Important examples of $\Pi^0_1$ classes of functions $f \in {}^\omega\omega$ are the classes of sets (elements of ω 2) which separate a given pair of disjoint r.e. sets: ${\mathsf S}_2(A_0, A_1) := \{f \in{}^\omega2 : (\forall i < 2)(\forall x \in A_i)f(x) \neq i\}$ . A wider class consists of the classes of functions f ∈ ω k which in a generalized sense separate a k-tuple of r.e. sets (not necessarily pairwise disjoint) for each k ∈ ω: ${\mathsf S}_k(A_0,\ldots,A_k-1) (...) := \{f \in {}^\omega k : (\forall i < k) (\forall x \in A_i) f(x) \neq i\}$ . We study the structure of the Medvedev degrees of such classes and show that the set of degrees realized depends strongly on both k and the extent to which the r.e. sets intersect. Let ${\mathcal S}^m_k$ denote the Medvedev degrees of those ${\mathsf S}_k(A_0,\ldots,A_{k-1})$ such that no m + 1 sets among A 0,...,A k-1 have a nonempty intersection. It is shown that each ${\mathcal S}^m_k$ is an upper semi-lattice but not a lattice. The degree of the set of k-ary diagonally nonrecursive functions $\mathsf{DNR}_k$ is the greatest element of ${\mathcal S}^1_k$ . If 2 ≤ l < k, then 0 M is the only degree in ${\mathcal S}^1_l$ which is below a member of ${\mathcal S}^1_k$ . Each ${\mathcal S}^m_k$ is densely ordered and has the splitting property and the same holds for the lattice ${\mathcal L}^m_k$ it generates. The elements of ${\mathcal S}^m_k$ are exactly the joins of elements of ${\mathcal S}^1_i$ for $\lceil{k \over m}\rceil \leq i \leq k$. (shrink)

The present "philological commentary" is directed at those who have decided to take time for reading the original text, at least in an English translation. The first volume "Textual Criticism" is intended to meet the difficulties caused by the fact that our text editions are based on manuscripts separated from the original by more than a millennium.

Let ≤r and ≤sbe two binary relations on 2ℕ which are meant as reducibilities. Let both relations be closed under finite variation and consider the uniform distribution on 2ℕ, which is obtained by choosing elements of 2ℕ by independent tosses of a fair coin.Then we might ask for the probability that the lower ≤r-cone of a randomly chosen set X, that is, the class of all sets A with A ≤rX, differs from the lower ≤s-cone of X. By c (...) osure under finite variation, the Kolmogorov 0-1 aw yields immediately that this probability is either 0 or 1; in case it is 1, the relations are said to be separable by random oracles.Again by closure under finite variation, for every given set A, the probability that a randomly chosen set X is in the upper ≤r-cone of A is either 0 or 1; let Almostr be the class of sets for which the upper ≤r-cone has measure 1. In the following, results about separations by random oracles and about Almost classes are obtained in the context of generalized reducibilities, that is, for binary relations on 2ℕ which can be defined by a countable set of total continuous functionals on 2ℕ in the same way as the usual resource-bounded reducibilities are defined by an enumeration of appropriate oracle Turing machines. The concept of generalized reducibility comprises a natura resource-bounded reducibilities, but is more general; in particular, it does not involve any kind of specific machine model or even effectivity. The results on generalized reducibilities yield corollaries about specific resource-bounded reducibilities, including several results which have been shown previously in the setting of time or space bounded Turing machine computations. (shrink)

We show, assuming weak large cardinals, that in the context of games of length \ with moves coming from a proper class, clopen determinacy is strictly weaker than open determinacy. The proof amounts to an analysis of a certain level of L that exists under large cardinal assumptions weaker than an inaccessible. Our argument is sufficiently general to give a family of determinacy separation results applying in any setting where the universal class is sufficiently closed; e.g., in third, (...) seventh, or \\)th order arithmetic. We also prove bounds on the strength of Borel determinacy for proper class games. These results answer questions of Gitman and Hamkins. (shrink)

Let X be a Polish space and a separable compact subset of the first Baire class on X. For every sequence dense in , the descriptive set-theoretic properties of the set are analyzed. It is shown that if is not first countable, then is -complete. This can also happen even if is a pre-metric compactum of degree at most two, in the sense of S. Todorčević. However, if is of degree exactly two, then is always Borel. A deep result (...) of G. Debs implies that contains a Borel cofinal set and this gives a tree-representation of . We show that classical ordinal assignments of Baire-1 functions are actually -ranks on . We also provide an example of a Ramsey-null subset A of for which there does not exist a Borel set BA such that the difference BA is Ramsey-null. (shrink)

The paper deals with the study of the existence of weak positive solutions for a new class of the system of elliptic differential equations with respect to the symmetry conditions and the right hand side which has been defined as multiplication of two separate functions by using the sub-supersolutions method.

