The right to justification is the central tenet of Rainer Forst's theory of justice: every person has the moral right to receive justification for morally relevant actions. This approach is instructive also from a jurisprudential perspective. One important purpose of the law is to force state authorities to provide justifications for their actions. Therefore, based on the right to legal equality, justification must be provided for any unequal treatment. Likewise, restrictions of fundamental rights in a constitutional state are subject to (...) accountability as well. Binding decisions and judgements must also be justified. Currently, the obligation of argument remains poorly developed on a transnational level. It is therefore important to implement obligations for justification in this context as well. (shrink)

This paper studies John Buridan's octagons of opposition for the de re modal propositions and the propositions of unusual construction. Both Buridan himself and the secondary literature have emphasized the strong similarities between these two octagons (as well as a third one, for propositions with oblique terms). In this paper, I argue that the interconnection between both octagons is more subtle than has previously been thought: if we move beyond the Aristotelian relations, and also take Boolean considerations into account, then (...) the strong analogy between Buridan's octagons starts to break down. These differences in Boolean structure can already be discerned within the octagons themselves; on a more abstract level, they lead to these two octagons having different degrees of Boolean complexity (i.e. Boolean closures of different sizes). These results are obtained by means of bitstring analysis, which is one of the key tools from contemporary logical geometry. Finally, I argue that this historical investigation is directly relevant for the theoretical framework of logical geometry, and discuss how it helps us to address certain open questions in this framework. (shrink)

Hendrik Lorenz presents a comprehensive study of Plato's and Aristotle's conceptions of non-rational desire. They see this as something that humans share with animals, and which aims primarily at the pleasures of food, drink, and sex. Lorenz explores the cognitive resources that both philosophers make available for the explanation of such desires, and what they take rationality to add to the motivational structure of human beings. In doing so, he finds conceptions of the mind that are coherent and (...) deeply integrated with both philosophers' views about such topics as the relation between body and soul, or the nature of the virtues. (shrink)

"Dialogisches Handeln" ist eine Sammlung von philosophischen BeitrAgen zum dialogischen Konstruktivismus im Sinne von Kuno Lorenz, der Philosophie als eine Kommunikation mit Selbstreflexion versteht. Dabei umfaAt der Dialog in diesem Buch verschiedene Disziplinen der Philosophie - von der Anthropologie A1/4ber die Grundlagen von Logik und Sprachphilosophie bis hin zur Wissenschafts- und Erkenntnistheorie. Die Reflexion auf die kommunikativen Grundlagen wissenschaftlichen Austauschs schlieAt dabei Denkperspektiven verschiedener Wissenschaften - von der Psychologie bis zu Physik und Chemie - ebenso ein wie die philosophischen (...) Disziplinen. (shrink)

What are the relationships between philosophy and the history of philosophy, the history of science and the philosophy of science? This selection of essays by Lorenz Krüger (1932-1994) presents exemplary studies on the philosophy of John Locke and Immanuel Kant, on the history of physics and on the scope and limitations of scientific explanation, and a realistic understanding of science and truth. In his treatment of leading currents in 20th century philosophy, Krüger presents new and original arguments for a (...) deeper understanding of the continuity and dynamics of the development of scientific theory. These result in significant consequences for the claim of the sciences that they understand reality in a rational manner. The case studies are complemented by fundamental thoughts on the relationship between philosophy, science, and their common history. (shrink)

For a set M, let \\) denote the set of all finite sequences which can be formed with elements of M, and let \ denote the set of all 2-element subsets of M. Furthermore, for a set A, let Open image in new window denote the cardinality of A. It will be shown that the following statement is consistent with Zermelo–Fraenkel Set Theory \: There exists a set M such that Open image in new window and no function Open image (...) in new window is finite-to-one. (shrink)

