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)

With the outbreak of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), global researchers were confronted with major challenges. The German National Pandemic Cohort Network (NAPKON) was launched in fall 2020 to effectively leverage resources and bundle research activities in the fight against the coronavirus disease 2019 (COVID-19) pandemic. We analyzed the setup phase of NAPKON as an example for multicenter studies in Germany, highlighting challenges and optimization potential in connecting 59 university and nonuniversity study sites. We examined the ethics application (...) process of 121 ethics submissions considering durations, annotations, and outcomes. Study site activation and recruitment processes were investigated and related to the incidence of SARS-CoV-2 infections. For all initial ethics applications, the median time to a positive ethics vote was less than two weeks and 30 of these study sites (65%) joined NAPKON within less than three weeks each. Electronic instead of postal ethics submission (9.5 days (Q1: 5.75, Q3: 17) vs. 14 days (Q1: 11, Q3: 26), p value = 0.01) and adoption of the primary ethics vote significantly accelerated the ethics application process. Each study center enrolled a median of 37 patients during the 14-month observation period, with large differences depending on the health sector. We found a positive correlation between recruitment performance and COVID-19 incidence as well as hospitalization incidence. Our analysis highlighted the challenges and opportunities of the federated system in Germany. Digital ethics application tools, adoption of a primary ethics vote and standardized formal requirements lead to harmonized and thus faster study initiation processes during a pandemic. (shrink)

Since 2020, the COVID-19 pandemic had an impact on education worldwide. There is increased discussion of possible negative effects on students’ learning outcomes and the need for targeted support. We examined fourth graders’ reading achievement based on a school panel study, representative on the student level, with N = 111 elementary schools in Germany. The students were tested with the Progress in International Reading Literacy Study instruments in 2016 and 2021. The analysis focused on total average differences in reading achievement (...) between 2016 and 2021, average differences controlling for student composition, and changes in achievement gaps between student subgroups. The methodological approach met international standards for the analysis of large-scale assessments. The results showed a substantial decline in mean reading achievement. The decline corresponds to one-third of a year of learning, even after controlling for changes in student composition. We found no statistically significant changes of achievement gaps between student subgroups, despite numerical tendencies toward a widening of achievement gaps between students with and without immigration background. It is likely that this sharp achievement decline was related to the COVID-19 pandemic. The findings are discussed in terms of further research needs, practical implications for educating current student cohorts, and educational policy decisions regarding actions in crises such as the COVID-19 pandemic. (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)

In an age where issues like climate change and the unintended consequences of technological innovation are high on the ethical and political agenda, questions about the nature and extent of our responsibilities to future generations have never been more important, yet simultaneously so difficult to answer. This book takes a unique approach to the problem by drawing on diverse traditions of thinking about care (including developmental psychology, phenomenology and feminist ethics) to explore the nature and meaning of our relationship with (...) the future. Demonstrating that many influential perspectives on intergenerational ethics, including positions advanced by John Rawls, Brian Barry, and Ulrich Beck are undermined by problems relating to uncertainty, it shows that an approach based on care ethics can confront the uncertain future successfully and give a viable account of the nature and scope of future-oriented responsibilities. (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 lieu of an abstract, here is a brief excerpt of the content:Philosophy, Psychiatry, & Psychology 10.2 (2003) 193-194 [Access article in PDF] A Comment on Christopher Ciocchetti:"The Responsibility of the Psychopathic Offender" Daniel W. Shuman Questions of responsibility for serious harm are complex and potentially divisive. The way in which we frame these questions and the criteria by which we assess answers to them are colored, in part, by the lens though which we view them. I am a (...) law professor whose writings includes issues relating to criminal responsibility of person with mentally disabilities and I come to this commentary from that perspective. For me questions of moral responsibility raise psychological and philosophical, as well as legal, questions.Ciocchetti's (2003) paper addressees the criminal responsibility of the psychopathic offender. Not sure quite what the author meant by this label, I began to try to understand the concept of psychopathic offender by picking up my DSM-IV-TR and found no reference to such a diagnostic category. Absent a handy reference in the DSM, I looked in the paper for some other definition or recognition of the category of psychopathic offender and found no satisfactory grounding in any body of sound research. Left with the author's unsupported assertions that such a category of persons exists and that they behave in ways the author asks me to accept without support, my lawyer instincts cautioned me to proceed carefully. The psychopathic offender the author describes might well exist, but I am not persuaded of that, nor would be my legal colleagues, absent proof.Next, to the author's thesis that "[t]he offender must have the capacities necessary to understand relationships to justify holding that offender responsible" (p. 176) and that because psychopaths lack this capacity they should not be held criminally responsible. Is the author proposing a new test for criminal responsibility? Certainly no test for criminal nonresponsibility based on mental disability (i.e., insanity defense) in Anglo-American jurisprudence, or any other with which I am familiar, makes status (i.e., a diagnostic category) exculpatory for criminal responsibility. Rather, every test addresses the impact of a mental illness or disorder on the cognitive or volitional capacity of the offender at the time of the offense. The author offers no reason to reject the approach taken for the last several hundred years [End Page 193] by Anglo-American jurisprudence. Moreover, to the extent that status or diagnostic category is relevant in the insanity defense in Anglo-American jurisprudence, character or personality disorders (the genus to which I take it that psychopathy belongs) are often specifically excluded from consideration. For example, the Texas Penal Code (Tex. Penal Code Ann. § 8.01 (Vernon 1994)) states that:It is an affirmative defense to prosecution that, at the time of the conduct charged, the actor, as a result of a severe mental disease or defect, did not know that his conduct was wrong. The term "mental disease or defect" does not include an abnormality manifested only by repeated criminal or otherwise antisocial conduct.The federal Insanity Defense Reform Act raises the same issue: The government relies on the legislative history of 18 U.S.C. § 17, which states: "The concept of severity was added to emphasize that non-psychotic behavior disorders or neuroses such as an 'inadequate personality,' 'immature personality,' or a pattern of 'antisocial tendencies' do not constitute the defense.".... The government argues that because Dr. Lorenz would have testified that Salava was not psychotic but suffered from antisocial personality disorder, his testimony was not relevant to either prong of the insanity defense under the statute.... As a general matter, the government's point is well-taken. It seems clear that Congress intended by its inclusion of the modifier "severe" to place certain "disorders" outside the purview of the insanity defense altogether. United States v. Salava, 978 F.2d 320, 323 (7th Cir. 1992) Nowhere in the paper does Chiochetti acknowledge or address these societal judgments of moral responsibility that have already rejected a far more modest version of his proposal in categorically ruling out character or personality disorders from inclusion as... (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)

