Whenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given (...) class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings ; integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories. (shrink)

We analyze the degree spectra of structures in which different types of immunity conditions are encoded. In particular, we give an example of a structure whose degree spectrum coincides with the hyperimmune degrees. As a corollary, this shows the existence of an almost computable structure of which the complement of the degree spectrum is uncountable.

We study the bi-embeddability and elementary bi-embeddability relation on graphs under Borel reducibility and investigate the degree spectra realized by these relations. We first give a Borel reduction from embeddability on graphs to elementary embeddability on graphs. As a consequence we obtain that elementary bi-embeddability on graphs is a $\boldsymbol {\Sigma }^1_1$ complete equivalence relation. We then investigate the algorithmic properties of this reduction. We obtain that elementary bi-embeddability on the class of computable graphs is $\Sigma ^1_1$ complete (...) with respect to computable reducibility and show that the elementary bi-embeddability and bi-embeddability spectra realized by graphs are related. (shrink)

We show that for every computable ordinal of the form β=δ+2n+1>1, where δ is zero or a limit ordinal and n∈ω, there exists a torsion-free abelian group having an X-computable copy if and only if X is nonlowβ.

A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a $\mathbf {0}'$ -computable low $_3$ compact Polish space which is not homeomorphic to a computable one, and that, for any natural number $n\geq 2$, there exists a Polish space $X_n$ such that exactly the high $_{n}$ -degrees are required to present the homeomorphism type of $X_n$. (...) Along the way we investigate the computable aspects of Čech homology groups. We also show that no compact Polish space has a least presentation with respect to Turing reducibility. (shrink)

We give some new examples of possible degree spectra of invariant relations on Δ 0 2 -categorical computable structures, which demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the sets of degrees that can be realized as degree spectra of such relations. In particular, we give a sufficient condition for a relation to have infinite degree spectrum that implies that every invariant computable (...) relation on a Δ 0 2 -categorical computable structure is either intrinsically computable or has infinite degree spectrum. This condition also allows us to use the proof of a result of Moses [23] to establish the same result for computable relations on computable linear orderings. We also place our results in the context of the study of what types of degree-theoretic constructions can be carried out within the degree spectrum of a relation on a computable structure, given some restrictions on the relation or the structure. From this point of view we consider the cases of Δ 0 2 -categorical structures, linear orderings, and 1-decidable structures, in the last case using the proof of a result of Ash and Nerode [3] to extend results of Harizanov [14]. (shrink)

We show that for every c.e. degree a > 0 there exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is {0, a}. This result can be extended in two directions. First we show that for every uniformly c.e. collection of sets S there exists an intrinsically c.e. relation on the domain of a computable structure whose degree spectrum is the set of degrees of elements of S. Then we show that (...) if α ∈ ω∪{ω } then for any α-c.e. degree a > 0 there exists an intrinsically α-c.e. relation on the domain of a computable structure whose degree spectrum is {0, a}. All of these results also hold for m-degree spectra of relations. (shrink)

Much previous study has been done on the degree spectra of prime models of a complete atomic decidable theory. Here we study the analogous questions for homogeneous models. We say a countable model A has a d-basis if the types realized in A are all computable and the Turing degree d can list $\Delta _{0}^{0}$ -indices for all types realized in A. We say A has a d-decidable copy if there exists a model B ≅ A such (...) that the elementary diagram of B is d-computable. Goncharov, Millar, and Peretyat'kin independently showed there exists a homogeneous A with a 0-basis but no decidable copy. We prove that any homogeneous A with a 0'-basis has a low decidable copy. This implies Csima's analogous result for prime models. In the case where all types of the theory T are computable and A is a homogeneous model with a 0-basis, we show A has copies decidable in every nonzero degree. A degree d is 0-homogeneous bounding if any automorphically nontrivial homogeneous A with a 0-basis has a d-decidable copy. We show that the nonlow₂ $\Delta _{2}^{0}$ degrees are 0-homogeneous bounding. (shrink)

We consider a recursive model and an additional recursive relation R on its domain, such that there are uncountably many different images of R under isomorphisms from to some recursive model isomorphic to . We study properties of the set of Turing degrees of all these isomorphic images of R on the domain of.