Separate and Dominate is Delphy's manifesto, lambasting liberal hypocrisy and calling for a fluid understanding of political identity that does not place different political struggles in a false opposition. She dismantles the absurd claim that Afghanistan was invaded to save women, and that homosexuals and immigrants alike should reserve their self-expression for private settings. She calls for a true universalism that sacrifices no one at the expense of others. In the aftermath of the Charlie Hebdo massacre, her arguments appear more (...) prescient and pressing than ever.--Provided by publisher. (shrink)

We prove two sets of results concerning computational complexity classes. First, we propose a new variation of the random oracle hypothesis, originally posed by Bennett and Gill after they showed that relative to a randomly chosen oracle, with probability 1. Their original hypothesis was quickly disproven in several ways, most famously in 1992 with the result that, in spite of the classes being shown unequal with probability 1. Here we propose a variation of what it means to be “large” using (...) the Ellentuck topology. In this new context, we demonstrate that the set of oracles separating and is not small, and obtain similar results for the separation of from along with the separation of from. We also show that the set of oracles equating with is large in this new sense. We demonstrate that this version of the hypothesis provides a sufficient condition for unrelativized relationships, at least in the cases considered here. Second, we examine the descriptive complexity of the classes of oracles providing the separations for these various classes, and determine their exact placement in the Borel hierarchy. (shrink)

In the early nineties of the previous century, leaf languages were introduced as a means for the uniform characterization of many complexity classes, mainly in the range between P and PSPACE . It was shown that the separability of two complexity classes can be reduced to a combinatorial property of the corresponding defining leaf languages. In the present paper, it is shown that every separation obtained in this way holds for every generic oracle in the sense of Blum and Impagliazzo. (...) We obtain several consequences of this result, regarding, e. g., universal oracles, simultaneous separations and type-2 complexity. (shrink)

The class forcing theorem, which asserts that every class forcing notion ${\mathbb {P}}$ admits a forcing relation $\Vdash _{\mathbb {P}}$, that is, a relation satisfying the forcing relation recursion—it follows that statements true in the corresponding forcing extensions are forced and forced statements are true—is equivalent over Gödel–Bernays set theory $\text {GBC}$ to the principle of elementary transfinite recursion $\text {ETR}_{\text {Ord}}$ for class recursions of length $\text {Ord}$. It is also equivalent to the existence of truth (...) predicates for the infinitary languages $\mathcal {L}_{\text {Ord},\omega }$, allowing any class parameter A; to the existence of truth predicates for the language $\mathcal {L}_{\text {Ord},\text {Ord}}$ ; to the existence of $\text {Ord}$ -iterated truth predicates for first-order set theory $\mathcal {L}_{\omega,\omega }$ ; to the assertion that every separative class partial order ${\mathbb {P}}$ has a set-complete class Boolean completion; to a class-join separation principle; and to the principle of determinacy for clopen class games of rank at most $\text {Ord}+1$. Unlike set forcing, if every class forcing notion ${\mathbb {P}}$ has a forcing relation merely for atomic formulas, then every such ${\mathbb {P}}$ has a uniform forcing relation applicable simultaneously to all formulas. Our results situate the class forcing theorem in the rich hierarchy of theories between $\text {GBC}$ and Kelley–Morse set theory $\text {KM}$. (shrink)

A characterization of the subdirectly irreducible separable dynamic algebras is presented. The notions develo- ped for this study were also suitable to describe the previously found class of simple separable dynamic algebras.

Linear temporal logic with Since and Until modalities is expressively equivalent, over the class of complete linear orders, to a fragment of first-order logic known as FOMLO. It turns out that LTL, under some basic assumptions, is expressively complete if and only if it has the property, called separation, that every formula is equivalent to a Boolean combination of formulas that each refer only to the past, present or future. Herein we present simple algorithms and their implementations to perform (...) separation of the LTL with Since and Until, over discrete and complete linear orders, and translation from FOMLO formulas into equivalent temporal logic formulas. We additionally show that the separation of a certain fragment of LTL results in at most a double exponential size growth. (shrink)

