Say that the property Φ of a cardinal λ strongly implies the property Ψ. If and only if for every λ,Φ implies that Ψ and that for some λ′<λ,Ψ. Frequently in the hierarchy of large cardinal axioms, stronger axioms strongly imply weaker ones. Some strong implications are proved between axioms of the form “there is an elementary embedding j:Lα[Vλ+1]→Lα[Vλ+1] with ”.

Let a be a Kleene's ordinal notation of a nonzero computable ordinal. We give a sufficient condition on a, so that for every \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of two embedded sets, i.e., two sets A, B, with A properly contained in B, the Rogers semilattice of the family is infinite. This condition is satisfied by every notation of ω; moreover every nonzero computable ordinal that is not sum of any two smaller ordinals has a notation that satisfies this condition. On (...) the other hand, we also give a sufficient condition on a, that yields that there is a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of two embedded sets, whose Rogers semilattice consists of exactly one element; this condition is satisfied by all notations of every successor ordinal bigger than 1, and by all notations of the ordinal ω + ω; moreover every computable ordinal that is sum of two smaller ordinals has a notation that satisfies this condition. We also show that for every nonzero n ∈ ω, or n = ω, and every notation a of a nonzero ordinal there exists a \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma ^{-1}_a$\end{document}‐computable family of cardinality n, whose Rogers semilattice consists of exactly one element. (shrink)

In spite of feminist recognition that hierarchical organizations are an important location of male dominance, most feminists writing about organizations assume that organizational structure is gender neutral. This article argues that organizational structure is not gender neutral; on the contrary, assumptions about gender underlie the documents and contracts used to construct organizations and to provide the commonsense ground for theorizing about them. Their gendered nature is partly masked through obscuring the embodied nature of work.jobs and hierarchies, common concepts in organizational (...) thinking, assume a disembodies and universal worker. This worker is actually a man; men's bodies, sexuality, and relationships to procreation and paid work are subsumed in the image of the worker. Images of men's bodies and masculinity pervade organizational processes, marginalizing women and contributing to the maintenance of gender segregation in organizations. The positing of gender-neutral and disembodied organizational structures and work relations is part of the larger strategy of control in industrial capitalist societies, which, at least partly, are built upon a deeply embedded substructure of gender difference. (shrink)

We introduce a new hierarchy of computably enumerable degrees. This hierarchy is based on computable ordinal notations measuring complexity of approximation of${\rm{\Delta }}_2^0$functions. The hierarchy unifies and classifies the combinatorics of a number of diverse constructions in computability theory. It does so along the lines of the high degrees (Martin) and the array noncomputable degrees (Downey, Jockusch, and Stob). The hierarchy also gives a number of natural definability results in the c.e. degrees, including a definable antichain.

Intersectional feminist scholars emphasize how overlapping systems of oppression structure gender inequality, but in focusing on the gendered, classed, and racialized bases of stratification, many often overlook disability as an important social category in determining economic outcomes. This is a significant omission given that disability severely limits opportunities and contributes to cumulative disadvantage. We draw from feminist disability and intersectional theories to account for how disability intersects with gender, race, and education to produce economic insecurity. The findings from our analyses (...) of 2015 American Community Survey data provide strong empirical support for hierarchies of disadvantage, where women and racial minority groups with disabilities and less education experience the highest poverty levels, report the lowest total income, and have a greater reliance on sources outside the labor market for economic security. By taking disability into account, our study demonstrates how these multiple characteristics lead to overlapping oppressions that become embedded and reproduced within the larger social structure. (shrink)

We study the computational content of various theorems with reverse mathematical strength around Arithmetical Transfinite Recursion (ATR_0) from the point of view of computability-theoretic reducibilities, in particular Weihrauch reducibility. Our main result states that it is equally hard to construct an embedding between two given well-orderings, as it is to construct a Turing jump hierarchy on a given well-ordering. This answers a question of Marcone. We obtain a similar result for Fraïssé's conjecture restricted to well-orderings.

We show that any partial order with a Σ3 enumeration can be effectively embedded into any partial order obtained by imposing a strong reducibility such as ≤tt on the c. e. sets. As a consequence, we obtain that the partial orders that result from imposing a strong reducibility on the sets in a level of the Ershov hiearchy below ω + 1 are co-embeddable.

Let CH be the class of compacta (i.e., compact Hausdorff spaces), with BS the subclass of Boolean spaces. For each ordinal α and pair $\langle K,L\rangle$ of subclasses of CH, we define Lev ≥α K,L), the class of maps of level at least α from spaces in K to spaces in L, in such a way that, for finite α, Lev ≥α (BS,BS) consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of quantifier rank (...) α. Maps of level ≥ 0 are just the continuous surjections, and the maps of level ≥ 1 are the co-existential maps introduced in [8]. Co-elementary maps are of level ≥α for all ordinals α; of course in the Boolean context, the co-elementary maps coincide with the maps of level ≥ω. The results of this paper include: (i) every map of level ≥ω is co-elementary; (ii) the limit maps of an ω-indexed inverse system of maps of level ≥α are also of level ≥α; and (iii) if K is a co-elementary class, k ≥ k (K,K) = Lev ≥ k+1 (K,K), then Lev ≥ k (K,K) = Lev ≥ω (K,K). A space X ∈ K is co-existentially closed in K if Lev ≥ 0 (K, X) = Lev ≥ 1 (K, X). Adapting the technique of "adding roots," by which one builds algebraically closed extensions of fields (and, more generally, existentially closed extensions of models of universal-existential theories), we showed in [8] that every infinite member of a co-inductive co-elementary class (such as CH itself, BS, or the class CON of continua) is a continuous image of a space of the same weight that is co-existentially closed in that class. We show here that every compactum that is co-existentially closed in CON (a co-existentially closed continuum) is both indecomposable and of covering dimension one. (shrink)

