The wedge problem, that is, the propagation of radiation or particles in the presence of a wedge, is examined in different contexts. Generally, the paper follows the historical order from Sommerfeld's early work to recent stochastic results—hindsights and new results being woven in as appropriate. In each context, identifying the relevant mathematical problem has been the key to the solution. Thus each section can be given both a physics and a mathematics title: Section 2: diffraction by reflecting wedge; boundary value (...) problem of differential equations; solutions defined on mutiply connected spaces. Section 3: geometrical theory of diffraction; identificiation of function spaces. Section 4: path integral solutions; path integration on multiply connected spaces; asymptotics on the boundaries of function spaces. Section 5: probing the shape of the wedge and the roughness of its surface; stochastic calculus. Several propagators and Green functions are given explicitly, some old ones and some new ones. They include the knife-edge propagator for Dirichlet and Neumann boundary conditions, the absorbing knife edge propagator, the wedge propagators, the propagator for a free particle on a μ-sheeted Riemann surface, the Dirichlet and the Neumann wedge Green function. (shrink)

For time-independent fields the Aharonov-Bohm effect has been obtained by idealizing the coordinate space as multiply-connected and using representations of its fundamental homotopy group to provide information on what is physically identified as the magnetic flux. With a time-dependent field, multiple-connectedness introduces the same degree of ambiguity; by taking into account electromagnetic fields induced by the time dependence, full physical behavior is again recovered once a representation is selected. The selection depends on a single arbitrary time (hence the so-called holonomies (...) are not unique), although no physical effects depend on the value of that particular time. These features can also be phrased in terms of the selection of self-adjoint extensions, thereby involving yet another question that has come up in this context, namely, boundary conditions for the wave function. (shrink)

Because of his profound sense of wonder and celebration, there is much in Chesterton's religious writings that is fresh and pertinent to contemporary discussion.

This paper obtains the weak completeness and decidability results for standard systems of modal logic using models built from formulas themselves. This line of work began with Fine (Notre Dame J. Form. Log. 16:229-237, 1975). There are two ways in which our work advances on that paper: First, the definition of our models is mainly based on the relation Kozen and Parikh used in their proof of the completeness of PDL, see (Theor. Comp. Sci. 113-118, 1981). The point is to (...) develop a general model-construction method based on this definition. We do this and thereby obtain the completeness of most of the standard modal systems, and in addition apply the method to some other systems of interest. None of the results use filtration, but in our final section we explore the connection. (shrink)

This article enlarges classical syllogistic logic with assertions having to do with comparisons between the sizes of sets. So it concerns a logical system whose sentences are of the following forms: Allxareyand Somexarey, There are at least as manyxasy, and There are morexthany. Herexandyrange over subsets (not elements) of a giveninfiniteset. Moreover,xandymay appear complemented (i.e., as$\bar{x}$and$\bar{y}$), with the natural meaning. We formulate a logic for our language that is based on the classical syllogistic. The main result is a soundness/completeness theorem. (...) There are efficient algorithms for proof search and model construction. (shrink)

We construct logical languages which allow one to represent a variety of possible types of changes affecting the information states of agents in a multi-agent setting. We formalize these changes by defining a notion of epistemic program. The languages are two-sorted sets that contain not only sentences but also actions or programs. This is as in dynamic logic, and indeed our languages are not significantly more complicated than dynamic logics. But the semantics is more complicated. In general, the semantics of (...) an epistemic program is what we call aprogram model. This is a Kripke model of ‘actions’,representing the agents' uncertainty about the current action in a similar way that Kripke models of ‘states’ are commonly used in epistemic logic to represent the agents' uncertainty about the current state of the system. Program models induce changes affecting agents' information, which we represent as changes of the state model, called epistemic updates. Formally, an update consists of two operations: the first is called the update map, and it takes every state model to another state model, called the updated model; the second gives, for each input state model, a transition relation between the states of that model and the states of the updated model. Each variety of epistemic actions, such as public announcements or completely private announcements to groups, gives what we call an action signature, and then each family of action signatures gives a logical language. The construction of these languages is the main topic of this paper. We also mention the systems that capture the valid sentences of our logics. But we defer to a separate paper the completeness proof. (shrink)

