We show that if M is an r‐maximal set, A is a major subset of M, B is an arbitrary set and, then. We prove that the c.e. ‐degrees are not dense. We also show that there exist infinite collections of ‐degrees and such that the following hold: (i) for every i, j,, and,(ii) each consists entirely of r‐maximal sets, and(iii) each consists entirely of non‐r‐maximal hyperhypersimple sets.

Euphorion SH 417 deserves to be better known. A curiosity in itself—an apparent poetic reference to number theory—it is also, potentially, one of our earliest references to Euclidean material. On the authority of a late commentator on Aristotle, Euphorion, a mid-third-century b.c. Euboean poet who was also active in Athens and Antioch, is said to have mentioned perfect numbers—i.e. numbers which equal the total of all their factors, including 1 . It is a pity that the context in Euphorion does (...) not survive, and that the line is only preserved, and indeed interpreted, by so late a source. But the wording of the fragment—if Westerink's restoration of its various corruptions is plausible—would strongly suggest a reference to the notion of perfect number. The fragment has been known since Westerink published it in 1960, and was included both in Van Groningen's edition of Euphorion in 1977 and in Supplementum Hellenisticum . But its implications have still not been discussed, and when David Fowler came to gather the evidence for references to Euclidean material in and after the third century b.c. in The Mathematics of Plato's Academy, his attention, unsurprisingly, was not drawn to it. Euphorion has had a bad press, as a poet of rebarbatively obscure myth and intractable vocabulary; yet he holds some interest, and we may be missing more insights into the intellectual life of the Hellenistic period which the perverse intelligence and baneful wit of the fragments display. (shrink)

Euphorion SH 417 deserves to be better known. A curiosity in itself—an apparent poetic reference to number theory—it is also, potentially, one of our earliest references to Euclidean material. On the authority of a late commentator on Aristotle, Euphorion, a mid-third-century b.c. Euboean poet who was also active in Athens and Antioch, is said to have mentioned perfect numbers—i.e. numbers which equal the total of all their factors, including 1. It is a pity that the context in Euphorion does not (...) survive, and that the line is only preserved, and indeed interpreted, by so late a source. But the wording of the fragment—if Westerink's restoration of its various corruptions is plausible—would strongly suggest a reference to the notion of perfect number. The fragment has been known since Westerink published it in 1960, and was included both in Van Groningen's edition of Euphorion in 1977 and in Supplementum Hellenisticum. But its implications have still not been discussed, and when David Fowler came to gather the evidence for references to Euclidean material in and after the third century b.c. in The Mathematics of Plato's Academy, his attention, unsurprisingly, was not drawn to it. Euphorion has had a bad press, as a poet of rebarbatively obscure myth and intractable vocabulary; yet he holds some interest, and we may be missing more insights into the intellectual life of the Hellenistic period which the perverse intelligence and baneful wit of the fragments display. (shrink)

The subjective effects and therapeutic potential of the shamanic practice of journeying is well known. However, previous research has neglected to provide a comprehensive assessment of the subjective effects of shamanic-like journeying techniques on non-shamans. Shamanic-like techniques are those that demonstrate some similarity to shamanic practices and yet deviate from what may genuinely be considered shamanism. Furthermore, the personality traits that influence individual susceptibility to shamanic-like techniques are unclear. The aim of the present study was, thus, to investigate experimentally the (...) effect of shamanic-like techniques and a personality trait referred to as "ego boundaries" on subjective experience including mood disturbance. Forty-three non-shamans were administered a composite questionnaire consisting of demographic items and a measure of ego boundaries (i.e., the Short Boundary Questionnaire; BQ-Sh). Participants were randomly assigned to one of three conditions: listening to monotonous drumming for 15 minutes coupled with one of two sets of journeying instructions; or sitting quietly with eyes closed for 15 minutes. Participants' subjective experience and mood disturbance were retrospectively assessed using the Phenomenology of Consciousness Inventory (PCI) and the Profile of Mood States-Short Form, respectively. The results indicated that there was a statistically significant difference between conditions with regard to the PCI major dimensions of visual imagery, attention and rationality, and minor dimensions of imagery amount and absorption. However, the shamanic-like conditions were not associated with a major reorganization of the pattern of subjective experience compared to the sitting quietly condition, suggesting that what is typically referred to as an altered state of consciousness effect was not evident. One shamanic-like condition and the BQ-Sh subscales need for order, childlikeness, and sensitivity were statistically significant predictors of total mood disturbance. Implications of the findings for the anthropology of consciousness are also considered. (shrink)

The ritual handling of serpents remains an unnoticed cultural form for the explanatory aims and theoretical insights desired by cognitive scientists of religion. In the current article, we introduce the Hood and Williams archives at The University of Tennessee at Chattanooga that contains data culled from Hood’s 40-plus year career of studying serpent handlers. The archives contain hundreds of hours of interviews and recordings of speaking in tongues, handling fire, drinking poison, and taking up serpents by different congregants and congregations. (...) The archive remains a rich but untapped source of data for building, testing, and refining cognitive theories of ritual in general, and serpent handling in specific. We connect Hood’s work to current cognitive theories and engage critically with research on the social functions of ritual. Finally, we discuss several further reasons to pay more attention to SHS communities and practices in cognitive theories of ritual. (shrink)

