The definition of a pseudofinite structure can be translated verbatim into continuous logic, but it also gives rise to a stronger notion and to two parallel concepts of pseudocompactness. Our purpose is to investigate the relationship between these four concepts and establish or refute each of them for several basic theories in continuous logic. Pseudofiniteness and pseudocompactness turn out to be equivalent for relational languages with constant symbols, and the four notions coincide with the standard pseudofiniteness in the case of (...) classical structures, but the details appear to be slightly more important here than in the usual translation of definitions from classical logic. We also prove that injective “formula-definable” endofunctions are surjective, and conversely, in strongly pseudofinite omega-saturated structures. (shrink)

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite-dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand's theorem can be used to characterize definable functions in absolutely ubiquitous structures from classical logic.

The Connes Embedding Problem asks whether every separable II1 factor embeds into an ultrapower of the hyperfinite II1 factor. We show that the CEP is equivalent to the statement that every type II1 tracial von Neumann algebra has a computable universal theory.

We show that any countable model of a model complete theory has an elementary extension with a “pseudofinite-like” quasi-dimension that detects dividing.

Let H be an infinite-dimensional (real or complex) Hilbert space, viewed as a metric structure in its natural signature. We characterize the definable linear operators on H as exactly the "scalar plus compact" operators.

Answering a question of Cifú Lopes, we give a syntactic characterization of those continuous sentences that are preserved under reduced products of metric structures. In fact, we settle this question in the wider context of general structures as introduced by the second author.

Answering a question of Cifú Lopes, we give a syntactic characterization of those continuous sentences that are preserved under reduced products of metric structures. In fact, we settle this question in the wider context of general structures as introduced by the second author.

The present paper shows how one might model Everettian quantum mechanics using hyperfinitely many worlds. A hyperfinite model allows one to consider idealized measurements of observables with continuous-valued spectra where different outcomes are associated with possibly infinitesimal probabilities. One can also prove hyperfinite formulations of Everett’s limiting relative-frequency and randomness properties, theorems he considered central to his formulation of quantum mechanics. Finally, this model provides an intuitive framework in which to consider no-collapse formulations of quantum mechanics more generally.

We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for discrete first-order real rosy theories.

We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form $$\phi ^\mathcal {M}\le r$$, and the open diagram, which encapsulates strict inequalities of the form $$\phi ^\mathcal {M}< r$$. We show (...) that the closed and open $$\Sigma _N$$ diagrams are $$\Pi ^0_{N+1}$$ and $$\Sigma ^0_N$$ respectively, and that the closed and open $$\Pi _N$$ diagrams are $$\Pi ^0_N$$ and $$\Sigma ^0_{N + 1}$$ respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal. (shrink)

For a locally compact group G, we show that it is possible to present the class of continuous unitary representations of G as an elementary class of metric structures, in the sense of continuous logic. More precisely, we show how nondegenerate ∗-representations of a general ∗-algebra A (with some mild assumptions) can be viewed as an elementary class, in a many-sorted language, and use the correspondence between continuous unitary representations of G and nondegenerate ∗-representations of L1(G). We relate the notion (...) of ultraproduct of logical structures, under this presentation, with other notions of ultraproduct of representations appearing in the literature, and we characterize property (T) for G in terms of the definability of the sets of fixed points of L1 functions on G. (shrink)

We study the relationship between fields of transseries and residue fields of convex subrings of non-standard extensions of the real numbers. This was motivated by a question of Todorov and Vernaeve, answered in this paper.

The present paper shows how one might model Everettian quantum mechanics using hyperfinitely many worlds. A hyperfinite model allows one to consider idealized measurements of observables with continuous-valued spectra where different outcomes are associated with possibly infinitesimal probabilities. One can also prove hyperfinite formulations of Everett’s limiting relative-frequency and randomness properties, theorems he considered central to his formulation of quantum mechanics. Finally, this model provides an intuitive framework in which to consider no-collapse formulations of quantum mechanics more generally.

