Keywords: Recursively Enumerable; Computably Enumerable; Friedberg Splitting; Congruence Relation; Ideal; Quotient; Lattice.

We investigate the pair-splitting number $\germ{s}_{pair}$ which is a variation of splitting number, pair-reaping number $\germ{r}_{pair}$ which is a variation of reaping number and cardinal invariants of ideals on ω. We also study cardinal invariants of F σ ideals and their upper bounds and lower bounds. As an application, we answer a question of S. Solecki by showing that the ideal of finitely chromatic graphs is not locally Katětov-minimal among ideals not satisfying Fatou's lemma.

Split brains that result in two simultaneous streams of consciousness cut off from each other are wrongly held to be grounds for doubting the existence of the divinely created soul. The mistake is based on two related errors: first, a failure to appreciate the soul's dependence upon neurological functioning; second, a fallacious belief that if the soul is simple, i.e. without parts, then there must be a unity to its thought, all of its thoughts being potentially accessible to reflection or (...) even unreflective causal interactions. But a soul theorist can allow neurological events to keep some conscious thoughts unavailable to others. (shrink)

We investigate the relationship of the degrees of splittings of a computably enumerable set and the degree of the set. We prove that there is a high computably enumerable set whose only proper splittings are low 2.

Reverse mathematics is a program in the foundations of mathematics founded by Friedman and developed extensively by Simpson and others. The aim of RM is to find the minimal axioms needed to prove a theorem of ordinary, that is, non-set-theoretic, mathematics. As suggested by the title, this paper deals with two RM-phenomena, namely, splittings and disjunctions. As to splittings, there are some examples in RM of theorems A, B, C such that A↔, that is, A can be split (...) into two independent parts B and C. As to disjunctions, there are examples in RM of theorems D, E, F such that D↔, that is, D can be written as the disjunction of two independent parts E and F. By contrast, we show in this paper that there is a plethora of splittings and disjunctions in Kohlenbach’s higher-order RM. (shrink)

This is a contribution to the study of the Muchnik and Medvedev lattices of non-empty Π01 subsets of 2ω. In both these lattices, any non-minimum element can be split, i. e. it is the non-trivial join of two other elements. In fact, in the Medvedev case, ifP > MQ, then P can be split above Q. Both of these facts are then generalised to the embedding of arbitrary finite distributive lattices. A consequence of this is that both lattices have decidible (...) ∃-theories. (shrink)

In this paper, I contend that there are at least two essential traits that commonly define being an I: self-identity and self-consciousness. I argue that they bear quite an odd relation to each other in the sense that self-consciousness seems to jeopardize self-identity. My main concern is to elucidate this issue within the range of the transcendental philosophies of Immanuel Kant and Edmund Husserl. In the first section, I shall briefly consider Kant’s own rendition of the problem of the Egosplitting. (...) My reading of the Kantian texts reveals that Kant himself was aware of this phenomenon but eventually deems it an unexplainable fact. The second part of the paper tackles the same problematic from the standpoint of Husserlian phenomenology. What Husserl’s extensive analyses on this topic bring to light is that the phenomenon of the Ego-splitting constitutes the bedrock not only of his thought but also of every philosophy that works within the framework of transcendental thinking. (shrink)

A common presupposition in the concepts literature is that concepts constitute a sin- gular natural kind. If, on the contrary, concepts split into more than one kind, this literature needs to be recast in terms of other kinds of mental representation. We offer two new arguments that concepts, in fact, divide into different kinds: (a) concepts split because different kinds of mental representation, processed independently, must be posited to explain different sets of relevant phenomena; (b) concepts split because different kinds (...) of mental representation, processed independently, must be posited to explain responses to different kinds of category. Whether these arguments are sound remains an open empirical question, to be resolved by future empirical and theoretical work. (shrink)