Information can be public among a group. Whether or not information is public matters, for example, for accounts of interdependent rational choice, of communication, and of joint intention. A standard analysis of public information identifies it with (some variant of) common belief. The latter notion is stipulatively defined as an infinite conjunction: for p to be commonly believed is for it to believed by all members of a group, for all members to believe that all members believe it, and so (...) forth. This analysis is often presupposed without much argument in philosophy. Theoretical entrenchment or intuitions about cases might give some traction on the question, but give little insight about why the identification holds, if it does. The strategy of this paper is to characterize a practical-normative role for information being public, and show that the only things that play that role are (variants of) common belief as stipulatively characterized. In more detail: a functional role for “taking a proposition for granted” in non-isolated decision making is characterized. I then present some minimal conditions under which such an attitude is correctly held. The key assumption links this attitude to beliefs about what is public. From minimal a priori principles, we can argue that a proposition being public among a group entails common commitment to believe among that group. Later sections explore partial converses to this result, the factivity of publicity and publicity from the perspective of outsiders to the group, and objections to the aprioricity of the result deriving from a posteriori existential presuppositions. (shrink)

Should we regard Jennifer Lackey’s ‘Creationist Teacher’ as understanding evolution, even though she does not, given her religious convictions, believe its central claims? We think this question raises a range of important and unexplored questions about the relationship between understanding, factivity and belief. Our aim will be to diagnose this case in a principled way, and in doing so, to make some progress toward appreciating what objectual understanding—i.e., understanding a subject matter or body of information—demands of us. Here is the (...) plan. After some ground clearing in §1, §2 outlines and motivates a plausible working model—moderate factivity—for characterising the sense in which objectual understanding should be regarded as factive. §3 shows how the datum that we can understand false theories can, despite initial suggestions to the contrary, be assimilated straightforwardly within the moderate factivity model. §4 highlights how the inverse kind of case to that explored in §3—viz., a variant of Lackey’s creationist teacher case—poses special problems for moderate factivity. With reference to recent work on moral understanding by Hills, §5 proposes a solution to the problem, and §6 attempts to diagnose why it is that we might originally have been led to draw the wrong conclusion. (shrink)

[1] Sefer Orḥot Ḥayim leha-Rosh ʻim beʼur "ha-Shem orḥotenu" ... 2. Ḳunṭres Be-Shaʻar ha-melekh ʻal haḳdamat ha-Rambam, kolel tsiyunim u-meḳorim u-veʼurim be-divre ha-Rambam ṿeha-Raʼabad ... 3. Ḳunṭres Tashlum yefeh ʻenayim ʻal Seder Zeraʻim u-Ṭehorot, ʻeduyot, Tamid, Midot, Ḳenim ṿe-ʻod ṿe-hu tsiyunim ʻal mas. elu mi-Yerushalmi u-midrashim ... 4. Ḳunṭres Marʼot maḳom, ṿe-hu tsiyunim ʻal ha-meḳomot she-Rashi ṿe-Tos. meviʼim mirdrash o Yerushalmi ṿe-Tosefta ṿe-khu. ṿe-lo tsuyan meḳoro ... 5. Ḳunṭres Ṭeʻama de-ḳara, ṿe-hu ḳetsat ḥidushim ʻal ha-Torah ṿe-ʻal Neviʼim u-Khetuvim -- ḥeleḳ (...) 3. Heʻarot ʻal seder ha-Shas (ṿe-kholel gam ezeh ḳunṭresim). (shrink)

Protagoras, of all the ancient philosophers, has perhaps attracted the most interest in modern times. His saying ‘Man is the measure of all things’ caused Schiller to adopt him as the patron of the Oxford pragmatists, and has generally earned him the title of the first humanist. Yet the exact delineation of his philosophcal position remains a baffling task. Neumann, writing on Die Problematik des ‘Homo-mensura’ Satzes in 1938,2 concludes that no certainty whatever can be reached on the meaning of (...) the dictum, since ‘man’, ‘measure’, and ‘things’ are all ambiguous, nor can we tell which of the three is the predicate. The time is past when it was believed that a man's philosophy could be understood apart from the events of his life and the circumstances of his age: yet in the case of Protagoras the historical method of approach has hardly been attempted, although the impact of environment was perhaps more decisive in the formation of philosophical concepts in the Greek world than anywhere else. The reality about which the Greek philosopher spoke had three aspects: it was either the one universe of physics, or the political unity, or God. For example, the book of Heracleitus Περ φσεως was divided into three chapters or logoi, About the Whole, a political, and a theological. Neither physics nor theology supplies from its own subject-matter the form in which this reality was described. (shrink)

The main result of this paper is Theorem 3.1 which is a criterion for weak o-minimality of a linearly ordered structure in terms of realizations of 1-types. Here we also prove some other properties of weakly o-minimal structures. In particular, we characterize all weakly o-minimal linear orderings in the signature $\{ . Moreover, we present a criterion for density of isolated types of a weakly o-minimal theory. Lastly, at the end of the paper we present some remarks on the Exchange (...) Principle for algebraic closure in a weakly o-minimal structure. (shrink)

We show that the e-degree 0'e and the map u ↦ u' are definable in the upper semilattice of all e-degrees. The class of total e-degrees ≥0'e is also definable.

Faults; religious aspects; Islam; Islamic daʻwah.

Muslim women; religious life; Islmic call.