This paper provides a foundation for the polarity marking technique introduced by David Dowty [3] in connection with monotonicity reasoning in natural language and in linguistic analyses of negative polarity items based on categorial grammar. Dowty's work is an alternative to the better-known algorithmic approach first proposed by Johan van Benthem [11], and elaborated by Víctor Sánchez Valencia [10]. Dowty's system internalized the monotonicity/polarity markings by generating strings using a categorial grammar whose categories already contain the markings that the earlier (...) system would obtain by separate steps working on an already-derived string. Despite the linguistic advantages of the internalized system, no soundness proof has yet been given for it. This paper offers an account. The leading mathematical idea is to interpret categorial types as preorders (in order to talk about monotonicity in the first place), and then to add a primitive operation to the type hierarchy of taking the opposite of a preorder (in order to capture monotone decreasing functions). At the same time, the use of internalized categories also raises issues. Although these will not be addressed in full, the paper points out possible approaches to them. (shrink)

The Heythrop Journal, Volume 63, Issue 4, Page 817-827, July 2022.

This paper adds comparative adjectives to two systems of syllogistic logic. The comparatives are interpreted by transitive and irreflexive relations on the underlying domain. The main point is to obtain sound and complete axiomatizations of the valid formulas in the logics.

There has been considerable controversy surrounding the routine circumcision of male infants in the United States. This is of particular concern, since the medical establishments of all the other countries of the developed world have abandoned this procedure as having dubious benefits. This article examines the medical pros and cons of neonatal male circumcision in a historical perspective and suggests that the circumstances that led to its establishment as a routine practice are largely absent today. Reasons for its continued use (...) and ethical and moral issues associated with its practice are then examined closely, especially in light of the current debate surrounding female circumcision. The article concludes that the medical and ethical issues surrounding this practice should be more broadly debated until there is some consensus as to its benefits. Prudence dictates that until such a consensus emerges, the practice should be suspended. (shrink)

The Heythrop Journal, Volume 63, Issue 4, Page 817-827, July 2022.

We present a bimodal logic suitable for formalizing reasoning about points and sets, and also states of the world and views about them. The most natural interpretation of the logic is in subset spaces , and we obtain complete axiomatizations for the sentences which hold in these interpretations. In addition, we axiomatize the validities of the smaller class of topological spaces in a system we call topologic . We also prove decidability for these two systems. Our results on topologic relate (...) early work of McKinsey on topological interpretations of S4 with recent work of Georgatos on topologic. Some of the results of this paper were presented at the 1992 conference on Theoretical Aspects of Reasoning about Knowledge. (shrink)

From its first appearance in 1844, Max Stirner’s major work, Der Einzige und sein Eigentum ,[1] has produced little agreement among its many interpreters. The very first of these interpreters was Friedrich Engels, who suggested that Stirner’s doctrines would be quite compatible with Benthamite utilitarianism, which he then admired, and even saw in these doctrines the potential of benefiting communism.[2] Marx, in short order, corrected this optimistic deviation, and then—with a surely repentant Engels—set forth the orthodox gospel for all future (...) generations of communists: Stirner, or “Sankt Max,” was but the speculative spokesman for the petty bourgeois, a decadent Hegelian boasting over the unrestraint of his self inflated ego.[3] Sidney Hook echoed Marx when he condemned Stirner’s work as but the “social defense mechanism of a petty bourgeois soul.”[4] Others, unsatisfied with this “petty” status, elevate him to that of the Grand Bourgeois, or Fascist.[5] Still others, taking an opposite stance, see in Stirner the most articulate defender of individual liberty.[6] In between, he has been called a nihilist, an anarchist, an existentialist, a solipsist, an anti Benthamite, an intemperate capitalist, or—as we might now suspect—an anti capitalist.[7] At least two commentators, lost in the confusion, have managed to escape the need to classify Stirner within the ongoing political and ethical categories and simply declare him to be insane.[8] In short, the list of radically diverse interpretations.. (shrink)

This book examines how, in a series of critical confrontations, Stirner rejected the efforts of his “Young Hegelian” contemporaries to recast Hegel as a revolutionary. For him, the various apocalyptic declarations of these “pious atheists” were only the expressions of adolescent dreams set upon the annihilation of real individuality.