Is it time to take a break from feminism? In this pathbreaking book, Janet Halley reassesses the place of feminism in the law and politics of sexuality. She argues that sexuality involves deeply contested and clashing realities and interests, and that feminism helps us understand only some of them. To see crucial dimensions of sexuality that feminism does not reveal--the interests of gays and lesbians to be sure, but also those of men, and of constituencies and values beyond the realm (...) of sex and gender--we might need to take a break from feminism. Halley also invites feminism to abandon its uncritical relationship to its own power. Feminists are, in many areas of social and political life, partners in governance. To govern responsibly, even on behalf of women, Halley urges, feminists should try taking a break from their own presuppositions. Halley offers a genealogy of various feminisms and of gay, queer, and trans theories as they split from each other in the United States during the 1980s and 1990s. All these incommensurate theories, she argues, enrich thinking on the left not despite their break from each other but because of it. She concludes by examining legal cases to show how taking a break from feminism can change your very perceptions of what's at stake in a decision and liberate you to decide it anew. (shrink)

A splitting of an r.e. set A is a pair A1, A2 of disjoint r.e. sets such that A1 A2 = A. Theorems about splittings have played an important role in recursion theory. One of the main reasons for this is that a splitting of A is a decomposition of A in both the lattice, , of recursively enumerable sets and in the uppersemilattice, R, of recursively enumerable degrees . Thus splitting theor ems have been used to obtain results (...) about the structure of , the structure of R, and the relationship between the two structures. Furthermore it is fair to say that questions about splittings have often generated important new technical developments in recursion theory. In this article we survey most of the results and techniques associated with splitting properties of r.e. sets in ordinary recursion theory. (shrink)

Rudolf Arnheim's great forte is his ability to illuminate the perceptual processes that go into the making and reception of artworks—painting, sculpture, architecture, and film. Over the years, his pioneering mode of "reading" art from a unique scientific/philosophic perspective has garnered him an established and devoted audience. That audience will take pleasure in Arnheim's most recent collection of essays, one that covers a range of topics and includes titles such as "Outer Space and Inner Space," "What Is an Aesthetic Fact?," (...) "As I Saw Children's Art," "Two Ways of Being Human," "Consciousness—an Island of Images," and "From Chaos to Wholeness." The notion of structure is Arnheim's guide in these explorations. Most of the essays examine the nature of structure affirmatively: how it comes about, its incentives and objectives, its celebration of perfection. He is interested in how artists grope for structure to shape powerful, enlightening images, and how a scientist's search for truth is a search for structure. Writing with enviable clarity, even when deploying complex arguments, Arnheim makes it easy and exciting to follow him as he thinks. America is not abundantly supplied with "public intellectuals" such as Rudolf Arnheim—to have his writings with us is cause for celebration. "The word 'structure' appears for good reason in the title of this collection.... Structure seems to be needed as an arbiter wherever this civilization of ours is split by selfish interests and fighting for either/or decisions. The essays want to speak with the voice of reason, because they want to show how the parts require the whole.". (shrink)

We investigate splitting number and reaping number for the structure (ω) ω of infinite partitions of ω. We prove that ${\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}}$ and ${\mathfrak{s}_{d}\geq\mathfrak{b}}$ . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}$ and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}$ . To prove the consistency ${\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}$ and ${\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})}$ we introduce new cardinal invariants ${\mathfrak{r}_{pair}}$ and ${\mathfrak{s}_{pair}}$ . We also study the relation between ${\mathfrak{r}_{pair}, \mathfrak{s}_{pair}}$ and other cardinal invariants. We show (...) that ${\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\ma thfrak{r}}$ and ${\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})}$. (shrink)

After the first callosotomy surgeries were performed, the general consensus among the medical community was that severing the corpus callosum had relatively little, if any, effect on an individual's behavior. Nearly twenty years later, researchers discovered that, under experimental conditions, the two hemispheres could simultaneously maintain very different interpretations of the same stimulus. These findings immediately called into question the unity of subjective experience, a fundamental characteristic of human consciousness. How could the split‐brain patient not experience any disruption in his (...) experience as a unified self when the two hemispheres are physically and functionally disconnected? In this chapter, we review the research that has led to a conceptualization of the split brain as two minds within one body and the implications of this research for the scientific study of consciousness. We argue that consciousness is a neural function that emerges from the integration of information across available functional modules. (shrink)