Recently two pairs of authors have aimed to vindicate the longstanding "orthodox" or conventional claim that a first-class constraint generates a gauge transformation in typical gauge theories such as electromagnetism, Yang-Mills and General Relativity, in response to the Lagrangian-equivalent reforming tradition, in particular Pitts, _Annals of Physics_ 2014. Both pairs emphasize the coherence of the extended Hamiltonian formalism against what they take to be core ideas in Pitts 2014, but both overlook Pitts 2014's sensitivity to ways that one might (...) rescue the claim in question, including an additive redefinition of the electrostatic potential. Hence the bulk of the paper is best interpreted as arguing that the longstanding claim about separate first-class constraints is _either false or trivial_---de-Ockhamization (using more when less suffices by splitting one quantity into the sum of two) being trivial. Unfortunately section 9 of Pitts 2014, a primarily verbal argument that plays no role in other works, is refuted. Pooley and Wallace's inverse Legendre transformation to de-Ockhamized electromagnetism with an additively redefined electrostatic potential, however, opens the door to a precisely analogous calculation introducing a photon mass, which shows that a _second-class primary_ constraint generates a 'gauge transformation' in the exactly same sense---a _reductio ad absurdum_ of the orthodox claim that a first-class constraint generates a gauge transformation _and a second-class constraint does not_. A _reductio_ using massive electromagnetism was presaged in Pitts 2014. Gauge freedom by de-Ockhamization does not require any constraints at all, first-class or second-class, because any dynamical variable in any Lagrangian can be de-Ockhamized into exhibiting trivial additive artificial gauge freedom by splitting one quantity into the sum of two. Physically interesting gauge freedom, however, is typically generated by a tuned sum of first-class constraints, not each first-class constraint alone. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Separating the Human from the Divine Cesáreo Bandera University ofNorth Carolina at Chapel Hill Myths are hard to die. One such myth concerns what happened with poetry in general, that is to say, imaginative literature or literary fiction, in the transition from the Middle Ages to the Renaissance and beyond. Its basic outline was developed during the nineteenth century. J. E. Spingarn, for example, echoes such a myth (...) in his History of Literary Criticism in the Renaissance, first published in 1899, which has gone through many editions and reprintings since then. In it we read the following: The first problem of Renaissance criticism was the justification of imaginative literature. The existence and continuity of the aesthetic consciousness, and perhaps, in a less degree, of the critical faculty, throughout the Middle Ages, can hardly be denied; yet distrust of literature was keenest among the very class of men in whom the critical faculty might be presupposed, and it was as the handmaid of philosophy, and most of all as the vassal of theology, that poetry was chiefly valued.... The Renaissance was thus confronted with the necessity ofjustifying its appreciation of the vast body of literature which the Revival of Learning had recovered for the modern world. (3) Speaking of Savonarola's De Divisione ac Utilititate Omnium Scientiarum, written about 1492, he concludes as follows: In fine, as a reformer, he represents for us the religious reaction against the paganization of culture by the humanists. But the forces against him were too strong. Even the Christianization of culture effected during the next century by the Council of Trent was hardly more than temporary. Humanism, which represents 74Cesáreo Bandera the revival of ancient pagan culture, and rationalism, which represents the growth of the modem spirit in science and art, were currents too powerful to be impeded by any reformer, however great. (14) This general picture of the historical standing and the fate of literary fiction as we move from the Middle Ages into the modern era, is still very much with us, even though it is ill at ease with, or flatly contradicted by, undeniable historical facts. For example, the incredible notion of a "paganization of culture by the humanists" would have horrified somebody as profoundly Christian as Petrarch and, later on, Erasmus, who criticized the scholastics "in the name of a purer and simpler mode of piety and religious devotion" (Mazzeo 17). Or the mistaken idea that modern rationalism was much more hospitable to poetry than the old scholastic one. For from the seventeenth century on, as J. F. West has pointed out, the whole tide of [scientific] opinion was running strongly against poetry, metaphor, and poetic prose. The philosophers generally saw them as an obstacle to truth. Hobbes regarded metaphor as one of the hindrances to straight thinking. Rousseau thought that the philosophy of Descartes had 'cut the throat of poetry'; and John Locke, late in the century, openly regarded poetry as made up of 'pleasant pictures and agreeable visions in the fancy,' but basically misleading. (114) It is indeed true that in medieval scholasticism poetry occupies the lowest rank in the hierarchical structure of the arts and sciences. But it does have a place there. It would have never occurred to Thomas Aquinas, for example, to expel it from that position as something dangerous, something to be avoided. In fact, as he tells us at the beginning of the Summa Theologica, theology itself, the highest of the sciences, does not consider it below its dignity to make use of metaphor and other poetic devices propter utilitatem ac necessitatem. And yet it is precisely that kind of violent expulsion ofpoetic fiction that will be attempted during the Renaissance. In the words of Russell Fraser, "it is in the Renaissance, and not the Middle Ages that the artist is driven from the commonwealth" (39). It was then that the "war against poetry" intensified to fever pitch. It is, therefore, an error to imagine that those in the sixteenth and seventeenth centuries who bitterly attacked poetic fiction in general and the theater in particular as a very dangerous threat to religion and morality, were fanatic hold-overs from a medieval past... (shrink)