We show that for every computably enumerable degree a > 0 there is an intrinsically c.e. relation on the domain of a computable structure of computable dimension 2 whose degree spectrum is { 0 , a } , thus answering a question of Goncharov and Khoussainov 55–57). We also show that this theorem remains true with α -c.e. in place of c.e. for any α∈ω∪{ω} . A modification of the proof of this result similar to what was done (...) in Hirschfeldt shows that for any α∈ω∪{ω} and any α -c.e. degrees a 0 ,…, a n there is an intrinsically α -c.e. relation on the domain of a computable structure of computable dimension n+1 whose degree spectrum is { a 0 ,…, a n } . These results also hold for m-degree spectra of relations. (shrink)

There has been increasing interest over the last few decades in the study of the effective content of Mathematics. One field whose effective content has been the subject of a large body of work, dating back at least to the early 1960s, is model theory. Several different notions of effectiveness of model-theoretic structures have been investigated. This communication is concerned withcomputablestructures, that is, structures with computable domains whose constants, functions, and relations are uniformly computable.In model theory, we identify isomorphic structures. (...) From the point of view of computable model theory, however, two isomorphic structures might be very different. For example, under the standard ordering of ω the success or relation is computable, but it is not hard to construct a computable linear ordering of type ω in which the successor relation is not computable. In fact, for every computably enumerable degree a, we can construct a computable linear ordering of type ω in which the successor relation has degree a. It is also possible to build two isomorphic computable groups, only one of which has a computable center, or two isomorphic Boolean algebras, only one of which has a computable set of atoms. Thus, for the purposes of computable model theory, studying structures up to isomorphism is not enough. (shrink)

We consider the Turing degrees of prime models of complete decidable theories. In particular we show that every complete decidable atomic theory has a prime model whose elementary diagram is low. We combine the construction used in the proof with other constructions to show that complete decidable atomic theories have low prime models with added properties. If we have a complete decidable atomic theory with all types of the theory computable, we show that for every degree d with 0 (...) 0, T has a d-decidable model omitting the nonprincipal types listed by L. (shrink)

Several researchers have recently established that for every Turing degree \, the real closed field of all \-computable real numbers has spectrum \. We investigate the spectra of real closed fields further, focusing first on subfields of the field \ of computable real numbers, then on archimedean real closed fields more generally, and finally on non-archimedean real closed fields. For each noncomputable, computably enumerable set C, we produce a real closed C-computable subfield of \ with no computable copy. (...) Then we build an archimedean real closed field with no computable copy but with a computable enumeration of the Dedekind cuts it realizes, and a computably presentable nonarchimedean real closed field whose residue field has no computable presentation. (shrink)

We establish that for every computably enumerable (c.e.) Turing degree b the upper cone of c.e. Turing degrees determined by b is the degree spectrum of the successor relation of some computable linear ordering. This follows from our main result, that for a large class of linear orderings the degree spectrum of the successor relation is closed upward in the c.e. Turing degrees.

The spectrum of a relation on a computable structure is the set of Turing degrees of the image of R under all isomorphisms between and any other computable structure . The relation is intrinsically computably enumerable if its image under all such isomorphisms is c.e. We prove that any computable partially ordered set is isomorphic to the spectrum of an intrinsically c.e. relation on a computable structure. Moreover, the isomorphism can be constructed in such a way that the image of (...) the minimum element of the partially ordered set is computable. This solves the spectrum problem. The theorem and modifications of its proof produce computably categorical structures whose expansions by finite number of constants are not computably categorical and, indeed, ones whose expansions can have any finite number of computable isomorphism types. They also provide examples of computably categorical structures that remain computably categorical under expansions by constants but have no Scott family. (shrink)

We construct a class of relations on computable structures whose degree spectra form natural classes of degrees. Given any computable ordinal and reducibility r stronger than or equal to m-reducibility, we show how to construct a structure with an intrinsically invariant relation whose degree spectrum consists of all nontrivial r-degrees. We extend this construction to show that can be replaced by either or.

With every new recursive relation R on a recursive model , we consider the images of R under all isomorphisms from to other recursive models. We call the set of Turing degrees of these images the degree spectrum of R on , and say that R is intrinsically r.e. if all the images are r.e. C. Ash and A. Nerode introduce an extra decidability condition on , expressed in terms of R. Assuming this decidability condition, they prove that R (...) is intrinsically r.e. if and only if a natural recursive-syntactic condition is satisfied. We show that, while a recursive non-intrinsically r.e. relation may have a two element degree spectrum, a non-intrinsically r.e. relation which satisfies the Ash–Nerode decidability condition has an infinite degree spectrum. We also study several related decidability conditions and their effects on the degree spectra, including some conditions which are sufficient to obtain every r.e. degree in a spectrum. (shrink)