The traditional scope theory of intensionality (STI) (see Russell 1905; Montague 1973; Ladusaw 1977; Ogihara 1992, 1996; Stowell 1993) is simple, elegant, and, for the most part, empirically adequate. However, a few quite troubling counterexamples to this theory have lead researchers to propose alternatives, such as positing null situation pronouns (Percus 2000) or actuality operators (Kamp 1971; Cresswell 1990) in the syntax of natural language. These innovative theories do correct the undergeneration of the original scope theory, but at a cost: (...) the situation pronoun and operator theories overgenerate, as argued extensively by Percus (2000) and Keshet (2008). This paper presents new data that supports the STI over other analyses, such as structures where DPs lose their de re readings in positions where syntactic movement is blocked. These data point the way to a new theory of intensionality. This new theory, called split intensionality, is a modification of the STI which aims to solve the problems raised for the original scope theory without overgenerating. The proposal calls for an additional intensional abstraction operator that creates an expression denoting an intension from an expression denoting an extension. When a DP moves to a position above this operator, it is interpreted de re; otherwise it is de dicto. The crucial part of the new proposal is that a DP may move above this operator and yet remain, for instance, below an intentional verb or inside an if-clause. Therefore, a DP within an island for syntactic movement may be de re and yet not move out of the island when the intensional abstraction operator is also within the island. (shrink)

Elizabeth Schechter explores the implications of the experience of people who have had the pathway between the two hemispheres of their brain severed, and argues that there are in fact two minds, subjects of experience, and intentional agents inside each split-brain human being: right and left. But each split-brain subject is still one of us.

A splitting A1A2 = A of an r.e. set A is called a Friedberg splitting if for any r.e. set W with W — A not r.e., W — Ai≠0 for I = 1,2. In an earlier paper, the authors investigated Friedberg splittings of maximal sets and showed that they formed an orbit with very interesting degree-theoretical properties. In the present paper we continue our investigations, this time analyzing Friedberg splittings and in particular their orbits and degrees for (...) various classes of r.e. sets. (shrink)

In this paper I argue that split-brain syndrome is best understood within an extended mind framework and, therefore, that its very existence provides support for an externalist account of conscious perception. I begin by outlining the experimental aberration model of split-brain syndrome and explain both: why this model provides the best account of split-brain syndrome; and, why it is commonly rejected. Then, I summarise Susan Hurley’s argument that split-brain subjects could unify their conscious perceptual field by using external factors to (...) stand-in for the missing corpus callosum. I next provide an argument that split-brain subjects do unify their perceptual fields via external factors. Finally, I explain why my account provides one with an experimental aberration model which avoids the problems typically levelled at such views, and highlight some empirical predictions made by the account. The nature of split-brain syndrome has long been considered mysterious by proponents of internalist accounts of consciousness. However, in this paper I argue that externalist theories can provide a straightforward explanation of the condition. I therefore conclude that the ability of externalist accounts to explain split-brain syndrome gives us strong reason to prefer them over internalist rivals. (shrink)

Negotiation is not only used to settle differences of interest but also to settle differences of opinion. Discussants who are unable to resolve their difference about the objective worth of a policy or action proposal may be willing to abandon their attempts to convince the other and search instead for a compromise that would, for each of them, though only a second choice yet be preferable to a lasting conflict. Our questions are: First, when is it sensible to enter into (...) negotiations and when would this be unwarranted or even fallacious? Second, what is the nature of a compromise? What does it mean to settle instead of resolve a difference of opinion, and what might be the dialectical consequences of mistaking a compromise for a substantial resolution? Our main aim is to contribute to the theory of argumentation within the context of negotiation and compromise formation and to show how arguing disputants can shift to negotiation in a dialectically virtuous way. (shrink)

