In 1941, Tarski published an abstract, finitely axiomatized version of the theory of binary relations, called the theory of relation algebras, He asked whether every model of his abstract theory could be represented as a concrete algebra of binary relations. He and Jonsson obtained some initial, positive results for special classes of abstract relation algebras. But Lyndon showed, in 1950, that in general the answer to Tarski's question is negative. Monk proved later that the answer remains negative even if one (...) adjoins finitely many new axioms to Tarski's system. In this paper we describe a far-reaching generalization of the positive results of Jonsson and Tarski, as well as of some later, related results of Maddux. We construct a class of concrete models of Tarski's axioms-called coset relation algebras-that are very close in spirit to algebras of binary relations, but are built using systems of groups and cosets instead of elements of a base set. The models include all algebras of binary relations, and many non-representable relation algebras as well, We prove that every atomic relation algebra satisfying a certain measurability condition-a condition generalizing the conditions imposed by Jonsson and Tarski-is essentially isomorphic to a coset relation algebra. The theorem raises the possibility of providing a positive solution to Tarski's problem by using coset relation algebras instead of the standard algebras of binary relations. (shrink)

An introduction to logic from the perspective of algebra.

It is shown that the automorphism group of a relation algebra ${\cal B}_P$ constructed from a projective geometry P is isomorphic to the collineation group of P. Also, the base automorphism group of a representation of ${\cal B}_P$ over an affine geometry D is isomorphic to the quotient of the collineation group of D by the dilatation subgroup. Consequently, the total number of inequivalent representations of ${\cal B}_P$ , for finite geometries P, is the sum of the numbers ${\mid Col(P)\mid\over (...) {\mid Col(D)\mid/\mid Dil(D)\mid}}$ where D ranges over a list of the non-isomorphic affine geometries having P as their geometry at infinity. This formula is used to compute the number of inequivalent representations of relation algebras constructed over projective lines of order at most 10. For instance, the relation algebra constructed over the projective line of order 9 has 56,700 mutually inequivalent representations. (shrink)

We investigate a notion called uniqueness in power κ that is akin to categoricity in power κ, but is based on the cardinality of the generating sets of models instead of on the cardinality of their universes. The notion is quite useful for formulating categoricity-like questions regarding powers below the cardinality of a theory. We prove, for universal theories T, that if T is κ-unique for one uncountable κ, then it is κ-unique for every uncountable κ; in particular, it is (...) categorical in powers greater than the cardinality of T. (shrink)

This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabhäuser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1-dimensional geometry.

Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable . This theorem and its analogues for quasi-polyadic algebras with and without equality are formulated in Henkin, Monk and Tarski [13]. We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasi-polyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend this result (...) to other classes of algebras of logic, for example quasi-polyadic algebras and substitution-cylindrification algebras with and without equality, relation algebras, and special Boolean monoids. The results of op. cit. mentioned above are special cases of our general theorems. We point out an error in the proof of the Henkin-Monk-Tarski representation theorem for atomic, equality-free, quasi-polyadic algebras with rectangular atoms. The error consists in the implicit assumption of a property that does not, in general, hold. We then give a correct proof of their theorem. Henkin and Tarski also introduced the notion of a rich cylindric algebra and proved in op. cit. that every rich cylindric algebra of finite dimension satisfying certain special identities is representable. We introduce a modification of the notion of a rich algebra that, in our opinion, renders it more natural. In particular, under this modification richness becomes a density notion. Moreover, our notion of richness applies not only to algebras with equality, such as cylindric algebras, but also to algebras without equality. We show that a finite dimensional algebra is rich iff it is rectangularly dense and quasi-atomic; moreover, each of these conditions is also equivalent to a very natural condition of point density . As a consequence, every finite dimensional rich algebra of logic is representable. We do not have to assume the validity of any special identities to establish this representability. Not only does this give an improvement of the Henkin-Tarski representation theorem for rich cylindric algebras, it solves positively an open problem in op. cit. concerning the representability of finite dimensional rich quasi-polyadic algebras without equality. (shrink)

A series of nonrepresentable relation algebras is constructed from groups. We use them to prove that there are continuum many subvarieties between the variety of representable relation algebras and the variety of coset relation algebras. We present our main construction in terms of polygroupoids.

