We show that, assuming the consistency of a supercompact cardinal, the first inaccessible cardinal can satisfy a strong form of a Löwenheim–Skolem–Tarski theorem for the equicardinality logic L, a logic introduced in [5] strictly between first order logic and second order logic. On the other hand we show that in the light of present day inner model technology, nothing short of a supercompact cardinal suffices for this result. In particular, we show that the Löwenheim–Skolem–Tarski theorem for the equicardinality logic at (...) κ implies the Singular Cardinals Hypothesis above κ as well as Projective Determinacy. (shrink)

We present a downward Löwenheim-Skolem theorem which transfers downward formulas from L ∞,ω to L κ +, ω . The simplest instance is: Theorem 1. Let $\lambda > \kappa$ be infinite cardinals, and let L be a similarity type of cardinality κ at most. For every L-structure M of cardinality λ and every $X \subseteq M$ there exists a model $N \prec M$ containing the set X of power |X| · κ such that for every pair of finite sequences a, (...) b ∈ N $\langle N, \mathbf{a}\rangle \equiv_{\| N \|^+,\omega} \langle N, \mathbf{b}\rangle \Leftrightarrow \langle M, \mathbf{a}\rangle \equiv_{\infty,\omega} \langle M, \mathbf{b}\rangle.$ The following theorem is an application: Theorem 2. Let $\lambda , and suppose χ is a Ramsey cardinal greater than λ. If T has the (χ, L κ +, ω -unsuperstability property, then T has the (χ, L λ +, ω )-unsuperstability property. (shrink)

In this paper we study abstract elementary classes with Löwenheim-Skolem number $\kappa$ , where $\kappa$ is cofinal with $\omega$ , which have finite character. We generalize results obtained by Kueker for $\kappa=\omega$ . In particular, we show that $\mathbb{K}$ is closed under $L_{\infty,\kappa}$ -elementary equivalence and obtain sufficient conditions for $\mathbb{K}$ to be $L_{\infty,\kappa}$ -axiomatizable. In addition, we provide an example to illustrate that if $\kappa$ is uncountable regular then $\mathbb{K}$ is not closed under $L_{\infty,\kappa}$ -elementary equivalence.

The Löwenheim-Hilbert-Bernays theorem states that, for an arithmetical first-order language L, if S is a satisfiable schema, then substitution of open sentences of L for the predicate letters of S...

THEOREM 1. (⋄ ℵ 1 ) If B is an infinite Boolean algebra (BA), then there is B 1 such that $|\operatorname{Aut} (B_1)| \leq B_1| = \aleph_1$ and $\langle B_1, \operatorname{Aut} (B_1)\rangle \equiv \langle B, \operatorname{Aut}(B)\rangle$ . THEOREM 2. (⋄ ℵ 1 ) There is a countably compact logic stronger than first-order logic even on finite models. This partially answers a question of H. Friedman. These theorems appear in §§ 1 and 2. THEOREM 3. (a) (⋄ ℵ 1 ) If (...) B is an atomic ℵ 1 -saturated infinite BA, ψ ε L ω 1ω and $\langle B, \operatorname{Aut} (B)\rangle \models\psi$ then there is B 1 such that $|\operatorname{Aut}(B_1)| \leq |B_1| = \aleph_1$ and $\langle B_1, \operatorname{Aut}(B_1)\rangle\models\psi$ . In particular if B is 1-homogeneous so is B 1 . (b) (a) holds for B = P(ω) even if we assume only CH. (shrink)

Die vorliegende Arbeit benutzt anstatt der üblichen Peano-Russellschen Zeichen die Peirce-Schröderschen Zeichen. Ich bedaure es aus mehr als einem Grunde schwer, daß man von denselben abgewichen is...

Working in a fixed Grothendieck topos Sh(C, J C ) we generalize \({\mathcal{L}_{{\infty},\omega}}\) to allow our languages and formulas to make explicit reference to Sh(C, J C ). We likewise generalize the notion of model. We then show how to encode these generalized structures by models of a related sentence of \({\mathcal{L}_{{\infty},\omega}}\) in the category of sets and functions. Using this encoding we prove analogs of several results concerning \({\mathcal{L}_{{\infty},\omega}}\) , such as the downward Löwenheim–Skolem theorem, the completeness theorem and (...) Barwise compactness. (shrink)

Edmund Gettier is Professor Emeritus at the University of Massachusetts, Amherst. This short piece, published in 1963, seemed to many decisively to refute an otherwise attractive analysis of knowledge. It stimulated a renewed effort, still ongoing, to clarify exactly what knowledge comprises.

This new edition of the classic Contemporary Theories of Knowledge has been significantly updated to include analyses of the recent literature in epistemology.