We propose a Condorcet-consistent voting method that we call Split Cycle. Split Cycle belongs to the small family of known voting methods satisfying the anti-vote-splitting criterion of independence of clones. In this family, only Split Cycle satisfies a new criterion we call immunity to spoilers, which concerns adding candidates to elections, as well as the known criteria of positive involvement and negative involvement, which concern adding voters to elections. Thus, in contrast to other clone-independent methods, Split Cycle mitigates both “spoiler (...) effects” and “strong no show paradoxes.”. (shrink)

In this paper, I contend that there are at least two essential traits that commonly define being an I: self-identity and self-consciousness. I argue that they bear quite an odd relation to each other in the sense that self-consciousness seems to jeopardize self-identity. My main concern is to elucidate this issue within the range of the transcendental philosophies of Immanuel Kant and Edmund Husserl. In the first section, I shall briefly consider Kant’s own rendition of the problem of the Ego-splitting. (...) My reading of the Kantian texts reveals that Kant himself was aware of this phenomenon but eventually deems it an unexplainable fact. The second part of the paper tackles the same problematic from the standpoint of Husserlian phenomenology. What Husserl’s extensive analyses on this topic bring to light is that the phenomenon of the Ego-splitting constitutes the bedrock not only of his thought but also of every philosophy that works within the framework of transcendental thinking. (shrink)

This article analyzes US drone operations through a historical and ethnographic analysis of the remote split paradigm used by the US Air Force. Remote split refers to the globally distributed command and control of drone operations and entails a network of human operators and analysts in the Middle East, Europe, and Southeast Asia as well as in the continental United States. Though often viewed as a teleological progression of “unmanned” warfare, this paper argues that historically specific technopolitical logics establish the (...) conditions of possibility for the work of war to be divisible into discreet and computationally mediated tasks that are viewed as effective in US military engagements. To do so, the article traces how new forms of authorized evidence and expertise have shaped developments in military operations and command and control priorities from the Cold War and the “electronic battlefield” of Vietnam through the Gulf War and the conflict in the Balkans to contemporary deployments of drone operations. The article concludes by suggesting that it is by paying attention to divisions of labor and human–machine configurations that we can begin to understand the everyday and often invisible structures that sustain perpetual war as a military strategy of the United States. (shrink)

Many have claimed that split-brain patients are actually two persons. I maintain that both the traditional separation argument and the more recent sophistication argument for the two persons interpretation are inadequate on conceptual grounds. An autonomy argument is inadequate on empirical grounds. Overall, theoretical and practical consequences weigh heavily in favor of adopting a one person interpretation.

Instagram serves as a powerful instrument for youth socialization, self-expression, and self-performance in visual online spaces. Using social semiotics and multimodal discourse analysis, this study examines the potential ideological meanings and implications of selfie-shooting and sharing on Instagram on young adults’ self-concept. A corpus of 110 questionnaires, including almost 85 captioned selfies, was surveyed as multimodal utterances. In doing so, this study argues that selfies can create young adults’ split-selves while constructing their multiple personas in visual online spaces. This marks (...) the significance of viewing selfie-creators not only as authors of their selfies, but rather as viewers of a three-fold self: an ideal-self, a projected-self, and an internal-self, to negotiate social and power relationships, while (re)positioning observer-observed roles. This study claims originality in unraveling how young adults use visual and textual mediated communication to represent and perform their split-selves. It suggests that selfie-shooting-sharing has become a key self-performance tactic and behavior in online cultures. Therefore, young adults deploy selfies and captions to posit a redefinition of certain social values, such as aesthetics and freedom, while deploying their selfies and captions. Challenging certain orthodox social allegiances, they conceive wildness, messiness, and exuberance as emerging neo-aesthetics components of appeal. This study contributes to the literature on personal visual communication with insights on how Egyptian young adults perform their self-concept via the semiotic practice of selfie-shooting-sharing. (shrink)

We investigate some notions of splitting families and estimate sizes of the corresponding cardinal coefficients. In particular we solve a problem of P. Simon.

