We say that a ring admits elimination of quantifiers, if in the language of rings, {0, 1, +, ·}, the complete theory of R admits elimination of quantifiers. Theorem 1. Let D be a division ring. Then D admits elimination of quantifiers if and only if D is an algebraically closed or finite field. A ring is prime if it satisfies the sentence: ∀ x ∀ y ∃ z (x = 0 ∨ y = 0 ∨ xzy (...) ≠ 0). Theorem 2. If R is a prime ring with an infinite center and R admits elimination of quantifiers, then R is an algebraically closed field. Let A be the class of finite fields. Let B be the class of 2 × 2 matrix rings over a field with a prime number of elements. Let C be the class of rings of the form $GF(p^n) \bigoplus GF(p^k)$ such that either n = k or g.c.d. (n, k) = 1. Let D be the set of ordered pairs (f, Q) where Q is a finite set of primes and f: Q → A ∪ B ∪ C such that the characteristic of the ring f(q) is q. Finally, let E be the class of rings of the form $\bigoplus_{q \in Q}f(q)$ for some (f, Q) in D. Theorem 3. Let R be a finite ring without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R belongs to E. Theorem 4. Let R be a ring with the descending chain condition of left ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R is an algebraically closed field or R belongs to E. In contrast to Theorems 2 and 4, we have Theorem 5. If R is an atomless p-ring, then R is finite, commutative, has no nonzero trivial ideals and admits elimination of quantifiers, but is not prime and does not have the descending chain condition. We also generalize Theorems 1, 2 and 4 to alternative rings. (shrink)

ABSTRACT: René Descartes' Cogito is an example of a paradigmatic thought experiment, herald of both subjectivism and new science in Europe's Modern Age, that seems to have escaped the attention of thought experiment philosophers. On deep analysis, the Cogito appears as universal instantiation. The Cogito has strong rhetorical effects for it narratively generalizes from I to all human kind, and its historical and philosophical success can be explained from its concise enthymematic structure that rings true in many possible senses. We (...) consider it a preeminent example of a thought experiment as it states the power of thinking as its very contents. From Descartes' methodology of doubt we can conclude that, e.g., on a Wittgensteinian interpretation, the Cogito is a logical thought experiment rather than a psychological one. RESUMO: O Cogito de René Descartes é um exemplo de experimento mental paradigmático, precursor tanto do subjetivismo quanto da nova ciência, na Idade Moderna europeia, o qual parece ter escapado à atenção dos filósofos que estudaram o experimento mental. Na análise profunda, o Cogito aparece como uma instanciação universal. O Cogito tem fortes efeitos retóricos por si mesmo, generalizando narrativamente desde o eu para toda a espécie humana, e seu sucesso histórico e filosófico pode ser explicado por sua estrutura entimemática concisa, que soa através de muitos sentidos possíveis. Consideramos que é um exemplo proeminente de um experimento mental, na medida em que afirma o poder de pensar como seus próprios conteúdos. A partir da metodologia da dúvida de Descartes, podemos concluir que, em uma interpretação wittgensteiniana, o Cogito é um experimento mental mais lógico que psicológico. (shrink)

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite, and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.

Reflecting Alan Robinson's fundamental contribution to computational logic, this book brings together seminal papers in inference, equality theories, and logic programming. It is an exceptional collection that ranges from surveys of major areas to new results in more specialized topics. Alan Robinson is currently the University Professor at Syracuse University. Jean-Louis Lassez is a Research Scientist at the IBM Thomas J. Watson Research Center. Gordon Plotkin is Professor of Computer Science at the University of Edinburgh. Contents: Inference. Subsumption, A Sometimes (...) Undervalued Procedure, Larry Wos, Ross Overbeek, and Ewing Lusk. The Markgraf Karl Refutation Procedure, Hans Jurgen Ohlbach and Jorg H. Siekmann. Modal Logic Should Say More than it Does, Melvin Fitting. Interactive Proof Presentation, W. W. Bledsoe. Intelligent Backtracking Revisited, Maurice Bruynooghe. A Science of Reasoning, Alan Bundy. Inductive Inference of Theories from Facts, Ehud Y. Shapiro. Equality. Solving Equations in Abstract Algebras: A Rule-based Survey of Unification, Jean-Pierre Jouannaud and Claude Kirchner. Disunification: A Survey, Hubert Comon. A Case Study of the Completion Procedure: Proving Ring Commutativity Problems, Deepak Kapur and Hantao Zhang. Computations in Regular Rewriting Systems I and II, Girard Huet and JeanJacques Levy. Unification and ML Type Reconstruction, Paris Kanellakis, Harry Mairson, and John Mitchell. Automatic Dimensional Analysis, Mitchell Wand. Logic Programming. Logic Programming Schemes and Their Implementations, Keith Clark. A Near-Horn Prolog for Compilation, Donald Loveland and David Reed. Unfold/Fold Transformations of Logic Programs, P. A. Gardner and J. C. Shepherdson. An Algebraic Representation of Logic Program Computations, Andrea Corradini and Ugo Montanari. Theory of Disjunctive Logic Programs, Jack Minker, Arcot Rajasekar, and Jorge Lobo. Bottom-Up Evaluation of Logic Programs, Jeffrey Naughton and Raghu Ramakrishnan. Absys, the First Logic Programming Language: A View of the Inevitability of Logic Programming, E. W. Elcock. (shrink)