We shall show that certain natural and interesting intervals in the lattice of varieties of representable relation algebras embed the lattice of all subsets of the natural numbers, and therefore must have a very complicated lattice-theoretic structure.

This work presents a systematic study of decision problems for equational theories of algebras of binary relations (relation algebras). For example, an easily applicable but deep method, based on von Neumann's coordinatization theorem, is developed for establishing undecidability results. The method is used to solve several outstanding problems posed by Tarski. In addition, the complexity of intervals of equational theories of relation algebras with respect to questions of decidability is investigated. Using ideas that go back to Jonsson and Lyndon, the (...) authors show that such intervals can have the same complexity as the lattice of subsets of the set of the natural numbers. Finally, some new and quite interesting examples of decidable equational theories are given. The methods developed in the monograph show promise of broad applicability. They provide researchers in algebra and logic with a new arsenal of techniques for resolving decision questions in various domains of algebraic logic. (shrink)

It is shown that Tarski’s set of ten axioms for the calculus of relations is independent in the sense that no axiom can be derived from the remaining axioms. It is also shown that by modifying one of Tarski’s axioms slightly, and in fact by replacing the right-hand distributive law for relative multiplication with its left-hand version, we arrive at an equivalent set of axioms which is redundant in the sense that one of the axioms, namely the second involution law, (...) is derivable from the other axioms. The set of remaining axioms is independent. Finally, it is shown that if both the left-hand and right-hand distributive laws for relative multiplication are included in the set of axioms, then two of Tarski’s other axioms become redundant, namely the second involution law and the distributive law for converse. The set of remaining axioms is independent and equivalent to Tarski’s axiom system. (shrink)

Alfred Tarski (1901--1983) is widely regarded as one of the two giants of twentieth-century logic and also as one of the four greatest logicians of all time (Aristotle, Frege and Gödel being the other three). Of the four, Tarski was the most prolific as a logician. The four volumes of his collected papers, which exclude most of his 19 monographs, span over 2500 pages. Aristotle's writings are comparable in volume, but most of the Aristotelian corpus is not about logic, whereas (...) virtually everything written by Tarski concerns logic more or less directly. There is no doubt that Tarski wrote more on logic than any other author; he started publishing on logic in 1921 at the age of 20 and continued until his death at the age of 82. Two of his works appeared posthumously [Hist. Philos. Logic 7 (1986), no. 2, 143--154; MR0868748 (88b:03010); Tarski and Givant, A formalization of set theory without variables, Amer. Math. Soc., Providence, RI, 1987; MR0920815 (89g:03012)]. Tarski's voluminous writings were widely scattered in numerous journals, some quite rare. It has been extremely difficult to study the development of Tarski's thought and to trace the interconnections and interdependence of his various papers. Thanks to the present collection all this has changed, and it is likely that the increased accessibility of Tarski's papers will have the effect of increasing Tarski's already enormous influence. (shrink)

Weinberg's 1972 work, in his description, had two purposes. The first was practical to bring together and assess the wealth of data provided over the previous decade while realizing that newer data would come in even as the book was being printed. He hoped the comprehensive picture would prepare the reader and himself to that new data as it emerged. The second was to produce a textbook about general relativity in which geometric ideas were not given a starring role for (...) (in his words) too great an emphasis on geometry can only obscure the deep connections between gravitation and the rest of physics. (shrink)

This paper presents a way of thinking about how to respond to the pluralism of modern societies that avoids any commitment to contractualist norms of political justification. The argument developed appeals to the notion of a constitutional settlement. Constitutional settlements are complex on-going social practices that both express certain values to which political societies are committed and establish procedures for resolving disputes among members of these societies. As such, they are a product of both moral commitment and the balance of (...) power. The paper shows how constitutional settlements relate to issues of stability and non-subjugation in politics, and explains how they can ground a distinction between justice and legitimacy. (shrink)

In the only two articles on the topic of which I am aware, Chad Carlson and Scott Aikin have leveled three objections against fantasy sports—namely, that participation in fantasy sports elicits...

