Preface to Volumes 1 and 2 Lorenz Krüger xv Introduction to Volume 1 Lorraine J. Daston 1 I Revolution 1 What Are Scientific Revolutions? Thomas S. Kuhn 7 2 Scientific Revolutions, Revolutions in Science, and a Probabilistic Revolution 1800-1930 I. Bernard Cohen 23 3 Was There a Probabilistic Revolution 1800-1930? Ian Hacking 45 II Concepts 4 The Slow Rise of Probabilism: Philosophical Arguments in the Nineteenth Century Lorenz Krüger 59 5 The Decline of the Laplacian Theory of Probability: (...) A Study of Stumpf, von Kries, and Meinong Andreas Kamlah 91 6 Fechner’s Indeterminism: From Freedom to Laws of Chance Michael Heidelberger 117 7 The Saint Petersburg Paradox 1713-1937 Gerard Jorland 157 8 Laplace and Thereafter: The Status of Probability Calculus in the Nineteenth Century Ivo Schneider 191 9 Emile Borel as a Probabilist Eberhard Knobloch 215 III Uncertainty 10 The Domestication of Risk: Mathematical Probability and Insurance 1650-1830 237 11 The Objectification of Observation: Measurement and Statistical Methods in the Nineteenth Century Zeno G. Swijtink 261 12 The Measurement of Uncertainty in Nineteenth-Century Social Science Stephen M. Stigler 287 IV Society 13 Rational Individuals versus Laws of Society: From Probability to Statistics Lorraine J. Daston 295 14 Decrire, Compter, Calculer: The Debate over Statistics during the Napoleonic Period Marie-Noelle Bourguet 305 15 Probability in Vital and Social Statistics: Quetelet, Farr, and the Bertillons Bernard-Pierre Lécuyer 317 16 Paupers and Numbers: The Statistical Argument for Social Reform in Britain during the Period of Industrialization Karl H. Metz 337 17 Lawless Society: Social Science and the Reinterpretation of Statistics in Germany, 1850-1880 Theodore M. Porter 351 18 Prussian Numbers 1860-1882 Ian Hacking 377 19 How Do Sums Count? On the Cultural Origins of Statistical Causality M. Norton Wise 395. (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)