Splitting livers allows two people (usually an adult and a child) to receive a liver transplant from one donated adult liver, but the risks to the adult recipient are greater than if they had received the equivalent whole liver. It has been suggested, therefore, that splitting livers harms adult recipients. Without liver splitting, however, there would be few livers available for children, and paediatric waiting time and waiting list mortality would significantly increase. In this paper, we argue that although splitting (...) livers makes adults worse off, this should be considered sub-optimal benefit rather than harm. We explore justifications for sub-optimally benefitting adults in this way and consider alternatives to the current approach. We argue that splitting livers masks the more fundamental problem of low paediatric donation rates and that increasing the number of paediatric donations would improve the situation for both adult and paediatric liver patients. (shrink)

We investigate some aspects of bounding, splitting, and almost disjointness. In particular, we investigate the relationship between the bounding number, the closed almost disjointness number, the splitting number, and the existence of certain kinds of splitting families.

This paper argues for a particular semantic decomposition of morphological definiteness. I propose that the meaning of ‘the’ comprises two distinct compositional operations. The first builds a set of witnesses that satisfy the restricting noun phrase. The second tests this set for uniqueness. The motivation for decomposing the denotation of the definite determiner in this way comes from split-scope intervention effects. The two components—the selection of witnesses on the one hand and the counting of witnesses on the other—may take effect (...) at different points in the composition of a constituent, and this has non-trivial semantic consequences when other operators inside the DP take action in between them. In particular, I analyze well-known examples of mutually recursive definite descriptions like ‘the rabbit in the hat’ as examples of definites whose referent-introducing and referent-testing components are interleaved rather than nested. I further demonstrate that this picture leads to a new theory of relative superlative descriptions like ‘the kid who climbed the highest tree’, which explains the previously mysterious role of the definite determiner in licensing such readings. (shrink)

The goal of this paper is to propose a unified approach to the split scope readings of negative indefinites, comparative quantifiers, and numerals. There are two main observations that justify this approach. First, split scope shows the same kinds of restrictions across these different quantifiers. Second, split scope always involves low existential force. In our approach, following Sauerland, natural language determiner quantifiers are quantifiers over choice functions, of type <<,t>,t>. In split readings, the quantifier over choice functions scopes above other (...) operators (such as intensional verbs like must or can). Determiner quantifiers leave a choice-function trace when they move and this trace combines with the noun restriction, which is interpreted low. That split scope always involves low existential force is derived, without stipulation, from Kratzer’s idea that low existential force can be achieved via binding (into the noun restriction). (shrink)

An old conjecture of modal logics states that every splitting of the major systems K4, S4, G and Grz has the finite model property. In this paper we will prove that all iterated splittings of G have fmp, whereas in the other cases we will give explicit counterexamples. We also introduce a proof technique which will give a positive answer for large classes of splitting frames. The proof works by establishing a rather strong property of these splitting frames namely (...) that they preserve the finite model property in the following sense. Whenever an extension Λ has fmp so does the splitting Λ/f of Λ by f. Although we will also see that this method has its limitations because there are frames lacking this property, it has several desirable side effects. For example, properties such as compactness, decidability and others can be shown to be preserved in a similar way and effective bounds for the size of models can be given. Moreover, all methods and proofs are constructive. (shrink)

Neither the George Washington University embryo splitting experiment nor the technique of embryo splitting itself has ethical flaws. The experiment harmed or wronged no one, and the investigators followed intramural review procedures for the experiment, although some might fault them for failing to seek extramural consultation or for not waiting until national guidelines for research on preembryos were developed. Ethical objections to such cloning on the basis of possible loss of individuality, possible lessening of individual worth, and concern about potential (...) harm to the resulting children are discussed and challenged, as are objections to the creation of embryos for the purpose of genetic diagnosis. Many of the ethical questions raised by the George Washington experiment are similar to those posed by existing reproductive technologies that allow the simultaneous production of several embryos. A multidisciplinary group should consider whether regulation of cloning is needed, and laws should be enacted to prohibit a commercial market for all frozen embryos. (shrink)

By splitting idempotent morphisms in the total and base categories of fibrations we provide an explicit elementary description of the Cauchy completion of objects in the categories Fib) of fibrations with a fixed base category \ and Fib of fibrations with any base category. Two universal constructions are at issue, corresponding to two fibered reflections involving the fibration of fibrations \.