In 1970, K. Kunen, working in the context of Kelley–Morse set theory, showed that the existence of a nontrivial elementary embedding j:V→V is inconsistent. In this paper, we give a finer analysis of the implications of his result for embeddings V→V relative to models of ZFC. We do this by working in the extended language , using as axioms all the usual axioms of ZFC , along with an axiom schema that asserts that j is a nontrivial elementary embedding. Without (...) additional axiomatic assumptions on j, we show that that the resulting theory is weaker than an ω-Erdös cardinal, but stronger than n-ineffables. We show that natural models of ZFC+BTEE give rise to Schindler’s remarkable cardinals. The approach to inconsistency from ZFC+BTEE forks into two paths: extensions of ZFC+BTEE+Cofinal Axiom and ZFC+BTEE+¬Cofinal Axiom, where Cofinal Axiom asserts that the critical sequence is cofinal in the ordinals. We describe near-minimal inconsistent extensions of each of these theories. The path toward inconsistency from ZFC+BTEE+¬Cofinal Axiom is paved with a sequence of theories of increasing large cardinal strength. Indeed, the extensions of the theory ZFC +“j is a nontrivial elementary embedding” form a hierarchy of axioms, ranging in strength from Con to the existence of a cardinal that is super-n-huge for every n, to inconsistency. This hierarchy is parallel to the usual hierarchy of large cardinal axioms, and can be used in the same way. We also isolate several intermediate-strength axioms which, when added to ZFC+BTEE, produce theories having strengths in the vicinity of a measurable cardinal of high Mitchell order, a strong cardinal, ω Woodin cardinals, and n-huge cardinals. We also determine precisely which combinations of axioms, of the form result in inconsistency. (shrink)

This paper establishes a conjecture of Steel [7] regarding the structure of elementary embeddings from a level of the cumulative hierarchy into itself. Steel’s question is related to the Mitchell order on these embeddings, studied in [5] and [7]. Although this order is known to be illfounded, Steel conjectured that it has certain large wellfounded suborders, which is what we establish. The proof relies on a simple and general analysis of the much broader class of extender embeddings and a (...) variant of the Mitchell order called the internal relation. (shrink)

We study large cardinal properties associated with Ramseyness in which homogeneous sets are demanded to satisfy various transfinite degrees of indescribability. Sharpe and Welch [25], and independently Bagaria [1], extended the notion of $\Pi ^1_n$ -indescribability where $n<\omega $ to that of $\Pi ^1_\xi $ -indescribability where $\xi \geq \omega $. By iterating Feng’s Ramsey operator [12] on the various $\Pi ^1_\xi $ -indescribability ideals, we obtain new large cardinal hierarchies and corresponding nonlinear increasing hierarchies of normal ideals. We provide (...) a complete account of the containment relationships between the resulting ideals and show that the corresponding large cardinal properties yield a strict linear refinement of Feng’s original Ramsey hierarchy. We isolate Ramsey properties which provide strictly increasing hierarchies between Feng’s $\Pi _\alpha $ -Ramsey and $\Pi _{\alpha +1}$ -Ramsey cardinals for all odd $\alpha <\omega $ and for all $\omega \leq \alpha <\kappa $. We also show that, given any ordinals $\beta _0,\beta _1<\kappa $ the increasing chains of ideals obtained by iterating the Ramsey operator on the $\Pi ^1_{\beta _0}$ -indescribability ideal and the $\Pi ^1_{\beta _1}$ -indescribability ideal respectively, are eventually equal; moreover, we identify the least degree of Ramseyness at which this equality occurs. As an application of our results we show that one can characterize our new large cardinal notions and the corresponding ideals in terms of generic elementary embeddings; as a special case this yields generic embedding characterizations of $\Pi ^1_\xi $ -indescribability and Ramseyness. (shrink)

Mechanism realists assert the existence of mechanisms as objective structures in the world, but their exact metaphysical commitments are unclear. We introduce Local Hierarchy Realism (LHR) as a substantive and plausible form of mechanism realism. The limits of LHR reveal a deep tension between two aspects of mechanists’ explanatory strategy. Functional decomposition identifies locally relevant entities and activities, while these same entities and activities are also embedded in a nested hierarchy of levels. In principle, a functional decomposition (...) may identify entities engaging in causal interactions that crosscut the hierarchical structure of composition relations, violating the mechanist’s injunction against interlevel causation. We argue that this possibility is realized in the example of ephaptic coupling, a subsidiary process of neural computation that crosscuts the hierarchy derived from synaptic transmission. These considerations undermine the plausibility of LHR as a general view, yet LHR has the advantages that (i) its metaphysical implications are precisely stateable; (ii) the structure it identifies is not reducible to mere aggregate causation; and (iii) it clearly satisfies intuitive and informal definitions of mechanism. We conclude by assessing the prospects for a form of mechanism realism weaker than LHR that nevertheless satisfies all three of these requirements. (shrink)