We investigate computability theoretic and topological properties of spaces of orders on computable orderable groups. A left order on a group G is a linear order of the domain of G, which is left-invariant under the group operation. Right orders and bi-orders are defined similarly. In particular, we study groups for which the spaces of left orders are homeomorphic to the Cantor set, and their Turing degree spectra contain certain upper cones of degrees. Our approach unifies and extends (...) Sikora’s [28] investigation of orders on groups in topology and Solomon’s [31] investigation of these orders in computable algebra. Furthermore, we establish that a computable free group Fn of rank n>1 has a bi-order in every Turing degree. (shrink)

We study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable $\Pi _{1}^{0}$ subsets of 2ω and Kolmogorov complexity play a major role in the proof.

We investigate effective categoricity for polymodal algebras. We prove that the class of polymodal algebras is complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations. In particular, this implies that every categoricity spectrum is the categoricity spectrum of a polymodal algebra.

We consider embeddings of structures which preserve spectra: if g: M → S with S computable, then M should have the same Turing degree spectrum (as a structure) that g(M) has (as a relation on S). We show that the computable dense linear order L is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph G. Such structures are said to be spectrally universal. We use (...) our results to answer a question of Goncharov, and also to characterize the possible spectra of structures as precisely the spectra of unary relations on G. Finally, we consider the extent to which all spectra of unary relations on the structure L may be realized by such embeddings, offering partial results and building the first known example of a structure whose spectrum contains precisely those degrees c with c′ ≥T 0″. (shrink)

Patients with psychotic disorders experience a range of reality distortions. These often include auditory-verbal hallucinations, and thought insertion to a lesser degree; however, their mechanisms and relationships between each other remain largely elusive. Here we attempt to establish a integrative model drawing from the phenomenology of both AVHs and TI and argue that they in fact can be seen as ‘spectra’ of experiences with varying degrees of agency and ownership, with ‘silent and internal own thoughts’ on one extreme (...) and ‘fully external and clearly audible voices’ in the absence of a speaker on the other. We believe a spectral model will add emphasis to the continuity of experience and help to better understand how one type of psychotic symptom may interact with another, and put forward the argument that the experience of TI itself is not sufficient to classify as a delusion. In addition we aim to discuss some of the conceptual issues surrounding AVHs and TI with first-person accounts and current philosophical and neuropsychological theories in mind. We propose that the mechanisms behind AVHs and TI are more complex than source-monitoring deficits; indeed, to understand such phenomena one must appreciate that their very ‘existence’ and ‘reality’ as experienced by the individual have much deeper implications and meaning, both philosophically and clinically. (shrink)

For a computable structure $\mathcal {M}$, the categoricity spectrum is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable copies of $\mathcal {M}$. If the spectrum has a least degree, this degree is called the degree of categoricity of $\mathcal {M}$. In this paper we investigate spectra of categoricity for computable rigid structures. In particular, we give examples of rigid structures without degrees of categoricity.

We study degree spectra of structures with respect to the bi-embeddability relation. The bi-embeddability spectrum of a structure is the family of Turing degrees of its bi-embeddable copies. To facilitate our study we introduce the notions of bi-embeddable triviality and basis of a spectrum. Using bi-embeddable triviality we show that several known families of degrees are bi-embeddability spectra of structures. We then characterize the bi-embeddability spectra of linear orderings and study bases of bi-embeddability spectra of (...) strongly locally finite graphs. (shrink)

We study the weak truth table degree spectra of first-order relational structures. We prove a dichotomy among the possible wtt degree spectra along the lines of Knight’s upward-closure theorem for Turing degree spectra. We prove new results contrasting the wtt degree spectra of finite- and infinite-signature structures. We show that, as a method of defining classes of reals, the wtt degree spectrum is, except for some trivial cases, strictly more expressive than (...) the Turing degree spectrum. (shrink)

The linear vector space for the quantum description of a physical system is formulated as the intersection of the domains of Hermiticity of the observables characterizing the system. It is shown that on a continuous interval of its spectrum every Hermitian operator on a Hilbert space of one degree of freedom is a generalized coordinate with a conjugate generalized momentum. Every continuous spectral interval of a Hermitian operator is the limit of a discrete spectrum in the same interval. This (...) result is applied to description of the state of a system by a statistical operator p( $\hat q$ ) following measurement of the probabilities of the eigenvalues of an observable $\hat q$ . Each continuous interval in the spectrum of $\hat q$ is replaced by a discrete spectrum in the same interval with a parameterK; the limitK → ∞ in which the spectrum becomes continuous is not attainable physically. In the linear vector space of a quantum mechanical system every continuous interval in the spectra of the observables is similarly replaced by a discrete spectrum with a finite parameter: it is a space of discrete bases and a set of such parameters. The case of degenerate spectrum is also discussed. (shrink)

