Let n be a positive integer. By a $\beta_{n}-model$ we mean an $\omega-model$ which is elementary with respect to $\sigma_{n}^{1}$ formulas. We prove the following $\beta_{n}-model$ version of $G\ddot{o}del's$ Second Incompleteness Theorem. For any recursively axiomatized theory S in the language of second order arithmetic, if there exists a $\beta_{n}-model$ of S, then there exists a $\beta_{n}-model$ of S + "there is no countable $\beta_{n}-model$ of S". We also prove a $\beta_{n}-model$ version of $L\ddot{o}b's$ Theorem. As a corollary, we obtain (...) a $\beta_{n}-model$ which is not a $\beta_{n+1}-model$. (shrink)

In this discussion note, I examine scattered comments about evil from John Dewey’s works. After a brief consideration of what critics like Reinhold Niebuhr have said about the weaknesses of Dewey’s theodicy, I will offer my own critique. In short, I argue that Dewey subverts his own theory of inquiry when he comes to the problem of the origin and genesis of evil.

The truth-table degree of the set of shortest programs remains an outstanding problem in recursion theory. We examine two related sets, the set of shortest descriptions and the set of domain-random strings, and show that the truth-table degrees of these sets depend on the underlying acceptable numbering. We achieve some additional properties for the truth-table incomplete versions of these sets, namely retraceability and approximability. We give priority-free constructions of bounded truth-table chains and bounded truth-table antichains inside the truth-table complete (...) degree by identifying an acceptable set of domain-random strings within each degree. (shrink)

In this paper, an incompleteness theorem for modal extensions of relevant logics is proved. The proof uses elementary methods and builds upon the work of Fuhrmann.

Upshot: According to its introduction, the aim of Enaction is to “present the paradigm of enaction as a framework for a far-reaching renewal of cognitive science as a whole.” While many of the chapters make progress towards this aim, the book as a whole does not present enactivism as a coherent framework, and it could be argued that enactivism’s embrace of phenomenology means it is no longer a theory of cognition.

We present an analogue of Gödel’s second incompleteness theorem for systems of second-order arithmetic. Whereas Gödel showed that sufficiently strong theories that are $\Pi ^0_1$ -sound and $\Sigma ^0_1$ -definable do not prove their own $\Pi ^0_1$ -soundness, we prove that sufficiently strong theories that are $\Pi ^1_1$ -sound and $\Sigma ^1_1$ -definable do not prove their own $\Pi ^1_1$ -soundness. Our proof does not involve the construction of a self-referential sentence but rather relies on ordinal analysis.

How to Human is a tender and irreverent take on one of life's most fundamental questions: how to be a better human in a world dead set against it. Alice Connor offers sage, no-nonsense wisdom, encouraging you to embrace the chaos and see life as an experiment -- not a quest for the right answers.

John Rawls’s use of the “fully cooperating assumption” has been criticized for hindering attempts to address the needs of disabled individuals, or non-cooperators. In response, philosophers sympathetic to Rawls’s project have extended his theory. I assess one such extension by Cynthia Stark, that proposes dropping Rawls’s assumption in the constitutional stage (of his four-stage sequence), and address the needs of non-cooperators via the social minimum. I defend Stark’s proposal against criticisms by Sophia Wong, Christie Hartley, and Elizabeth Edenberg and Marilyn (...) Friedman. Nevertheless, I argue that Stark’s proposal is crucially incomplete. Her formulation of the social minimum lacks accompanying criteria with which the adequacy of the provisions for non-cooperators may be assessed. Despite initial appearances, Stark’s proposal does not fully address the needs of non-cooperators. I conclude by considering two payoffs of identifying this lack of criteria. (shrink)