The aim is to defend the starting?point of Marx's theory of class, which is located in a definition of the working class in the Communist Manifesto. It is a definition solely in terms of separation from productive resources and a need to sell one's labour power, and it is closely connected with Marx's thesis that the population in capitalism has a tendency to polarize. That thesis conflicts with the widely?held belief in the growth of a large middle (...) class='Hi'>class, unaccounted for by Marx. Moreover, recent critics such as Elster, Roemer, and Cohen have argued that this definition fails even in its own terms. The definition is refurbished so as to withstand these objections. But is there any point in using it? Does it serve to pick out the exploited producers as Marx intended? It does, once due attention is given to the idea of the collective worker, which is central in the volume of Capital which Marx himself published. That idea makes plain that it is an irreducibly corporate entity which is productive and subject to exploitation. The structural conditions for membership of that entity remove Marx's view from any simple identification of working?class membership with manual or lowly labour. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Journal of Aesthetic Education 37.1 (2003) 54-63 [Access article in PDF] Movement Class as an Integrative Experience:Academic, Cognitive, and Social Effects Svetlana Nikitina I believe the benefits of this type of course reach beyond the obvious possibilities of professional and academic achievement. The degree of personal discovery, creativity, self-development and insight are immeasurable. I am particularly referring to my experience here at Harvard. Claire Mallardi, from course (...) syllabus Performing arts courses at research institutions such as Harvard University are often seen as extracurricular with only tangential or imperceptible effect on students' academic interests. Yet, when I interviewed biology, math, or neuroscience concentrators who took Claire Mallardi's Movement for Actors and Directors course at Harvard in the spring of 2000, most of them told me that it was one of the most meaningful experiences in their academic development, and, according to one psychology concentrator, "the best psychology course" she ever took. In spite of boundaries separating the arts courses from hard and soft sciences, in students' experience the connections run deep. What were those connections that students made from their dance class experience to their academic and social development? And, what in the teaching and nature of the course supported this highly integrative experience that allowed students to reach beyond mastering an artistic medium and become better learners in general? These were the two major questions that guided this study.Why am I interested in the integrative quality of arts education? I am highlighting the academic importance of performing arts programs to suggest to university administrators and instructors in other disciplines the "immeasurable" personal and academic value of such courses whose place is often seen as marginal. Perhaps also, pedagogy used for integration in this course into the academic fabric of the institution could provide models for other courses and curricula that are trying to be integrative or interdisciplinary in a more self-conscious way. And, there are more and more such [End Page 54] courses on every college campus these days. Moriel Blaisdell in Preparing Faculty for the New Conceptions of Scholarship for example, cites "integrative studies" programs among most important trends in higher education today with their goal "to synthesize, to look for new relationships between the parts and the whole, to relate the past and future to the present, and to ferret out patterns of meaning that cannot be seen through traditional disciplinary lenses." 1One of the most daring experiments in universities now, and the one to which the integrative dance course could speak directly, is interdisciplinary teaching in neuroscience, for example. Neuroscience proceeds from the belief in the inherent unity of mind and body and attempts to discover the nature of the relationship between them. In a different way and with a different goal than arts courses, neurosciences also try to cross the boundary of subjective and objective experience, mind and body, physical technique and personal interpretation, which are at the heart of teaching and learning in the performing arts. While not being prescriptive, I would like to suggest that perhaps close attention to the pedagogy and integrative experiences of students in the performing arts courses could provide some insights for pioneering bridge-making in other academic areas as well.I would like to propose that what lies at the core of integrative capacity of a performing arts course, like the one Claire Mallardi teaches at Harvard, is the inherent tendency of the arts to transcend the boundary of body with the mind, and to seek unity of self as both the subject and the object. In a dance, in the words of Margaret H'Doubler, the body is seen "as the outer aspect of personality, for it is the agent through which we receive impressions from the external world and by which we communicate our meaning." 2 This basic translation between body and mind, mental and physical, artistic and academic that went on in Mallardi's class, resulted in integrative and transformational experiences for her students. "I never until this class — comments one student — equated emotions with movement or... (shrink)

LeSniewski?s systems of Ontology and Mereology, considered from a purely formal point of view, possessstriking algebraic parallels, ascan be seen in their respective relations to Boolean algebra. But there are alsoimportant divergences, above all that general Mereology is silent, where Ontology is not, on the existenceof ?atoms? (individuals). By employing plural terms, LeSniewski sought to accommodate talk of (distributive)classes, without according these an autonomous ontological status. His logic also ? like predicate logic? has no place for mass predication in its (...) raw state. It is argued that reference both to pluralities and tomasses is ineliminable, and that one must therefore separate the grammatical distinction singular/pluralfrom the ontological distinction individual/class/mass. A kind of language, modelled on kiniewski?s, issuggested which enables these distinctions to be held apart and at the same time, building on the work ofRichard Sharvy, allows us to express the most important relationships between individuals, classes andmasses. (shrink)

A full separation theorem for the derivable rules of intuitionistic linear logic without bounds, 0 and exponentials is proved. Several structural consequences of this theorem for subreducts of (commutative) residuated lattices are obtained. The theorem is then extended to the logic LR+ and its proof is extended to obtain the finite embeddability property for the class of square increasing residuated lattices.

"You have to come to my wedding," Kavita told me, turning to face me where I sat next to her on the couch. "You can come with the other people from the street. You will get everything you need for your *research* there." "I will come, I will come!" I replied enthusiastically. I had only met Kavita and her two younger sisters, Arthi and Deepti (see Figure 2.1), mere minutes before this invitation was extended. I had initially come to Pulan (...) that day in October 2012 to meet another woman, Heena, whose family rents a room on the third story of Kavita's family's home. Heena and I had been sitting in the furniture refurbishing store she operates with her husband on the main street of Pulan when Deepti, Kavita's youngest sister, passed by. Heena introduced us and told me to go with Deepti to meet her family. When we reached the family's three-story house-the largest in the gali-Deepti led me past the empty rooms on the ground floor, which I would eventually begin renting, to the second-story living room. There, we found Kavita and Arthi organizing clothing and jewelry they had purchased earlier in the day for the upcoming wedding festivities. Kavita made room for me to sit next to her on the couch and began asking me about myself. I immediately warmed to her because of her open, friendly smile and sharp, staccato Hindi, which I delighted in being able to understand. I explained that I had come to India to study how women's lives are different in rural and urban areas, and Kavita assured me that she and her family could help. She noted that her parents had come to Udaipur from Ram Nagar, a large village thirty-five kilometers north of the city, and that the family would be returning for her and her older brother Krishna's weddings the following month. Their weddings would be held five days apart to help reduce the difficulties of family members traveling from outside Udaipur. Prompted by the description of my research, Kavita commented on differences that she recognized between the village and the city. The biggest difference, she suggested, was the experience of caste, namely that in the village, people from different jatis live separately, whereas in the city, people are "mixed." As I would come to learn when visiting Ram Nagar for various functions, there is a fair amount of caste and religious diversity in the village. Although spatial and ritual segregation was rather strictly maintained during religious observances, it is likely more flexible in everyday life. The segregation during ritual functions-the occasions for which Kavita also traveled to the village-likely informed her sense of a lack of "mixing" in the village as. The majority of residents in the area of Ram Nagar where the family maintains a home were also from the Mali (lit: gardener) jati, although Mali was not a majority jati in Pulan. (shrink)