We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure A as the family of Turing degrees that compute embeddings between any computable bi-embeddable copies of A; the degree of bi-embeddable categoricity of A is the least degree in this spectrum (if it exists). We extend many known results about categoricity spectra to the case of bi-embeddability. In particular, we exhibit structures without (...) class='Hi'>degree of bi-embeddable categoricity, and we show that every degree d.c.e above 0(α) for α a computable successor ordinal and 0(λ) for λ a computable limit ordinal is a degree of bi-embeddable categoricity. We also give examples of families of degrees that are not bi-embeddable categoricity spectra. (shrink)

The impossibility of an indeterministic evolution for standard relativistic quantum field theories, that is, theories in which all fields satisfy the condition that the generators of space-time translation have spectra in the forward light-cone, is demonstrated. The demonstration proceeds by arguing that a relativistically invariant theory must have a stable vacuum and then showing that stability of the vacuum, together with the requirements imposed by relativistic causality, entails deterministic evolution, if all degrees of freedom are standard degrees of freedom.

We prove that there is a structure, indeed a linear ordering, whose degree spectrum is the set of all non-hyperarithmetic degrees. We also show that degree spectra can distinguish measure from category.

In this paper we investigate computable isomorphisms of Boolean algebras with operators (BAOs). We prove that there are examples of polymodal Boolean algebras with finitely many computable isomorphism types. We provide an example of a polymodal BAO such that it has exactly one computable isomorphism type but whose expansions by a constant have more than one computable isomorphism type. We also prove a general result showing that BAOs are complete with respect to the degree spectra of structures, computable (...) dimensions, expansions by constants, and the degree spectra of relations. (shrink)

Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and bi-reducibility. These spectra provide a natural way of measuring the complexity of reductions between equivalence relations. We prove that any upward closed collection of Turing degrees with a countable basis can be realised as a reducibility spectrum or as a bi-reducibility spectrum. We show also that there (...) is a reducibility spectrum of computably enumerable equivalence relations with no countable basis and a reducibility spectrum of computably enumerable equivalence relations which is downward dense, thus has no basis. (shrink)

In the last five years there have been a number of results about the computable content of the prime, saturated, or homogeneous models of a complete decidable theory T in the spirit of Vaught's "Denumerable models of complete theories" combined with computability methods for degrees d ≤ 0′. First we recast older results by Goncharov, Peretyat'kin, and Millar in a more modern framework which we then apply. Then we survey recent results by Lange, "The degree spectra of homogeneous (...) models," which generalize the older results and which include positive results on when a certain homogeneous model of T has an isomorphic copy of a given Turing degree. We then survey Lange's "A characterization of the 0-basis homogeneous bounding degrees" for negative results about when does not have such copies, generalizing negative results by Goncharov, Peretyat'kin, and Millar. Finally, we explain recent results by Csima, Harizanov, Hirschfeldt, and Soare, "Bounding homogeneous models," about degrees d that are homogeneous bounding and explain their relation to the PA degrees. (shrink)

Fraïssé studied countable structures S through analysis of the age of S i.e., the set of all finitely generated substructures of S. We investigate the effectiveness of his analysis, considering effectively presented lists of finitely generated structures and asking when such a list is the age of a computable structure. We focus particularly on the Fraïssé limit. We also show that degree spectra of relations on a sufficiently nice Fraïssé limit are always upward closed unless the relation is (...) definable by a quantifier-free formula. We give some sufficient or necessary conditions for a Fraïssé limit to be spectrally universal. As an application, we prove that the computable atomless Boolean algebra is spectrally universal. (shrink)