This book is available either individually, or as part of the specially-priced Arguments of the Philosphers Collection.

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)

Suppose we believe that God created the world. Then surely we want it to be the case that he intended, in some sense at least, to create THIS world. Moreover, most theists want to hold that God didn't just guess or hope that the world would take one course or another; rather, he KNEW precisely what was going to take place in the world he planned to create. In particular, of each person P, God knew that P was to exist. (...) Call this the "standard" conception. Most theists find the standard conception appealing. Unfortunately, the view seems to conflict with the equally appealing idea — call it "temporal actualism" — that there are no "future" individuals beyond those that already exist in the present moment. For, on this view, for any historical person P, prior to creation, there was no such person as P and hence nothing about P for God to know. Hence, in particular, God couldn't have known that P was to exist. This is of course not a new problem. But past solutions to it are highly problematic. In this paper, after canvassing previous approaches, I will propose a solution that seems to preserve both temporal actualism and a suitably robust form of the standard conception while avoiding the pitfalls of the past. (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)

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)

It is often thought that the correct metaphysics and epistemology of reasons will be broadly unified across different kinds of reason: reasons for belief, and reasons for action. This approach is sometimes thought to be undermined by the contrasting natures of belief and of action: whereas belief appears to have the ‘constitutive aim’ of truth (or knowledge), action does not appear to have any such constitutive aim. I develop this disanalogy into a novel challenge to metanormative approaches by thinking about (...) disagreement. The constitutive aim of belief can play a role in adjudicating epistemic disagreements for which there is no analogue in practical disagreements. Consequently, we have more reason, all else being equal, to expect convergence in epistemic judgment than in practical judgment. This represents a prima facie challenge to the metanormative theorist because the extent of (suitably specified) disagreement in an area of thought is of prima facie significance for the metaphysics of that area of thought. (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)

Aristotle attaches particular significance to the homonymy of many central concepts in philosophy and science: that is, to the diversity of ways of being common to a single general concept. His preoccupation with homonymy influences his approach to almost every subject that he considers, and it clearly structures the philosophical methodology that he employs both when criticizing others and when advancing his own positive theories. Where there is homonymy there is multiplicity: Aristotle aims to find the order within this multiplicity, (...) and believes that doing so is crucial to scientific inquiry and philosophical progress.Christopher Shields investigates and evaluates Aristotle's approach to questions about homonymy, characterizing the metaphysical and semantic commitments necessary to establish the homonymy of a given concept. Then, in a series of case-studies, Shields examines in detail some of Aristotle's principal applications of homonymy--to the body, sameness and oneness, life, goodness, and being. Shields's aim is not only to give a fuller understanding of Aristotle's methodology and to illuminate his specific doctrines in a variety of areas, but to show that this methodology remains fruitful today. (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)

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)

Christopher Hookway presents a series of studies of themes from the work of the great American philosopher and pragmatist, Charles S. Peirce (1839-1913). These themes center on the question of how we are to investigate the world rationally. Hookway shows how Peirce's ideas about this continue to play an important role in contemporary philosophy.

What is required for an individual to entertain a singular thought about an object they have encountered before but that is currently no longer within their perceptual range? More specifically, does the individual have to think about the object as having been encountered in the past? I consider this question against the background of the assumption that non-human animals are cognitively ‘stuck in the present’. Does this mean that, for them, ‘out of sight is out of mind’, as, e.g., Schopenhauer (...) seems to have thought? I suggest an alternative answer, also drawing on some empirical work on animal cognition. (shrink)

Moral error theories are often rejected by appeal to ‘companions in guilt’ arguments. The most popular form of companions in guilt argument takes epistemic reasons for belief as a ‘companion’ and proceeds by analogy. I show that this strategy fails. I claim that the companions in guilt theorist must understand epistemic reasons as evidential support relations if her argument is to be dialectically effective. I then present a dilemma. Either epistemic reasons are evidential support relations or they are not. If (...) they are not, then the companions in guilt argument fails. If they are, then a reduction of epistemic reasons to evidential support relations becomes available and, consequently, epistemic reasons cease to be a viable ‘companion’ for moral reasons. I recommend this structure of argument over existing strategies within the literature and defend my claims against recent objections from companions in guilt theorists. (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)

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)

This original new book argues for a reassessment of Plato's challenge to the arts. Plato was the first great figure in Western philosophy to assess the value of the arts; he argued in the Republic that traditionally accepted forms of poetry, drama, and music are unsound. While this view has been widely rejected, Janaway argues that Plato's hostile case is a more coherent and profound challenge to the arts than has sometimes been supposed. Denying that Plato advocates "good art" in (...) any modern sense, Janaway seeks both to understand Plato's critique in the context of his own philosophy and to locate him in today's philosophy of art, showing how issues in aesthetics arise from responses to his charges. (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)

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)

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)

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)