Mechanism realists assert the existence of mechanisms as objective structures in the world, but their exact metaphysical commitments are unclear. We introduce Local Hierarchy Realism (LHR) as a substantive and plausible form of mechanism realism. The limits of LHR reveal a deep tension between two aspects of mechanists’ explanatory strategy. Functional decomposition identifies locally relevant entities and activities, while these same entities and activities are also embedded in a nested hierarchy of levels. In principle, a functional decomposition (...) may identify entities engaging in causal interactions that crosscut the hierarchical structure of composition relations, violating the mechanist’s injunction against interlevel causation. We argue that this possibility is realized in the example of ephaptic coupling, a subsidiary process of neural computation that crosscuts the hierarchy derived from synaptic transmission. These considerations undermine the plausibility of LHR as a general view, yet LHR has the advantages that (i) its metaphysical implications are precisely stateable; (ii) the structure it identifies is not reducible to mere aggregate causation; and (iii) it clearly satisfies intuitive and informal definitions of mechanism. We conclude by assessing the prospects for a form of mechanism realism weaker than LHR that nevertheless satisfies all three of these requirements. (shrink)

We consider iterations of general elementary embeddings and, using this notion, point out helices of consistency-wise implications between large large cardinals.Up to now, large cardinal properties have been considered as properties which cannot be accessed by any weaker properties and it has been known that, with respect to this relation, they form a proper hierarchy. The helices we point out significantly change this situation: the same sequence of large cardinal properties occurs repeatedly, changing only the parameters.As results of our (...) investigation of this helical structure, we have new characterizations of extendible cardinals and of Vopěnkan cardinals in terms of elementary embeddings from the universe V, a new large large cardinal property which looks like Shelahness properly between Vopěnkaness and almost hugeness, and a more direct relation between Vopěnkaness and Woodinness than already known.We also consider limits of the helices and relations with the axioms I1–I3. (shrink)

We prove that the existential theory of the Turing degrees, in the language with Turing reduction, 0, and unary relations for the classes in the generalized high/low hierarchy, is decidable. We also show that every finite poset labeled with elements of (where is the partition of induced by the generalized high/low hierarchy) can be embedded in preserving the labels. Note that no condition is imposed on the labels.

Dougherty, R., Critical points in an algebra of elementary embeddings, Annals of Pure and Applied Logic 65 211-241.Given two elementary embeddings from the collection of sets of rank less than λ to itself, one can combine them to obtain another such embedding in two ways: by composition, and by applying one to the other. Hence, a single such nontrivial embedding j generates an algebra of embeddings via these two operations, which satisfies certain laws . Laver has shown, among other things, (...) that this algebra is free on one generator with respect to these laws.The set of critical points of members of this algebra is the subject of this paper. This set contains the critical point κ0 of j, as well as all of the other ordinals κn in the critical sequence of j ). But the set includes many other ordinals as well. The main result of this paper is that the number of critical points below κn grows so quickly with n that it dominates any primitive recursive function. In fact, it grows faster than the Ackermann function, and even faster than a slow iterate of the Ackermann function. Further results show that, even just below κ4, one can find so many critical points that the number is only expressible using fast-growing hierarchies of iterated functions. (shrink)

We investigate the expressive power of fragments of first-order logic that are defined in terms of prefixes. The main result establishes a strict hierarchy among these fragments over the signature consisting of a single binary relation. It implies that for each prefix p, there is a sentence in prenex normal form with prefix p, over a single binary relation, such that for all sentences θ in prenex normal form, if θ is equivalent to , then p can be (...) class='Hi'>embedded in the prefix of θ. This strengthens a theorem of Walkoe. (shrink)

This article seeks to analyse the rise of ‘resilience ethics’, in terms of the shift in ethical approaches away from the hierarchical liberal internationalist constructions of the 1990s and towards broader and more inclusive understandings of ethical responsibility for global problems. This shift in ethical attention away from the formal international politics of inter-state relations and towards the unintended consequences of both institutional structures and the informal market choices of individuals has diversified understandings of global ethical responsibilities. It is argued (...) that the recasting of ethical responsibility in the increasingly sociological terms of unintended and indirect consequences of socio-material embeddedness constructs new ethical differentials and hierarchies of responsibility. These framings have facilitated new policy practices, recasting interventionist policy-making in terms of the growing self-awareness and reflexivity of Western actors, reframing ethical foreign policy as starting with the choices of individual citizens, and, at the same time, operating to reify the relations of the market. (shrink)

I prove that there is a recursive function T that does the following: Let X be transitive and rudimentarily closed, and let X ′ be the closure of X ∪ {X } under rudimentary functions. Given a Σ0-formula φ and a code c for a rudimentary function f, T is a Σω-formula such that for any equation image ∈ X, X ′ ⊧ φ [f ] iff X ⊧ T [equation image]. I make this precise and show relativized versions of (...) this. As an application, I prove that under certain conditions, if Y is the Σω extender ultrapower of X with respect to some extender F that also is an extender on X ′, then the closure of Y ∪ {Y } under rudimentary functions is the Σ0 extender ultrapower of X′ with respect to F, and the ultrapower embeddings agree on X. (shrink)

By convening a multidisciplinary team(the Monitoring Team) that included farmers,university and agency researchers, andnon-profit staff; a small group of farmers insoutheast Minnesota, U.S.A., bolstered thelegitimacy of the sustainable agriculturemovement. Through the experience of forming ateam and working with individuals who operatedwithin the mainstream knowledge paradigm,farmers gained validation of their knowledgeabout farming, while researchers came to valuealternative knowledge systems. In the contextof a socially embedded movement, farmers wereempowered by sharing their knowledge withresearchers, and ultimately contributed to thesustainable agriculture movement by (...) challengingtraditional patterns of knowledge exchange. (shrink)