Mereological theories are based on the binary relation “being a part of”. The systematic investigations of mereology were initiated by Leśniewski. More recent authors (including Simons, Casati and Varzi, Hovda) formulated a series of first‐order mereological axioms. These axioms give rise to a plenitude of theories, which are of great philosophical interest. The paper considers first‐order mereological theories from the point of view of computable (or effective) algebra. Following the approach of Hirschfeldt, Khoussainov, Shore, and Slinko, we isolate two important (...) computability‐theoretic properties P (namely, degree spectra of structures, and effective dimensions), and consider the following problem: for a given mereological theory T, is it true that its models can realize every possible variant of the property P? If the answer is positive, then we say that the theory T is ‐universal. We obtain the following results about known mereological theories. Any theory T which is weaker than Extensional Closure Mereology (CEM) is ‐universal. A similar fact is true for the theory GM2. On the other hand, any theory stronger that CEM + (C) + (G) is not ‐universal. In particular, General Extensional Mereology is not ‐universal. (shrink)

We introduce a hierarchy of sets which can be derived from the integers using countable collections. Such families can be coded into countable algebraic structures preserving their algorithmic properties. We prove that for different finite levels of the hierarchy the corresponding algebraic structures have different classes of possible degree spectra.

ABSTRACTIn this paper, I address the question of what we are really after when we seek Smithian mutual sympathy; I also show how the answer I propose can be used to illuminate a crucial feature of Smith's moral philosophy. The first section develops a Smithian response to egoistic interpretations of the desire for mutual sympathy. The second section identifies a number of different self- and other-relevant ways in which one could desire mutual sympathy. Some of these different ways of desiring (...) mutual sympathy comprise a spectrum of degrees of self-centredness; others comprise a spectrum of degrees of other-centredness. The third section shows that the spectra of ways of desiring mutual sympathy can be used to explain the kind of sincere, motivating attachment to morality characteristic of fully developed Smithian moral agents. (shrink)

Sensory deficits are a feature of autism and schizophrenia, as well as the upper end of their non-clinical spectra. The mismatch negativity, an index of pre-attentive auditory processing, is particularly sensitive in detecting such deficits; however, little is known about the relationship between the visual MMN to facial emotions and autism and schizophrenia spectrum symptom domains. We probed the vMMN to happy, sad, and neutral faces in 61 healthy adults, and evaluated their degree of autism and schizophrenia spectrum (...) traits using the Autism Spectrum Quotient and Schizotypal Personality Questionnaire. The vMMN to happy faces was significantly larger than the vMMNs to sad and neutral faces. The vMMN to happy faces was associated with interpersonal difficulties as indexed by AQ Communication and Attention to Detail subscales, and SPQ associated with more interpersonal difficulties. These data suggest that pre-attentive processing of positive affect might be more specific to the interpersonal features associated with autism and schizophrenia. These findings add valuable insights into the growing body of literature investigating symptom-specific neurobiological markers of autism and schizophrenia spectrum conditions. (shrink)

In many applications of information theory, information measures the reduction of uncertainty that results from the knowledge that an event has occurred. Even so, an item of information learned need not be the occurrence of an event but, rather, the change in probability distribution associated with an ensemble of events. This paper examines the basic account of information, which focuses on events, and reviews how it may be naturally generalized to probability distributions/measures. The resulting information measure is special case of (...) the Rényi information divergence (also known as the Rényi entropy). This information measure, herein dubbed the variational information, meaningfully assigns a numerical bit-value to arbitrary state transitions of physical systems. The information topology of these state transitions is characterized canonically by a right and left continuity spectrum defined in terms of the Kantorovich- Wasserstein metric. These continuity spectra provide a theoretical framework for characterizing the informational continuity of evolving systems and for rigorously assessing the degree to which such systems exhibit, or fail to exhibit, continuous change. (shrink)

We present the historical development of Nonlinear Dynamical Astronomy with emphasis on the “third integral” and its applications. The new era started with the use of computers, and of formal analytical developments in the spirit of Poincaré. Most dynamical systems were found to contain both ordered and chaotic orbits. The transition from order to chaos is discussed. Recent developments refer to the dynamical spectra, integrals of notion in self-consistent models, systems of 3 or more degrees of freedom, chaos in (...) relativity and cosmology, and chaos in quandum mechanics. (shrink)