For an integer \, Ramsey Choice\ is the weak choice principle “every infinite setxhas an infinite subset y such that\ has a choice function”, and \ is the weak choice principle “every infinite family of n-element sets has an infinite subfamily with a choice function”. In 1995, Montenegro showed that for \, \. However, the question of whether or not \ for \ is still open. In general, for distinct \, not even the status of “\” or “\” is known. (...) In this paper, we provide partial answers to the above open problems and among other results, we establish the following:1.For every integer \, if \ is true for all integers i with \, then \ is true for all integers i with \.2.If \ are any integers such that for some prime p we have \ and \, then in \: \ and \.3.For \, \\\ implies \, and \ implies neither \ nor \ in \.4.For every integer \, \ implies “every infinite linearly orderable family of k-element sets has a partial Kinna–Wagner selection function” and the latter implication is not reversible in \ ). In particular, \ strictly implies “every infinite linearly orderable family of 3-element sets has a partial choice function”.5.The Chain-AntiChain Principle implies neither \ nor \ in \, for every integer \. (shrink)

This paper studies the Lockean thesis from the perspective of contemporary epistemic logic. The Lockean thesis states that belief can be defined as ‘sufficiently high degree of belief’. Its main problem is that it gives rise to a notion of belief which is not closed under conjunction. This problem is typical for classical epistemic logic: it is single-agent and static. I argue that from the perspective of contemporary epistemic logic, the Lockean thesis fares much better. I briefly mention that it (...) can successfully be extended from single-agent to multi-agent settings. More importantly, I show that accepting the Lockean thesis (and a more sophisticated version for conditional beliefs) leads to a significant and unexpected unification in the dynamic behavior of (conditional) belief and high (conditional) probability with respect to public announcements. This constitutes a methodological argument in favor of the Lockean thesis. Furthermore, if one accepts Baltag’s Erlangen program for epistemology, this technical observation has even stronger philosophical implications: because belief and high probability display the same dynamic behavior, it is plausible that they are indeed one and the same epistemic notion. (shrink)

The extensive research in logic conducted by using concepts and methods of game theory as documented in this collection of papers, allows to see dialogue logic in a number of new perspectives. This situation may gain further clarity by looking back to the inception of dialogue logic in the late fifties and early sixties.

Duality in Logic and Language [draft--do not cite this article] Duality phenomena occur in nearly all mathematically formalized disciplines, such as algebra, geometry, logic and natural language semantics. However, many of these disciplines use the term ‘duality’ in vastly different senses, and while some of these senses are intimately connected to each other, others seem to be entirely … Continue reading Duality in Logic and Language →.

In this paper, we consider certain cardinals in ZF (set theory without AC, the axiom of choice). In ZFC (set theory with AC), given any cardinals C and D, either C ≤ D or D ≤ C. However, in ZF this is no longer so. For a given infinite set A consider $\operatorname{seq}^{1 - 1}(A)$ , the set of all sequences of A without repetition. We compare $|\operatorname{seq}^{1 - 1}(A)|$ , the cardinality of this set, to |P(A)|, the cardinality of (...) the power set of A. What is provable about these two cardinals in ZF? The main result of this paper is that $ZF \vdash \forall A(|\operatorname{seq}^{1 - 1}(A)| \neq|\mathscr{P}(\mathscr{A})|)$ , and we show that this is the best possible result. Furthermore, it is provable in ZF that if B is an infinite set, then $|\operatorname{fin}(B)| <|\mathscr{P}(B)|$ even though the existence for some infinite set B* of a function f from $\operatorname{fin}(B^\ast)$ onto P(B*) is consistent with ZF. (shrink)