This article focuses on the concept of common sense in order to shed new light on the radical and pluralist democracy developed by Ernesto Laclau and Chantal Mouffe in Hegemony and Socialist Strategy. It is argued that their move via Antonio Gramsci away from both Marxism and traditional liberal democracy cannot be fully understood without reference to the role common sense plays in it. Focusing on common sense reveals crucial aspects of the relations between intellectuals and ordinary people in Laclau (...) and Mouffe's political theory, and how they deal with problems of embedded hierarchy, characteristic of modern political thought. First I reconstruct Gramsci's concept of common sense, trace its origins and analyze its egalitarian aspect. Next I present the concept's novelty with regard to other modern political theories and their presuppositions, and discuss Gramsci's eventual commitment to the latter. I then turn to discussing both the reasons why Laclau and Mouffe cannot accept the traditional concept of common sense and why they assert that constructing some sort of common sense is necessary for the formation of democratic hegemony. Finally, I argue that Laclau and Mouffe can be seen as having dealt with the problems that result from the need for common sense and the inability to accept it only if we interpret their concept of common sense as a development of Gramsci's, namely as a heterogeneous matrix of various incompatible "common senses". (shrink)

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. Motivated by results of Bagaria, Magidor and Väänänen, we study characterizations of large cardinal properties through reflection principles for classes of structures. More specifically, we aim to characterize notions from the lower end of the large cardinal hierarchy through the principle [math] introduced by Bagaria and Väänänen. Our results isolate a narrow interval in the large cardinal hierarchy that is bounded from below by total indescribability and from above (...) by subtleness, and contains all large cardinals that can be characterized through the validity of the principle [math] for all classes of structures defined by formulas in a fixed level of the Lévy hierarchy. Moreover, it turns out that no property that can be characterized through this principle can provably imply strong inaccessibility. The proofs of these results rely heavily on the notion of shrewd cardinals, introduced by Rathjen in a proof-theoretic context, and embedding characterizations of these cardinals that resembles Magidor’s classical characterization of supercompactness. In addition, we show that several important weak large cardinal properties, like weak inaccessibility, weak Mahloness or weak [math]-indescribability, can be canonically characterized through localized versions of the principle [math]. Finally, the techniques developed in the proofs of these characterizations also allow us to show that Hamkin’s weakly compact embedding property is equivalent to Lévy’s notion of weak [math]-indescribability. (shrink)

Attribution of mental states is fundamental to our engagement with fiction. Crucially, its social content depends on mental states recursively “embedded” within each other; for instance, when a person doesn’t want other people to know about her intentions. Given that some characters seem to be consistently capable of embedding mental states on a higher level than others, this essay reviews factors that may influence authors’ constructions of such mindreading hierarchies as well as their reversals. The argument focuses on the (...) reversal scenes in films Goodbye Lenin, The Lives of Others, and Never Look Away, and on prosocial emotions evoked by their depiction of a more equitable distribution of a presumably valuable and scarce resource, that is, access to other people's minds. (shrink)

We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(\tau )$ be the set of countable structures with universe $\omega $ in vocabulary $\tau $ topologized by the Scott topology. We show that an invariant set $X\subseteq Mod(\tau )$ is $\Pi ^0_\alpha $ in the Borel hierarchy of this topology if and only if it is definable by a $\Pi ^p_\alpha $ -formula, a positive $\Pi ^0_\alpha $ formula in the infinitary logic $L_{\omega _1\omega }$. (...) As a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $\mathcal {K}$ be positively computably embeddable in $\mathcal {K}'$ by $\Phi $, then for every $\Pi ^p_\alpha $ formula $\xi $ in the vocabulary of $\mathcal {K}'$ there is a $\Pi ^p_\alpha $ formula $\xi ^{*}$ in the vocabulary of $\mathcal {K}$ such that for all $\mathcal {A}\in \mathcal {K}$, $\mathcal {A}\models \xi ^{*}$ if and only if $\Phi (\mathcal {A})\models \xi $. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings. (shrink)

An area in the theory of action that has received little attention is how mental agency and world-directed agency interact. The purpose of the present contribution is to clarify the rational conditions of such interaction, through an analysis of the central case of acceptance. There are several problems with the literature about acceptance. First, it remains unclear how a context of acceptance is to be construed. Second, the possibility of conjoining, in acceptance, an epistemic component, which is essentially mind-to-world, and (...) a utility component, which requires a world-to-mind direction of fit, is merely posited rather than derived from the rational structure of acceptance. Finally, the norm of acceptance is generally seen as related to truth, which turns out to be inapplicable in a number of cases. We will argue, first, that the specific context-dependence of acceptances is derived from their being mental actions, each embedded in a complex hierarchy of acceptances composing, together, a planning sequence. Second, that acceptances come in several varieties, corresponding to the specific epistemic norm(s) that constitute them. The selection of a particular norm for accepting answers to considerations of utility – to the association of an epistemic goal with an encompassing world-directed action. Once a type of acceptance is selected, however, the epistemic norm constitutive for that acceptance strictly applies. Third, we argue that context-dependence superimposes a decision criterion on the output of the initial epistemic acceptance. Strategic acceptance is regulated by instrumental norms of expected utility, which may rationally lead an agent to screen off her initial epistemic acceptance. (shrink)