Matter is pictured as a primitive fluid substratum having the fundamental property of fluctuating at a constant frequency. From this are derived the discrete properties of space and time, and it follows that, at the microlevel, talk of pure space and pure time involves us in ambiguities. A new interpretation of Planck's constant emerges according to which it is a quantum of matter-time combination. Thus, a quantum of matter-space combination should exist. On pursuing further the hydrodynamic model, such a constant (...) is in fact discovered as the drag-quantum of the quantum fluid. A fourth-degree differential equation is considered which, with the help of this new constant, generates spectra of frequency, mass, and fine structure constants. The theory seems to answer some important fundamental questions. (shrink)

continent. 1.4 (2011): 253—278. A Sense of French Politics Politics itself is not the exercise of power or struggle for power. Politics is first of all the configuration of a space as political, the framing of a specific sphere of experience, the setting of objects posed as "common" and of subjects to whom the capacity is recognized to designate these objects and discuss about them.(1) On April 14, 2011, France implemented its controversial ban of the niqab and burqa , commonly (...) referred to as the Islamic veil, in public places. On the coattails of the 2004 prohibition of religious artifacts, including the less concealing hijab or Islamic headscarves, in schools, the veil ban sparked equal outrage across both political and theological spectra, and many on both sides of the debate were quick to inquire about enforcement. As Steven Erlanger reports, “The police do not have the authority under the law to remove full veils, only to fine or require citizenship lessons for those who violate the new law.”(2) While the legal and social implications of this interdiction are ripe for study, the reprisals for breaking the law, notably compulsory citizenship classes, illuminate the strategic governmentality underlying the French government's targeting of the Muslim community, particularly those elements it considers affiliated with fundamentalist factions and/or terrorist organizations, if only through fashion—an integral component of French citizenship. Indeed, in allowing the (fashion) police to mandate instructional courses in citizenship for offenders, France implies that any Muslim choosing to wear the niqab or burqa , perhaps even for those who are already citizens, must not be aware of the subtle nuances of dress indicative of the French citizenry within public space. Furthermore, the degree of instruction requisite to citizenship, which clearly can and must be updated as the state sees fit, cements the imagined relation between citizens and states within the discourses on the veil ban. As Sharma contends, “Concepts of citizenship are the ideological glue that bonds the nation to the state. Citizenship provides the legal framework through which the state performs its role as ruler for the nation.”(3) Speaking for/as the state, French President Nicolas Sarkozy declared in a 2009 presentation before members of the government and other citizens that the burqa “is not welcome on the territory of France.”(4) Although there is nothing unique, especially recently, about a nation-state taking extreme measures to guard itself against what it deems to be threats to its sovereignty, the way in which the ongoing discourses on hijab have (d)evolved, which coincide with recent pronouncements by governmental leaders in Germany, Britain, and France that policies of multiculturalism have “failed,”(5) points toward nationalist reterritorializations, which is also to say physical and ideological re-appropriations, of public space within democratic milieux, which remain predicated upon the mediation of spaces. As Panagia deduces from Rancière, “Democracy […] is not an institutional form of government […], but an appearance whose visibilities and audibilites arise out of the dissonant blur of the everyday.”(6) Ultimately, the veil ban represents a calculated move to regulate the every day, which is to say the micropolitics of, that which is visible and invisible within French society, which makes it nothing short of a struggle to manage (the sensation of) bodies. Re-situating the political within the context of sense, Rancière argues that the fundamental essence of politics lies within the distinction between consensus and dissensus , and whereas most consider politics proper as fundamentally concerned with the fluid operations of the state— consensus , Rancière actually locates the political within events that disrupt the normative sensory order of things— dissensus . He explains, “Politics revolves around what is seen and what can be said about it, around who has the ability to see and the talent to speak, around the properties of spaces and the possibilities of time.”(7) This reformulation of politics centers on agency and, albeit perhaps unintended, representationality and offers a useful formulation for spatializing the discourses surrounding the Islamic veil, which are mired within seemingly senseless policies and practices of objectification. Instituting a normative “regime of visibility,” the French veil ban encapsulates the political economy of agency, now both mediated and representational, under the rule of nationalist governmentalities in increasingly transnational contexts.(8) Adjudicating presence within public space signals a concern with the sensory aspects of agency itself—what Rancière terms the “distribution of the sensible,” which is “a matrix that defines a set of relations between sense and sense: that is, between a form of sensory experience and an interpretation which makes sense of it.”(9) To be concerned with sense is to focus upon the body, and this somatic repositioning of the (fashionable) citizen within the territory of the state augurs the resurgence of nationalist bio-politics within Europe, since, as Hage explicates, “Nationalism, before being an explicit practice or a mode of classification, is a state of the body. It is a way of imagining one's position within the nation and what one can aspire to as a national.”(10) The French government's attempt to make (bodily) sense of public space for its citizens by and through policies aimed against a burqa-clad Muslim minority, which might only apply to “less then 400 to fewer than 2,000” women(11), reveals the necessary, yet mercurial, transmutation of public space into national space through bio-political strategies of sense-making, which, as Hage observes, denotes that “nationalists perceive themselves as spatial managers.”(12) At the center of this concern over public space is the equally contentious dimension of governing female Islamic identity, which has become one of the main critiques levied against Islam from outsiders who view the veil in its various forms as a clear sign of the religion's actual and symbolic violence against women, even if such critiques remain ensconced within nationalist discourses. As Halim and Meyers argue, “Although Western news media may focus their coverage of Muslim women on the debate over whether and the extent to which the veil may by oppressive to the women who wear it, they have exhibited little interest in the physical oppression of violence against women in Islamic countries.”