Erwin Panofsky explicitly states that the first half of the opening chapter of Studies in Iconology—his landmark American publication of 1939—contains ‘the revised content of a methodological article published by the writer in 1932’, which is now translated for the first time in this issue of Critical Inquiry.1 That article, published in the philosophical journal Logos, is among his most important works. First, it marks the apogee of his series of philosophically reflective essays on how to do art history,2 that (...) reach back, via a couple of major pieces on Alois Riegl, to the 1915 essay on Heinrich Wölfflin.3 Under the influence of his colleague at Hamburg Ernst Cassirer, the principal interpreter of Kant in the 1920s, Panofsky from 1915 on exhibits in his work ever more Kantian thinking and language.4 But Logos was not an art-historical review or one dedicated to aesthetics but a principal mainstream journal of the philosophy of culture. So ‘On the Problem of Describing and Interpreting Works of the Visual Arts’ has a good claim to be the culmination of Panofsky's philosophical thinking in his German period under the Weimar Republic. · 1. Erwin Panofsky, Studies in Iconology: Humanistic Themes in the Art of the Renaissance , p. xv; hereafter abbreviated SI. See Panofsky, ‘Zum Problem der Beschreibung und Inhaltsdeutung von Werken der bildenden Kunst’, Logos 21 : 103–19; trans. Jaś Elsner and Katharina Lorenz under the title ‘On the Problem of Describing and Interpreting Works of the Visual Arts’, Critical Inquiry 38 : 467–82; hereafter abbreviated ‘P’.· 2. See the discussion in Carlo Ginzburg, ‘From Aby Warburg to E. H. Gombrich: A Problem of Method’, Myths, Emblems, Clues, trans. John and Anne C. Tedeschi , pp. 17–59, esp. pp. 36–41.· 3. See Panofsky, ‘Das Problem des Stils in der bildenden Kunst’, Deutschsprachige Aufsätze, ed. Karen Michels and Martin Warnke, 2 vols. , 2:1009–18; ‘Der Begriff des Kunstwollens’,Deutschsprachige Aufsätze, 2:1019–34, trans. Kenneth J. Northcott and Joel Snyder under the title ‘The Concept of Artistic Volition’, Critical Inquiry 8 : 17–33; and ‘Über das Verhältnis der Kunstgeschichte zur Kunsttheorie: Ein Beitrag zu der Erörterung über die Möglichkeit kunstwissenschaftlicher Grundbegriffe’, Deutschsprachige Aufsätze, 2: 1035–63, trans. Lorenz and Elsner under the title ‘On the Relationship of Art History and Art Theory: Towards the Possibility of a Fundamental System of Concepts for a Science of Art’, Critical Inquiry 35 : 43–71.· 4. On neo-Kantianism in pre-Nazi Germany, see Michael Friedman, A Parting of the Ways: Carnap, Cassirer, and Heidegger , pp. 25–37; Éric Dufour and T. Z. R. Créteil, ‘Le Statue du singulier: Kant et le néokantisme de l’École de Marbourg', Kantstudien 93 : 324–50; Edward Skidelsky, Ernst Cassirer: The Last Philosopher of Culture , pp. 22–51; and Peter E. Gordon, Continental Divide: Heidegger, Cassirer, Davos , pp. 52–86. Specifically on the Cassirerian Kantianism of Panofsky, see Michael Podro, The Critical Historians of Art , pp. 181–82; Michael Ann Holly, Panofsky and the Foundations of Art History , pp. 91–92, 147–52; Silvia Ferretti, Cassirer, Panofsky, and Warburg: Symbol, Art, and History, trans. Richard Pierce , pp. 174–77, 182–84; David Summers, ‘Meaning in the Visual Arts as a Humanistic Discipline’, in Meaning in the Visual Arts: Views from the Outside, ed. Irving Lavin , pp. 9–24; Mark A. Cheetham, Kant, Art, and Art History: Moments of Discipline , pp. 68–77; Paul Crowther, The Transhistorical Image: Philosophizing Art and Its History , pp. 70–73; Allister Neher, ‘“The Concept of Kunstwollen”, Neo-Kantianism, and Erwin Panofsky's Early Art Theoretical Essays', Word and Image 20 : 41–51; Georges Didi-Huberman,Confronting Images: Questioning the Ends of a Certain History of Art, trans. John Goodman , pp. 4–6, 90–138; and Lorenz and Elsner, ‘Translators’ Introduction', Critical Inquiry35 : 33–42, esp. pp. 38, 40–42. (shrink)