We show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those of level $\omega _1$ and iteratively applying the operations of expansion and separated differences. The proof is essentially due to Louveau, and it yields at the same time a new proof of a theorem of Van Wesep. The exposition is self-contained, except for facts from classical descriptive set theory.

The separation of Avicenna between Essence and Existence influenced logic and Arab and Muslim logicians in the Middle Ages among them Ibn al-Nafīs (1208-1288). Under this influence he contributed to the development of logic and especially the theory of the universal term. By means of the consequences of this analysis:-It has become possible to make a distinction between abstract concepts and formal concepts independent of any sensible reality, and hence the questioning of Aristotelian categories, that is to say the ability (...) to conceive for Ibn al-Nafīs like Avicenna, subjects and predicates which are not necessarily words expressing or belonging to one of the Aristotelian categories.-The analysis of the universal proposition as a conditional proposition, which made easier the distinction between absolute proposition which has no existential implication, and the other which differ from the first by an affirmation of explicit existence. (shrink)

In continuation of our study of HST, Hrbaek set theory (a nonstandard set theory which includes, in particular, the ZFC Replacement and Separation schemata in the st--language, and Saturation for well-orderable families of internal sets), we consider the problem of existence of elementary extensions of inner "external" subclasses of the HST universe.We show that, given a standard cardinal , any set R * generates an "internal" class S(R) of all sets standard relatively to elements of R, and an "external" (...) class L[S(R)] of all sets constructible (in a sense close to the Gödel constructibility) from sets in S(R). We prove that under some mild saturation-like requirements for R the class L[S(R)] models a certain -version of HST including the principle of +-saturation; moreover, in this case L[S(R)] is an elementary extension of L[S(R)] in the st--language whenever sets R R satisfy the requirements. (shrink)

We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property, We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation subclasses whose axiomatisations are recursively enumerable in our second-order fragment can also be recursively axiomatised in their original first-order language. We pin down the expressive power of this formalism with respect to first-order logic, and investigate some questions relating to decidability and computational complexity. (...) As applications of these results, by showing that certain classes can be straightforwardly defined as separation subclasses, we obtain first-order axiomatisability results for these classes. In particlar, we apply this technique to graph colourings and a class of partial algebras arising from separation logic. (shrink)

We study hybrid logics in topological semantics. We prove that hybrid logics of separation axioms are complete with respect to certain classes of finite topological models. This characterisation allows us to obtain several further results. We prove that aforementioned logics are decidable and PSPACE-complete, the logics of T 1 and T 2 coincide, the logic of T 1 is complete with respect to two concrete structures: the Cantor space and the rational numbers.

The partial ordering of Medvedev reducibility restricted to the family of Π0 1 classes is shown to be dense. For two disjoint computably enumerable sets, the class of separating sets is an important example of a Π0 1 class, which we call a ``c.e. separating class''. We show that there are no non-trivial meets for c.e. separating classes, but that the density theorem holds in the sublattice generated by the c.e. separating classes.

As a means to challenge and diminish the hold of mainstream curriculum's claim of being a colorblind, politically neutral text, we will address two particular features that partially, though significantly, constitute the hidden curriculum in the United States—race and class—historically studied as separate social issues. Race and class have been embedded within the institutional curriculum from the beginning in the US; though rarely acknowledged as intertwined issues. We illustrate how the theoretical and interpretive structure of French philosopher and (...) sociologist Pierre Bourdieu can productively subsume the insights of critical race theory into its framework in a way that provides a more robust understanding of how race and class continue to be socially reproduced in schools. To perform this task we examine, through Bourdieu's constructs of habitus, field, capital, symbolic violence and misrecognition, the ways in which race, in general, and whiteness, specifically, influences pedagogical and curricular existence within the institutional superstructure of school. (shrink)

Like modal logic, temporal logic, and description logic, separation logic has become a popular class of logical formalisms in computer science, conceived as assertion languages for Hoare-style proof systems with the goal to perform automatic program analysis. In a broad sense, separation logic is often understood as a programming language, an assertion language and a family of rules involving Hoare triples. In this survey, we present similarities between separation logic as an assertion language and modal and temporal logics. Moreover, (...) we propose a selection of landmark results about decidability, complexity and expressive power. (shrink)