The randomization of a complete first-order theory [Formula: see text] is the complete continuous theory [Formula: see text] with two sorts, a sort for random elements of models of [Formula: see text] and a sort for events in an underlying atomless probability space. We study independence relations and related ternary relations on the randomization of [Formula: see text]. We show that if [Formula: see text] has the exchange property and [Formula: see text], then [Formula: see text] has a strict independence (...) relation in the home sort, and hence is real rosy. In particular, if [Formula: see text] is o-minimal, then [Formula: see text] is real rosy. (shrink)

In the paper Randomizations of Scattered Sentences, Keisler showed that if Martin’s axiom for aleph one holds, then every scattered sentence has few separable randomizations, and asked whether the conclusion could be proved in ZFC alone. We show here that the answer is “yes”. It follows that the absolute Vaught conjecture holds if and only if every \-sentence with few separable randomizations has countably many countable models.

I present and explain the Bohmian account of collapse in quantum mechanics.

I propose an account of possible worlds. According to the account, possible worlds are pluralities of sentences in an extremely large language. This account avoids a problem, relating to the total number of possible worlds, that other accounts face. And it has several additional benefits.

We present a novel model of individual people, online posts, and media platforms to explain the online spread of epistemically toxic content such as fake news and suggest possible responses. We argue that a combination of technical features, such as the algorithmically curated feed structure, and social features, such as the absence of stable social-epistemic norms of posting and sharing in social media, is largely responsible for the unchecked spread of epistemically toxic content online. Sharing constitutes a distinctive communicative act, (...) governed by a dedicated norm and motivated to a large extent by social identity maintenance. But confusion about this norm and its lack of inherent epistemic checks lead readers to misunderstand posts, attribute excess or insufficient credibility to posts, and allow posters to evade epistemic accountability—all contributing to the spread of epistemically toxic content online. This spread can be effectively addressed if people and platforms add significantly more context to shared posts and platforms nudge people to develop and follow recognized epistemic norms of posting and sharing. (shrink)

For the past fifty years anxiety over naturalism has driven debates in social theory. One side sees social science as another kind of natural science, while the other rejects the possibility of objective and explanatory knowledge. _Interpretation and Social Knowledge_ suggests a different route, offering a way forward for an antinaturalist sociology that overcomes the opposition between interpretation and explanation and uses theory to build concrete, historically specific causal explanations of social phenomena.

My aim in this paper is to give a philosophical analysis of the relationship between contingently available technology and the knowledge that it makes possible. My concern is with what specific subjects can know in practice, given their particular conditions, especially available technology, rather than what can be known “in principle” by a hypothetical entity like Laplace’s Demon. The argument has two parts. In the first, I’ll construct a novel account of epistemic possibility that incorporates two pragmatic conditions: responsibility and (...) practicability. For example, whether subjects can gain knowledge depends in some circumstances on whether they have the capability of gathering relevant evidence. In turn, the possibility of undertaking such investigative activities depends in part on factors like ethical constraints, economical realities, and available technology. In the second part of the paper, I’ll introduce “technological possibility” to analyze the set of actions made possible by available technology. To help motivate the problem and later test my proposal, I’ll focus on a specific historical case, one of the earliest uses of digital electronic computers in a scientific investigation. I conclude that the epistemic possibility of gaining access to scientific knowledge about certain subjects depends (in some cases) on the technological possibility for making responsible investigations. (shrink)

There has recently been increasing interest in the possibility and ethics of lethal autonomous weapons systems (LAWS), which would combine sophisticated AI with machinery capable of deadly force. One objection to LAWS is that their use will create a troubling responsibility gap, where no human agent can properly be held accountable for the outcomes that they create. While some authors have attempted to show that individual agents can, in fact, be responsible for the behaviour of LAWS in various circumstances, this (...) will not be possible in all cases. This article argues, however, that by assigning responsibilities to group agents operating within the military‐industrial complex, as well as the military‐industrial complex as a whole, we may be able to close the responsibility gap, or at least reduce its costs. Insofar as the potential for responsibility gaps poses an ethical barrier to the development and deployment of LAWS, this argument, if sound, may show their use to be morally permissible in a greater range of cases. (shrink)

Presents Newton's unifying idea of gravitation and explains how he converted physics from a science of explanation into a general mathematical system.