Die Frage, wie sich Aussagen uber Fahigkeiten von Personen zu Aussagen uber Moglichkeiten von Zustanden in der Welt verhalten, ist fur unser menschliches Selbstverstandnis zentral. Der vorliegende Band versammelt unter dieser Frage durchweg Originalbeitrage in historisch-systematischer Absicht; sie behandeln die Geschichte der Metaphysik und Ontologie von Parmenides bis Heidegger.INHALT: VORWORT - Klaus Jacobi: Das Konnen und die Moglichkeiten. Potentialitat und Possibilitat - Mischa von Perger: Moglichkeit, Parmenideisch - Ulrich Nortmann: 'Das Saatkorn ist dem Vermogen nach eine Pflanze'. Uber ontologische und (...) logische Aspekte Aristotelischer Moglichkeitssatze - Sang-Jin Kang: Anselm und die modallogische Betrachtung der gottlichen Notwendigkeit - Michael Astroh: Petrus Abaelardus on Modalities de re and de dicto - Christopher Martin: Peter Abaelard on Modality. Some Possibilities and Some Puzzles - Allan Back: Avicenna and Averroes: Modality and Theology - Seung-Chan (Elias) Park: Die Differenz von personlicher und unpersonlicher Moglichkeit bei Thomas von Aquin - Charles Lohr: 'Art' and Possibility: The Rule Concerning Possibility in the Ars lulliana - Peter King: Duns Scotus on Possibilities, Powers, and the Possible - Matthias Kaufmann: Gottes Allmacht und die Wahrheit modaler Satze. Potentialitat und Possibilitat bei Wilhelm von Ockham - Simo Knuuttila: On the History of Theory of Modality as Alternativeness - Stephan Meier-Oeser: Potentia versus Possibilitas? Posse! Zur cusanischen Konzeption der Moglichkeit - Dominik Perler: Cartesische Moglichkeiten - Hans Poser: Leibniz'sche Handlungsmodi zwischen Ontologie und Deontologie - Wilhelm Metz: Der transzendentale Moglichkeitsbegriff bei Kant und Fichte - Rainer Marten: Heidegger: Die eigenste eigentliche Moglichkeit - Tilman Borsche: Moglichkeiten des Seins, Moglichkeiten des Denkens - Thomas Buchheim: Zwischen Antinomie und Kompatibilitat: Versuch uber die naturliche Einbettung unserer Handlungsfreiheit - Kuno Lorenz: Dynamis und Energeia. Zur Aktualitat eines begrifflichen Werkzeugs von Aristoteles. (shrink)

Although Lorenz Oken is a classic example of Naturphilosophie as applied to biology, his views have been imperfectly understood. He is best viewed as a follower of Schelling who consistently attempted to apply Schelling's ideas to biological data. His version of Naturphilosophie, however, was strongly influenced by older pseudoscience traditions, especially alchemy and numerology as they had been presented by Robert Fludd, whose works were current in Jena and available to him. According to those influences, parts of Oken's philosophical (...) conception were communicable even in a non-idealistic scientific culture, for example in Paris, where Oken met Étienne Geoffroy Saint-Hilaire. Geoffroy however was embedded in a French intellectual tradition, and the correspondence between his views and those of Oken was only superficial. The English anatomist Richard Owen attempted to incorporate the views of Oken and Geoffroy within his own, idiosyncratic system. Although Darwin knew of Oken's ideas, it was Geoffroy who really affected his evolutionary biology, and any influence of Oken must have been attenuated to the point of triviality. (shrink)

Das Bose hat Konjunktur - und bleibt doch unberechenbar. Lassen sich dennoch angemessene Strategien des Umgangs entwickeln, die weder die prekare Faktizitat des Bosen noch seine historische Tiefendimension ausblenden? Ausgehend von dieser Leitfrage nahert sich der vorliegende, interdisziplinar breit angelegte Band dem vielschichtigen Phanomen des Bosen in praxeologischer Perspektive. So fokussiert er Moglichkeiten eines - asthetisch wie wissenschaftlich reflektierten -, guten' Umgangs mit dem Bosen. Dabei zeigt sich, dass dies nur unter Beachtung der unaufhebbaren Verwiesenheit von Gut und Bose gelingen (...) kann. Im facherubergreifenden Dialog der Beitrage entstehen so Reflexionen auf Dialektik, Differenz und Paradoxien der Fundamentalkategorien Gut und Bose von wissenschaftsethischer wie grundlagentheoretischer Brisanz.Der Band bietet einen gesonderten Schwerpunkt zu Marcel Beyers 'Kaltenburg', einem Schlusselroman zu Konrad Lorenz, dem Autor von 'Das sogenannte Bose'. (shrink)

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 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)