In this paper, I build on Wittgenstein’s metaphor of a toolbox to introduce the metaphor of ‘tool confusion’ – how differing conceptual constructs may be applied, or misapplied, to one another and the effect that such applications have on the advancement of management theory. Moving beyond metaphor, I investigate a theory of management through two specific philosophical lenses (Popper and Lyotard). This analysis tests both the theory and the philosophies with regard to how each philosophy may be applied as a (...) tool to advance theory towards more effective application. Preliminary conclusions confirm that the application of partial philosophies is not as useful as the application of complete philosophies. Deeper contemplation, however, suggests that there is no upper limit to the completeness of philosophies. Thus, the problem of completeness is inescapable. In place of completeness, I explore the use of perceptual tools that are more specific, foundational and concise. Engaging in a second investigation, I use structures of logic (circular, linear, branching and co-causal) to investigate the subject theory. This investigation suggests at least two important insights relating to the structure of theory and the fuzziness of theory. Combined, these investigations and related conversations suggest rigorous methods for advancing theories and a more normative role for the philosophy of management that will support the accelerated advancement of management theory and practice. (shrink)

Are liberalism and perfectionism compatible? In this study Steven Wall presents and defends a perfectionist account of political morality that takes issue with many currently fashionable liberal ideas but retains the strong liberal commitment to the ideal of personal autonomy. He begins by critically discussing the most influential version of anti-perfectionist liberalism, examining the main arguments that have been offered in its defence. He then clarifies the ideal of personal autonomy, presents an account of its value and shows that (...) a strong commitment to personal autonomy is fully compatible with an endorsement of perfectionist political action designed to promote valuable pursuits and discourage base ones. (shrink)

In a number of publications, Gerald Gaus has presented an ambitious account of political morality that gives the ideal of public justification pride of place. This article critically discusses Gaus’s characterization and defense of the ideal of public justification in politics. It also presents an account and an argument in support of first-person political justification.

While examining the important role of imagination in making moral judgments, John Dewey and Moral Imagination focuses new attention on the relationship between American pragmatism and ethics. Steven Fesmire takes up threads of Dewey's thought that have been largely unexplored and elaborates pragmatism's distinctive contribution to understandings of moral experience, inquiry, and judgment. Building on two Deweyan notions—that moral character, belief, and reasoning are part of a social and historical context and that moral deliberation is an imaginative, dramatic rehearsal (...) of possibilities—Fesmire shows that moral imagination can be conceived as a process of aesthetic perception and artistic creativity. Fesmire's original readings of Dewey shed new light on the imaginative process, human emotional make-up and expression, and the nature of moral judgment. This original book presents a robust and distinctly pragmatic approach to ethics, politics, moral education, and moral conduct. [The downloadable sample is Chapter Seven, "The Moral Artist."]. (shrink)

In ancient times, when persuasion played the role of public force, eloquence was necessary. Of what use would it be today, when public force has replaced persuasion. One needs neither art nor metaphor to say such is my pleasure. Jean Jacques Rousseau.

