Abstract As Paul B. Thompson suggests in his recent seminal paper, “‘There’s an App for That’: Technical Standards and Commodification by Technological Means,” technical standards restructure property (and other social) relations. He concludes with the claim that the development of technical standards of commodification can serve purposes with bad effects such as “the rise of the factory system and the deskilling of work” or progressive effects such as how “technical standards for animal welfare… discipline the unwanted consequences of market (...) forces.” In this reply, we want to append several points to his argument and suggest that he rightly points out that standards can promote various goods; however, there are peculiar powers wielded by standardization processes that might profitably be unpacked more systematically than Thompson's article seems to suggest. First, the concealment of the technopolitics around standards is largely due to their peculiar ontological status as recipes for reality. Second, technical standards can and do commit violence against persons, but such violence is often suffered not in the formation of class consciousness, as Marx might have put it, but as a failure to conform to the laws of nature . Content Type Journal Article Category Commentary Pages 1-6 DOI 10.1007/s13347-011-0048-1 Authors Lawrence Busch, Department of Sociology, Michigan State University, 429A Berkey Hall, East Lansing, MI 48824, USA Kyle Powys Whyte, Department of Philosophy, Michigan State University, 503 S. Kedzie Hall, East Lansing, MI 48824, USA Journal Philosophy & Technology Online ISSN 2210-5441 Print ISSN 2210-5433. (shrink)

Justice is a natural virtue. Well-functioning humans are just, as are well-ordered human societies. Roughly, this means that in a well-ordered society, just humans internalize the laws and social norms (the nomoi)--they internalize lawfulness as a disposition that guides the way they relate to other humans. In societies that are mostly well-ordered, with isolated zones of substantial dysfunction, the nomoi are limited to those norms that are not clearly inconsistent with the function of law--to create the conditions for human flourishing. (...) In a radically dysfunctional society, humans are thrown back on their own resources--doing the best they can in circumstances that may require great practical wisdom to avoid evil and achieve good. Justice is naturally good for humans--it is part and partial of human flourishing. All of these are natural ethical facts. "Natural Justice" develops these claims in four stages. Part I contextualizes the claim that justice is a natural virtue in relationship to David Hume's famous argument about deriving ought from is, G. E. Moore's open-question argument, and the so-called fact-value distinction. The upshot of the discussion in part I is the claim that there are no clearly decisive objections to existence of natural ethical facts. Part II traces the movement from neo-Aristotelian virtue ethics to virtue jurisprudence by articulating a theory of the judicial virtues. Among these are the virtues of practical wisdom and of justice. Practical wisdom or phronesis is best understood on the model of moral vision, which in the context of law is legal vision or situation sense. The virtue of justice is best understood as lawfulness. Just humans are law-abiding or nomimos--in that they internalize the widely shared and deeply held social norms of their social groups. This part concludes with the claim that a legally correct decision is the decision that characteristically would be rendered by a fully virtuous judge under the circumstances of the case. Part III argues that natural justice can be understood on the model of natural goodness as articulated in the work of Philippa Foot and Michael Thompson. The intuitive idea is that justice as lawfulness is naturally good for reason--using social creatures in human circumstances. This part also articulates and responds to a variety of objections. Part IV concludes by articulating the sense in which an aretaic theory of law that incorporates a natural virtue of justice as lawfulness can be viewed as an expression of the natural law tradition. The natural law idea that an unjust enactment is not a true law corresponds to two senses in which positive laws can fail to be nomoi (in the technical sense specified by virtue jurisprudence). First, a given enactment may contravene deeply held and widely shared social norms. Second, such enactments may be fundamentally inconsistent with the purpose of law--the promotion of human flourishing. (shrink)

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)

Equality before the law is one of the fundamental guarantees citizens expect in a just and fair society. We argue that recent trend toward mass incarceration, which has had vastly disproportionate impact on African Americans, is undermining this claim to fairness and raises a serious legitimacy problem for the legal system as a whole. Using original data from the Race, Crime and Public Opinion study we show that African Americans view the 'War on Drugs" as racially biased in its implementation. (...) This perception of bias consequentially undermines legal system legitimacy by lowering blacks' expectations for police performance in their communities and encouraging receptivity to appeals for jury nullification based on race. (shrink)

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

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.

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

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.

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)

