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.

Quantifier variance is a well-known view in contemporary metaontology, but it remains very widely misunderstood by critics. Here we briefly and clearly explain the metasemantics of quantifier variance and distinguish between modest and strong forms of variance (Section I), explain some key applications (Section II), clear up some misunderstandings and address objections (Section III), and point the way toward future directions of quantifier-variance-related research (Section IV).

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...

In this paper, we defend two main claims. The first is a moderate claim: we have a negative duty to not use binary gender-specific pronouns he or she to refer to genderqueer individuals. We defend this with an argument by analogy. It was gravely wrong for Mark Latham to refer to Catherine McGregor, a transgender woman, using the pronoun he; we argue that such cases of misgendering are morally analogous to referring to Angel Haze, who identifies as genderqueer, as he (...) or she. The second is a radical claim: we have a negative duty to not use any gender-specific pronouns to refer to anyone, regardless of their gender identity. We offer three arguments in favor of this claim (which appeal to concerns about inegalitarianism and risk, invasions of privacy, and reinforcing essentialist ideologies). We also show why the radical claim is compatible with the moderate claim. Before concluding, we examine common concerns about incorporating either they or a neologism such as ze as a third-person singular gender-neutral pronoun. These concerns, we argue, do not provide sufficient reason to reject either the moderate or radical claim. (shrink)

Based on a framework that distinguishes several types, roles and functions of values in science, we discuss legitimate applications of values in the validation of computer simulations. We argue that, first, epistemic valuesEpistemic values, such as empirical accuracyAccuracy and coherence with background knowledgeBackground knowledge, have the role to assess the credibilityCredibility of simulation results, whereas, second, cognitive valuesCognitive values, such as comprehensiveness of a conceptual modelConceptual model or easy handling of a numerical model, have the role to assess the usefulness (...) of a model for investigating a hypothesisHypothesis. In both roles, values perform what we call first-order functions. In addition, cognitive values may also serve an auxiliary function by facilitating the assessment of credibility. As for a third type of values, i.e. social valuesSocial values, their legitimate role consists in specifying and weighing epistemic and cognitive values with respect to practical uses of a simulation, which is considered a second-order function. Rational intersubjective agreementAgreement on how to specify and weigh the different values is supposed to ensure objectivity in simulation validation. (shrink)

We live in a world saturated in both racial and gendered divisions. Our focus is on one place where attitudes about these divisions diverge: language. We suspect most everyone would be horrified at the idea of adding race-specific pronouns, honorifics, generic terms, and so on to English. And yet gender-specific terms of the same sort are widely accepted and endorsed. We think this asymmetry cannot withstand scrutiny. We provide three considerations against incorporating additional race-specific terms into English, and argue that (...) these considerations also support eliminating the analogous gender-specific terms. With respect to these parts of speech, English should be no more gender-specific than it already is race-specific. (shrink)

Gender classifications often are controversial. These controversies typically focus on whether gender classifications align with facts about gender kind membership: Could someone really be nonbinary? Is Chris Mosier really a man? I think this is a bad approach. Consider the possibility of ontological oppression, which arises when social kinds operating in a context unjustly constrain the behaviors, concepts, or affect of certain groups. Gender kinds operating in dominant contexts, I argue, oppress trans and nonbinary persons in this way: they marginalize (...) trans men and women, and exclude nonbinary persons. As a result, facts about membership in dominant gender kinds should not settle gender classification practices. (shrink)

What’s important about ‘coming out’? Why do we wear business suits or Star Trek pins? Part of the answer, we think, has to do with what we call agential identity. Social metaphysics has given us tools for understanding what it is to be socially positioned as a member of a particular group and what it means to self-identify with a group. But there is little exploration of the general relationship between self-identity and social position. We take up this exploration, developing (...) an account of agential identity—the self-identities we make available to others. Agential identities are the bridge between what we take ourselves to be and what others take us to be. Understanding agential identity not only fills an important gap in the literature, but also helps us explain politically important phenomena concerning discrimination, malicious identities, passing, and code-switching. These phenomena, we argue, cannot be understood solely in terms of self-identity or social position. (shrink)

Alex Byrne’s article, “Are Women Adult Human Females?”, asks a question that Byrne treats as nearly rhetorical. Byrne’s answer is, ‘clearly, yes’. Moreover, Byrne claims, 'woman' is a biological category that does not admit of any interpretation as (also) a social category. It is important to respond to Byrne’s argument, but mostly because it is paradigmatic of a wider phenomenon. The slogan “women are adult human females” is a political slogan championed by anti-trans activists, appearing on billboards, pamphlets, and anti-trans (...) online forums. In this paper, I respond to Byrne’s argument, revealing significant problems with its background assumptions, content, and methodology. (shrink)

This book reflects on the variety of ways in which mourning affects political and social life. Through the narrative of the contributors, the book demonstrates how mourning is intertwined with politics and how politics involves a struggle over which losses and whose lives can, or should, be mourned.

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)