(13) While it falls outside of the scope of this analysis to arbitrate all of the dimensions of this debate, understanding some of the issues related to female identity and hijab within public space are crucial to this study. As these topics are extraordinarily complex, even and perhaps especially within the Muslim community, it is necessary to provide some context concerning the value, purpose, and meaning of the Islamic veil, which has a rich and arduous history. The Art of the Veil At present, there are only two countries, Iran and Saudi Arabia, that require women to wear hijab , which is a decidedly problematic term and concept whose locus is itself caught within a struggle of sensation. Noting the strategic reterritorialization of the term, Berger explains: The choice of the name, hijab , now widely adopted to designate the “Islamic veil,” it itself interesting, although the semantic shift it operates is not directly commented upon by its users. Although the design (both the cut and the symbolic function) of the hijab and the Iranian chador are indeed the same, as opposed, for instance, to the traditional Algerian haik , the word hijab has come to replace the term chador popularized by the Iranian revolution, as a properly Arabic (hence Koranic) designation. This discursive shift points to the successful reclaiming of the national revolution in Iran by a transnational pan-Islamic movement, whose language of reference can only be Koranic Arabic.(14) While the hijab is certainly a point of contention within various countries dealing with immigrant Islamic populations (of which France has the largest in Europe as part of its colonial legacy in Algeria), it is clear that the veil is a historically contextualized artifact of Muslim identity with a variety of manifestations, even as forces outside of and within Islam seek to codify its actual and symbolic value. Ultimately, the varieties of hijab , including the full-faced burqa and niqab , are complex assemblages of meaning that speak to the multitudinous flows of Islamic identities over time and space. As Gökariksel and McLarney observe, “Wearing a certain style of veil may simultaneously be a disciplinary practice crucial to the cultivation of piety (Mahmood 2005; Gokariksel 2009) and a gendered performance of social distinction in terms of class, taste, and urbanity (White 1999; Navaro-Yashin 2002; Gokariksel and Secor 2009) or of ethnicity and race (Dwyer 1999).”(15) The ideological marketplace from which the varieties of hijab ebb and flow serves both as a reflection of the ever-present materiality and dynamic functionality of spiritual economies, which are derivative of the equally ubiquitous flows of transnational capital driving agency and representationality within our historical moment. Keeping these dual, yet deeply interconnected, capital ecologies as the site of exploration for how the veil is represented within public space, one might begin to sense what lurks behind such veiled agendas. Examining the symbolic and representational nature of the Islamic “veil” in its various forms, this project situates the decidedly political contestations of public space at stake in the French ban alongside recent condemnations of multiculturalism as calculated efforts to re-distribute the sensible so as to valorize national identity within increasingly transnational and globalized socio-economic spaces, which collapse exchange values between distinct, yet interwoven, economies of (cultural/material/spiritual) capital. As the discourses on the veil rely upon representational imag(in)ings, it seems fitting to explore the political economy and art of the veil through the phenomenon of street art, which, much like hijab , is a complex assemblage of meaning of sensation mired in representationality and capital exchange. As street artists have taken up hijab in various forms, the movement offers a lens with which to situate critical responses to the veil debate across both spiritual and material economies. Situating this street art both aesthetically and historically, Lewisohn explains, “'Street art' is a sub-genre of graffiti writing and owes much to its predecessor. Though there is a good deal of crossover between the genres, they are distinct and separate in their own right.”(16) Evolving out of the graffiti explosion of the late 1970's and 1980's, street art broke the unspoken rules of the artistic underground by challenging and re-examining one's sensory experience of art in public space by experimenting with both content and form in ways that harken back to the Situationist International, which as a collective challenged one's experience of urban space through various media and experiential phenomena. Again, Lewisohn observes, “It's important to note street art's break with the tradition of the tag, and its focus on visual symbols that embrace a much wider range of media than graffiti writers would use.”(17) From stencils to prints and murals to one-liners, street artists developed a reputation for critical social and political commentary, and while it is certainly the case that not all street art is intentionally political, in challenging the normative “regime of visibility,” as Rancière puts it, of public space, street artists—even if unconsciously—call into question the common sensory experience of the politics underlying the formation of (nationalist) space.(18) Indeed, Rancière refers to the partisan sensory managers as “police,” who ultimately forge “the rules governing the circulation of appearances, of their visibility and audibility, and the proper distribution of bodies therein.”(19) As such, one can sense that street art's relationship with the police is tenous for a variety of reasons. Writing about his own experience as a street artist, Shepard Fairey observes, “With street art, there's no committee deciding whether I can put my work up on the street, there's no censorship, and I have total freedom of expression, and that concept of freedom is expressed just by using the street as a medium.”(20) As Fairey and other street artists demonstrate, the medium, to paraphrase Marshall McLuhan, is the most glaring aspect of street art's message, and even as it now finds itself in galleries and usurped by private collectors, the ethos of the movement remains relavent to politics as it is, first and foremost, concerned with the configuration and experience of space. Just as examining the punishments for breaking the veil ban can map the terrain of French politics, a representational genealogy of street artist's deployment of hijab and female Muslim identity can assist in mining the depths of public space as a site of bio-political control and contestation in light of recent events in France and the rising tide of anti-multiculturalist rhetoric across Europe, even though the Islamic presence there is radically heterogenous. Using the work of Shepard Fairey, Princess Hijab, br1, and Banksy, this paper sets out to answer a number of key questions concerning the deployment of hijab, and female Islamic identity in particular, within street art as a movement now integral to and critical of the economies of exchange underlying global capitalism, which as a force of influence cannot be underestimated as a driver in the resurgence of nationalist biopolitical governmentalities. Although there exists a litany of artists to choose from, the aforementioned quartet of designers integrate hijab into their work in a seminal and critical fashion, and all have been working with the veil in some way prior to the French ban, which grants some additional context to their usage of it as a symbol in regards to the breakdown of multiculturalism across Europe, even if unconsciously. Although each artist depicts various amalgamations of hijab in their respective corpus, they share common tropes that raise significant questions concerning the veil debate, namely: what are the conditions of possibility for hijab to become a marker of Islamic identity within public space? How can we make sense of the representational topologies within street art's utilization of the hijab and female Islamic identity in light of the French ban? How might these aesthetic imag(in)ings simultaneously inhabit and combat normative discourses concerning hijab , multiculturalism, female Islamic identity, and public space? With these considerations as a backdrop, it is first necessary to outline more fully the hijab as a representative marker of female Islamic identity. The (Veiled) Writing on the Wall