A split preorder is a preordering relation on the disjoint union of two sets, which function as source and target when one composes split preorders. The paper presents by generators and equations the category SplPre, whose arrows are the split preorders on the disjoint union of two finite ordinals. The same is done for the subcategory Gen of SplPre, whose arrows are equivalence relations, and for the category Rel, whose arrows are the binary relations between finite ordinals, and which has (...) an isomorphic image within SplPre by a map that preserves composition, but not identity arrows. It was shown previously that SplPre and Gen have an isomorphic representation in Rel in the style of Brauer.The syntactical presentation of Gen and Rel in this paper exhibits the particular Frobenius algebra structure of Gen and the particular bialgebraic structure of Rel, the latter structure being built upon the former structure in SplPre. This points towards algebraic modelling of various categories motivated by logic, and related categories, for which one can establish coherence with respect Rel and Gen. It also sheds light on the relationship between the notions of Frobenius algebra and bialgebra. The completeness of the syntactical presentations is proved via normal forms, with the normal form for SplPre and Gen being in some sense orthogonal to the composition-free, i.e. cut-free, normal form for Rel. The paper ends by showing that the assumptions for the algebraic structures of SplPre, Gen and Rel cannot be extended with new equations without falling into triviality. (shrink)

It is a common assumption in contemporary cognitive neuroscience that discovering a putative realized kind to be dissociably realized (i.e., to be realized in each instance by two or more distinct realizers) mandates splitting that kind. Here I explore some limits on this inference using two deceptively similar examples: the dissociation of declarative and procedural memory and Ramachandran's argument that the self is an illusion.

An argument of A. Borel [Bor—61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.

In this paper we extend to various classes of subreducts of hoops some results about splitting algebras. In particular we prove that every finite chain in the purely implicational fragment of basic hoops is splitting and that every finite chain in the \ fragment of hoops is splitting. We also produce explicitly the splitting equations in most cases.

Ray splitting is a universal phenomenon that occurs with appreciable amplitude in all wave systems when the properties of the system change on a scale smaller than the wave length. We study the quantum implications of ray splitting theoretically and experimentally with the help of ray-splitting billiards in one and two dimensions. We show that Gutzwiller's trace formula works even in the context of ray-splitting systems provided reflection and transmission of waves at ray-splitting boundaries is properly included.

To assess the ability of the disconnected cerebral hemispheres to recognize images of the self, a split-brain patient was tested using morphed self-face images presented to one visual hemifield at a time while making “self/other” judgments. The performance of the right and left hemispheres of this patient as assessed by a signal detection method was not significantly different, though a measure of bias did reveal hemispheric differences. The right and left hemispheres of this patient independently and equally possessed the ability (...) to self-recognize, but only the right hemisphere could successfully recognize familiar others. This supports a modular concept of self-recognition and other-recognition, separately present in each cerebral hemisphere. (shrink)

Some philosophers have argued that so long as two neural events, within a subject, are both of the same type and both carry the same content, then these events may jointly constitute a single mental token, regardless of the sort of causal relation to each other that they bear. These philosophers have used this claim—which I call the “singularity-through-redundancy” position—in order to argue that a split-brain subject normally has a single stream of consciousness, disjunctively realized across the two hemispheres. This (...) paper argues, against this position, that the kind of causal relations multiple neural events bear to each other constrains the mental tokens with which functionalists who are realists can identify them. (shrink)

We study a generalization of the splitting number s to uncountable cardinals. We prove that $\mathfrak{s}(\kappa) > \kappa^+$ for a regular uncountable cardinal κ implies the existence of inner models with measurables of high Mitchell order. We prove that the assumption $\mathfrak{s}(\aleph_\omega) > \aleph_{\omega + 1}$ has a considerable large cardinal strength as well.