Recognition that biological systems are stabilized far from equilibrium by self-organizing, informed, autocatalytic cycles and structures that dissipate unusable energy and matter has led to recent attempts to reformulate evolutionary theory. We hold that such insights are consistent with the broad development of the Darwinian Tradition and with the concept of natural selection. Biological systems are selected that re not only more efficient than competitors but also enhance the integrity of the web of energetic relations in which they are (...) class='Hi'>embedded. But the expansion of the informational phase space, upon which selection acts, is also guaranteed by the properties of open informational-energetic systems. This provides a directionality and irreversibility to evolutionary processes that are not reflected in current theory.For this thermodynamically-based program to progress, we believe that biological information should not be treated in isolation from energy flows, and that the ecological perspective must be given descriptive and explanatory primacy. Levels of the ecological hierarchy are relational parts of ecological systems in which there are stable, informed patterns of energy flow and entropic dissipation. Isomorphies between developmental patterns and ecological succession are revealing because they suggest that much of the encoded metabolic information in biological systems is internalized ecological information. The geneological hierarchy, to the extent that its information content reflects internalized ecological information, can therefore be redescribed as an ecological hierarchy. (shrink)

I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical point of an elementary embedding j:V→M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${j: V \to M}$$\end{document} such that M is closed under sequences of length sup{j|f:κ→κ}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sup\{{j\,|\,f: \kappa \to \kappa}\}}$$\end{document}. Some of the (...) other cardinals analyzed include the super-high-jump cardinals, almost-high-jump cardinals, Shelah-for-supercompactness cardinals, Woodin-for-supercompactness cardinals, Vopěnka cardinals, hypercompact cardinals, and enhanced supercompact cardinals. I organize these cardinals in terms of consistency strength and implicational strength. I also analyze the superstrong cardinals, which are weaker than supercompact cardinals but are related to high-jump cardinals. Among the results, I highlight the following. Vopěnka cardinals are the same as Woodin-for-supercompactness cardinals.There are no excessively hypercompact cardinals.Furthermore, I prove some results relating high-jump cardinals to forcing, as well as analyzing Laver functions for super-high-jump cardinals. (shrink)

Philosophical discussions of mechanisms and mechanistic explanation have often been framed by contrast to laws and deductive-nomological explanation. A more adequate conception of lawfulness and nomological necessity, emphasizing the role of modal considerations in scientific reasoning, circumvents such contrasts and enhances understanding of mechanisms and their scientific significance. The first part of the paper sketches this conception of lawfulness, drawing upon Haugeland, Lange, and Rouse. This conception emphasizes the role of lawful stability under relevant counterfactual suppositions in scientific reasoning across (...) the sciences, in place of traditional conceptions of law that are primarily confined to the physical sciences. It also extends lawful stability beyond verbally or mathematically expressed law-statements, to encompass other ways of conjoining patterns in the world with scientific pattern recognition. The remainder of this paper shows how and why mechanisms constructively exemplify this conception of lawfulness in scientific practice: • Mechanisms are robust, counterfactually stable and inductively projectible patterns, even though they are not exceptionless “laws of nature”. • Mechanistic explanations often take non-verbal forms, which consequently resist philosophical inclinations to semantic ascent, but understanding lawfulness in terms of counterfactually stable pattern recognition accounts for these ways in which scientific understanding outruns the expressive capacities of natural languages; • Mechanisms are sometimes characterized as real patterns in the world, and sometimes as epistemic representations; understanding mechanisms as modal patterns shows why both conceptions are needed, as mutually supportive. • Mechanisms are typically open-ended, and only partially specified, in ways open to and directive toward further articulation and revision. Understanding mechanisms as modal patterns incorporates this aspect of mechanistic understanding within a broader conception of scientific understanding as embedded in research practice, rather than in bodies of knowledge extracted from it. • Mechanistic explanation has often been placed on the causal side of an opposition between causal and nomological explanation, but understanding mechanisms as modal patterns helps overcome that opposition, and contributes to a pluralist conception of causal relations and their characteristic forms of counterfactual invariance. • The recognition of mechanisms as modal patterns allows for a new way to think about the relations among distinct levels of a mechanistic hierarchy, and the broader scientific significance of mechanistic understanding. (shrink)

We establish the hierarchy among twelve equivalence relations on the class of relational structures: the equality, the isomorphism, the equimorphism, the full relation, four similarities of structures induced by similarities of their self-embedding monoids and intersections of these equivalence relations. In particular, fixing a language L and a cardinal κ, we consider the interplay between the restrictions of these similarities to the class ModL of all L-structures of size κ. It turns out that, concerning the number of different similarities (...) and the shape of the corresponding Hasse diagram, the class of all structures naturally splits into three parts: finite structures, infinite structures of unary languages, and infinite structures of non-unary languages. (shrink)

On 27 March 2014, Loreal Tsingine, a 27-year-old Navajo woman was shot and killed by Austin Shipley, a white male police officer, also 27 years old, who said he was trying to apprehend her for a suspected shoplifting. Shipley was never charged, and the Department of Justice declined to investigate the Winslow police on the matter. This article explores Shipley’s killing of Loreal Tsingine and the police investigation of the shooting as quotidian events in settler colonial states. Police shootings of (...) Indigenous people and the legal response to police use of force (along with everyday settler violence) are a part of the racial terror that is a central part of settler colonialism. Both the shooting and the official narratives of it as a justifiable use of force reveal the psychic and material underpinnings of a settler state, a state that continually imagines and consolidates itself as a community of whites imperiled by Indians among others. White settler violence directed at those imagined as threats lives just beneath the surface of everyday settler life, and importantly, flows through institutions such as policing, embedding itself in everyday professional routines. The extractive relations that are the basis of settler colonialism require and produce white subjects for whom Indigenous lands and bodies are the resource for white identity; policing is one site where white men and women (as well as those aspiring to whiteness), can enact racial hierarchy on behalf of the colonial state with impunity. (shrink)