We present a rendering of some common grammatical formalisms in terms of evolving algebras. Though our main concern in this paper is on constraint-based formalisms, we also discuss the more basic case of context-free grammars. Our aim throughout is to highlight the use of evolving algebras as a specification tool to obtain grammar formalisms.

Papers delivered at the joint meeting of the Hegel Society of America and the Hegel Society of Great Britain held at Merton College, Oxford, Sept. 1-4, 1981, to mark the 150th anniversary of Hegel's death. Includes bibliographical references and index.

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Hegel’s Ethics of Recognition by Robert R. WilliamsLawrence S. StepelevichRobert R. Williams. Hegel’s Ethics of Recognition. Los Angeles: University of California Press, 1998. Pp. xviii +433. Cloth, $60.00.The eminent Hegel scholar, Vittorio Hoesle, perceived the major weakness of Hegel’s philosophy in its seeming failure to adequately deal with the issue of interpersonal relations. Hardly a new objection, as Hoesle’s critique has a lineage that reaches at least as (...) far back as Schleiermacher. Against this thesis, Robert Williams had earlier developed a counter-argument which he presented in his work, Recognition: Fichte and Hegel on the Other (1992). In this foundational work, Williams drew out and clarified Hegel’s concept of recognition from its inchoate source in Fichte; and by so doing defended “Hegel against charges that his thought violates intersubjectivity and difference by reducing the other to the same.”(1) In the present work, the conceptual and textual grounds established by Williams in his first work on the topic allow for a more extensive extrapolation of Hegel’s doctrine, and argues that “The story of recognition is a story about Fichte and Hegel. Fichte introduced the concept but did not make it the basis of either his ethics or his politics. Hegel appropriated and transformed the concept of recognition and regarded it as the fundamental intersubjective structure of ethical life.” (26)After a brief introductory critical survey of some of the more or less misleading contemporary views of both Hegelians and non-Hegelians regarding Hegel’s concept of interpersonal recognition, Williams proceeds to present its historical genesis, of how it emerged from its early and unfocused appearances in the philosophies of Fichte and Schelling. Williams then traces the further development and elaboration of this concept as it gains expression in the course of Hegel’s own ethical theory, not only as it is found in such major works as the Phenomenology of Mind and The Philosophy of Spirit, but also as it appears in his earlier and lesser known Jena Manuscripts of 1805. After this preliminary doctrinal and historical introduction, Williams turns to the most substantial section of his study, that work of Hegel’s in which the concept of recognition would be expected to and does indeed play its most evident ethical role: The Philosophy of Right.Almost every category within The Philosophy of Right is examined not only for the appearance of the concept, but for the defining role which it plays in the construction of that category. Such an examination is necessary for Williams, since “Reciprocal recognition in its various determinate types and instances is the general structure of ethical life and the embodiment of social reason that underlies and supports the concepts of law and state.” (364) Among the categories considered, categories that Williams discusses within the context of contemporary issues and each of which is taken up in a separate chapter of his work, are Family, Crime and Punishment, Poverty, and the State. In every instance considered he argues that an understanding of the defining role of recognition is a necessary prerequisite not only for a clear and correct comprehension [End Page 174] of Hegel’s ethical theory but of all ethical theory. Certainly, after this scholarly tour de force Hegel’s concept of recognition will have to be taken into account in any credible discussion of his ethical theory. In a final chapter Williams turns to some contemporary ethical theorists who have not taken that concept into consideration, or if they have, have either misunderstood or have rejected it. For Williams, such thinkers as Kojeve, Sartre, Deleuze, Derrida and Levinas, in their misunderstanding or rejection of the ethically mediating concept of recognition, have all contributed toward the “disrupted and fragmented cultural situation at the end of the twentieth century.” (412)In sum, Williams’ presentation and support of Hegel’s ethical employment of the concept of intersubjective recognition cannot but encourage theorists to reconsider the subject, and to see in it not only a conceptual tool which can serve to lucidly contextualize Hegel’s ethical theory into the matrix of his total philosophical system, but also as a needed ethical concept able to transcend both... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Hegel’s Ladder, Volume I: The Pilgrimage of Reason by Henry Silton HarrisLawrence S. StepelevichHenry Silton Harris. Hegel’s Ladder, Volume I: The Pilgrimage of Reason. Pp. xvi+ 658. Volume II: The Odyssey of Spirit. Pp. xiii + 909. Indianapolis: Hackett Publishing Company, 1997. Cloth, $150.00, the set.This commentary upon Hegel’s Phenomenology of Spirit is the concentrated result of over three decades of sustained study by one of the most (...) renowned of contemporary Hegelian scholars.The Phenomenology stands in the course of philosophic history as (depending upon one’s point of view) either an opportunity or an obstacle—an unavoidable Sphinx. Both admirers and detractors of this famous work will often assign themselves or, more readily, their students, the task of reading it as a introduction to Hegel’s philosophy. However, what occurs is not usually what is intended. After a short perusal, the reader finds that the Phenomenology itself needs an introduction. This need has not been lost on scholars, and their various introductory observations, when expanded and duly annotated, are entered into the ever-growing list of commentaries upon the Phenomenology. Needless to say, given the hundreds of studies which, either in whole or in part, are devoted to commenting upon this work, there are some grand failures, as Lowenberg’s elegantly vapid Dialogues on the Phenomenology, or Heidegger’s inconclusive “critical debate with Hegel.” There are also a few successes, such as Jean Hyppolite’s Genesis and Structure. Without doubt, Harris’s commentary is a success, and surely the most successful of all Anglophone commentaries upon the Phenomenology. [End Page 473]Hegel’s Ladder, which assumes a learned readership, embarks upon the same long and tortured path facing any reader of the Phenomenology. It is no small thing to follow where one thinks Hegel leads, as he had a somewhat perverse tendency to avoid both clarifying examples and a tendency to, as J. N. Findlay described it, “wanton obscurity.” To make matters worse, there is also the complex ordering of the text itself, a palimpsest character which has led more than a few commentators to take the work as a disorderly juvenalia, best left ignored in favor of the more lucid Encyclopaedia. And finally, there is also the extratextual matter of whether or not one “buys into” Hegel’s thought—which Harris does. In any case, a superficial reading will not do, and readers will be reminded of Hegel’s own “path of despair” when they follow the course of Harris’s 1,500-page commentary. It is not an “easy read,” nor could it be.Professor Harris is well aware of the problems confronting any commentator, but addresses them in an way which would assure the reader that the commentary is directly linked to the text, and does not simply follow “paths of one’s own, offering variations, elaborations, and applications instead of faithful analysis” (xii). To this end, a “logical commentary” is developed, one which follows a serial paragraph-by-paragraph analysis of the text. Each paragraph-analysis is referenced to the Phenomenology, which allows the reader a direct reference to Hegel’s own words. These “safe” and constrained analytic paragraphs are each followed by an extensive and free-ranging commentary. The intention of this procedure is not only to assure the reader that Harris’s commentary is textually grounded, but to convince the reader that Harris’s contention regarding the nature of the Phenomenology is correct—that it is a profoundly cohesive and extended logical argument. In short, for Harris, the Phenomenology is exactly what what Hegel claimed it to be: a science of experience. Most commentators have not accepted this claim of either the unity or the logical course of the text, but this “received view” is, for Harris, the “view that I want to challenge and, if possible, to overthrow. If I am right, an acceptably continuous chain of argument, paragraph by paragraph, ought to be discoverable in the text” (xii).In the total context of Hegel’s philosophy, Harris understands the Phenomenology to be the “necessary preamble to speculative logic.” Once this is accepted, then “we ought to know at once, by the simplest of... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Hegel’s Ladder, Volume I: The Pilgrimage of Reason by Henry Silton HarrisLawrence S. StepelevichHenry Silton Harris. Hegel’s Ladder, Volume I: The Pilgrimage of Reason. Pp. xvi+ 658. Volume II: The Odyssey of Spirit. Pp. xiii + 909. Indianapolis: Hackett Publishing Company, 1997. Cloth, $150.00, the set.This commentary upon Hegel’s Phenomenology of Spirit is the concentrated result of over three decades of sustained study by one of the most (...) renowned of contemporary Hegelian scholars.The Phenomenology stands in the course of philosophic history as (depending upon one’s point of view) either an opportunity or an obstacle—an unavoidable Sphinx. Both admirers and detractors of this famous work will often assign themselves or, more readily, their students, the task of reading it as a introduction to Hegel’s philosophy. However, what occurs is not usually what is intended. After a short perusal, the reader finds that the Phenomenology itself needs an introduction. This need has not been lost on scholars, and their various introductory observations, when expanded and duly annotated, are entered into the ever-growing list of commentaries upon the Phenomenology. Needless to say, given the hundreds of studies which, either in whole or in part, are devoted to commenting upon this work, there are some grand failures, as Lowenberg’s elegantly vapid Dialogues on the Phenomenology, or Heidegger’s inconclusive “critical debate with Hegel.” There are also a few successes, such as Jean Hyppolite’s Genesis and Structure. Without doubt, Harris’s commentary is a success, and surely the most successful of all Anglophone commentaries upon the Phenomenology. [End Page 473]Hegel’s Ladder, which assumes a learned readership, embarks upon the same long and tortured path facing any reader of the Phenomenology. It is no small thing to follow where one thinks Hegel leads, as he had a somewhat perverse tendency to avoid both clarifying examples and a tendency to, as J. N. Findlay described it, “wanton obscurity.” To make matters worse, there is also the complex ordering of the text itself, a palimpsest character which has led more than a few commentators to take the work as a disorderly juvenalia, best left ignored in favor of the more lucid Encyclopaedia. And finally, there is also the extratextual matter of whether or not one “buys into” Hegel’s thought—which Harris does. In any case, a superficial reading will not do, and readers will be reminded of Hegel’s own “path of despair” when they follow the course of Harris’s 1,500-page commentary. It is not an “easy read,” nor could it be.Professor Harris is well aware of the problems confronting any commentator, but addresses them in an way which would assure the reader that the commentary is directly linked to the text, and does not simply follow “paths of one’s own, offering variations, elaborations, and applications instead of faithful analysis” (xii). To this end, a “logical commentary” is developed, one which follows a serial paragraph-by-paragraph analysis of the text. Each paragraph-analysis is referenced to the Phenomenology, which allows the reader a direct reference to Hegel’s own words. These “safe” and constrained analytic paragraphs are each followed by an extensive and free-ranging commentary. The intention of this procedure is not only to assure the reader that Harris’s commentary is textually grounded, but to convince the reader that Harris’s contention regarding the nature of the Phenomenology is correct—that it is a profoundly cohesive and extended logical argument. In short, for Harris, the Phenomenology is exactly what what Hegel claimed it to be: a science of experience. Most commentators have not accepted this claim of either the unity or the logical course of the text, but this “received view” is, for Harris, the “view that I want to challenge and, if possible, to overthrow. If I am right, an acceptably continuous chain of argument, paragraph by paragraph, ought to be discoverable in the text” (xii).In the total context of Hegel’s philosophy, Harris understands the Phenomenology to be the “necessary preamble to speculative logic.” Once this is accepted, then “we ought to know at once, by the simplest of... (shrink)