The calculus of relations was created and developed in the second half of the nineteenth century by Augustus De Morgan, Charles Sanders Peirce, and Ernst Schröder. In 1940 Alfred Tarski proposed an axiomatization for a large part of the calculus of relations. In the next decade Tarski's axiomatization led to the creation of the theory of relation algebras, and was shown to be incomplete by Roger Lyndon's discovery of nonrepresentable relation algebras. This paper introduces the calculus of relations and (...) the theory of relation algebras through a review of these historical developments. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Hume Studies Volume XXIII, Number 2, November 1997, pp. 315-335 Reid: Conception, Representation and Innate Ideas ROGER D. GALLIE Section I of this paper begins with a presentation of Thomas Reid's doctrine of the signification of words, of what words signify or represent. That presentation serves to introduce a problem of interpretation, namely, what Reid thinks the connection is between conceiving something and grasping what a term for (...) it signifies. It is pointed out that even if Reid maintains that what a word signifies is conceived by both speaker and hearer when that word is understood, this is a far cry from the position that he takes the stronger view, whether for all or only for some things, that their being conceived involves the understanding of a term or terms. The position that he does indeed take this stronger, although moderate, view is defended in sections II, III and IV. In sections II and III it is argued that, in Reid's view, for an adequate conception of an individual, real or imaginary, a conception of its attributes is needed. In section IV it is argued that the possession of general conceptions (in distinction from their formation), without which conception of attributes is impossible, requires appropriate understanding of general terms. So since for the conception of an individual a conception of its attributes is needed, and since for the conception of attributes appropriate understanding of general terms is required, we arrive at a textually well-founded position: for the conception of most things, in Reid's view, understanding of terms is indispensable. Roger D. Gallie is at the Department of Philosophy, Nottingham University, University Park, Nottingham NG7 2RD UK. email: Roger[email protected] 316 Roger D. Gallie The remainder of the paper considers the following question: to what extent, if any, Reid is committed to acceptance of nativist views on concept possession? To answer, in section V we consider to what extent Reid's moderate position on the possession of conceptions commits him to accepting Jerry Fodor's representational theory of mind (RTM) for propositional attitudes. It is clear that the view that a propositional attitude state such as belief involves some relation between a person and a representation such as a significant word, phrase or sentence of a natural language such as English can fairly be attributed to Reid. But it is also clear that Reid cannot be an adherent of the view that persons have a functional cum computational relation to Fodorean mental representations, so he cannot be an adherent of any nativism ex- plicitly contained in such a view. But it is to be noted that in his Inquiry, Chapter IV, Reid argues by a different route that in order to acquire a language such as English prior mastery of a system of significant signs is required. In section VI Keith Lehrer's argument in criticism of a reductionist interpretation of a position such as Fodor's is presented: that for a sign to signify something to someone presupposes that that person has a conception of what is signified. It is however pointed out that since there is a case for not viewing Fodor's position as reductionist overall Fodor can accept Lehrer's argument. Lehrer's innatist position, which he both claims to follow easily from this critical argument and to be a correct interpretation of Reid, is then presented: that we must presuppose an innate understanding of some signs to account for our learning the meaning of any signs. The objection that Reid too must accept the alleged innatist implications of Lehrer's critical argument in view of the implications of his moderate position on the possession of conceptions is considered and rebutted. In section VIII argue that while Lehrer makes a respectable case for innate principles of formation of conceptions such as that of hardness, something that Reid clearly endorses in his Inquiry, Chapter V, he does not succeed in making a case for there being any innate possession of conceptions. Finally, in section VIII, Reid's mature views in the Intellectual Powers, Essay VI, on how conceptions come to be possessed are expounded, and are fleshed out with some... (shrink)

This book is intended as an equivalent to or substitute for that "more reflective reading" which Rousseau considered essential to an understanding of his ideas. It is designed to complement perusal of the texts themselves, and the arrangement is such that chapters on each of Rousseau's major writings can be consulted separately or the commentary may be read through in sequence. The author's purpose is not to present a "key" to Rousseau's political philosophy, but rather to explore the works themselves (...) in an effort to reveal Rousseau's "system," from which the reader may then draw his own conclusions. Originally published in 1976. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905. (shrink)