This paper explores the interplay between logic-sensitivity and bitstring semantics in the square of opposition. Bitstring semantics is a combinatorial technique for representing the formulas that appear in a logical diagram, while logic-sensitivity entails that such a diagram may depend, not only on the formulas involved, but also on the logic with respect to which they are interpreted. These two topics have already been studied extensively in logical geometry, and are thus well-understood by themselves. However, the precise details of their (...) interplay turn out to be far more complicated. In particular, the paper describes an elegant and natural interaction between bitstrings and logic-sensitivity, which makes perfect sense when bitstrings are viewed as purely combinatorial entities. However, when we view bitstrings as semantically meaningful entities (which is actually the standard perspective, cf. the term ‘bitstring semantics’!), this interaction does not seem to have a full and equally natural counterpart. The paper describes some attempts to address this situation, but all of them are ultimately found wanting. For now, it thus remains an open problem to capture this interaction between bitstrings and logic-sensitivity from a semantic (rather than merely a combinatorial) perspective. (shrink)

If we assume the axiom of choice, then every two cardinal numbers are comparable, In the absence of the axiom of choice, this is no longer so. For a few cardinalities related to an arbitrary infinite set, we will give all the possible relationships between them, where possible means that the relationship is consistent with the axioms of set theory. Further we investigate the relationships between some other cardinal numbers in specific permutation models and give some results provable without using (...) the axiom of choice. (shrink)

This paper studies Aumann’s agreeing to disagree theorem from the perspective of dynamic epistemic logic. This was first done by Dégremont and Roy (J Phil Log 41:735–764, 2012) in the qualitative framework of plausibility models. The current paper uses a probabilistic framework, and thus stays closer to Aumann’s original formulation. The paper first introduces enriched probabilistic Kripke frames and models, and various ways of updating them. This framework is then used to prove several agreement theorems, which are natural formalizations of (...) Aumann’s original result. Furthermore, a sound and complete axiomatization of a dynamic agreement logic is provided, in which one of these agreement theorems can be derived syntactically. These technical results are used to show the importance of explicitly representing the dynamics behind the agreement theorem, and lead to a clarification of some conceptual issues surrounding the agreement theorem, in particular concerning the role of common knowledge. The formalization of the agreement theorem thus constitutes a concrete example of the so-called dynamic turn in logic. (shrink)

Martin Heidegger zählt zu den wirkmächtigsten Denkern des 20. Jahrhunderts - zugleich ist er einer der umstrittensten, nicht zuletzt aufgrund seiner Parteinahme für den Nationalsozialismus 1933. Basierend auf neuesten Quellen erzählt Lorenz Jäger das Leben des Philosophen, der den Menschen und sein Dasein in der Welt auf ganz neue Weise gedacht hat - von der katholischen Kindheit in Messkirch und den geistigen Auseinandersetzungen der zwanziger Jahre über den Nationalsozialismus bis weit in die Jahre des Wiederaufbaus hinein. Dabei begegnen uns (...) Lehrer wie Edmund Husserl, dem 1936 die Lehrerlaubnis entzogen wurde, Vertraute wie Karl Jaspers und Hannah Arendt, deren so schwieriges wie intensives Verhältnis zu Heidegger über historische Brüche hinweg anhielt, Intellektuelle und Dichter wie Ernst Jünger und Paul Celan, die ihn in seiner Schwarzwaldhütte besuchten, bis hin zu späten Interpreten wie Lacan und Derrida. Warum Heidegger jede Generation aufs Neue fasziniert und polarisiert, sein Denken auch heute nichts an Bedeutung eingebüsst hat: Auch das zeigt Lorenz Jäger in dieser Biographie, die meisterhaft das Leben Heideggers erzählt - und zugleich ein deutsches Jahrhundert"--Publisher's website. (shrink)