Mathematical Essays in Honor of Gian-Carlo Rota, Boston, Basel, Berlin, ... Crapo, H. (1993), On the Anick-Rota Representation of the Bracket Ring of the ...

Perhaps we can all agree that in order for intelligent activity to be produced by embodied nervous systems, those nervous systems have to have things in them that are about other things in the following minimal sense: there is information about these other things not just present but usable by the nervous system in its modulation of behavior. (There is information about the climatic history of a tree in its growth rings--the information is present, but not usable by the tree.) (...) The disagreements set in when we start trying to characterize what these things-aboutthings are--are they "just" competences or dispositions embodied somehow (e.g., in connectionist networks) in the brain, or are they more properly mental representations, such as sentences in a language of thought, images, icons, maps, or other data structures? And if they are "symbols", how are they "grounded"? What, more specifically, is the analysis of the aboutness that these things must have? Is it genuine intentionality or mere as if intentionality? These oft-debated questions are, I think, the wrong questions to be concentrating on at this time, even if, "in the end", they make sense and deserve answers. These questions have thrived in the distorting context provided by two ubiquitous idealizing assumptions that we should try setting aside: an assumption about how to capture content and an assumption about how to isolate the vehicles of content from the "outside" world. (shrink)

In this article, we present results from a literature review of intrinsic, instrumental, and relational values of nature conducted for the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services, as part of the Methodological Assessment of the Diverse Values and Valuations of Nature. We identify the most frequently recurring meanings in the heterogeneous use of different value types and their association with worldviews and other key concepts. From frequent uses, we determine a core meaning for each value type, which is (...) sufficiently inclusive to serve as an umbrella over different understandings in the literature and specific enough to help highlight its difference from the other types of values. Finally, we discuss convergences, overlapping areas, and fuzzy boundaries between different value types to facilitate dialogue, reduce misunderstandings, and improve the methods for valuation of nature's contributions to people, including ecosystem services, to inform policy and direct future research. © 2023 The Author(s). Published by Oxford University Press on behalf of the American Institute of Biological Sciences. (shrink)

(Also titled "A Place for Peirce's Categories?"in Meaning without Analyticity.) This book arose from the author’s recent dissertation written under the Gerhard Schönrich at Munich. It focuses on Peirce’s theory of categories and his epistemology. According to Baltzer, what is distinctive in Peirce’s theory of knowledge is that he reconstrues objects as “knots in networks of relations.” The phrase may ring a bell. It suggests a structuralist interpretation of Peirce, influenced by the Munich environs. The study aims to shows (...) how Peirce’s theory of categories supports his theory of knowledge and how “question concerning a priori structures of knowledge” are transformed within this relational framework. A chief critical target is David Savan’s semiotics, specifically the idea that “the multiplicity of development of the categories” is “conditioned by nothing but the indefiniteness of the categories.” But in contrast with this, if there is any indefiniteness in the categories, they cannot fully direct their own application, and this is to say regarding them “that our knowledge is never absolute but always swims, as it were, in a continuum...” If the doctrine of continuity applies to the categories, they also have a continuum to swim in. (shrink)

G. C. J. Midgley; XIV—Linguistic Rules, Proceedings of the Aristotelian Society, Volume 59, Issue 1, 1 June 1959, Pages 271–290, https://doi.org/10.1093/aristot.