This edited volume represents a unique addition to the available literature on animal ethics, animal studies, and neuroethics. Its goal is to expand discussions on animal ethics and neuroethics by weaving together different threads: philosophy of mind and animal minds, neuroscientific study of animal minds, and animal ethics. Neuroethical questions concerning animals’ moral status, animal minds and consciousness, animal pain, and the adequacy of animal models for neuropsychiatric disease have long been topics of debate in philosophy and ethics, and more (...) recently also in neuroscientific research. The book presents a transdisciplinary blend of voices, underscoring different perspectives on the broad questions of how neuroscience can contribute to our understanding of nonhuman minds, and on debates over the moral status of nonhuman animals. All chapters were written by outstanding scholars in philosophy, neuroscience, animal behavior, biology, neuroethics, and bioethics, and cover a range of issues and species/taxa. Given its scope, the book will appeal to scientists and students interested in the debate on animal ethics, while also offering an important resource for future researchers. Chapter 13 of this book is available open access under a CC BY 4.0 license at link.springer.com. (shrink)

Le philosophe Jean Baudrillard (1929-2007) a-t-il vraiment existé? Que reste-t-il de lui? "Une élégance certaine de la pensée", affirme l'un de ses meilleurs interprètes, François L'Yvonnet. Le philosophe de La Société de consommation, des Stratégies fatales et des Cool Memories s'attachait à l'idée du fragment comme mode de pensée : car dans le détail, tout est parfait, c'est dans sa reproduction que tout se complique. François L'Yvonnet explore cinq fragments de la philosophie de Jean Baudrillard et de sa biographie pour (...) mieux saisir l'éclat et la jubilation d'une pensée toujours vivante et contemporaine. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the (...) traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)

Excerpt from Der Echte: Und der Xenophontische Sokrates Der xen0phontische Sokrates selbst wieder zwang, die Fuh rung der Untersuchung weit mehr, als bisher geschehen, zu ver breitem. Die Memorabilien sind das Gegentheil eines selbst herrlichen Kunstwerks, weisen an allen Ecken und Enden uber sich hinaus, stehen als ein schwaches Glied in der Kette der sokratischen Literatur und zunachst in der der xenophontischen Schriften. Es galt, sie zunachst als solches zu begreifen und das volle Licht der Parallelen bei Xenophon auf sie (...) einwirken zu lassen. Dabei enthullten sich die Mem. Oft nur als durftiger Abklatsch anderer xenophontischen Schriften, von denen sie erst die farbige Beleuchtung, die Erklarung ihrer Motive, die Auf hellung ihrer Ordnung erhalten. So ward nun moglichst psycho logisch aus Xenophon erklart, was sich aus ihm erklaren liess. Indessen fur die Hauptsache, das Philosophische, stellte Xeno phon's notorische Unselbstandigkeit neue Fragen und Aufgaben, und hier ergab nun eine weitschichtige Vergleichung seine fast vollige philosophische Abhangigkeit im Ganzen wie im Einzelnen vom Kynismus, die in Grad und Art und plausiblen Motiven genau der Abhangigkeit der romischen Praktiker von den stoischen Nachfolgern der Kyniker_ entspricht. Nicht nur die Memorabilien, auch die andern Socratica Xenophon's, sowie vor Allem Cyropadie Hiero Agesilaus, de rep. Lac. Prolog und Epilog des Oynegeticus offenbarten sich mehr und mehr als durchaus kynisirende Schriften, und auch hier liegt dem Verf. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

First published in 1995. Routledge is an imprint of Taylor & Francis, an informa company.

Ingthorsson, McTaggart’s Paradox and the R-theory of Time L. Nathan Oaklander University of Michigan-Flint, USA [email protected] his provocative book, McTaggart’s Paradox, R.D. Ingthors- son argues that McTaggart’s argument for the unreality of time rests on the principle of temporal parity according to which all times or events in time exist equally or co-exist in a sense that is compatible with their being successive. Moreover, since temporal parity is also an essential tenet of the B-theory, McTaggart’s argument against the reality of (...) time can also be used to undermine the B-theory. Ingthorsson argues further that only by adopting an ontologically frugal presentist metaphysics can one avoid McTaggart’s paradox and account for identity through time and change. The aim of this paper is to clarify Russell’s authentic view of time in con- trast to the B-theory which is McTaggart’s misrepresentation of Russell and argue that temporal parity it is not a fundamental tenet of the Rus- sellian theory. For that reason, the R-theory is immune to objections that are based on temporal parity. I shall then offer my own interpretation of McTaggart’s paradox that renders Ingthorsson’s version of presentism subject to it. (shrink)

We present here an approach to the fine structure of L based solely on elementary model theoretic ideas, and illustrate its use in a proof of Global Square in L. We thereby avoid the Lévy hierarchy of formulas and the subtleties of master codes and projecta, introduced by Jensen [3] in the original form of the theory. Our theory could appropriately be called ”Hyperfine Structure Theory”, as we make use of a hierarchy of structures and hull operations which refines the (...) traditional Lα -or Jα-sequences with their Σn-hull operations.§1. Introduction. In 1938, K. Gödel defined the model L of set theory to show the relative consistency of Cantor's Continuum Hypothesis. L is defined as a unionof initial segments which satisfy: L0 = ∅, Lλ = ∪α<λLα for limit ordinals λ, and, crucially, Lα + 1 = the collection of 1st order definable subsets of Lα. Since every transitive model of set theory must be closed under 1st order definability, L turns out to be the smallest inner model of set theory. Thus it occupies the central place in the set theoretic spectrum of models.The proof of the continuum hypothesis in L is based on the very uniform hierarchical definition of the L-hierarchy. The Condensation Lemma states that if π : M → Lα is an elementary embedding, M transitive, then some ; the lemma can be proved by induction on α. If a real, i.e., a subset of ω, is definable over some Lα,then by a Löwenheim-Skolem argument it is definable over some countable M as above, and hence over some, < ω1. This allows one to list the reals in L in length ω1 and therefore proves the Continuum Hypothesis in L. (shrink)

Zagadochnyĭ starik.--T︠S︡iolkovskiĭ v Peterburge.