible, is Donald Davidson. Critics of Davidson object that the the kinds of difficulties that his account runs into shows that the Socratic version of the analogy must be false; not only is this account phenomenologically implausible, it also exaggerates the degree of irrationality of the weak agent. In this study I suggest the possibility of a weaker version of the Davidsonian account. The main source of difficulty for the Davidsonian account, I argue, may not be its reliance on the Socratic version of the analogy between practical and theoretical reasoning, but rather its particular understanding of this analogy in terms of an analogy between practical and inductive-statistical reasoning. This understanding may lead one to accept too “rationalistic” a picture of the reasoning processes behind weakness of the will which, in effect, may make the account vulnerable to objections that it is phenomenologically implausible and exaggerates the irrationality of the weak agent. The weaker version of the Davidsonian account that I propose, ties in the account of weakness with the notions of “satisficing” and “default reasoning”. Although it preserves the central idea of the stronger version that intentions are unconditional evaluative judgements, it proposes a different view of the contents of these judgements, as well as of the reasoning process that lie behind them. It is not difficult to motivate this weaker view: given our human finitary predicament, both satisficing and default reasoning seem to be likely results of an optimal trade-off between the speed and reliability of our computational processes. The question is how these notions may shape the concepts we use to understand weakness of the will. I argue that if we modify the account usually ascribed to Davidson in the light of these notions, the result is a less rationalistic picture of weakness of the will. The gains are more phenomenological credibility as well as a less extreme view of the degree of irrationality involved. (shrink)

Localism is the view that the unit of evaluation in the scientific realism debate is a single scientific discipline, sub-discipline, or claim, whereas individualism is the view that the unit of evaluation is a single scientific theory. Localism is compatible, while individualism is not, with a local pessimistic induction and a local selective induction. Asay presents several arguments to support localism and undercut globalism, according to which the unit of evaluation is the set of all scientific disciplines. I (...) argue that some of his arguments clash with localism as well as with globalism and support individualism, and that individualism goes hand in hand, while localism does not, with the basic rule of how to evaluate an argument. (shrink)

Weak Quantum Theory (WQT) and the Model of Pragmatic Information (MPI) are two psychophysical concepts developed on the basis of quantum physics. The present study contributes to their empirical examination. The issue of the study is whether WQT and MPI can not only explain ‘psi’-phenomena theoretically but also prove to be consistent with the empirical phenomenology of extrasensory perception (ESP). From the main statements of both models, 33 deductions for psychic readings are derived. Psychic readings are defined as settings, (...) in which psychics support or counsel clients by using information not mediated through the five senses. A qualitative approach is chosen to explore how the psychics experience extrasensory perceptions. Eight psychics are interviewed with a half-structured method. The reports are examined regarding deductive and inductive aspects, using a multi-level structured content analysis. The vast majority of deductions is clearly confirmed by the reports. Even though the study has to be seen as an explorative attempt with many aspects to be specified, WQT and MPI prove to be coherent and helpful concepts to explain ESP in psychic readings. (shrink)

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)

By , we denote the system of second-order arithmetic based on recursive comprehension axioms and Σ10 induction. is defined to be plus weak König's lemma: every infinite tree of sequences of 0's and 1's has an infinite path. In this paper, we first show that for any countable model M of , there exists a countable model M′ of whose first-order part is the same as that of M, and whose second-order part consists of the M-recursive sets and sets (...) not in the second-order part of M. By combining this fact with a certain forcing argument over universal trees, we obtain the following result : if proves X!Y with arithmetical, so does . We also discuss several improvements of this results. (shrink)

Can a collective be an agent in its own right? Can it be the bearer of moral and other properties that we have traditionally reserved for individual agents? The answer, as one might expect, is ‘In some ways yes, in other ways no.’ The way in which the answer is ‘Yes’ has been described recently by Copp; I intend to discuss his position and defend it against objections. This describes a fairly weak form of autonomy that I will claim (...) does not require the abandonment of methodological individualism or our commonplace intuitions about individual responsibility. I will also discuss, and reject, a stronger conception of autonomy suggested especially by the work of Pettit that would result in methodological individualism. (shrink)

We show that length initial submodels of S12 can be extended to a model of weak second order arithmetic. As a corollary we show that the theory of length induction for polynomially bounded second order existential formulae cannot define the function division.