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Supervaluationism is often described as the most popular semantic treatment of indeterminacy. There’s little consensus, however, about how to fill out the bare-bones idea to include a characterization of logical consequence. The paper explores one methodology for choosing between the logics: pick a logic thatnorms beliefas classical consequence is standardly thought to do. The main focus of the paper considers a variant of standard supervaluational, on which we can characterizedegrees of determinacy. It applies the methodology above to focus ondegree logic. (...) This is developed first in a basic, single-premise case; and then extended to the multipremise case, and to allowdegreesof consequence. The metatheoretic properties of degree logic are set out. On the positive side, the logic is supraclassical—all classical valid sequents are degree logic valid. Strikingly, metarules such as cut and conjunction introduction fail. (shrink)

The central aim of this book is to understand modules and the categories they form through associated structures and dimensions, which reflect the complexity of these, and similar, categories.

We regularly make graded normative judgements in the epistemic domain. Recent work in the literature examines degrees of justification, degrees of rationality, and degrees of assertability. This paper addresses a different dimension of the gradeability of epistemic normativity, one that has been given little attention. How should we understand degrees of epistemic criticizability? In virtue of what sorts of factors can one epistemic failing be worse than another? The paper develops a dual-factor view of degrees of epistemic criticizability. According to (...) the view, degrees of epistemic criticizability are (i) an inverse function of degrees of doxastic justification and (ii) a function of degrees of agent culpability. The paper develops an account of each factor, and explains how they should be weighted. The paper also addresses the importance of modelling degrees of epistemic criticizability in a broader context. I focus on the role that such a model can play in the ethics of epistemic criticism. (shrink)