This paper introduces a new Ehrenfeucht-Fraïssé type game that is played on two classes of models rather than just two models. This game extends and generalizes the known Ajtai-Fagin game to the case when there are several alternating moves played in different models. The game allows Duplicator to delay her choices of the models till the very end of the game, making it easier for her to win. This adds on the toolkit of winning strategies for Duplicator in Ehrenfeucht-Fraïssé type (...) games and opens up new methods for tackling some open problems in descriptive complexity theory. As an application of the class game, it is shown that, if m is a power of a prime, then first order logic augmented with function quantifiers F1n of arity 1 and height n < m can not express that the size of the model is divisible by m. This, together with some new expressibility results for Henkin quantifiers H1n gives some new separation results on the class of finite models among various F1n on one hand and between F1n and H1n on the other. Since function quantifiers involve a bounded type of second order existential quantifiers, the class game solves an open problem raised by Fagin, which asks for some inexpressibility result using a winning strategy by Duplicator in a game that involves more than one coloring round. (shrink)

"Parts of Classes" tries to separate the unproblematic part of set theory (mereology) from the problematic part (singletons). In the process several things get lost: an empty set which is really empty; a satisfying account of the paradoxes; and the motivation for the iterative conception of set. Lewis' attack on the coherence of singletons makes it puzzling what he sees his book as doing. Nor is it clear that mereology is as ontologically innocent as Lewis would have us believe.

In Dirac-Bergmann constrained dynamics, a first-class constraint typically does not _alone_ generate a gauge transformation. By direct calculation it is found that each first-class constraint in Maxwell's theory generates a change in the electric field E by an arbitrary gradient, spoiling Gauss's law. The secondary first-class constraint p^i,_i=0 still holds, but being a function of derivatives of momenta, it is not directly about E. Only a special combination of the two first-class constraints, the Anderson-Bergmann -Castellani gauge (...) generator G, leaves E unchanged. This problem is avoided if one uses a first-class constraint as the generator of a _canonical transformation_; but that partly strips the canonical coordinates of physical meaning as electromagnetic potentials and makes the electric field depend on the smearing function, bad behavior illustrating the wisdom of the Anderson-Bergmann Lagrangian orientation of interesting canonical transformations. The need to keep gauge-invariant the relation dot{q}- dH/dp= -E_i -p^i=0 supports using the primary Hamiltonian rather than the extended Hamiltonian. The results extend the Lagrangian-oriented reforms of Castellani, Sugano, Pons, Salisbury, Shepley, _etc._ by showing the inequivalence of the extended Hamiltonian to the primary Hamiltonian even for _observables_, properly construed in the sense implying empirical equivalence. Dirac and others have noticed the arbitrary velocities multiplying the primary constraints outside the canonical Hamiltonian while apparently overlooking the corresponding arbitrary coordinates multiplying the secondary constraints _inside_ the canonical Hamiltonian, and so wrongly ascribed the gauge quality to the primaries alone, not the primary-secondary team G. Hence the Dirac conjecture about secondary first-class constraints rests upon a false presupposition. The usual concept of Dirac observables should also be modified to employ the gauge generator G, not the first-class constraints separately, so that the Hamiltonian observables become equivalent to the Lagrangian ones such as the electromagnetic field F. (shrink)

In [20] Krajíček and Pudlák discovered connections between problems in computational complexity and the lengths of first-order proofs of finite consistency statements. Later Pudlák [25] studied more statements that connect provability with computational complexity and conjectured that they are true. All these conjectures are at least as strong as $\mathsf {P}\neq \mathsf {NP}$ [23–25].One of the problems concerning these conjectures is to find out how tightly they are connected with statements about computational complexity classes. Results of this kind had been (...) proved in [20, 22].In this paper, we generalize and strengthen these results. Another question that we address concerns the dependence between these conjectures. We construct two oracles that enable us to answer questions about relativized separations asked in [19, 25] (i.e., for the pairs of conjectures mentioned in the questions, we construct oracles such that one conjecture from the pair is true in the relativized world and the other is false and vice versa). We also show several new connections between the studied conjectures. In particular, we show that the relation between the finite reflection principle and proof systems for existentially quantified Boolean formulas is similar to the one for finite consistency statements and proof systems for non-quantified propositional tautologies. (shrink)

The combinatorial parity principle states that there is no perfect matching on an odd number of vertices. This principle generalizes the pigeonhole principle, which states that for a fixed bipartition of the vertices, there is no perfect matching between them. Therefore, it follows from recent lower bounds for the pigeonhole principle that the parity principle requires exponential-size bounded-depth Frege proofs. Ajtai previously showed that the parity principle does not have polynomial-size bounded-depth Frege proofs even with the pigeonhole principle as an (...) axiom schema. His proof utilizes nonstandard model theory and is nonconstructive. We improve Ajtai's lower bound from barely superpolynomial to exponential and eliminate the nonstandard model theory. Our lower bound is also related to the inherent complexity of particular search classes . In particular, oracle separations between the complexity classes PPA and PPAD, and between PPA and PPP also follow from our techniques. (shrink)

This paper presents the algebraic and Kripke modelsoundness and completeness ofa logic over Boolean monoids. An additional axiom added to thelogic will cause the resulting monoid models to be representable as monoidsof relations. A star operator, interpreted as reflexive, transitiveclosure, is conservatively added to the logic. The star operator isa relative modal operator, i.e., one that is defined in terms ofanother modal operator. A further example, relative possibility,of this type of operator is given. A separate axiom,antilogism, added to the logic (...) causes the Kripke models to support acollection of abstract topological uniformities which become concretewhen the Kripke models are dual to monoids of relations. The machineryfor the star operator is shownto be a recasting of Scott-Montague neighborhood models. An interpretationof the Kripke frames and properties thereof is presented in terms ofcertain CMOS transister networks and some circuit transformation equivalences.The worlds of the Kripke frame are wires and the Kripke relation is a specializedCMOS pass transistor network. (shrink)