What are the semantic representations that underlie language production? We use structural priming to distinguish between two competing theories. Thematic roles define semantic structure in terms of atomic units that specify event participants and are ordered with respect to each other through a hierarchy of roles. Event structures instead instantiate semantic structure as embedded sub‐predicates that impose an order on verbal arguments based on their relative positioning in these embeddings. Across two experiments, we found that priming for datives (...) depended on the degree of overlap in event structures. Specifically, while all dative structures showed priming, due to common syntax, there was a boost for compositional datives priming other compositional datives. Here, the two syntactic forms have distinct event structures. In contrast, there was no boost in priming for dative light verbs, where the two forms map onto a single event representation. On the thematic roles hypothesis, we would have expected a similar degree of priming for the two cases. Thus, our results support event structural approaches to semantic representation and not thematic roles. (shrink)

TEST is a novel taxonomy of knowledge representations based on three distinct hierarchically organized representational features: Tropism, Embodiment, and Situatedness. Tropic representational features reflect constraints of the physical world on the agent's ability to form, reactivate, and enrich embodied (i.e., resulting from the agent's bodily constraints) conceptual representations embedded in situated contexts. The proposed hierarchy entails that representations can, in principle, have tropic features without necessarily having situated and/or embodied features. On the other hand, representations that are situated (...) and/or embodied are likely to be simultaneously tropic. Hence, although we propose tropism as the most general term, the hierarchical relationship between embodiment and situatedness is more on a par, such that the dominance of one component over the other relies on the distinction between offline storage versus online generation as well as on representation-specific properties. (shrink)

For each natural number n, let C(n) be the closed and unbounded proper class of ordinals α such that Vα is a Σn elementary substructure of V. We say that κ is a C(n)-cardinal if it is the critical point of an elementary embedding j : V → M, M transitive, with j(κ) in C(n). By analyzing the notion of C(n)-cardinal at various levels of the usual hierarchy of large cardinal principles we show that, starting at the level of (...) superstrong cardinals and up to the level of rank-into-rank embeddings, C(n)-cardinals form a much finer hierarchy. The naturalness of the notion of C(n)-cardinal is exemplified by showing that the existence of C(n)-extendible cardinals is equivalent to simple reflection principles for classes of structures, which generalize the notions of supercompact and extendible cardinals. Moreover, building on results of Bagaria et al. (2010), we give new characterizations of Vopeňka’s Principle in terms of C(n)-extendible cardinals. (shrink)

One of the numerous characterizations of a Ramsey cardinal κ involves the existence of certain types of elementary embeddings for transitive sets of size κ satisfying a large fragment of ZFC. We introduce new large cardinal axioms generalizing the Ramsey elementary embeddings characterization and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. These new axioms serve to further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent (...) with V = L. (shrink)

This book offers an analysis of experimental psychology that is embedded in a general understanding of human behavior. It provides methodological self-awareness for researchers who study and use the experimental method in psychology. The book critically reviews key research areas, examining their scope, limits, ambiguities, and implicit theoretical commitments. Topics featured in this text include: Methods of critique in experimental research Goal hierarchies and organization of a task Rule-following and rule-breaking behavior Sense of agency Free-choice tasks Mind wandering Experimental (...) Psychology and Human Agency will be of interest to researchers and undergraduate and graduate students in the fields of experimental psychology, cognitive psychology, theoretical psychology, and critical psychology, as well as various philosophical disciplines. (shrink)