This book draws on fields as diverse as biochemistry, physiology, pharmacology, psychology, psychiatry, and ethology, to form a fascinating synthesis of information on the nature of fear and of panic and anxiety disorders. Dr. Marks offers both a detailed discussion of the clinical aspects of fear-related syndromes and a broad exploration of the sources and mechanisms of fear and defensive behavior. Dealing first with normal fear, he establishes a firm, scientific basis for understanding it. He then presents a thorough analysis (...) of the development, symptoms and treatment of fear-related syndromes. Phobic and obsessive-compulsive disorders are examined in detail. The book is illustrated with examples of fear and defensive behavior in other living organisms. By drawing provocative analogies between animal and human behavior, it sheds new light on the origins of fears, phobias, and obsessive-compulsive problems, as well as on their treatment by drugs and psychological means. Clinical psychologists, ethologists, and anyone interested in the mechanisms of behavior will be fascinated by this authoritative study. The text is intriguing and informative, and the bibliography of over 2,100 entries makes it an invaluable reference. (shrink)

This major work challenges some widely held positions in epistemology - those of Peirce and Popper on the one hand and those of Quine and Kuhn on the other.

The Mathematical Principles of Natural Philosophy.

David Chalmers thinks his iPhone exemplifies the extended mind thesis by meeting the criteria ‎that he and Andy Clark established in their well-known 1998 paper. Andy Clark agrees. We take ‎this proposal seriously, evaluating the case of the GPS-enabled smartphone as a potential mind ‎extender. We argue that the “trust and glue” criteria enumerated by Clark and Chalmers are ‎incompatible with both the epistemic responsibilities that accompany everyday activities and the ‎practices of trust that enable users to discharge them. Prospects (...) for revision of the original ‎criteria are dim. We therefore call for a rejection of the trust criterion and a reevaluation of the ‎extended mind thesis.‎. (shrink)

Reproduces the text of Newton's dissertation on the nature and properties of light.

Criminal justice systems have traditionally relied heavily on human decision-making, but new technologies are increasingly supplementing the human role in this sector. This paper considers what general limits need to be placed on the use of algorithms in sentencing decisions. It argues that, even once we can build algorithms that equal human decision-making capacities, strict constraints need to be placed on how they are designed and developed. The act of condemnation is a valuable element of criminal sentencing, and using algorithms (...) in sentencing – even in an advisory role – threatens to undermine this value. The paper argues that a principle of “meaningful public control” should be met in all sentencing decisions if they are to retain their condemnatory status. This principle requires that agents who have standing to act on behalf of the wider political community retain moral responsibility for all sentencing decisions. While this principle does not rule out the use of algorithms, it does require limits on how they are constructed. (shrink)

This article draws on the conceptual link between power and causality to develop an account of the relational, discursive, and performative dimensions of power. Each proposed dimension of power is grounded in a different understanding of social causes: relational-realist, discursive-hermeneutic, and performative-pragmatic. For the purposes of empirical analysis, this dimensional schema crosscuts the classic sources of power typology developed by Michael Mann and others, thus rendering the conceptual apparatus for pursuing sociological research on power more complex and explanatorily effective. The (...) schema is illustrated by an example from comparative-historical sociology: explaining the storming of the Bastille and its effects. A series of research questions for investigating the relative autonomy of performative power is proposed. Finally, the current schema is situated vis-à-vis classic sociological theories of power, including the arguments of Steven Lukes, Michel Foucault, and Pierre Bourdieu, among others. (shrink)

This comprehensive discussion of the problem of rational belief develops the subject on the pattern of Bayesian decision theory. The analogy with decision theory introduces philosophical issues not usually encountered in logical studies and suggests some promising new approaches to old problems."We owe Professor Levi a debt of gratitude for producing a book of such excellence. His own approach to inductive inference is not only original and profound, it also clarifies and transforms the work of his predecessors. In short, the (...) book deserves to become a classic....There is a great deal of interest in the book besides these basic matters [forumlating rules of acceptance]. Some of the most interesting chapters are those that examine the implications of such rules. The discussions of probability, generalization, and various forms of inference are brilliant and enlightening. Indeed, the problems and methods elaborated by Professor Levi in his book serve as a new foundation for the study of inductive inference."--Keith Lehrer, Nous"Levi's book is an extremely interesting report on 'tentative and speculative first steps' toward a decision-theoretic approach to inductive inference....Professor Levi is to be congratulated on his ingenious development and application of this approach...."--Richard C. Jeffrey, The Journal of Philosophy. (shrink)