In this paper we find an algebraic equivalent of the Hallden property in modal logics, namely, we prove that the Hallden-completeness in any normal modal logic is equivalent to the so-called super-embedding property of a suitable class of modal algebras. The joint embedding property of a class of algebras is equivalent to the Pseudo-Relevance Property. We consider connections of the above-mentioned properties with interpolation and amalgamation. Also an algebraic equivalent of of the principle of variable separation in superintuitionistic (...) logics will be found. (shrink)

l investigate versions of the Maximality Principles for the classes of forcings which are <κ-closed. <κ-directed-closed, or of the form Col (κ. <Λ). These principles come in many variants, depending on the parameters which are allowed. I shall write MPΓ(A) for the maximality principle for forcings in Γ, with parameters from A. The main results of this paper are: • The principles have many consequences, such as <κ-closed-generic $\Sigma _{2}^{1}(H_{\kappa})$ absoluteness, and imply. e.g., that ◇κ holds. I give an application (...) to the automorphism tower problem, showing that there are Souslin trees which are able to realize any equivalence relation, and hence that there are groups whose automorphism tower is highly sensitive to forcing. • The principles can be separated into a hierarchy which is strict, for many κ. • Some of the principles can be combined, in the sense that they can hold at many different κ simultaneously. The possibilities of combining the principles are limited, though: While it is consistent that MP<κ-closed(HκT) holds at all regular κ below any fixed α. the "global" maximality principle, stating that MP<κ-closed(Hκ ∪ {κ}) holds at every regular κ. is inconsistent. In contrast to this, it is equiconsistent with ZFC that the maximality principle for directed-closed forcings without any parameters holds at every regular cardinal. It is also consistent that every local statement with parameters from HκT that's provably <κ-closed-forceably necessary is true, for all regular κ. (shrink)

The purpose of the present study was to profile high school students’ achievement as a function of their demographic characteristics, parent attributes, and school behaviors. Students were nested within schools in the Saudi Arabia Kingdom. Out of a large sample of 500k, participants involved 3 random samples of 2,000 students measured during the years 2016, 2017, and 2018. Randomization was conducted at the student level to ensure that all school units will be represented and at their respective frequency. Students were (...) nested within 50 high schools. We adopted the multilevel latent profile analysis protocol put forth by Schmiege et al. and Mäkikangas et al. that account for nested data and tested latent class structure invariance over time. Results pointed to the presence of a 4-profile solution based on BIC, the Bayes factor, and several information criteria put forth by Masyn. Latent profile separation was mostly guided by parents’ education and the number of student absences. Two models tested whether the proportions of level 1 profiles to level 2 units are variable and whether level 2 profiles vary as a function of level 1 profiles. Results pointed to the presence of significant variability due to schools. (shrink)

Based on a multiyear study, this article analyzes the reproduction of adult gender segregation in two youth-sports organizations in which most men volunteers become coaches and most women volunteers become “team moms.” We use interviews and participant observation to explore how these gender divisions are created. While most participants say the divisions result from individual choices, our interviews show how gendered language, essentialist beliefs, and analogies with gendered divisions of labor in families and work-places naturalize this division of labor. Observation (...) reveals how patterned, informal interactions reproduce it as well. We show how nonreflexive informal interactions at the nexus of three gender regimes—youth sports, families, and workplaces—produce a gender formation with two interrelated characteristics: an ascendant professional class gender ideology that we call “soft essentialism” and a “gender category sorting system” that channels most men into coaching and most women into being “team moms.”. (shrink)

In this note a T1 formal space is a formal space whose points are closed as subspaces. Any regular formal space is T1. We introduce the more general notion of a formal space, and prove that the class of points of a weakly set-presentable formal space is a set in the constructive set theory CZF. The same also holds in constructive type theory. We then formulate separation properties for constructive topological spaces , strengthening separation properties discussed elsewhere. Finally we (...) relate the properties for ct-spaces with corresponding properties of formal spaces. (shrink)

The United States Patent Office of the 1850s offers a rare opportunity to analyze the early gendering of science. In its crowded rooms, would-be scientists shared a workplace with women earning equal pay for equal work. Scientific men worked as patent examiners, claiming this new occupation as scientific in opposition to those seeking to separate science and technology. At the same time, in an unprecedented and ultimately unsuccessful experiment, female clerks were hired to work alongside male clerks. This article examines (...) the controversies surrounding these workers through the lens of manners and deportment. In the unique context of a workplace combining scientific men and working ladies, office behavior revealed the deep assumption that the emerging American scientist was male and middle class. (shrink)

We prove that, given a planar embedding of a graph in the sphere, the expansion of the graph structure by predicates encoding vertex separation by simple graph cycles is dp-minimal. This provides a rich natural class of examples of unstable, dp-minimal, and also monadically NIP theories. We also show how to infer the existence of a distal expansion of the theory of the Farey graph.

