We show, for any ordinal γ ≥ 3, that the class RaCAγ is pseudo-elementary and has a recursively enumerable elementary theory. ScK denotes the class of strong subalgebras of members of the class K. We devise games, Fⁿ (3 ≤ n ≤ ω), G, H, and show, for an atomic relation algebra A with countably many atoms, that Ǝ has a winning strategy in Fω(At(A)) ⇔ A ∈ ScRaCAω, Ǝ has a winning strategy in Fⁿ(At(A)) ⇐ A ∈ ScRaCAn, Ǝ (...) has a winning strategy in G(At(A)) ⇐ A ∈ RaCAω, Ǝ has a winning strategy in H(At(A)) ⇒ A ∈ RaRCAω for 3 ≤ n < ω. We use these games to show, for γ ≥ 5 and any class K of relation algebras satisfying RaRCAγ ⊆ K ⊆ ScRaCA₅, that K is not closed under subalgebras and is not elementary. For infinite γ, the inclusion RaCAγ ⊂ ScRaCAγ is strict. For infinite γ and for a countable relation algebra A we show that A has a complete representation if and only if A is atomic and Ǝ has a winning strategy in F(At(A)) if and only if A is atomic and A ∈ ScRaCAγ. (shrink)

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)

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finite relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an (...) algebra is ω-categorical. We have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this approach are looked at, and include the step by step method. (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)

We study relation algebras with n-dimensional relational bases in the sense of Maddux. Fix n with 3nω. Write Bn for the class of non-associative algebras with an n-dimensional relational basis, and RAn for the variety generated by Bn. We define a notion of relativised representation for algebras in RAn, and use it to give an explicit equational axiomatisation of RAn, and to reprove Maddux's result that RAn is canonical. We show that the algebras in Bn are precisely those that have (...) a complete relativised representation of this type. Then we prove that whenever 4nshrink)

We prove, for each 4⩽ n ω , that S Ra CA n+1 cannot be defined, using only finitely many first-order axioms, relative to S Ra CA n . The construction also shows that for 5⩽n S Ra CA n is not finitely axiomatisable over RA n , and that for 3⩽m S Nr m CA n+1 is not finitely axiomatisable over S Nr m CA n . In consequence, for a certain standard n -variable first-order proof system ⊢ m (...) , n of m -variable formulas, there is no finite set of m -variable schemata whose m -variable instances, when added to ⊢ m , n as axioms, yield ⊢ m , n +1. (shrink)

We prove that everyn-modal logic betweenKnandS5nis undecidable, whenever n ≥ 3. We also show that each of these logics is non-finitely axiomatizable, lacks the product finite model property, and there is no algorithm deciding whether a finite frame validates the logic. These results answer several questions of Gabbay and Shehtman. The proofs combine the modal logic technique of Yankov–Fine frame formulas with algebraic logic results of Halmos, Johnson and Monk, and give a reduction of the (undecidable) representation problem of finite (...) relation algebras. (shrink)

We characterise the class S Ra CA n of subalgebras of relation algebra reducts of n -dimensional cylindric algebras by the notion of a ‘hyperbasis’, analogous to the cylindric basis of Maddux, and by representations. We outline a game–theoretic approximation to the existence of a representation, and how to use it to obtain a recursive axiomatisation of S Ra CA n.

A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.

A cylindric algebra atom structure is said to be strongly representable if all atomic cylindric algebras with that atom structure are representable. This is equivalent to saying that the full complex algebra of the atom structure is a representable cylindric algebra. We show that for any finite n >3, the class of all strongly representable n-dimensional cylindric algebra atom structures is not closed under ultraproducts and is therefore not elementary. Our proof is based on the following construction. From an arbitrary (...) undirected, loop-free graph Γ, we construct an n-dimensional atom structure E(Γ), and prove, for infinite Γ, that E(Γ) is a strongly representable cylindric algebra atom structure if and only if the chromatic number of Γ is infinite. A construction of Erdős shows that there are graphs (k < ω)with infinite chromatic number, but having a non-principal ultraproduct $\prod _D\Gamma _k $ whose chromatic number is just two. It follows that $E(\Gamma _k )$ is strongly representable (each k < ω) but $\Pi _D E(\Gamma _k)$ is not. (shrink)

A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension are elementary.

Using a variation of the rainbow construction and various pebble and colouring games, we prove that RRA, the class of all representable relation algebras, cannot be axiomatised by any first-order relation algebra theory of bounded quantifier depth. We also prove that the class At(RRA) of atom structures of representable, atomic relation algebras cannot be defined by any set of sentences in the language of RA atom structures that uses only a finite number of variables.