Conjecture (1) of [Ma83] is confirmed here by the following result: if $3 \leq \alpha < \omega$, then there is a finite relation algebra of dimension α, which is not a relation algebra of dimension α + 1. A logical consequence of this theorem is that for every finite α ≥ 3 there is a formula of the form $S \subseteq T$ (asserting that one binary relation is included in another), which is provable with α + 1 variables, but not (...) provable with only α variables (using a special sequent calculus designed for deducing properties of binary relations). (shrink)

The set of equations which use only one variable and hold in all representable relation algebras cannot be derived from any finite set of equations true in all representable relation algebras. Similar results hold for cylindric algebras and for logic with finitely many variables. The main tools are a construction of nonrepresentable one-generated relation algebras, a method for obtaining cylindric algebras from relation algebras, and the use of relation algebras in defining algebraic semantics for first-order logic.

Sound and complete semantics for classical propositional logic can be obtained by interpreting sentences as sets. Replacing sets with commuting dense binary relations produces an interpretation that turns out to be sound but not complete for R. Adding transitivity yields sound and complete semantics for RM, because all normal Sugihara matrices are representable as algebras of binary relations.

In the United States, a longstanding legal rule exists against patenting natural phenomena. The Supreme Court recently had an opportunity to help define the boundaries and clarify the implications of this “natural phenomenon doctrine” in Laboratory Corporation of America v. Metabolite Labs., dismissed as improvidently granted. This article argues that the natural phenomenon doctrine renders both the patent claim at issue in LabCorp, and the patents that directly or indirectly claim biological correlations between genotypes and medical phenotypes, invalid or unenforceable (...) under U.S. patent law. (shrink)

Molecular genetic testing has increasingly been incorporated into clinical medicine, and this trend is likely to accelerate in the future. The introduction of genetic testing into medical practice is beginning to collide head on with patents that claim ownership of correlations between human genetic variants and predisposition to disease, response to therapeutic drugs, and susceptibility to pharmacologic side effects. Patent holders or licensees of genes, genetic variants, and their genotype-phenotype correlations are already using the threat of litigation to monopolize genetic (...) tests for important well-known syndromes like Duchenne Muscular Dystrophy and inherited breast and ovarian cancer, in addition to a host of less commonly discussed conditions. (shrink)

Most people are familiar with the traditional view of the role of ethics in the auditing profession - the need for auditors with integrity and objectivity. This essay addresses a second dimension of ethics in the auditing profession - the demand for auditors to assess the integrity and ethical values of clients. This second dimension is a difficult task for auditors in practice and demands a deep and robust understanding of ethics, ethical infrastructures, and the products of those infrastructures. The (...) essay proposes how educators and researchers might facilitate that understanding. (shrink)

This is a supplement to the paper “Finitary Algebraic Logic” [1]. It includes corrections for several errors and some additional results. MSC: 03G15, 03G25.

If K is a class of semiassociative relation algebras and K contains the relation algebra of all binary relations on a denumerable set, then the word problem for the free algebra over K on one generator is unsolvable. This result implies that the set of sentences which are provable in the formalism Lwx is an undecidable theory. A stronger algebraic result shows that the set of logically valid sentences in Lwx forms a hereditarily undecidable theory in Lwx. These results generalize (...) similar theorems, due to Tarski, concerning relation algebras and the formalism Lx. (shrink)

Relation algebras were invented by Tarski and his collaborators in the middle of the 20th century. The concept of integrality arose naturally early in the history of the subject, as did various constructions of finite integral relation algebras. Later the concept of finite-dimensionality was introduced for classifying nonrepresentable relation algebras. This concept is closely connected to the number of variables used in proofs in first-order logic. Some results on these topics are presented in chronological order.

