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)

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.

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)

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.

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 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)

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)

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.

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)

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.

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.

Robin McKenna argues that we need to make space for an approach to epistemology that avoids the idealizations typical of the field. He applies this approach to topics in applied and social epistemology, such as what to do about science denial, whether we should try to be intellectually autonomous, and what our obligations are to other inquirers.

Introduction: the emergence of the proportionate sentence -- Sentence proportionality sketched briefly -- Why should the criminal sanction exist? -- Why punish proportionately? -- Ordinal and cardinal proportionality -- Seriousness, severity and the living-standard -- The role of previous convictions -- Proportionate non-custodial sanctions -- A "modified" desert model? -- The politics of the desert model -- Proportionate sentences for juveniles -- Appendix: the desert model's evolution : a brief chronology.

_The Routledge Companion to Philosophy of Psychology, Second Edition_ is an invaluable guide and major reference source to the major topics, problems, concepts and debates in philosophy of psychology and is the first companion of its kind. A team of renowned international contributors provide forty-nine chapters organised into six clear parts: Historical background to Philosophy of Psychology Psychological Explanation Cognition and Representation The biological basis of psychology Perceptual Experience Personhood. _The Companion_ covers key topics such as the origins of experimental (...) psychology; folk psychology; behaviorism and functionalism; philosophy, psychology and neuroscience; the language of thought, modularity, nativism and representational theories of mind; consciousness and the senses; dreams emotion and temporality; personal identity and the philosophy of psychopathology. For the_ second edition_ many of the current chapters have been updated, and seven new chapters added on important new topics such predictive processing, comparative cognition, learning, and group cognition, as well as a new introductory chapter by the editors on the demarcation between philosophy and psychology. Essential reading for all students of philosophy of mind, science and psychology, _The Routledge Companion to Philosophy of Psychology _will also be of interest to anyone studying psychology and its related disciplines. (shrink)

The Routledge Companion to Metaphysics is an outstanding, comprehensive and accessible guide to the major themes, thinkers, and issues in metaphysics. The Companion features over fifty specially commissioned chapters from international scholars which are organized into three clear parts: History of Metaphysics Ontology Metaphysics and Science. Each section features an introduction which places the range of essays in context, while an extensive glossary allows easy reference to key terms and definitions. The Routledge Companion to Metaphysics is essential reading for students (...) of philosophy and anyone interested in surveying the central topics and problems in metaphysics from causation to vagueness and from Plato and Aristotle to the present-day. (shrink)

While many of the phenomenological currents in philosophy allegedly utilize a peculiar method, the type under consideration here is characterized by Franz Brentano s ambition to make philosophy scientific by adopting no other method but that of natural science. Brentano became particularly influential in teaching his students (such as Carl Stumpf, Anton Marty, Alexius Meinong, and Edmund Husserl) his descriptive psychology, which is concerned with mind as intentionally directed at objects. As Brentano and his students continued in their investigations in (...) descriptive psychology, another side of Austrian phenomenology, namely object theory, became more and more prominent. The philosophical orientation under consideration in this collection of essays is accordingly a two-sided discipline, concerned with both mind and objects, and applicable to various areas of philosophy such as epistemology, philosophy of language, value theory, and ontology.". (shrink)

In The Gay Science §341, the thought of eternal return is introduced as the announcement of a “demon.” Two possible hearers are described: one is crushed by the demon’s speech, while the other is overjoyed. This article argues that these responses are different because they are responses to different messages. One is conveyed in plain words by the demon’s speech; the other is implied by a final reference to “this ultimate eternal confirmation and sealing.” While that confirmation is provided by (...) what the demon says, not everyone will be aware of it: rather, only someone for whom the value of life has become problematic will be aware of it. Further, it seems that the demon does not understand his own message, since he personifies a will to knowledge that is indifferent to such concerns. Thus, only a human being can gain insight into the value and meaning of life from the thought of eternal return. (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.

One of the most important students of Franz Brentano was Anton Marty, who made it his task to develop a philosophy of language on the basis of Brentano’s analysis of mind. It is most unfortunate that Marty does not receive the attention he deserves, primarily due to his detailed and distracting polemics. In the analysis presented here his philosophy of language and other aspects of his thought, such as his ontology , are examined first and foremost in their positive rather (...) than critical character. The analysis is moreover supplemented by translations of four important works by Marty, including his entire work On the Origin of Language. These are in fact the first English translations of any substantial writings by him. The resulting picture that emerges from the analysis and translations is that Marty has much to say that proves to be of enduring interest for the philosophy of language on a range of topics, especially the meanings of statements, of emotive expressions, and of names as regards both their communicative and their ontological aspects. The volume will be of interest not only to philosophers and historians of philosophy, but also to historians of linguistics and psychology. (shrink)

This essay defends a novel interpretation of the term xìng 性 as it occurs in Chinese texts of the late Warring States period (roughly 320–221 BCE). The term played an important role both in the famous controversy over the goodness or badness of people’s xìng and elsewhere in the intellectual discourse of the period. Extending especially the work of A.C. Graham, the essay stresses the importance for understanding xìng of early Chinese assumptions about spontaneity, continuity, health, and (in the human (...) case) motivation. These assumptions make xìng fundamentally different from the contemporary nature concepts with which it is often equated. In particular, people’s xìng is not a near-equivalent of human nature or (in modern Chinese) of rénxìng 人性. (shrink)

This essay concerns the disputed nature of the telos in Stoicism and argues that Michel Foucault’s description of the Stoic telos plausibly constitutes an accurate characterization, despite the frequent criticism it has received and the fact that it apparently neglects the important role of nature or physics in Stoicism. To advance this claim, the essay draws upon a neglected set of observations made by Foucault in The Hermeneutics of the Subject, in which the telos is characterized in terms of the (...) elimination of stultitia, or “indecision.” Once we analyze indecision and its causes from the perspective of physics, we can see that indecision is incompatible with the Stoic telos as standardly described, for example as “life in accord with nature.” It can be concluded that the Foucault’s view is not only plausible but is in fact supported by Stoic physics. The essay concludes with a discussion of some possible objections. (shrink)