For \(n\in \omega \), the weak choice principle \(\textrm{RC}_n\) is defined as follows: _For every infinite set_ _X_ _there is an infinite subset_ \(Y\subseteq X\) _with a choice function on_ \([Y]^n:=\{z\subseteq Y:|z|=n\}\). The choice principle \(\textrm{C}_n^-\) states the following: _For every infinite family of_ _n_-_element sets, there is an infinite subfamily_ \({\mathcal {G}}\subseteq {\mathcal {F}}\) _with a choice function._ The choice principles \(\textrm{LOC}_n^-\) and \(\textrm{WOC}_n^-\) are the same as \(\textrm{C}_n^-\), but we assume that the family \({\mathcal {F}}\) is linearly orderable (...) (for \(\textrm{LOC}_n^-\) ) or well-orderable (for \(\textrm{WOC}_n^-\) ). In the first part of this paper, for \(m,n\in \omega \) we will give a full characterization of when the implication \(\textrm{RC}_m\Rightarrow \textrm{WOC}_n^-\) holds in \({\textsf {ZF}}\). We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part, we will show some generalizations. In particular, we will show that \(\textrm{RC}_5\Rightarrow \textrm{LOC}_5^-\) and that \(\textrm{RC}_6\Rightarrow \textrm{C}_3^-\), answering two open questions from Halbeisen and Tachtsis (Arch Math Logik 59(5):583–606, 2020). Furthermore, we will show that \(\textrm{RC}_6\Rightarrow \textrm{C}_9^-\) and that \(\textrm{RC}_7\Rightarrow \textrm{LOC}_7^-\). (shrink)

The aim of this paper is to initiate a systematic exploration of the model theory of epistemic plausibility models (EPMs). There are two subtly different definitions in the literature: one by van Benthem and one by Baltag and Smets. Because van Benthem's notion is the most general, most of the paper is dedicated to this notion. We focus on the notion of bisimulation, and show that the most natural generalization of bisimulation to van Benthem-type EPMs fails. We then introduce parametrized (...) bisimulations, and prove various bisimulationimplies- equivalence theorems, a Hennessy-Milner theorem, and several (un)definability results. We discuss the problems arising from the fact that these bisimulations are syntax-dependent (and thus not fully structural), and we present and compare two different ways of coping with this issue: adding a modality to the language, and putting extra constraints on the models. We argue that the most successful solution involves restricting to uniform and locally connected (van Benthem-type) EPMs: for this subclass the intuitively most natural notion of bisimulation and the technically sound notion coincide. Such EPMs turn out to correspond exactly with Baltag/Smets-type EPMs, which can be interpreted as constituting a methodological argument, favoring Baltag and Smets's definition of EPM over that of van Benthem. (shrink)

Although universities have undergone changes since the dawn of their existence, the speed of change started to accelerate remarkably in the 1960s. Spectacular growth in the number of students and faculty was immediately followed by administrative reforms aimed at managing this growth and managing the demands of students for democratic reform and societal relevance. Since the 1980s, however, an entirely different wind has been blowing along the academic corridors. The fiscal crisis of the welfare states and the neoliberal course of (...) the Reagan and Thatcher governments made the battle against budget deficits and against government spending into a political priority. Education, together with social security and health care, were targeted directly. As the eighties went on, the neoliberal agenda became more radical—smaller state and bigger market—attacking the public sector itself through efforts to systematically reduce public expenditure by privatizing public services and introducing market incentives. At the same time the societal relevance of the universities demanded by critical students was turned on its head to become economic relevance to business and industry in the knowledge society. (shrink)

In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type (...) of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams and four kinds of metalogical decorations. (shrink)