A $\Pi^0_1$ class can be defined as the set of infinite paths through a computable tree. For classes $P$ and $Q$, say that $P$ is Medvedev reducible to $Q$, $P \leq_M Q$, if there is a computably continuous functional mapping $Q$ into $P$. Let $\mathcal{L}_M$ be the lattice of degrees formed by $\Pi^0_1$ subclasses of $2^\omega$ under the Medvedev reducibility. In "Non-branching degrees in the Medvedev lattice of $\Pi \sp{0}\sb{1} classes," I provided a characterization of nonbranching/branching and a classification (...) of the nonbranching degrees. In this paper, I present a similar classification of the branching degrees. In particular, $P$ is separable if there is a clopen set $C$ such that $P \cap C \neq \emptyset \neq P \cap C^c$ and $P \cap C \perp_M P \cap C^c$. By the results in the first paper, separability is an invariant of a Medvedev degree and a degree is branching if and only if it contains a separable member. Further define $P$ to be hyperseparable if, for all such $C$, $P \cap C \perp_M P \cap C^c$ and totally separable if, for all $X,Y \in P$, $X \perp_T Y$. I will show that totally separable implies hyperseparable implies separable and that the reverse implications do not hold, that is, that these are three distinct types of branching degrees. Along the way I will show some related results and present a combinatorial framework for constructing $\Pi^0_1$ classes with priority arguments. (shrink)

Atoms and molecules are particular kinds of restricted n-body systems, which generally behave as quasi-separable, unlike other n-body systems, e.g., Newtonian ones. The Coulomb repulsion and the Pauli exclusion principle in atoms and molecules are responsible for that separability. Additionally, chemical bonds, especially covalent bonds, enhance the separability of molecules. Independent particle models do not describe atoms and molecules since first-order energy corrections are high. However, these corrections obtained by the first-order perturbation or mean-field strongly converge, implying a one-electron effective (...) potential description. Consequently, stable states of atoms and molecules can be reasonably described by one-electron effective potentials, which strongly differ from other n-body problems. We discuss the peculiarities of the correlation motion of generic systems in the context of the four fundamental forces. In particular, we have shown that the two components (attraction and repulsion) of the electromagnetic force confer a relatively low correlation motion to atoms and molecules. We discuss the physical and chemical nature of atoms and molecules, comparing the degrees of separability between different systems. For example, the separability of Newtonian systems is generally possible in particular classes of restricted systems due to relative mass differences. However, for atoms and molecules, the separability is much broader. (shrink)

We investigate a certain system of intuitionistic set theory from three points of view: an elementary set theory with bounded separation, a topos with distinguished inclusions, and a category of classes with a system of small maps. The three presentations are shown to be equivalent in a strong sense.

We prove the Decomposability Conjecture for functions of Baire class $2$ from a Polish space to a separable metrizable space. This partially answers an important open problem in descriptive set theory.

Michel Foucault’s analysis of pastoral power has generated a large body of work in many different disciplines. Much of it has considered the paradox of the power of “each and all” or has seen pastoral power as an extension of the disciplinary gaze into welfare state policy. The political economy of the pastorate and the mutual dependence of sovereign and pastoral power, by contrast, are both relatively neglected. This article focuses on the exercise of pastoral power in a moral and (...) political economy and examines the “arts of government” through which the Catholic Church attempted to claim that pastors lived from the flock only to live for it. While there is heuristic value in Foucault’s diagram of pastoral power, in practice that power cannot be separated from class relations and political sovereignty. Empirical material is drawn from the novel attempt of Britain to govern its Quebec colony through the Catholic Church. (shrink)

This essay argues that what separates poor mothers from their middle class counterparts are many factors — not simply couples' ability to plan, the power of their choices, and agency. For example, power, social clout, and access to health care influence status and parenting. Historically, for white women choice has been about abortion; for many women of color, choice is about being able to be a mother. This essay begs the question whether real parallels can be drawn on questions (...) of contraception, abortion, and ART simply by examining class and to what extent race and sexual orientation matter in research on parenting. (shrink)

We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

An unresolved issue in the occupational health literature that is of both theoretical and practical importance is whether interpersonal conflicts, aggression and bullying at work are distinct or overlapping phenomena for exposed workers. In this study, we addressed this question empirically by employing a Latent Class (LC) analysis using cross-industry data from 6,175 Belgian workers. We found that a two-factor solution with a conflict-aggression factor and a bullying factor had the best fit. Employees with low exposure to workplace conflicts-aggression (...) and bullying perceived the phenomena as mostly overlapping. Reversely, employees who were exposed more frequently to the phenomena perceived them as more distinct - especially so for workplace bullying. Conflicts-aggression and bullying seem to be perceived as separate phenomena given their distinctive relationships with criterion variables in the domain of well-being and strain. These findings entail that a simple unifying approach or a single label for the three phenomena is not appropriate, at least from a measurement point of view and from the different perspectives of those exposed. Our results have important implications for the theoretical understanding of the concepts of interpersonal conflicts, aggression and bullying, and for practitioners who have to assist those involved with how to handle these problems and who have to develop legal as well as internal policies for the prevention and management of these issues at the workplace. (shrink)

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL0. We study analogies and differences between WKL0 and the (...) class='Hi'>class of Sep-computable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the Hahn-Banach Extension Theorem is Sep-complete. (shrink)