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)

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)

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)

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 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)

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)

Although it has become increasingly evident that an adequate theory of obligation must rest on evolutionary biology and human ethology, attempts toward this end need to explore the full range of personal, cultural, and political obligations observed in our species. The new naturalism reveals the complexity of social behavior and the defects of reductionist models that oversimplify the foundations of human duties and rights. Ultimately, this approach suggest a return to the Aristotelian concept of natural justice.

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)

This book explains why contemporary liberal democracies are based on historical templates rather than revolutionary reforms; why the transition in Europe occurred during a relatively short period in the nineteenth century; why politically and economically powerful men and women voluntarily supported such reforms; how interests, ideas, and pre-existing institutions affected the reforms adopted; and why the countries that liberalized their political systems also produced the Industrial Revolution. The analysis is organized in three parts. The first part develops new rational choice (...) models of governance, the balance of authority between parliaments and kings, constitutional exchange, and suffrage reform. The second part provides historical overviews and detailed constitutional histories of six important countries. The third part provides additional evidence in support of the theory, summarizes the results, contrasts the approach taken in this book with that of other scholars, and discusses methodological issues. (shrink)