The most common way of proving decidability in propositional modal logic is to shew that the system in question has the finite model property. This is not however the only way. Gabbay in [4] proves the decidability of many modal systems using Rabin's result in [8] on the decidability of the second-order theory of successor functions. In particular [4, pp. 258-265] he is able to prove the decidability of a system which lacks the finite model property. Gabbay's system is however (...) complete, in the sense of being characterized by a class of frames, and the question arises whether there is a decidable modal logic which is not complete. Since no incomplete modal logic has the finite model property [9, p. 33], any proof of decidability must employ some such method as Gabbay's. In this paper I use the Gabbay/Rabin technique to prove the decidability of a finitely axiomatized normal modal propositional logic which is not characterized by any class of frames. I am grateful to the referee for suggesting improvements in substance and presentation.The terminology I am using is standard in modal logic. By aframeis understood a pair 〈W, R〉 in whichWis a class (of “possible worlds”) andR⊆W2. To avoid confusion in what follows, a frame will henceforth be referred to as aKripkeframe. By contrast, ageneral frameis a pair 〈,Π〉 in whichis a Kripke frame andΠis a collection of subsets ofWclosed under the Boolean operations and satisfying the condition that ifAis inΠthen so isR−1“A. A model on a frame (of either kind) is obtained by adding a functionVwhich assigns sets of worlds to propositional variables. In the case of a general frame we require thatV(p) ∈Π. (shrink)

The relevant modal logic G is a simple extension of the logic RT, the relevant counterpart of the familiar classically based system T. Using the Routley-Meyer semantics for relevant modal logics, this paper proves three main results regarding G: (i) G is semantically complete, but only with a non-standard interpretation of necessity. From this, however, other nice properties follow. (ii) With a standard interpretation of necessity, G is semantically incomplete; there is no class of frames that characterizes G. (iii) (...) The class of frames for G characterizes the classically based logic T. (shrink)

Recent work in epistemology has focused increasingly on the social dimensions of knowledge and inquiry. Education is one important social arena in which knowledge plays a leading role, and in which knowledge-claims are presented, analyzed, evaluated, and transmitted. Philosophers of education have long attended to the epistemological issues raised by the theory and practice of education . While historically philosophical issues concerning education were treated alongside other philosophical issues, in recent times the former set of issues have been largely neglected (...) by philosophers working in the core areas of the discipline. Interestingly, the rise of social epistemology has been accompanied by a renewed interest by mainstream philosophers in philosophical questions concerning education. Whether or not this accompaniment is accidental, or is legitimately explainable in terms of broad intellectual, philosophical, or social/political currents and movements, I will not endeavor to address here. The increasing respectability of and philosophical interest in both social epistemology and philosophy of education are in any case salutary developments, each signaling both a broadening of the set of interests and issues deemed legitimate by practitioners of the parent discipline, and an increased willingness to take seriously the philosophical problems raised by the ubiquitous social/communal effort to transmit/transform culture by way of education. (shrink)

This essay (written by Mark in dialogue with Stephan and including supporting commentary by Stanley Krippner and others1) is not a comprehensive survey of psi and anthropology, nor does it contain a thorough overview of Castaneda's work. Rather, it seeks to explore two primary questions. First, whatever became of the interest in psi and anthropology that led to the founding of the Society for the Anthropology of Consciousness? Second, what is the relationship between the anthropology of consciousness and transpersonal psychology? (...) Answers to these questions offer a glimpse into the historical origins of SAC as well as a historical understanding of the relationship between SAC and the Association for Transpersonal Psychology2. (shrink)

The paper reports an investigation on whether valid results can be achieved in analyzing the structure of datasets although a large percentage of data is missing without replacement. Two types of confirmatory factor analysis models were employed for this purpose: the missing data CFA model with an additional latent variable for representing the missing data and the semi-hierarchical CFA model that also includes the additional latent variable and reflects the hierarchical structure assumed to underlie the data. Whereas, the missing data (...) CFA model assumes that the model is equally valid for all participants, the semi-hierarchical CFA model is implicitly specified differently for subgroups of participants with and without omissions. The comparison of these models with the regular one-factor model in investigating simulated binary data revealed that the modeling of missing data prevented negative effects of missing data on model fit. The investigation of the accuracy in estimating the factor loadings yielded the best results for the semi-hierarchical CFA model. The average estimated factor loadings for items with and without omissions showed the expected equal sizes. But even this model tended to underestimate the expected values. (shrink)

Drawing upon art history, archaeology, literature, numismatics, philosophy, and film, this book explores the significance of the fragment and addresses the ...