The first part of this paper argues that if Craver’s ([2007a], [2007b]) popular mutual manipulability account (MM) of mechanistic constitution is embedded within Woodward’s ([2003]) interventionist theory of causation--for which it is explicitly designed--it either undermines the mechanistic research paradigm by entailing that there do not exist relationships of constitutive relevance or it gives rise to the unwanted consequence that constitution is a form of causation. The second part shows how Woodward’s theory can be adapted in such a way that (...) (MM) neither undermines the mechanistic paradigm nor reduces constitution to causation. However, it turns out that this modified theoretical embedding of (MM) makes it impossible to produce empirical evidence for constitutive relations. The paper ends by suggesting an additional criterion, the fat-handedness criterion, which, when combined with (MM), generates indirect empirical evidence for constitutive relevance. (shrink)

Die vorliegende Arbeit hat den Leibnizschen metaphysischen Optimismus und einige der weniger bekannten Stationen und Zuge der Rezeption der "Essais de Theodicee" und des Theorems von der besten Welt im deutschen 18. Jahrhundert zum Gegenstand. Sie erhebt nicht den Anspruch, eine auch nur annahernd vollstandige Darstellung des Verlaufes der Debatte um die "beste Welt" zu geben. Angesichts der Fulle der zeitgenossischen Einlassungen und Ausserungen zum Thema war eine Auswahl bestimmter Rezeptionsmomente vorzunehmen. Auf die Prozesse der Verwandlung einer strikt metaphysischen Theodizee (...) hin zu einem populareren, dafur aber auch anfechtbareren Optimismus will diese Arbeit ebenso hinweisen wie auf den Umstand, dass nicht erst die Kantische Schrift "Uber das Misslingen aller philosophischen Versuche in der Theodizee" von 1791 die (philosophische) Problematik der Lehre von der besten Welt dingfest macht. (shrink)

Inspired by the work of Wilfrid Sellars, Michael Williams launches an all-out attack on what he calls "phenomenalism," the idea that our knowledge of the world rests on a perceptual or experiential foundation.

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)

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)

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)

In the recent debate on future contingents and the nature of the future, authors such as G. A. Boyd, W. L. Craig, and E. Hess have made use of various logical notions, such as the Aristotelian relations of contradiction and contrariety, and the ‘open future square of opposition.’ My aim in this paper is not to enter into this philosophical debate itself, but rather to highlight, at a more abstract methodological level, the important role that Aristotelian diagrams can play in (...) organizing and clarifying the debate. After providing a brief survey of the specific ways in which Boyd and Hess make use of Aristotelian relations and diagrams in the debate on the nature of the future, I argue that the position of open theism is best represented by means of a hexagon of opposition. Next, I show that on the classical theist account, this hexagon of opposition ‘collapses’ into a single pair of contradictory statements. This collapse from a hexagon into a pair has several aspects, which can all be seen as different manifestations of a single underlying change. (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)

Judges in multiple US states, such as New York, Pennsylvania, Wisconsin, California, and Florida, receive a prediction of defendants’ recidivism risk, generated by the COMPAS algorithm. If judges act on these predictions, they implicitly delegate normative decisions to proprietary software, even beyond the previously documented race and age biases. Using the ProPublica dataset, we demonstrate that COMPAS predictions favor jailing over release. COMPAS is biased against defendants. We show that this bias can largely be removed. Our proposed correction increases overall (...) accuracy, and attenuates anti-black and anti-young bias. However, it also slightly increases the risk that defendants are released who commit a new crime before tried. We argue that this normative decision should not be buried in the code. The tradeoff between the interests of innocent defendants and of future victims should not only be made transparent. The algorithm should be changed such that the legislator and the courts do make this choice. (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)

The german „Politik“ is very comprehensive. Different theorists had a fundamentally different understandig of the term. We can distinguish between a realistic and a idealistic approach. The first is represented especially by Carl Schmitt and Max Weber, the second by Hannah Arendt and ancient greek thinkers (Plato, Aristotle). These different views can’t be synthesized. „Politik“ ist characterized by its generality. Politically is decided, what - within a certain territory - is binding for everyone. Therefore „Politik“ is inseparably connected with power. (...) Moreover, „Politik“ has got a creative element. In sum, „Politik“ means guiding the public affairs in a generally binding and powerful way. (shrink)