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)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Dialectic and Gospel in the Development of Hegel’s Thinking by Stephen CritesLawrence S. StepelevichStephen Crites. Dialectic and Gospel in the Development of Hegel’s Thinking. University Park: Pennsylvania State University Press, 1998. Pp. xvii + 572. Cloth, $65.00Unlike either Wittgenstein or Heidegger, or his contemporary, Schelling, there is really no “Early” or “Later” Hegel. The fundamentals of his system were, if not always fully articulated, nevertheless present from the (...) very beginning of his career. There is, in Hegel, a remarkable consistency in the development of his thought, in which the elements that comprised his final system can be detected in his earliest efforts—and it is only in this sense that one can speak of an “Early” or a “Later” Hegel. That his final expression was indeed the result of an effort, a struggle to fuse and clarify seemingly disparate elements, is the thesis of Professor Crites. In this case, the specific disparate elements that are considered are those which confronted Hegel from his earliest days as a student of theology, those elements expressed in the “Christian Drama.” Crites sets out the eight revered acts of this drama: the creation, the fall, God’s covenant with Israel, the incarnation, the death and resurrection of Christ, the Holy Spirit, the mission of the Church, and the last judgment. It is his contention that Hegel “critically engaged the claims of this mythos and the social practices associated with it.” And so, the essence of Crites’s well-written and well-researched study is to set out how this drama served as a “template that affected the way he framed other topics in the ever-widening range of his philosophical inquiry.” It is to his credit that Crites does not consider his own explication of Hegel’s preoccupation with the gospels as some sort of a “secret” key that could unlock the full meaning of Hegel’s philosophy, but nevertheless, as he rightly stresses, this preoccupation cannot be ignored—for it ultimately exercised a crucial role in the formation of that which is now taken as Hegel’s mature thought.Unlike most studies of Hegel’s development—and there are more than a few—Crites avoids taking a retrospective posture, a concluding a priori stance which would allow him to point out where and why this or that particular element was rejected or retained by Hegel, His method is historical, intent upon following Hegel’s own sometimes tortured development, until, finally in this course, we reach a point where we suddenly see that Hegel “shows clear signs of becoming what we can only call a Hegelian—or better still, he shows signs of being Hegel!”The source of this inner development begins, and is traced from, Hegel’s initial appearance as “The Model Pupil of Stuttgart” to that moment in when he stands revealed as embodying that Absolute Knowledge which concludes the Phenomenology. This path from Stuttgart to Jena, which Hegel himself might have had in mind when he spoke of the philosophical “highway of despair” is detailed by Crites with a sure hand. Every stumbling block is noted, and every new resolution ıs marked, from those first essays composed during the bitter days as Hauslehrer, through the steady advances [End Page 540] of “the Jena ‘Philosophy’ ” until the passage completes itself in the Phenomenology and its termination in the “Recollection and the Golgotha of Absolute Spirit.”More than half of Crites’ extensive study is devoted to explicating how the subject of religion is dealtwith in the course of the Phenomenology. In this survey of Hegel’s most well-known work, its fundamental leitmotif emerges clearly into focus: the identification of the “portentous power of the negative” with the Death of God. This death, which is philosophically reflected in dialectical negativity, is the central moment of the gospel drama. In this act, Absolute Spirit must itself suffer and die, for “Not only the Christian God incarnate, but the absolute spirit suffers, suffers its Golgotha and dies.” It is only from this sacrifice that new life can arise—for out of it comes “the resurrection of a new human-divine totality.” What Hegel has retained of the gospel... (shrink)

The course of Western philosophy has been profoundly altered by the philosophy of Hegel. The first of those who set about the transforming and revisioning of the world according to Hegel's dialectical theory were called "The Young Hegelians." Today, the most recognized names among them are Karl Marx and Friedrich Engels, but in their own age each of the Young Hegelians shared an equal notoriety. Each in turn, from Strauss with his reduction of the historical jesus into a Messianic myth, (...) to Stirner with his uncompromising egoism, shocked every cultural convention of their age. The aftershocks of their unrestrained criticism have forever altered the topography of our own. The Young Hegelians retrieves some of the central writings of that troubling generation. (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 recent profusion of studies directed to uncovering the “Young Marx” has also provoked some renewed interest in his contemporary, Johann Caspar Schmidt, better known as Max Stirner. With a few exceptions, the most important being William Brazill’s The Young Hegelians, Stirner has been retained in his traditional role as Marx’s first critic, the harried “Sankt Max” of The German Ideology. This perspective, established firmly by Sidney Hook and continued by David McLellen, does cast light upon Marx’s development, but it (...) cannot be expected to do justice to Stirner, who is more than a mere anti-Marxist. (shrink)

The recent profusion of studies directed to uncovering the “Young Marx” has also provoked some renewed interest in his contemporary, Johann Caspar Schmidt, better known as Max Stirner. With a few exceptions, the most important being William Brazill’s The Young Hegelians, Stirner has been retained in his traditional role as Marx’s first critic, the harried “Sankt Max” of The German Ideology. This perspective, established firmly by Sidney Hook and continued by David McLellen, does cast light upon Marx’s development, but it (...) cannot be expected to do justice to Stirner, who is more than a mere anti-Marxist. (shrink)

Since its foundation in 1969, The Hegel Society of America has sponsored an ongoing series of biennial conferences which have provided a regular forum for some of the finest displays of scholarship ever directed toward the explication and development of Hegelianism. The fourteen essays in this distinguished collection have been carefully selected from these biennial conferences. Each essay has been chosen for its profound scholarship, philosophical acumen, and literary excellence. All of the authors have attained international recognition for their studies (...) of Hegel, with almost half having been, at one time or another, elected to the Presidency of the Hegel Society. (shrink)

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.