A number of general points behind the story of this paper may be worth setting out separately, now that we have come to the end.There is perhaps one obvious omission to be addressed right away. Although the word “information” has occurred throughout this paper, it must have struck the reader that we have had nothing to say on what information is. In this respect, our theories may be like those in physics: which do not explain what “energy” is (a notion (...) which seems quite similar to “information” in several ways), but only give some basic laws about its behaviour and transmission.The eventual recommendation made here has been to use a broad type-theoretic framework for studying various more classical and more dynamic notions of proposition in their interaction. This is not quite the viewpoint advocated by many current authors in the area, who argue for a whole-sale switch from a ‘static’ to a ‘dynamic’ perspective on propositions. This is not the place, however, to survey the conceptual arguments for and against such a more radical move.This still leaves many questions about possible reductions from one perspective to another. For instance, it would seem that classical systems ought to serve as a ‘limiting case’, which should still be valid after procedural details of some cognitive process have been forgotten. There are various ways of implementing the desired correspondence: e.g. by considering extreme cases with ⫅ equal to identity, or, in the pure relational algebra framework by considering only pairs (x, x). What we shall want then are reductions of dynamic logics, in those special cases, to classical logic. But perhaps also, more sophisticated views are possible. How do we take a piece of ‘dynamic’ prose, remove control instructions and the like, and obtain a piece of ‘classical’ text, suitable for inference ‘in the light of eternity’?There is also a more technical side to the matter of ‘reduction’. By now, Logic has reached such a state of ‘inter-translatability’ that almost all known variant logics can be embedded into each other, via suitable translations. In particular, once an adequate semantic has been given for a new system, this usually induces an embedding into standard logic: as we know, e.g., for the case of Modal Logic. Like-wise, all systems of dynamic interpretation or inference proposed so far admit of direct embedding into an ordinary ‘static’ predicate logic having explicit transition predicates (cf. van Benthem 1988b). Thus, our moral is this. The issue is not whether the new systems of information structure or processing are essentially beyond the expressive resources of traditional logical systems: for, they are not. The issue is rather which interesting phenomena and questions will be put into the right focus by them.The next broad issue concerns the specific use of the perspective proposed here, vis-à-vis concrete proposals for information-oriented or dynamic semantics. The general strategy advocated here is to locate some suitable base calculus and then consider which ‘extras’ are peculiar to the proposal. For instance, this is the spirit in which modal S4 would be a base logic of information models, and intuitionistic logicthe special theory devoting itself to upward persistent propositions. Or, with the examples in Section 4.1, the underlying base logic is our relational algebra, whereas, say, ordinary updates then impose special properties, such as ‘idempotence’: $$xRy \Rightarrow yRy$$ Does this kind of application presuppose the existence of one distinguished base logic, of which all others are extensions? This would be attractive-and some form of relational algebra or linear logic might be a reasonable candidate. Nevertheless, the enterprise does not rest on this outcome. What matters is an increased sensitivity to the ‘landscape’ of dynamic logics, just as with the ‘Categorial Hierarchy’ in Categorial Grammar (cf. van Benthem 1989a, 1991) where the family of logics with their interconnections seems more important than any specific patriarch.Finally, perhaps the most important issue in the new framework is the possibility of new kinds of questions arising precisely because of its differences from standard logic. Notably, given the option of regarding propositions as programs, it will be of interest to consider systematically which major questions about programming languages now make sense inside logic too.EXAMPLE. Correctness. When do we have $$\left[\kern-0.15em\left[ \pi \right]\kern-0.15em\right](\left[\kern-0.15em\left[ A \right]\kern-0.15em\right]) \subseteq \left[\kern-0.15em\left[ B \right]\kern-0.15em\right]$$ for (s, t) propositions A, B and a dynamic (s, (s, t)) proposition π?Program Synthesis. Which dynamic proposition will take us from an information state satisfying A to one satisfying B? (This question needs refinement, lest there be trivial answers.)Determinism. Which propositions as programs are deterministic, in the sense of defining single-valued functions from states to states?Querying. What does it mean to ask for information in the present setting? (Again, individual types referring to e will be crucial here.)This is not merely an agenda for wishful thinking. Within Logic, there are various ways of introducing such concerns into semantics, especially, using tools from Automata Theory. (See van Benthem 1989c for further discussion of such computational perspectives in ‘cognitive programming’.)At least if one believes that ‘dynamics’ is of the essence in cognition (rather than a mere interfacing problem between the halls of eternal truth and the noisy streets of reality), the true test for the present enterprise is the development of a significant new research program not merely copying the questions of old. (shrink)

ABSTRACTThe pragmatist reform and opening-up in 1978 has revolutionised the way China communicates internally and engages with the outside world. Firmly embedded within this broader historical context, the interpreter-mediated and televised Premier-Meets-the-Press conferences are a high-profile institutional event in China. At this discursive event, the Chinese premier – ranked second in China’s political hierarchy – is put in the international media limelight, answering journalists’ questions on a range of topics. The section involving the interpreters’ rendering of journalists’ questions (...) is triadic and dynamic and represents a particularly interesting site of ideological contestation, which can be conceptualised profitably using Bakhtin’s concept dialogised heteroglossia. Drawing on a corpus containing 20 years’ press conference data between 1998 and 2017, this CDA study interrogates the interpreters’ agency, particularly in constructing the Chinese government’s image when rendering journalists’ questions. Despite the commonplace assumptions of interpreters being impartial with little agency, the government-affiliated interpreters are found to actively engage in facework and image construction. This leads to a discursive pattern described in Van Dijk’s ideological square, which involves further emphasising and foregrounding the positive elements yet de-emphasising and mitigating the negative elements about Beijing. (shrink)

A theory of the transfinite Tarskian hierarchy of languages is outlined and compared to a notion of partial truth by Kripke. It is shown that the hierarchy can be embedded into Kripke's minimal fixed point model. From this results on the expressive power of both approaches are obtained.

Like most goods‐producing sectors in the West, modern health‐care systems have been profoundly changed by globalization and the neoliberal policies that attend it. Since the 1970s, the role of the welfare state has been considerably reduced; funding and management of health systems have been subjected to wave upon wave of reorganization and assimilated to the private sector. At the same time, neoliberal policy has imposed the notion of patient empowerment, thus turning patients into consumers of health. The literature on nursing (...) has accordingly reported on the significant repercussions on all aspects of the profession, from delivery of care and treatment, through training for new nurses, to legislated policy reforms regarding the role and responsibilities of modern nurses. In light of these developments, this paper analyses and theorizes about the way the injection of neoliberal policy is linked to and affects the practice of nursing. Drawing on a number of Nietzschean arguments, we begin with an exploration of the complex effects of neoliberalism, bureaucratization, and technocratization on the health system and the practice of nursing. Our main theoretical point here is that neoliberal policy engenders and promotes a neoliberal tide, which results in the conversion of the values that drive modern nursing practice. We then examine this tide in the light of Nietzsche's concepts. Starting with an analysis based on the ontology of the will to power, we show that nurses are dominated by neoliberal values embedded in technocratic and bureaucratic ideologies. Finally, we argue that the application of neoliberal policy constitutes a form of domestication from which one might potentially be freed through the Nietzschean concept of transvaluation of values. This transvaluation, as its freeing from some of the neoliberal tide, may be accomplished in accordance with a hierarchy of specific life‐affirming values for nursing culture and practice. (shrink)

