It is proved that the (2p)-c. e. e-degrees are not elementarily equivalent to the (2p + 1)-c. e. e-degrees for each nonzero p ∈ ω. It follows that m-c. e. e-degrees are not elementarily equivalent to the n-c. e. e-degrees if 1 < m < n.

It is proved that if 1 $\langle \mathscr{D}_{2n}, \leq, P\rangle$ and $\langle \mathscr{D}_{2n}, \leq, P\rangle$ are not elementary equivalent where P is the predicate P(a) = "a is a Π 0 1 e-degree".

The variety \(\mathbb{DHMSH}\) of dually hemimorphic semi-Heyting algebras was introduced in 2011 by the second author as an expansion of semi-Heyting algebras by a dual hemimorphism. In this paper, we focus on the variety \(\mathbb{DHMSH}\) from a logical point of view. The paper presents an extensive investigation of the logic corresponding to the variety of dually hemimorphic semi-Heyting algebras and of its axiomatic extensions, along with an equally extensive universal algebraic study of their corresponding algebraic semantics. Firstly, we present a (...) Hilbert-style axiomatization of a new logic called "Dually hemimorphic semi-Heyting logic" (\(\mathcal{DHMSH}\), for short), as an expansion of semi-intuitionistic logic \(\mathcal{SI}\) (also called \(\mathcal{SH}\)) introduced by the first author by adding a weak negation (to be interpreted as a dual hemimorphism). We then prove that it is implicative in the sense of Rasiowa and that it is complete with respect to the variety \(\mathbb{DHMSH}\). It is deduced that the logic \(\mathcal{DHMSH}\) is algebraizable in the sense of Blok and Pigozzi, with the variety \(\mathbb{DHMSH}\) as its equivalent algebraic semantics and that the lattice of axiomatic extensions of \(\mathcal{DHMSH}\) is dually isomorphic to the lattice of subvarieties of \(\mathbb{DHMSH}\). A new axiomatization for Moisil's logic is also obtained. Secondly, we characterize the axiomatic extensions of \(\mathcal{DHMSH}\) in which the "Deduction Theorem" holds. Thirdly, we present several new logics, extending the logic \(\mathcal{DHMSH}\), corresponding to several important subvarieties of the variety \(\mathbb{DHMSH}\). These include logics corresponding to the varieties generated by two-element, three-element and some four-element dually quasi-De Morgan semi-Heyting algebras, as well as a new axiomatization for the 3-valued Łukasiewicz logic. Surprisingly, many of these logics turn out to be connexive logics, only a few of which are presented in this paper. Fourthly, we present axiomatizations for two infinite sequences of logics namely, De Morgan Gödel logics and dually pseudocomplemented Gödel logics. Fifthly, axiomatizations are also provided for logics corresponding to many subvarieties of regular dually quasi-De Morgan Stone semi-Heyting algebras, of regular De Morgan semi-Heyting algebras of level 1, and of JI-distributive semi-Heyting algebras of level 1. We conclude the paper with some open problems. Most of the logics considered in this paper are discriminator logics in the sense that they correspond to discriminator varieties. Some of them, just like the classical logic, are even primal in the sense that their corresponding varieties are generated by primal algebras. (shrink)

Faults; religious aspects; Islam; Islamic daʻwah.

Muslim women; religious life; Islmic call.

[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)

Now in a third edition, Introduction to Philosophy: Classical and Contemporary Readings is a highly acclaimed, topically organized collection that covers five major areas of philosophy--theory of knowledge, philosophy of religion, philosophy of mind, freedom and determinism, and moral philosophy. Editor Louis P. Pojman enhances the text's topical organization by arranging the selections into a pro/con format to help students better understand opposing arguments. He also includes accessible introductions to each chapter, subsection, and individual reading, a unique feature for an (...) anthology of this depth. While the book focuses on a compelling sampling of classical material--including selections from Plato, Aristotle, Descartes, Locke, Berkeley, Hume, and Kant--it also incorporates some of philosophy's best twentieth-century and contemporary work, featuring articles by Bertrand Russell, Richard Taylor, John Searle, Thomas Nagel, and others. This third edition contains an expanded glossary, more extensive section introductions, and twelve new selections: Karl Popper: "Epistemology without a Knowing Subject" Richard Rorty: "Dismantling Truth: Solidarity Versus Objectivity" Daniel Dennett: "Postmodernism and Truth" Bruce Russell: "The Problem of Evil: Too Much Suffering" David Chalmers: "Against Materialism: Can Consciousness Be Reductively Explained?" Baron Paul Henri d'Holbach: "A Defense of Determinism" Michael Levin: "A Compatibilist Defense of Moral Responsibility" Plato: Socratic Morality: "Crito" Herodotus: "Custom Is King" J. L. Mackie: "The Subjectivity of Values" Louis P. Pojman: "A Critique of Mackie's Theory of Moral Subjectivism" Thomas Nagel: "Moral Luck". (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.