According to the prevailing sentiment, changing the past is logically impossible. The prevailing sentiment is wrong. In this paper, I argue that the claim that changing the past entails a contradiction ultimately rests upon an empirical assumption, and so the conclusion that changing the past is logically impossible is to be resisted. I then present and discuss a model of time which drops the empirical assumption and coherently models changing the past. Finally, I defend the model, and changing the past, (...) against objections. (shrink)

Some philosophers argue that any attempt to model changing the past will either be contradictory or really model avoiding the past. Using Nicholas Smith's (1997) argument as a basis, I formulate a generic version of this Avoidance Argument. I argue that the Avoidance Argument fails because (i) it involves an equivocation of what is meant by ‘bifurcation of the time of an event’ and (ii) resolving the equivocation results in the falsity of at least one of the premises. Hence, the (...) Avoidance Argument will not support the claim that changing the past is logically impossible. (shrink)

Quantum mechanics, which describes the behavior of subatomic particles, seems to challenge common sense. Waves behave like particles; particles behave like waves. You can tell where a particle is, but not how fast it is moving--or vice versa. An electron faced with two tiny holes will travel through both at the same time, rather than one or the other. And then there is the enigma of creation ex nihilo, in which small particles appear with their so-called antiparticles, only to disappear (...) the next instant in a tiny puff of energy. Since its inception, physicists and philosophers have struggled to work out the meaning of quantum mechanics. Some, like Niels Bohr, have responded to quantum mechanics' mysteries by replacing notions of position and velocity with probabilities. Others, like Einstein and Penrose, have disagreed and think that the entire puzzle reflects not a fundamental principle of nature but our own ignorance of basic scientific processes. Sneaking a Look at God's Cards offers the general reader a deep and real understanding of the problems inherent to the interpretation of quantum mechanics, from its inception to the present. The book presents a balanced overview of current debates and explores how the theory of quantum mechanics plays itself out in the real world. Written from the perspective of a leading European physicist, the book looks extensively at ideas from both sides of the Atlantic and also considers what philosophers have contributed to the scientific discussion of this field. Sneaking a Look at God's Cards sets out what we know about the endlessly fascinating quantum world, how we came to this understanding, where we disagree, and where we are heading in our quest to comprehend the seemingly incomprehensible. (shrink)

One of Hume's most well-known works and a masterpiece of philosophy, A Treatise of Human Nature is indubitably worth taking the time to read.

We apply the distinction between parameter independence and outcome independence to the linear and nonlinear models of a recent nonrelativistic theory of continuous state vector reduction. We show that in the nonlinear model there is a set of realizations of the stochastic process that drives the state vector reduction for which parameter independence is violated for parallel spin components in the EPR-Bohm setup. Such a set has an appreciable probability of occurrence (≈ 1/2). On the other hand, the linear model (...) exhibits only extremely small parameter dependence effects. We investigate some specific features of the models and we recall that, as has been pointed out recently, if one wants to be able to speak of definite outcomes (or equivalently of possessed objective elements of reality) at finite times, one has to slightly change the criteria for their attribution to physical systems. The concluding section is devoted to a detailed discussion of the difficulties which one meets when one tries to take, as a starting point for the formulation of a relativistic theory, a nonrelativistic scheme which exhibits parameter dependence. Here we derive a theorem which identifies the precise sense in which the occurrence of parameter dependence forbids a genuinely relativistic generalization. Finally we show how the appreciable parameter dependence of the nonlinear model gives rise to problems with relativity, while the extremely weak parameter dependence of the linear model does not give rise to any difficulty, provided one takes into account the appropriate criteria for the attribution of definite outcomes. (shrink)

We analyse a recent paper in which an alleged devastating criticism of the so called GRW proposal to account for the objectification of the properties of macroscopic systems has been presented and we show that the author has not taken into account the precise implications of the GRW theory. This fact makes his conclusions basically wrong. We also perform a survey of measurement theory aimed to focus better on the physical and the conceptual aspects of the so-called macro-objectification problem.

Numerous informal logi- cians and argumentation theorists restrict their theorizing to what they call “real” arguments. But is there a clear distinction to be made between “real” and “non-real” arguments? Here I explore four possible accounts of the alleged distinction and argue that none can serve the theoretical uses to which the distinction is most often put. Résumé: Plusieurs logiciens construction formels et théoriciens de l’argument- ation limitent leur non de théories à ce qu’ils appellent des arguments « authentiques ». (...) Mais y-a- t’il une distinction claire entre des arguments« authentiques » et « inau- thentiques » ? Ici j’explore quatre descriptions possibles de cette préten- due distinction et je soutiens qu’- aucune ne répond aux besoins théori- ques auxquels elles sont censées ré- pondre. (shrink)

Despite a growing number of models argument for the logical possibility of changing the past there continues to be resistance to and confusion surrounding the possibility of changing the past. In this paper I shall attempt to mitigate the resistance and alleviate at least some of the confusion by distinguishing changing the past from what Richard Hanley calls ‘annulling’ the past and distinguishing both from what I shall call ‘otherwising’ the past.

