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)

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.

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)

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.

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.

There are eighteen isomorphism types of finite relation algebras with eight or fewer elements, and all of them are representable. We determine all the cardinalities of sets on which these algebras have representations.

We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language L with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of (...) which may occur arbitrarily many times), X has a proof containing exactly α + 1 variables, but X has no proof containing only α variables. This solves a problem posed by Tarski and Givant in 1987. (shrink)

Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared between relevance logics (...) and relation algebras. Some details of certain incompleteness results, however, pinpoint where relevance logics and relation algebras diverge. To carry out these semantic investigations, we define a new tableaux formalization and new sequent calculi (with the single cut rule admissible) for various relevance logics. (shrink)

For every finite n ≥ 4 there is a logically valid sentence φ n with the following properties: φ n contains only 3 variables (each of which occurs many times); φ n contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol): φ n has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n - 1 variables. This result was first proved using the machinery of (...) algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that φ n has a proof with only n variables. To show that φ n has no proof with only n - 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic. (shrink)

It is known that for all finite n ≥ 5, there are relation algebras with n-dimensional relational bases but no weak representations. We prove that conversely, there are finite weakly representable relation algebras with no n-dimensional relational bases. In symbols: neither of the classes RA n and wRRA contains the other.

This book grew out of previously published papers of mine composed over a period of years; they have been reworked (sometimes beyond recognition) so as to form a reasonably coherent whole. Part One treats of informative inference. I argue (Chapter 2) that the traditional principle of induction in its clearest formulation (that laws are confirmed by their positive cases) is clearly false. Other formulations in terms of the 'uniformity of nature' or the 'resemblance of the future to the past' seem (...) to me hopelessly unclear. From a Bayesian point of view, 'learning from experience' goes by conditionalization (Bayes' rule). The traditional stum bling block for Bayesians has been to fmd objective probability inputs to conditionalize upon. Subjective Bayesians allow any probability inputs that do not violate the usual axioms of probability. Many subjectivists grant that this liberality seems prodigal but own themselves unable to think of additional constraints that might plausibly be imposed. To be sure, if we could agree on the correct probabilistic representation of 'ignorance' (or absence of pertinent data), then all probabilities obtained by applying Bayes' rule to an 'informationless' prior would be objective. But familiar contra dictions, like the Bertrand paradox, are thought to vitiate all attempts to objectify 'ignorance'. BuUding on the earlier work of Sir Harold Jeffreys, E. T. Jaynes, and the more recent work ofG. E. P. Box and G. E. Tiao, I have elected to bite this bullet. In Chapter 3, I develop and defend an objectivist Bayesian approach. (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)

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)

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)

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)

This monograph introduces the functional characteristics and conceptual significance of a simulation software system called CyberRat (Ray, 1996a, 2003a, 2012a, 2012b). CyberRat expands upon prior illustrations (Ray & Delprato, 1989; Ray, 1992) of how such computer-based simulations can serve to formatively enhance, and eventually validate, the descriptive research methodology upon which their development relies. To illustrate this process I also review highlights of previous publications (cf. Ray & Brown, 1975, 1976; Ray & Delprato, 1989), detailing the unique research methodology used (...) to collect data that guided CyberRat’s development. This methodology integrates interbehavioral psychology (Kantor, 1959) and general systems analysis (von Bertalanffy, 1968), and thus is referred to as interbehavioral systems analysis (IBSA). CyberRat’s validation of IBSA methods involves a process analogous to Turing’s (1950) famous test for simulation authenticity, in that it relies upon “phenomenological equivalence” criteria for observers to compare experiences of real vs. simulated events. And because IBSA stresses three convergent strategies for research, including structural analysis, functional analysis, and operations analysis, my organizing theme addresses how closely CyberRat comes to passing a trilogy of hypothetical Turing tests—one for each of these three analytic strategies. (shrink)