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.

God loves to provide well for His people. But He understands human nature. He knows how attractive material things can be. The more people have, the greater their temptation to use more and more for selfish reasons. So God gives His people outlets for the excess. "Take heed, and beware of covetousness: for a man's life consisteth not in the abundance of the things which he possesseth" (Luke 12:15). "Charge them that are rich-- that they be rich in good works, (...) ready to distribute,--laying up in store for themselves a good foundation" (1 Timothy 6:17-19). The real enjoyment of riches is the ability to give and share freely with others. That is one way we can lay up treasures in heaven.--Back cover. (shrink)

How we can rethink our lives and reality to remove our filters and realize the wholeness that is inherent in ourselves and in our world.

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.

This article will argue that the church is the mystic body of Christ that believers must guard from purveyors bend on twisting the truth. There is no doubt that the Catholic social teaching on medical and moral matters has proven to be pertinent and applicable to the ever-changing circumstances of health care and its delivery. In response to today’s challenges, these same moral principles of Catholic teaching provide the rationale and direction for the community of faith. In times of coronavirus (...) characterized by falsehoods, the truth is a virtue the church must strive for: “… If you abide in My word, you are My disciples indeed. And you shall know the truth, and the truth shall make you free.”(John 8: 31-32). The truth is Christ; it is Christological. Christians must understand the clash of worldviews that is changing the face of Zimbabwean society. A worldview is simply the sum total of our beliefs about the world; Christianity is a worldview that directs our daily decisions and actions. Worldviews affect the way we think. A worldview is a well-reasoned framework of beliefs and convictions that gives a true and unified perspective on the meaning of human existence. To paraphrase from the book of First Samuel (16: 7) – to see as God sees and not how the world sees. Beliefs are not simple propositions held inside us which drive our outward behaviour in predictable ways, they are more of acts of commitment than anything else. There is no doubt that the future of Catholicism lies in the hands of the church’s leadership and the believing Christians. Catholics look to their faith leaders for guidance on big decisions as regards COVID-19 issues. Conspiracy theories around COVID-19 deny that the COVID-19 pandemic is a biological virus. Catholics require direction from the Pope down to the episcopal leadership to chart the way forward. A worldview not only impacts how one understands and makes sense of the world but also influences how one expresses oneself in the world. The constellation of personal values, beliefs, assumptions, attitudes, and ideas that make up our worldview have an effect on our understanding of reality. Coronavirus has caused uncertainty among believers and confusion as to how to cope with the complex problems that arise in COVID-19 related issues. This article will aver that the nature of drawing the contours of catholicity regarding the biblical foundations of the health falls in the hands of orthodox Catholic teachings. (shrink)

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

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)

Richard Rorty's discussions of "romanticism," a term by which he means a set of general philosophical themes, not merely a body of literary and philosophical work of the eighteenth and nineteenth century, are not univocal in their approach. Rorty endorses romanticism within an overall antirealistic view that he interprets as "pragmatism." In some respects, Richard Rorty's view of romanticism is diametrically opposed to Shelley's, for although Rorty invokes Shelley's appeal to poetry as "center and circumference," he has no interest in (...) providing any access to a noumenal world. With logical positivism of the twentieth century, Rorty seems to presuppose that "eliminating metaphysics," where metaphysics would be any appeal to a reality beyond appearances, is not only possible, but highly desirable from an intellectual point of view. Given Rorty's ultimate aim, it is not surprising that Rorty's romantic idol will have his metaphysics up his sleeve. (shrink)

The Mathematical Principles of Natural Philosophy.

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

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)

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)

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.

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)

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)

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)

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)