David Hitchcock, in his recent “Informal Logic and the Concept of Argument”, defends a recursive definition of ‘argument.’ I present and discuss several problems that arise for his definition. I argue that refining Hitchcock’s definition in order to resolve these problems reveals a crucial, but minimally explicated, relation that was, at best, playing an obscured role in the original definition or, at worst, completely absent from the original definition.

In previous work I argued against (i) the likelihood of finding a theoretically sound foundation for the linked/convergent distinction and (ii) the utility of the distinction even if a sound theoretical basis could be found. Here I subject Douglas Walton’s comprehensive discussion of the linked/convergent distinction found in Argument Structure: A Pragmatic Theory to careful scrutiny and argue that at best Walton’s theory remains incomplete and that attempts to fill out the details will run afoul of at least one of (...) the problems adduced above—i.e., result in either a theoretically unsound distinction or a theoretically sound, but unnecessary distinction. (shrink)

In this paper I argue that the debate over the purported distinction between deductive and inductive arguments can be bypassed because making the distinction is unnecessary for successfully evaluating arguments. I provide a foundation for doing logic that makes no appeal to the distinction and still performs all the relevant tasks required of an analysis of arguments. I also reply to objections to the view that we can dispense with the distinction. Finally, I conclude that the distinction between inductive and (...) deductive arguments is not one of the most important and fundamental ideas in logic, but rather is unnecessary. (shrink)

One popular test for distinguishing linked and convergent argument structures is Robert Yanal's Ordinary Summing Test. Douglas Walton, in his comprehensive survey of possible candidates for the linked/convergent distinction, advocates a particular version of Yanal's test. In a recent article, Alexander Tyaglo proposes to generalize and verifY Yanal's algorithm for convergent arguments, the basis for Yanal's Ordinary Summing Test. In this paper I will argue that Yanal's ordinary summing equation does not demarcate convergence and so his Ordinary Summing Test fails. (...) Hence, despite Walton's recommendation or Tyaglo's generalization, the Ordinary Summing Test should not be used for distinguishing linked argument structures from convergent argument structures. (shrink)

A brief review of the conceptual difficulties met by the quantum formalism is presented. The main attempts to overcome these difficulties are considered and their limitations are pointed out. A recent proposal based on the assumption of the occurrence of a specific type of wave function collapse is discussed and its consequences for the above-mentioned problems are analyzed.

In his recent paper, “What a Real Argument is”, Ben Hamby attempts to provide an adequate theoretical account of “real” arguments. In this paper I present and evaluate both Hamby’s motivation for distinguishing “real” from non-“real” arguments and his articulation of the distinction. I argue that neither is adequate to ground a theoretically significant class of “real” arguments, for the articulation fails to pick out a stable proper subclass of all arguments that is simultaneously both theoretically relevant and a proper (...) subclass of all arguments. (shrink)

With reference to recently proposed theoretical models accounting for reduction in terms of a unified dynamics governing all physical processes, we analyze the problem of working out a worldview accommodating our knowledge about natural phenomena. We stress the relevant conceptual differences between the considered models and standard quantum mechanics. In spite of the fact that both theories describe systems within a genuine Hilbert space framework, the peculiar features of the spontaneous reduction models limit drastically the states which are dynamically stable. (...) This fact by itself allows one to work out an interpretation of the formalism which makes it possible to give a satisfactory description of the world in terms of the values taken by an appropriately defined mass density function in ordinary configuration space. A topology based on this function and which is radically different from the one characterizing the Hilbert space is introduced, and in terms of it the idea of similarity of macroscopic situations is precisely defined. Finally, the formalism and the interpretation are shown to yield a natural criterion for establishing the psychophysical parallelism. The conclusion is that, within the considered theories and at the nonrelativistic level, one can satisfy all sensible requirements for a completely satisfactory macro-objective description of reality. (shrink)

This is a major contribution to post-Keynesian thought. With studies of the key pioneers - Keynes himself, Kalecki, Kahn, Goodwin, Kaldor, Joan Robinson, Sraffa and Pasinetti - G. C. Harcourt emphasizes their positive contributions to theories of distribution, pricing, accumulation, endogenous money and growth. The propositions of earlier chapters are brought together in an integrated narrative and interpretation of the major episodes in advanced capitalist economics in the post-war period, leading to a discussion of the relevance of post-Keynesian ideas to (...) both our understanding of economics and to policy-making. The appendices include biographical sketches of the pioneers and analysis of the conceptual core of their discontent with orthodox theories. Drawing on the author's experience of teaching and researching over fifty years, this book will appeal to undergraduate and graduate students interested in alternative approaches to theoretical, applied and policy issues in economics, as well as to teachers and researchers in economics. (shrink)