We prove that algebras of binary relations whose similarity type includes intersection, union, and one of the residuals of relation composition form a nonfinitely axiomatizable quasivariety and that the equational theory is not finitely based. We apply this result to the problem of the completeness of the positive fragment of relevance logic with respect to binary relations.

In this article we establish the undecidability of representability and of finite representability as algebras of binary relations in a wide range of signatures. In particular, representability and finite representability are undecidable for Boolean monoids and lattice ordered monoids, while representability is undecidable for Jónsson's relation algebra. We also establish a number of undecidability results for representability as algebras of injective functions.

Let S be a signature of operations and relations definable in relation algebra, let R be the class of all S-structures isomorphic to concrete algebras of binary relations with concrete interpretations for symbols in S, and let F be the class of S-structures isomorphic to concrete algebras of binary relations over a finite base. To prove that membership of R or F for finite S-structures is undecidable, we reduce from a known undecidable problem—here we use the tiling problem, the partial (...) group embedding problem and the partial group finite embedding problem to prove undecidability of finite membership of R or F for various signatures S. It follows that the equational theory of R is undecidable whenever S includes the boolean operators and composition. We give an exposition of the reduction from the tiling problem and the reduction from the group embedding problem, and summarize what we know about the undecidability of finite membership of R and of F for different signatures S. (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.

We outline a simple approach to axiomatising the class of representable relation algebras, using games. We discuss generalisations of the method to cylindric algebras, homogeneous and complete representations, and atom structures of relation algebras.

We define a finite relation algebra and show that the network satisfaction problem is undecidable for this algebra.

We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras and a similar schema for cylindric algebras are derived. Finite relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an algebra is $\omega$-categorical. We (...) have a simple proof that the perfect extension of a representable relation algebra is completely representable. An important open problem from algebraic logic is addressed by devising another two-player game, and using it to derive equational axiomatisations for the classes of all representable relation algebras and representable cylindric algebras. Other instances of this approach are looked at, and include the step by step method. (shrink)

Demonic composition, demonic refinement and demonic union are alternatives to the usual ‘angelic’ composition, angelic refinement (inclusion) and angelic (usual) union defined on binary relations. We first motivate both the angelic and the demonic via an analysis of the behaviour of non-deterministic programs, with the angelic associated with partial correctness and demonic with total correctness, both cases emerging from a richer algebraic model of non-deterministic programs incorporating both aspects. Zareckiĭ has shown that the isomorphism class of algebras of binary relations (...) under angelic composition and inclusion is finitely axiomatized as the class of ordered semigroups. The proof can be used to establish that the same axiomatization applies to binary relations under demonic composition and refinement, and a further modification of the proof can be used to incorporate a zero element representing the empty relation in the angelic case and the full relation in the demonic case. For the signature of angelic composition and union, it is known that no finite axiomatization exists, and we show the analogous result for demonic composition and demonic union by showing that the same axiomatization holds for both. We show that the isomorphism class of algebras of binary relations with the ‘mixed’ signature of demonic composition and angelic inclusion has no finite axiomatization. As a contrast, we show that the isomorphism class of partial algebras of binary relations with the partial operation of constellation product and inclusion (also a ‘mixed’ signature) is finitely axiomatizable. (shrink)

The concept of yinyang lies at the heart of Chinese thought and culture. The relationship between these two opposing, yet mutually dependent, forces is symbolized in the familiar black and white symbol that has become an icon in popular culture across the world. The real significance of yinyang is, however, more complex and subtle. This brilliant and comprehensive analysis by one of the leading authorities in the field captures the richness and multiplicity of the meanings and applications of yinyang, including (...) its visual presentations. Through a vast range of historical and textual sources, the book examines the scope and role of yinyang, the philosophical significance of its various layers of meanings and its relation to numerous schools and traditions within Chinese philosophy. By putting yinyang on a secure and clear philosophical footing, the book roots the concept in the original Chinese idiom, distancing it from Western assumptions, frameworks and terms, yet also seeking to connect its analysis to shared cross-cultural philosophical concerns. (shrink)

The trial of Socrates -- Socrates in court -- How the system worked -- The charge of impiety -- The war years -- Alcibiades, Socrates, and the aristocratic milieu -- Pestilence and war -- The rise and fall of Alcibiades -- The end of the war -- Critias and Civil War --- Crisis and conflict -- Symptoms of change -- Reactions to intellectuals -- The condemnation of Socrates -- Socratic politics -- A cock for Asclepius.

This book treats Chinese philosophy today as a global project, presenting the work of both Chinese and Western philosophers.