C. Stephen Layman contends that an argument supporting theism over naturalism can be constructed based on three defensible, non–question-begging premises about the moral order. Previous critics of Layman’s argument have challenged the truth of these premises. We stipulate them arguendo but go on to show that there is a deeper problem: a fourth premise introduced to complete the argument—the “completion premise,” as we call it—is true only if we assume that God exists or we concede that there is no afterlife. (...) We close with suggestions for how Layman’s argument must be strengthened to meet with dialectical success. (shrink)

A new history of liberalism which argues that liberalism has been predicated on definite morality and should be viewed as an attempt to encompass both fear and hope. Liberalism, argues Alan Kahan, is the search for a society in which people need not be afraid. Freedom from fear is the most basic freedom. If we are afraid, we are not free. These insights, found in Montesquieu and Judith Shklar, are the foundation of liberalism. What liberals fear has changed over time (...) (revolution, reaction, totalitarianism, religious fanaticism, poverty, and now populism) but the great majority of liberal thinkers have relied on three pillars to ward off their fears and to limit the concentrated power that causes fear: freedom, markets, and morals, or, to put it another way, politics, economics, and religion or morality. Most liberal thinkers emphasize one or two pillars more than another, but it is typical of liberalism down to the Second World War to rely on all three, although there were always minority voices who preferred to stand on only one leg. After WWII, "thin" procedural/market liberals, who wanted to strip any moral or religious basis or purpose from liberalism, dominated "thick" liberal moralists, who thought liberalism needed a moral basis and/or goal. It is the political contention of this book that liberalism is most convincing as program, language, and social analysis when it relies on all three pillars, and that the relative weakness of liberalism at the end of the twentieth century had much to do with neglect of the moral pillar of liberalism. Its historical contention is that for much of the past two centuries it did rely on all three pillars. But Kahan also argues that liberalism is not only a party of fear. It is also a party of hope, or the party of progress. Many of the contradictions typical of liberalism derive from the seemingly contradictory effort to encompass both hope and fear. If in case of conflict fear often trumps hope for liberals (loss aversion applies in politics as much as in economics), and utopia is subject to indefinite postponement, progress in personal autonomy and development has always been at the heart of liberalism. Liberals typically support their hopes on the same three pillars of freedom, markets, and morals which they use to ward off their fears. Nevertheless, in one respect those historians and political theorists who identify liberalism with laissez-faire economics are not wrong. It is characteristic of liberalism then that it bases its hopes not on the state but on civil society, which for liberals is the common source of a free politics, a free market, and of morals. Alan S. Kahan is Professor of History at the Université de Versailles. His previous books include Tocqueville, Democracy, and Religion: Checks and Balances for Democratic Souls (Oxford 2015), Alexis de Tocqueville (Continuum Books) and Mind vs Money: The War Between Intellectuals and Capitalism (Transaction Publishing, 2010). (shrink)

ABSTRACT:The paper begins by surveying a range of possible views on the metaphysics of romantic love, organizing them as responses to a single question. It then outlines a position, constructionist functionalism, according to which romantic love is characterized by a functional role that is at least partly constituted by social matters, although this role may be realized by states that are not socially constructed.

Alan Kahan’s productive professional career has been devoted primarily to translating and writing about European liberals. A partial list would include: Aristocratic Liberalism: The Social and Poli...

Suppose the principles explaining how the human mind (brain) reaches logical conclusions and judgments were different from – and independent of – thoseinvolved innormatively valid reasoning. Then such principles should affect both conclusion generation and recognition that particular conclusions are or are not justified. People, however, demonstrate a discrepancy between impaired performance in generating logical conclusions as opposed to rather impressive competence in recognizing rational (versus irrational) ones. This discrepancy is hypothesized to arise from often generating an incomplete specification (...) of a logical or judgmental problem when attempting to solve it – versus a recognition of such incompleteness when it is pointed out. The basic argument is developed, with common examples, in the context of specifying or failing to specify all possible combinations in simple logical arguments and is then extended to probabilistic reasoning, where complete versus incomplete specification corresponds to attending to all or to only some components of Bayes theorem-based reasoning. (shrink)