Robotization is an increasingly pervasive feature of our lives. Robots with high degrees of autonomy may cause harm, yet in sufciently complex systems neither the robots nor the human developers may be candidates for moral blame. John Danaher has recently argued that this may lead to a retribution gap, where the human desire for retribution faces a lack of appropriate subjects for retributive blame. The potential social and moral implications of a retribution gap are considerable. I argue that the retributive (...) intuitions that feed into retribution gaps are best understood as deontological intuitions. I apply a debunking argument for deontological intuitions in order to show that retributive intuitions cannot be used to justify retributive punishment in cases of robot harm without clear candidates for blame. The fundamental moral question thus becomes what we ought to do with these retributive intuitions, given that they do not justify retribution. I draw a parallel from recent work on implicit biases to make a case for taking moral responsibility for retributive intuitions. In the same way that we can exert some form of control over our unwanted implicit biases, we can and should do so for unjustifed retributive intuitions in cases of robot harm. (shrink)

Governments around the world have faced the challenge of how to respond to the recent outbreak of a novel coronavirus disease. Some have reacted by greatly restricting the freedom of citizens, while others have opted for less drastic policies. In this paper, I draw a parallel with vaccination ethics to conceptualize two distinct approaches to COVID-19 that I call altruistic and lockdown. Given that the individual measures necessary to limit the spread of the virus can in principle be achieved voluntarily (...) as well as through enforcement, the question arises of how much freedom governments ought to give citizens to adopt the required measures. I argue that an altruistic approach is preferable on moral grounds: it preserves important citizen freedoms, avoids a number of potential injustices, and gives people a much-needed sense of meaning in precarious times. (shrink)

Bioethics, Volume 36, Issue 6, Page 687-698, July 2022.

`Fluid, readable and accessible ... I found the overall quality of the book to be excellent. It provides an overview of major (and preceding) developments in the field of science studies. It examines landmark works, authors, concepts and approaches ... I will certainly use this book as one of the course texts' Eileen Crist, Associate Professor, Science & Technology in Society, Virginia Tech Science is at the heart of contemporary society and is therefore central to the social sciences. Yet science (...) studies has often encountered resistance from social scientists. This book attempts to remedy this by giving the most extensive, thorough and best argued account of the field and explaining to social scientists why science matters to them. This is a landmark book that demystifies science studies and successfully bridges the divide between social theory and the sociology of science. Illustrated with relevant, illuminating examples, it provides the ideal guide to science studies and social theory. (shrink)

In an unusual instance of lay participation in biomedical research, U.S. AIDS treatment activists have constituted themselves as credible participants in the process of knowledge construction, thereby bringing about changes in the epistemic practices of biomedical research. This article examines the mechanisms or tactics by which these lay activists have constructed their credibility in the eyes of AIDS researchers and government officials. It considers the inwlications of such interventions for the conduct of medical research; examines some of the ironies, tensions, (...) and limitations in the process; and argues for the importance of studying social movements that engage with expert knowledge. (shrink)

Moralization is a social-psychological process through which morally neutral issues take on moral significance. Often linked to health and disease, moralization may sometimes lead to good outcomes; yet moralization is often detrimental to individuals and to society as a whole. It is therefore important to be able to identify when moralization is inappropriate. In this paper, we offer a systematic normative approach to the evaluation of moralization. We introduce and develop the concept of ‘mismoralization’, which is when moralization is metaethically (...) unjustified. In order to identify mismoralization, we argue that one must engage in metaethical analysis of moralization processes while paying close attention to the relevant facts. We briefly discuss one historical example and two contemporary cases related to COVID-19 that we contend to have been mismoralized in public health. We propose a remedy of de-moralization that begins by identifying mismoralization and that proceeds by neutralizing inapt moral content. De-moralization calls for epistemic and moral humility. It should lead us to pull away from our tendency to moralize—as individuals and as social groups—whenever and wherever moralization is unjustified. (shrink)

An introduction to and critique of the latest trends in critical theory.

Editors provide a substantive introduction to the history and theories of perfectionism and neutrality, expertly contextualizing the essays and making the collection accessible.

The distinction between presentism and eternalism is usually sought in some formula like ‘Only presently existing things exist’ or ‘Past, present, and future events are equally real’. I argue that ambiguities in the copula prevent these slogans from distinguishing significant opposed positions. I suggest in addition that one can find a series of significant distinctions if one takes spacetime structure into account. These presentisms and eternalisms are not contradictory. They are complementary elements of a complete naturalistic philosophy of time.