The first main topic of this paper is a weak second-order theory that sits between firstorder Peano Arithmetic PA1 and axiomatized second-order Peano Arithmetic PA2 – namely, that much-investigated theory known in the trade as ACA0. What I’m going to argue is that ACA0, in its standard form, lacks a cogent conceptual motivation. Now, that claim – when the wraps are off – will turn out to be rather less exciting than it sounds. It isn’t that all the work (...) that has been done on ACA0 has been hopelessly misplaced: that would be a quite absurd suggestion. The mistake, if that’s what it is, has been a relatively small one. Still, we really ought to try to put things into conceptual good order here. That’s part of what philosophers are for. Here’s the structure of my main claim. On the one hand, interesting work on ACA0 actually only uses part of the strength of the theory: or as we might put it, the interesting work is actually carried on in a cut-down theory I’ll call ACA!. This theory, I’ll be claiming, does have a good conceptual motivation – it is in fact the theory that the putative conceptual grounding for ACA0 actually underpins. On the other hand, I’ll be arguing that original-strength ACA0 inductively inflates. I mean, to put it more carefully, that anyone who accepts ACA0 as a cogent theory can have no reason not to accept a certain significantly stronger theory, with a stronger induction principle. This stronger theory is standardly known as plain ACA. So, my claim comes to this: you can either go for the cut-down theory ACA!; or you can go for the much richer theory ACA. What you can’t do is – I mean, what you can’t have a stable conceptual motivation for doing – is to rest content with the intermediate strength ACA0 in its standard presentation. Yet in much of the literature, in particular in Simpson’s encyclopedic book Subsystems of Second-Order Arithmetic (1991), neither ACA! nor full ACA gets so much as a mention, and the conceptually unstable theory ACA0 gets all the glory. Why is my claim at all interesting? For at least two reasons.. (shrink)

The transformers that drive chatbots and other AI systems constitute large language models (LLMs). These are currently the focus of a lively discussion in both the scientific literature and the popular media. This discussion ranges from hyperbolic claims that attribute general intelligence and sentience to LLMs, to the skeptical view that these devices are no more than “stochastic parrots”. I present an overview of some of the weak arguments that have been presented against LLMs, and I consider several of (...) the more compelling criticisms of these devices. The former significantly underestimate the capacity of transformers to achieve subtle inductive inferences required for high levels of performance on complex, cognitively significant tasks. In some instances, these arguments misconstrue the nature of deep learning. The latter criticisms identify significant limitations in the way in which transformers learn and represent patterns in data. They also point out important differences between the procedures through which deep neural networks and humans acquire knowledge of natural language. It is necessary to look carefully at both sets of arguments in order to achieve a balanced assessment of the potential and the limitations of LLMs. (shrink)

The inductive logic developed in the second and third essays is limited in important ways. For example: (a) the logic makes no provision for missing or misleading data; (b) it gives the scientist no control over the evidence reaching him; (c) revision-based scientist must work with theories written in the cramped idiom of firstorder logic; (d) the idea of efficient induction is only weakly expressed (in terms of “dominance”).

My purpose in this chapter is to survey some of the principal approaches to inductive inference in the philosophy of science literature. My first concern will be the general principles that underlie the many accounts of induction in this literature. When these accounts are considered in isolation, as is more commonly the case, it is easy to overlook that virtually all accounts depend on one of very few basic principles and that the proliferation of accounts can be understood as (...) efforts to ameliorate the weaknesses of those few principles. In the earlier sections, I will lay out three inductive principles and the families of accounts of induction they engender. In later sections I will review standard problems in the philosophical literature that have supported some pessimism about induction and suggest that their import has been greatly overrated. In the final sections I will return to the proliferation of accounts of induction that frustrates efforts at a final codification. I will suggest that this proliferation appears troublesome only as long as we expect inductive inference to be subsumed under a single formal theory. If we adopt a material theory of induction in which individual inductions are licensed by particular facts that prevail only in local domains, then the proliferation is expected and not problematic. (shrink)

In this paper we continue the study of the theories IΔ n+1 (T), initiated in [7]. We focus on the quantifier complexity of these fragments and theirs (non)finite axiomatization. A characterization is obtained for the class of theories such that IΔ n+1 (T) is Π n+2 –axiomatizable. In particular, IΔ n+1 (IΔ n+1 ) gives an axiomatization of Th Π n+2 (IΔ n+1 ) and is not finitely axiomatizable. This fact relates the fragment IΔ n+1 (IΔ n+1 ) to induction (...) rule for Δ n+1 –formulas. Our arguments, involving a construction due to R. Kaye (see [9]), provide proofs of Parsons’ conservativeness theorem (see [16]) and (a weak version) of a result of L.D. Beklemishev on unnested applications of induction rules for Π n+2 and Δ n+1 formulas (see [2]). (shrink)

In this paper I offer a straight solution to Hume's problem of induction by defusing the assumptions on which it is based. I argue that Hume's problem only arises if we accept (i) that there is no necessity but logical necessity, or (ii) that it is unreasonable to believe that there is any form of necessity in addition to logical necessity. I show that Hume's arguments in favour of (i) and (ii) are unsound. I then offer a suggestion as to (...) how the weakness of his arguments has escaped detection. Finally, having claimed that there remains a surmountable problem with inductive arguments, I end by characterising that problem and a possible approach to its solution. (shrink)