The dissertation gives a second-order-logic-based explanation of modal incompleteness. The leading concept is that modal incompleteness is to be explained in terms of the incompleteness of standard second-order logic, since modal language is basically a second-order language. The development of Kripke-style semantics for modal logic has been underpinned by the conjecture that all modal systems are characterizable by classes of frames defined by first-order conditions on a binary relation. However, the discovery of certain incomplete modal systems has undermined the (...) all-encompassing feature of Kripke-style approach. There are logics not determined by any class of Kripke frames at all. ;In the dissertation I investigate a normal incomplete sentential modal system due to J. F. A. K. Van Benthem, and I address both the formal and the philosophical facets of modal incompleteness from the vantage point that modal logic is essentially second-order in its nature. Modal systems can be analyzed in terms of structures with a domain of second-order individuals that are assigned under an interpretation to propositional variables within languages of sentential modal logic. Hence, the phenomenon of there being incomplete modal systems can be accounted for through the incompleteness of standard second-order logic. ;Since second-order logic plays a crucial role in the explanation, Quine's animadversions upon second-order logic and in particular his views that second-order logic is 'Set Theory in Sheep's Clothing' are examined. Against Quine's stance I will seek to show that a vindication of second-order logic can be gotten provided a proper due is given to the sharp distinction between the logical notion of set which is the concern of second-order logic and the iterative notion of set which lies within the realm of set theory. (shrink)

An open formalism for arithmetic is presented based on first-order logic supplemented by a very strictly controlled constructive form of the omega-rule. This formalism (which contains Peano Arithmetic) is proved (nonconstructively, of course) to be complete. Besides this main formalism, two other complete open formalisms are presented, in which the only inference rule is modus ponens. Any closure of any theorem of the main formalism is a theorem of each of these other two. This fact is proved constructively for the (...) stronger of them and nonconstructively for the weaker one. There is, though, an interesting counterpart: the consistency of the weaker formalism can be proved finitarily. (shrink)

Critics and champions alike have fussed and fretted for well over fifty years about whether Russell’s treatment is compatible with certain alleged acceptable uses of incomplete definite descriptions,[2] where a description (the F( is incomplete just in case more than one object satisfies its nominal F, as in (1).

The archaeological record is very limited and its analysis has been contentious. Hence, molecular biologists have shifted their attention to molecular dating techniques. Recently on April 2013, the prestigious Cell Press Journal Current Biology published an article (Fu et al. 2013) entitled “A Revised Timescale for Human Evolution Based on Ancient Mitochondrial Genomes”. This paper has twenty authors and they are researchers from the world’s top institutes like Max Planck Institute, Harvard, etc. Respected authors of this paper have emphatically accepted (...) that the fossil record is inadequate and unreliable. These statements clearly substantiate that now biologists are agreeing that fossil records do not provide any significant evidence at all for conventional evolution theory. Despite the well-recorded fact of the continual grand propaganda of Darwinism based on fossil evidence for more than 150 years, in recent times biologists are surprisingly coming up with such statements, based on their confidence that evolution can be explained purely by the genealogical/genomic record provided by modern molecular biology. Still many respected journals (for example the article in Nature, Retallack, 2013) continue to publish articles on fossil evidence to support Darwinian evolution. These incoherently diverse claims prove that Darwinists are struggling with unscientific ideological approaches to explain biodiversity. Darwinian evolutionary theory is not only the basis of modern biology, but also acts as the guiding principle of science and intellectual reasoning for modern civilization. Hence, a scientific understanding of the breakdown of the Darwinian theory of objective evolution is very important for overcoming the traditional scientific temper of mechanistic intellectualism that characterizes this ideology. In my article “21st Century Biology Refutes Darwinian Abiology” (published in two parts in November and December 2012 issues of The Harmonizer: www.mahaprabhu.net/satsanga/harmonizer) it was noted that several recent findings challenge the credibility that random mutations and natural selection can provide a valid basis for justifying the naturalistic evolution of species. The present article summarizes the problems associated with the fossil record and dating techniques, and its implication on the neo-Darwinian mechanistic misconception of biological life as mere molecular chemistry or abiology. An alternative approach based on the Vedāntic view for explaining biodiversity in the light of 21st century biology is also discussed in the end of the article. (shrink)