Two main claims are defended in this paper: first, that typical disputes in the literature about the ontology of physical objects are merely verbal; second, that the proper way to resolve these disputes is by appealing to common sense or ordinary language. A verbal dispute is characterized not in terms of private idiolects, but in terms of different linguistic communities representing different positions. If we imagine a community that makes Chisholm's mereological essentialist assertions, and another community that makes Lewis's four‐dimensionalist (...) assertions, the members of each community speak the truth in their respective languages. This follows from an application of the principle of interpretive charity to the two communities. (shrink)

A number of jurisdictions, including England and Wales after their adoption of the 1991 Criminal Justice Act, require that sentences be `proportionate' to the severity of the crime. This book, written by the leading architect of `just deserts' sentencing theory, discusses how sentences may be scaled proportionately to the gravity of the crime. Topics dealt with include how the idea of a penal censure justifies proportionate sentences; how a penalty scale should be `anchored' to reduce overall punishment levels; how non-custodial (...) penalties should be graded and used; and how political pressures impinge on sentencing policies. (shrink)

Milton Friedman's famous methodological essay contains, along with much else, some strands that look as though they were taken from the “empirical-scientific” fabric described by Karl Popper. Think, for example, of Friedman's conviction that the way to test a hypothesis is to compare its implications with experience. Or of his more or less explicit espousal of the view that while no amount of facts can ever prove a hypothesis true, a single “fact” may refute it. Or of his assertion that (...) hypotheses are to be accepted only as provisionally true, and then only after repeated efforts to refute them have failed. The appearance of these Popperian ideas is not surprising. (shrink)

The principle that a sentence should be proportionate to the seriousness of the offence remains at the centre of penal practice and scholarly debate. This volume explores highly topical aspects of proportionality theory that require examination and further analysis. von Hirsch and Ashworth explore the relevance of the principle of proportionality to the sentencing of young offenders, the possible reasons for departing from the principle when sentencing dangerous offenders, and the application of the principle to socially deprived offenders. They (...) examine the claim that the principle tends to be associated with greater severity in sentencing, and explore the relevance of penance and of restorative justice to proportionality theory. Their examination of arguments and counter-arguments culminates in a re-statement of the main criteria for proportionate sentencing.The authors are well known for their previous writings on proportionality theory, and this volume broadens the theory to deal with important contemporary issues in crime and punishment. (shrink)

them. Though Aristotle does not say so, presumably the questioner who conceals in this way must be prepared, when challenged, to show that the conclusion...

What is the role of psychoanalysis in today's world? _Yesterday, Today and Tomorrow _presents a selection of papers written by Hanna Segal. The collection introduces the reader to a wide spectrum of insights into psychoanalysis, ranging from current thoughts on the nature of dreaming to new ideas about vision and disillusionment. Her long interest in factors affecting war is pursued in her examination of the psychotic factors, symbolic significance and psychological impact of the events of September the 11th, and the (...) ensuing war on Iraq. The second half of the book discusses Segal's presentations to conferences and symposia from 1969-2000, this material is split into six sections: Models of the mind and mental processes Psychoanalytic technique Segal on Klein Segal on Bion Envy and narcissism Interviews. _Yesterday, Today and Tomorrow_ is a masterly contribution to the field. Segal's clarity of thought and striking clinical illustrations make the book accessible to those new to the field as well as those acquainted with her seminal work. (shrink)

The struggle which Plato has Socrates recommend to his interlocutors in Gorgias - and to his readers - is the struggle to overcome the temptations of worldly success and to concentrate on genuine morality. Ostensibly an enquiry into the value of rhetoric, the dialogue soon becomes an investigation into the value of these two contrasting ways of life. In a series of dazzling and bold arguments, Plato attempts to establish that only morality can bring a person true happiness, and to (...) demolish alternative viewpoints. It is not surprising that Gorgias is one of Plato's most widely read dialogues. Philosophers read it for its coverage of central moral issues; others enjoy its vividness, clarity and occasional bitter humour. This new translation is accompanied by explanatory notes and an informative introduction. (shrink)

The first English-language biography of the well-known traditionalist metaphysican René Guénon, including a separate section assessing the impact of his work in the Western world, and an extensive annotated bibliography.

What responsibility do individuals bear for structural injustice? Iris Marion Young has offered the most fully developed account to date, the Social Connections Model. She argues that we all bear responsibility because we each causally contribute to structural processes that produce injustice. My aim in this article is to motivate and defend an alternative account that improves on Young’s model by addressing five fundamental challenges faced by any such theory. The core idea of what I call the “Role-Ideal Model” is (...) that we are each responsible for structural injustice through and in virtue of our social roles, i.e. our roles as parents, colleagues, employers, citizens, etc., because roles are the site where structure meets agency. In short, the Role-Ideal Model explains how individual action contributes to structural change, justifies demands for action from each particular agent, specifies what kinds of acts should be undertaken, moderates between demanding too much and too little of individual agents, and provides an account of the critical responses appropriate for holding individuals accountable for structural injustice. (shrink)