Ch. 1: Inadequate approaches to the question of God -- 1.1. Initial clarifications -- 1.2 Wholly unsystematic direct approaches -- 1.3. Semi-systematic indirect approaches -- 1.4. A wholly anti-systematic, anti-theoretical, and direct approach: Ludwig Wittgenstein -- 1.5. A characteristic example of a failed critique: Thomas Nagel's objections to God as "last point" -- Ch. 2. Heidegger's thinking of Being: the flawed development of a significant approach -- 2.1. Heidegger's failed and distorting interpretation and critique of the Christian metaphysics of Being (...) -- 2.2. Heidegger's four approaches to "retrieving" the "question of being" -- 2.3. What is unthought in Heidegger's thinking of Being I: Being-as-Ereignis -- 2.4. What is unthought in Heidegger's "thinking of Being" II: Being and being(s)- Ereignis and Ereignete(s) -- 2.5. The "overcoming [Überwinding] of metaphysics" as "transformational recovering [Verwindung]" of metaphysics and "the end of the history of Being" -- 2.6. The status of Heideggerian thinking I: thinking of Being as thinking within Ereignis, thinking that reaches its destination with Ereignis (Denken, das in das Ereignis einkehrt) -- 2.7. The status of Heideggerian thinking II: absolute claim, provisionality, the poverty of language, the language of thinking, the finitude of thinking -- 2.8. Heidegger's thinking and the topic "God" -- 2.9. Heidegger's "thinking": a fundamentally deficient and confused form of thinking -- Ch. 3:The structural-systematic approach to a theory of Being and God -- 3.1. The systematic context: the theoretical framework of the structural-systematic philosophy -- 3.2. The unrestricted universe of discourse as the universal dimension of primordial Being -- 3.3. Explication of the dimension of Being I: theory of Being as such -- 3.4. Explication of the dimension of Being II: theory of Being as a whole -- 3.5. Explication of the relation between absolutely necessary Being and the contingent dimension of Being as key to a conception of absolutely necessary Being as minded (as personal) -- 3.6. Absolutely necessary minded (personal) Being as creator of the world (as absolute creating) -- 3.7. The clarified relation between Being and God and the task of developing an integral theory about God -- Ch. 4: Critical examination of two counterpositions: Emmanuel Levinas and Jean-Luc Marion -- 4.1. Levinas's misguided conception of transcendence "beyond B/being" -- 4.2. Jean-Luc Marion's failed conception of "radical and non-metaphysical transcendence" and of "God without Being". (shrink)

"Presents, and in part develops, a systematic philosophy as the universal science, or the theorization of the unrestricted universe of discourse, explicitly including being as such and as a whole. Argues that complete exploration of the theoretical domain requires such a science"--Provided by publisher.

I PSYCHOLOGY 5 The Probabilistic Revolution in Psychology--an Overview Gerd Gigerenzer 7 1 Probabilistic Thinking and the Fight against Subjectivity Gerd Gigerenzer 11 2 Statistical Method and the Historical Development of Research Practice in American Psychology Kurt Danziger 35 3 Survival of the Fittest Probabilist: Brunswik, Thurstone, and the Two Disciplines of Psychology Gerd Gigerenzer 49 4 A Perspective for Viewing the Integration of Probability Theory in Psychology David J. Murray 73 II SOCIOLOGY 101 5 The Two Empirical Roots of (...) Social Theory and the Probability Revolution Anthony Oberschall 103 III ECONOMICS 133 The Probabilistic Revolution in Economics--an Overview Mary S. Morgan 135 6 Why Was There No Probabilistic Revolution in Economic Thought? Claude Mènard 139 7 The Rise of Macroeconomic Calculations in Economic Statistics Robert A. Horváth 147 8 Statistics without Probability and Haavelmo’s Revolution in Econometrics Mary S. Morgan 171 IV PHYSIOLOGY 199 9 Experimental Physiology and Statistical Inference: The Therapeutic Trial in Nineteenth-Century Germany William Coleman 201 V EVOLUTIONARY BIOLOGY 227 The Probabilistic Revolution in Evolutionary Biology--an Overview John H. Beatty 229 10 Natural Selection as a Causal, Empirical, and Probabilistic Theory M. J. S. Hodge 233 11 Dobzhansky and Drift: Facts, Values, and Chance in Evolutionary Biology John H. Beatty 271 12 Random Genetic Drift, R. A. Fisher, and the Oxford School of Ecological Genetics John R. G. Turner 313 13 On the Prior Probability of the Existence of Life Bernd-Olaf Küppers 355 VI PHYSICS 371 The Probabilistic Revolution in Physics--an Overview Lorenz Krüger 373 14 Probabilistic Physics the Classical Way Jan von Plato 379 15 Max Born and the Reality of Quantum Probabilities Nancy Cartwright 409 16 Philosophical Problems of Quantum Theory: The Response of American Physicists Nancy Cartwright. (shrink)