This article describes a specific pedagogical context for an advanced logic course and presents a strategy that might facilitate students’ transition from the object-theoretical to the metatheoretical perspective on logic. The pedagogical context consists of philosophy students who in general have had little training in logic, except for a thorough introduction to syllogistics. The teaching strategy tries to exploit this knowledge of syllogistics, by emphasizing the analogies between ideas from metalogic and ideas from syllogistics, such as existential import, the distinction (...) between contradictories and contraries, and the square of opposition. This strategy helps to improve students’ understanding of metalogic, because it allows the students to integrate these new ideas with their previously acquired knowledge of syllogistics. (shrink)

"Cover " -- "Title Page " -- "Copyright " -- "Dedication" -- "Contents" -- "Introduction: Moral Globalization and Its Discontents" -- "1. Jackson Heights, New York: Diversity Plaza" -- "2. Los Angeles: The Moral Operating Systems of Global Cities" -- "3. Rio de Janeiro: Order, Corruption, and Public Trust" -- "4. Bosnia: War and Reconciliation" -- "5. Myanmar: The Politics of Moral Narrative" -- "6. Fukushima: Resilience and the Unimaginable" -- "7. South Africa: After the Rainbow" -- "Conclusion: Human Rights, (...) Global Ethics, and the Ordinary Virtues" -- "Notes" -- "Acknowledgments. (shrink)

The Sure-Thing Principle famously appears in Savage’s axiomatization of Subjective Expected Utility. Yet Savage introduces it only as an informal, overarching dominance condition motivating his separability postulate P2 and his state-independence postulate P3. Once these axioms are introduced, by and large, he does not discuss the principle any more. In this note, we pick up the analysis of the Sure-Thing Principle where Savage left it. In particular, we show that each of P2 and P3 is equivalent to a dominance condition; (...) that they strengthen in different directions a common, basic dominance axiom; and that they can be explicitly combined in a unified dominance condition that is a candidate formal statement for the Sure-Thing Principle. Based on elementary proofs, our results shed light on some of the most fundamental properties of rational choice under uncertainty. In particular they imply, as corollaries, potential simplifications for Savage’s and the Anscombe- Aumann axiomatizations of Subjective Expected Utility. Most surprisingly perhaps, they reveal that in Savage’s axiomatization, P3 can be weakened to a natural strengthening of so-called Obvious Dominance. (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)

Tye's book develops a persuasive and, in many respects, original argument for the view that the qualitative side of our mental life is representational in..

Burgess-Jackson has recently suggested that the debate between theism and atheism can be represented by means of a classical square of opposition. However, in light of the important role that the position of agnosticism plays in Burgess-Jackson’s analysis, it is quite surprising that this position is not represented in the proposed square of opposition. I therefore argue that the square of opposition should be extended to a slightly larger, more complex Aristotelian diagram, viz., a hexagon of opposition. Since this hexagon (...) does represent the position of agnosticism, it arguably yields a more helpful representation of the theism/atheism debate. It would be naïve to presume that Aristotelian diagrams can, by themselves, lead to a comprehensive solution of debates as intricate as that between theism and atheism. Nevertheless, this paper aims to show that these diagrams — especially if they are chosen carefully — have an important methodological role to play, by systematically organizing and clarifying the debate. (shrink)

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)

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)

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)

In order to elucidate his logical analysis of modal quantified propositions (e.g. ‘all men are necessarily mortal’), the 14th century philosopher John Buridan constructed a modal octagon of oppositions. In the present paper we study this modal octagon from the perspective of contemporary logical geometry. We argue that the modal octagon contains precisely six squares of opposition as subdiagrams, and classify these squares based on their logical properties. On a more abstract level, we show that Buridan’s modal octagon precisely captures (...) the interaction between two classical squares of opposition, viz. one for the quantifiers and one for the modalities. Finally, we argue that several aspects of our contemporary formal analyses were already hinted at by Buridan himself. (shrink)