The C (n)-cardinals were introduced recently by Bagaria and are strong forms of the usual large cardinals. For a wide range of large cardinal notions, Bagaria has shown that the consistency of the corresponding C (n)-versions follows from the existence of rank-into-rank elementary embeddings. In this article, we further study the C (n)-hierarchies of tall, strong, superstrong, supercompact, and extendible cardinals, giving some improved consistency bounds while, at the same time, addressing questions which had been left open. In addition, we (...) consider two cases which were not dealt with by Bagaria; namely, C (n)-Woodin and C (n)-strongly compact cardinals, for which we provide characterizations in terms of their ordinary counterparts. Finally, we give a brief account on the interaction of C (n)-cardinals with the forcing machinery. (shrink)

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish space and E a Borel equivalence relation on X, a classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c = c. To be of any value we would expect I and c to be (...) “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ fFf. Then if is an embedding of X/E into Y/F, which is “Borel”.Intuitively, E ≤BF might be interpreted in any one of the following ways: The classi.cation problem for E is simpler than that of F: any invariants for F work as well for E. One can classify E by using as invariants F-equivalence classes. The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F. (shrink)

The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form Vκ where κ is a strongly inaccessible cardinal and Vκ denotes the κth level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of inaccessible set axioms heavily depend on the context (...) in which they are embedded. The context here will be the theory CZF− of constructive Zermelo–Fraenkel set theory but without -Induction . Let INAC be the statement that for every set there is an inaccessible set containing it. CZF−+INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF−+INAC also has a realizability interpretation in type theory which gives its theorems a direct computational meaning. The main result presented here is that the proof theoretic ordinal of CZF−+INAC is a small ordinal known as the Feferman–Schütte ordinal Γ0. (shrink)

Fisher [10] and Baur [6] showed independently in the seventies that if T is a complete first-order theory extending the theory of modules, then the class of models of T with pure embeddings is stable. In [25, 2.12], it is asked if the same is true for any abstract elementary class $(K, \leq _p)$ such that K is a class of modules and $\leq _p$ is the pure submodule relation. In this paper we give some instances where this is true:Theorem.Assume (...) R is an associative ring with unity. Let $(K, \leq _p)$ be an AEC such that $K \subseteq R\text {-Mod}$ and K is closed under finite direct sums, then: •If K is closed under pure-injective envelopes, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda \geq \operatorname {LS}(\mathbf {K})$ such that $\lambda ^{|R| + \aleph _0}= \lambda $.•If K is closed under pure submodules and pure epimorphic images, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda $ such that $\lambda ^{|R| + \aleph _0}= \lambda $.•Assume R is Von Neumann regular. If $\mathbf {K}$ is closed under submodules and has arbitrarily large models, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda $ such that $\lambda ^{|R| + \aleph _0}= \lambda $.As an application of these results we give new characterizations of noetherian rings, pure-semisimple rings, Dedekind domains, and fields via superstability. Moreover, we show how these results can be used to show a link between being good in the stability hierarchy and being good in the axiomatizability hierarchy.Another application is the existence of universal models with respect to pure embeddings in several classes of modules. Among them, the class of flat modules and the class of $\mathfrak {s}$ -torsion modules. (shrink)

This paper presents an ethical framework designed to support the development of critical data literacy for research methods courses and data training programmes in higher education. The framework we present draws upon our reviews of literature, course syllabi and existing frameworks on data ethics. For this research we reviewed 250 research methods syllabi from across the disciplines, as well as 80 syllabi from data science programmes to understand how or if data ethics was taught. We also reviewed 12 data ethics (...) frameworks drawn from different sectors. Finally, we reviewed an extensive and diverse body of literature about data practices, research ethics, data ethics and critical data literacy, in order to develop a transversal model that can be adopted across higher education. To promote and support ethical approaches to the collection and use of data, ethics training must go beyond securing informed consent to enable a critical understanding of the techno-centric environment and the intersecting hierarchies of power embedded in technology and data. By fostering ethics as a method, educators can enable research that protects vulnerable groups and empower communities. (shrink)

This article strongly argues the priority of symbolic, especially discursive, action over the material order in the genesis of social things. What turns a piece of stuff into a social object is its embedment in a narrative construction. The attribution of an active or a passive role to things in relation to persons is thus essentially story-relative: nothing happens or exists in the social world unless it is framed by human performative activity. Drawing on Gibson's notion of `affordance', Harré affirms (...) that material things may be disposed towards many different usages, and may acquire multiple identities according to different narrative constructions, even though the range of their possible `existences' is constrained by certain material features. Objects acquire their full significance only if one takes account of their double role in both the `practical' order, which includes social arrangements for maintaining life, and the `expressive' order, which creates hierarchies of honour and status, and which enjoys priority over the former. Reasoning from a microsociological constructionist perspective, Harré restates his view that there is nothing else to social life but symbolic exchanges and joint management of meaning, including the meaning of things; the illusion that some thing is real is merely an effect of certain interpretational grammars which remain stable across the generations or even the centuries. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which (...) refines the traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)