Das heutige, mehr und mehr vom Aspekt des Werdens beherrschte Wirklichkeitsverständnis (Geschichtlichkeit, Evolution) verlangt nach einem ihm entsprechenden, von Grund auf dynamisch bestimmten Seinsbegriff, damit es philosophisch aufgearbeitet werden kann. Ein solcher Seinsbegriff findet sich in der sog. Prozess- bzw. Organismusphilo- sophie von A.N. Whitehead (1861-1947). Es ist das Ziel der vorliegen- den Arbeit, diesen herauszuarbeiten und für die weitere philosophische und theologische Diskussion bereitzustellen.

The theory of Participatory Discrepancies (or PDs) claims that minute temporal asynchronies (microtiming) in music performance are crucial for prompting bodily entrainment in listeners, which is a fundamental effect of the “groove” experience. Previous research has failed to find evidence to support this theory. The present study tested the influence of varying PD magnitudes on the beat-related body movement behavior of music listeners. 160 participants (79 music experts, 81 non-experts) listened to twelve music clips in either Funk or Swing style. (...) These stimuli were based on two audio recordings (one in each style) of expert drum and bass duo performances. In one series of six clips, the PDs were downscaled from their originally performed magnitude to complete quantization in steps of 20%. In another series of six clips, the PDs were upscaled from their original magnitude to double magnitude in steps of 20%. The intensity of the listeners’ beat-related head movement was measured using video-based motion capture technology and Fourier analysis. A mixed-design four-factor ANOVA showed that the PD manipulations had a significant effect on the expert listeners’ entrainment behavior. The experts moved more when listening to stimuli with PDs that were downscaled by 60% compared to completely quantized stimuli. This finding offers partial support for PD theory: PDs of a certain magnitude do augment entrainment in listeners. But the effect was found to be small to moderately sized, and it affected music expert listeners only. (shrink)

Aristotle takes practical wisdom and arts or crafts to be forms of knowledge which, we argue, can usefully be thought of as ‘empiricist’. This empiricism has two key features: knowledge does not rest on grasping unobservable natures or essences; and knowledge does not rest on grasping logical relations that hold among propositions. Instead, knowledge rests on observation, memory, experience and everyday uses of reason. While Aristotle’s conception of theoretical knowledge does require grasping unobservable essences and logical relations that hold among (...) suitable propositions, his conception of practical and productive knowledge avoids such requirements and is consistent with empiricism. (shrink)

Die "Gottesfrage" ist anerkanntermaßen eine äußerst komplexe, vieldeutige und obskure Frage. In ausführlicher Auseinandersetzung mit Heidegger, in dessen Schriften eine sehr einflussreiche einseitige und entstellende Interpretation und Kritik der großen metaphysischen Tradition immer wieder präsentiert wird, zeigt Lorenz B. Puntel, dass die Gottesfrage klar, rational und argumentativ, kurz: mit theoretischer Rigorosität, behandelt werden kann. Dies aber setzt einen umfassenden philosophischen Theorierahmen voraus, der die Entfaltung einer systematischen Konzeption des Seins als solchen und im Ganzen ermöglicht und in nuce enthält. (...) Anschließend wird eine solche Konzeption im Grundriss zur Darstellung gebracht. In einem ausführlichen Kapitel werden die Positionen von É. Lévinas und J.-L. Marion, den zwei wohl radikalsten postmodernen Kritikern und Gegnern jeder Form eines positiven Verhältnisses zwischen dem "Sein(sdenken)" und (dem Thema) "Gott", eingehend analysiert und deren radikale Unhaltbarkeit demonstriert. (shrink)