The value of Abraham Kuyper’s thought is presently being discovered by a growing body of readers. Herman Dooyeweerd’s thought is appreciated by a smaller number. Yet he was trained in the Kuyperian tradition and conceived many of his most characteristic insights by building on Kuyper’s. What did he learn from Abraham and what can be learned by reflecting on their shared ideas? A view of the coherence, the gum and wire holding society and the cosmos together is the first of (...) their common themes, considered in this essay. The second is religion and its root, namely, the human heart in its encounter with the Eternal. This involves a discussion of time and whether human existence can be said to extend beyond time in some respect. Observations are made and questions are raised about the contrasting and comparative accessibility of the two authors’ thoughts and writings. The second half of the essay focuses on the earliest statements of Dooyeweerd’s germinating philosophy — in relationship to Kuyper’s thought. At the most fundamental level these two thinkers share basic principles, and in light of these they developed their most distinctive ideas. An implicit reassessment of this intellectual tradition is offered here with the hope of finding hints and motivation to open new vistas on some key insights and principles. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS 373 in the analysis of the "artificial" virtue of justice. Though he uses the term "faculties" as synonymous with energies or powers, he warns against the "faculty psychology" that uses faculties as explanations or causes. Hume writes: "By will I mean nothing but the internal impression we feel.., when we knowingly give rise to any new motion of our body or new perception of our mind." A (...) voluntary action, being "knowingly" aroused, is not a mere "direct " passion arising immediately from sensation; nevertheless, it belongs to the realm of passions rather than of judgments. It is included in the "bundle of impressions" psychology. Hume attempts to explain how the passion of sympathy serves to communicate impressions. Emotional expressions require the operation of the "imagination" to generate the social controls of conscience and to explain the origin of responsibility in terms of approvals and disapprovals. In this connection, /~rdal points out the differences between Hume's and Adam Smith's psychology of sympathy, ttume is content to rely largely on "imagination" to explain social sensitivities, whereas Adam Smith develops the theory that by sympathy men are led to put themselves in the place of others and thus to get a "disinterested spectator" view of passions. This seems to me a minor difference, and Hume, though he knew of Adam Smith's theory, and though he criticized him on other points, does not directly obiect to this particular doctrine of Adam Smith. It is, in fact, related to Hume's own analysis of justice as an "artificial" virtue, not directly related to the self-regarding passions. What separates Smith from Hume is his discussion of sympathy in the context of pity and benevolence which the Scottish moralists had made basic, whereas ttume keeps sympathy as a passion distinct from the virtues. In general, I have the "impression" that Hume, after he made this elaborate effort to apply experimental method to the passions and to valuational expressions, left the further development of this psychology to his close friend, Adam Smith, while he turned to "political essays" and to the more practical problems of the Enquiry. In his political essays, which brought him into public recognition in England and on the Continent while the Treatise continued to be basic for the Scottish social psychologists, Hume became especially concerned to call attention to the social operation of "opinions" and faiths. Therefore, it seems to me to be a mistake to regard the Enquiry as merely a "popular" edition of the Treatise. Hume shifted his interest from the critique of Locke's psychology to the practical problems of politics and morals. His History deals less with the psychological "origins" of customs, beliefs, and habits, more with their power in shaping the course of events. In other words, for Hume's ideas about judgments of value and the motivation of policies and principles, one must turn to Hume's later writings. The reader of ~rdal's excellent analysis of the Treatise cannot refrain from hoping that the author will make a similar examination of Hume's later "moral science." For, in addition to clarifying Hume, it will give.Ardal an opportunity to continue his own critical insights on present-day discussion of emotivist and intuitional ethics, which are not the least valuable feature of this volume. HERBERT W. SCHNEIDER Claremont, California Jean-Jacques Rousseau. By Jean Gu~henno. Translated from the French by John and Doreen Weightman. (2 vols. New York: Columbia University Press, 1967. Vol. I, pp. xvi + 460. Vol. II, pp. xi + 316. set $17.50.) Rousseau and Jean-Jacques It has been Rousseau's peculiar fate to exercise a double fascination. A major philosopher, especially in the fields of education, politics, and morality, Jean-Jacques has also been read, loved, and hated as a man. In reading again Jean Gu~henno's impressive biography, recently published in an excellent English translation, one wonders why so much attention has been paid to Rousseau's private life. Surely it is not because he lacks the credentials of a serious thinker; the Second Discourse alone, not to mention the Emile and Social Contract, would presumably ensure Rousseau's place in any historical survey of Western... (shrink)