Though written corporate codes of ethics have been touted as a panacea for the embarrassments and uncertainties of the past two decades, the absence of clear evaluation procedures severely compromises their usefulness. An ethnographic study comparing development processes and compliance outcomes in large health care facilities and energy companies shows that neither of the two industries has encountered much success with a codes of ethics program. Companies that distribute copies of their code of ethics seldom ensure the process is completed (...) or that employees understand the purpose of the document, and staff responsible for the code give it a low priority relative to their overall responsibilities. Contrary to expectations, health care facilities are no more likely to develop or implement codes of ethics effectively than are energy companies. More extensive research is needed in order to generate the data necessary for the development of realistic standards for the evaluation of codes of ethics. (shrink)

Typicality is routinely invoked in everyday contexts: bobcats are typically short-tailed; people are typically less than seven feet tall. Typicality is invoked in scientific contexts as well: typical gases expand; typical quantum systems exhibit probabilistic behaviour. And typicality facts like these support many explanations, both quotidian and scientific. But what is it for something to be typical? And how do typicality facts explain? In this paper, I propose a general theory of typicality. I analyse the notion of a typical property. (...) I provide a formalism for typicality explanations, and I give an account of why typicality explanations are explanatory. Along the way, I show how typicality facts explain a variety of phenomena, from everyday phenomena to the statistical mechanical behaviour of gases. Finally, I argue that typicality is not the same thing as probability. (shrink)

Isaac Levi's new book is concerned with how one can justify changing one's beliefs. The discussion is deeply informed by the belief-doubt model advocated by C. S. Peirce and John Dewey, of which the book provides a substantial analysis. Professor Levi then addresses the conceptual framework of potential changes available to an inquirer. A structural approach to propositional attitudes is proposed, which rejects the conventional view that a propositional attitude involves a relation between an agent and either a linguistic (...) entity or some other intentional object such as a proposition or set of possible worlds. The last two chapters offer an account of change in states of full belief understood as changes in commitments rather than changes in performance; one chapter deals with adding new information to a belief state, the other with giving up information. The book builds upon topics discussed in some of Levi's earlier work. It will be of particular interest to discussion theorists, epistemologists, philosophers of science, computer scientists, and cognitive psychologists. (shrink)

Sir Isaac Newton (1642-1727) left a voluminous legacy of writings. Despite his influence on the early modern period, his correspondence, manuscripts, and publications in natural philosophy remain scattered throughout many disparate editions. In this volume, Newton's principal philosophical writings are for the first time collected in a single place. They include excerpts from the Principia and the Opticks, his famous correspondence with Boyle and with Bentley, and his equally significant correspondence with Leibniz, which is often ignored in favor of (...) Leibniz's later debate with Samuel Clarke. Newton's exchanges with Leibniz place their different understandings of natural philosophy in sharp relief. The volume also includes 'De Gravitatione', offered here in a corrected translation, which is crucial for understanding Newton's relation to his great predecessor Descartes. In a historical and philosophical introduction, Andrew Janiak examines Newton's philosophical positions and his relations to canonical figures in early modern philosophy. (shrink)

Why would God institute the practice of efficacious petitionary prayer? Why would God not simply give us what we need before we ask? I examine recently proposed solutions to this puzzle and argue that they are inadequate to explain why an omniscient and perfectly good God would act differently in response to prayer. I propose that God has reasons to not always maximize a creature’s good, even in a sinless world, and that petitionary prayer functions as a means to reward (...) those who trust God, to enable us to actively love those we cannot otherwise help, and to give the petitioner personal evidence of God’s existence and care for her, creating a virtuous cycle of increasing faith. I refine this proposal by responding to several objections involving human responsibility and the epistemology of divine action. Along the way, I offer several ways petitioners can recognize God’s having answered a prayer and how God might help us with some common obstacles to prayer. (shrink)