In logic, Aristotelian diagrams are almost always assumed to be closed under negation, and are thus highly symmetric in nature. In linguistics, by contrast, these diagrams are used to study lexicalization, which is notoriously not closed under negation, thus yielding more asymmetric diagrams. This paper studies the interplay between logical symmetry and linguistic asymmetry in Aristotelian diagrams. I discuss two major symmetric Aristotelian diagrams, viz. the square and the hexagon of opposition, and show how linguistic considerations yield various asymmetric versions (...) of these diagrams. I then discuss a pentagon of opposition, which occupies an uneasy position between the square and the hexagon. Although this pentagon belongs neither to the symmetric realm of logic nor to the asymmetric realm of linguistics, it occurs several times in the literature. The oldest known occurrence can be found in the cosmological work of the 14th-century author Nicole Oresme. (shrink)

Hugely controversial on publication, this is an insightful and characteristically entertaining survey of animal behaviour and the evolution of aggression throughout the animal world.

Solomon, the legend goes, had a magic ring which enabled him to speak to the animals in their own language. Konrad Lorenz was gifted with a similar power of understanding the animal world. He was that rare beast, a brilliant scientist who could write beautifully. He did more than any other person to establish and popularize the study of how animals behave, receiving a Nobel Prize for his work. King Solomon's Ring , the book which brought him worldwide recognition, (...) is a delightful treasury of observations and insights into the lives of all sorts of creatures, from jackdaws and water-shrews to dogs, cats and even wolves. Charmingly illustrated by Lorenz himself, this book is a wonderfully written introduction to the world of our furred and feathered friends, a world which often provides an uncanny resemblance to our own. A must for any animal-lover! (shrink)

For a ⊆ b ⊆ ω with b\ a infinite, the set D = {x ∈ [ω]ω : a ⊆ x ⊆ b} is called a doughnut. A set S ⊆ [ω]ω has the doughnut property [MATHEMATICAL SCRIPT CAPITAL D] if it contains or is disjoint from a doughnut. It is known that not every set S ⊆ [ω]ω has the doughnut property, but S has the doughnut property if it has the Baire property ℬ or the Ramsey property ℛ. (...) In this paper it is shown that a finite support iteration of length ω1 of Cohen forcing, starting from L, yields a model for CH + equation image + equation image + equation image. (shrink)

Several authors have recently studied Aristotelian diagrams for various metatheoretical notions from logic, such as tautology, satisfiability, and the Aristotelian relations themselves. However, all these metalogical Aristotelian diagrams focus on the semantic (model-theoretical) perspective on logical consequence, thus ignoring the complementary, and equally important, syntactic (proof-theoretical) perspective. In this paper, I propose an explanation for this discrepancy, by arguing that the metalogical square of opposition for semantic consequence exhibits a natural analogy to the well-known square of opposition for the categorical (...) statements from syllogistics, but that this analogy breaks down once we move from semantic to syntactic consequence. I then show that despite this difficulty, one can indeed construct metalogical Aristotelian diagrams from a syntactic perspective, which have their own, equally elegant characterization in terms of the categorical statements. Finally, I construct several metalogical Aristotelian diagrams that incorporate both semantic and syntactic consequence (and their interaction), and study how they are influenced by the underlying logical system’s soundness and/or completeness. All of this provides further support for the methodological/heuristic perspective on Aristotelian diagrams, which holds that the main use of these diagrams lies in facilitating analogies and comparisons between prima facie unrelated domains of investigation. (shrink)

In this article we give a forcing characterization for the Ramsey property of Σ 1 2 -sets of reals. This research was motivated by the well-known forcing characterizations for Lebesgue measurability and the Baire property of Σ 1 2 -sets of reals. Further we will show the relationship between higher degrees of forcing absoluteness and the Ramsey property of projective sets of reals.