Duality in Logic and Language [draft--do not cite this article] Duality phenomena occur in nearly all mathematically formalized disciplines, such as algebra, geometry, logic and natural language semantics.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 →.

Religious sects and their practices have become highly topical in the recent past in Germany. The call for criminal sanctions in order to tackle actual or presumed misconduct is becoming louder and louder. The author, a criminal law professor, analyses the present discussion and puts it into a constitutional law perspective. Any civil society that aims to optimise the scope of the personal freedoms of its members must withstand the temptation of creating a special regime of criminal law rules just (...) to combat religious sects and their practices. Criminal law has tobe applied uniformly in the whole society. Therfore, criminal sanctions can only be handed out, when practices infringe intolerably with the individual' s rights protected by the law. (shrink)

Plus que tout autre philosophe antique, Plotin semble mettre ses interprètes au défi de décrypter le fond de sa pensée. En effet, elle a l’air d’un système, mais le lecteur des Ennéades ne se voit pas livrer les structures conceptuelles principales qui doivent en assurer la cohérence. Emilsson s’est proposé le décryptage de certaines structures qui ont la fonction de principe conceptuel soit tout à fait général soit spécifique pour un champ thématique déterminé. Après avoir proposé une étude...

It is a remarkable fact that Hilbert's programmatic papers from the 1920s still shape, almost exclusively, the standard contemporary perspective of his views concerning (the foundations of) mathematics; even his own, quite different work on the foundations of geometry and arithmetic from the late 1890s is often understood from that vantage point. My essay pursues one main goal, namely, to contrast Hilbert's formal axiomatic method from the early 1920s with his existential axiomatic approach from the 1890s. Such a contrast illuminates (...) the circuitous beginnings of the finitist consistency program and connects the complex emergence of existential axiomatics with transformations in mathematics and philosophy during the 19th century; the sheer complexity and methodological difficulties of the latter development are partially reflected in the well known, but not well understood correspondence between Frege and Hilbert. Taking seriously the goal of formalizing mathematics in an effective logical framework leads also to contemporary tasks, not just historical and systematic insights; those are briefly described as “one direction” for fascinating work. (shrink)

The Hygiene Hypothesis has been recognized as an important cornerstone to explain the sudden increase in the prevalence of asthma and allergic diseases in modernized culture. The recent epidemic of allergic diseases is in contrast with the gradual implementation of Homo sapiens sapiens to the present-day forms of civilization. This civilization forms a gradual process with cumulative effects on the human immune system, which co-developed with parasitic and commensal Helminths. The clinical manifestation of this epidemic, however, became only visible in (...) the second half of the twentieth century. In order to explain these clinical effects in terms of the underlying IgE-mediated reactions to innocuous environmental antigens, the low biodiversity of antigens in the domestic environment plays a pivotal role. The skewing of antigen exposure as a cumulative effect of reducing biodiversity in the immediate human environment as well as in changing food habits, provides a sufficient and parsimonious explanation for the rise in allergic diseases in a highly developed and helminth-free modernized culture. Socio-economic tendencies that incline towards a further reduction of environmental biodiversity may provide serious concern for future health. This article explains that the “Hygiene Hypothesis”, the “Old Friends Hypothesis”, and the “Skewed Antigen Exposure Hypothesis” are required to more fully explain the rise of allergy in modern societies. (shrink)

In Hume’s Of Miracles the person movecl by faith is put in a dilemma between faith and reason. Can one resolve this dilemma as a compleat Humean? The answer is yes. Within the Humean context different approaches can be developed ta overcome Hume’s dilemma. One uses Hume’s theory of utility to defend the belief in the afterlife. The other requires Hume to place faith on a par with beauty and therefore among the passions to which reason must be a slave. (...) Historically there was at least one compleat Humean who was also moved by faith --- Johann Georg Hamann [1730- 1788]. (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)

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)

In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one.

In his classic paper On Computable Numbers Turing analyzed what can be done by a human computor in a routine, “mechanical” way. He argued that mechanical op-erations obey locality conditions and are carried out on configurations satisfying boundedness conditions. Processes meeting these restrictive conditions can be shown to be computable by a Turing machine. Turing viewed memory limitations of computors as the ultimate reason for the restrictive conditions. In contrast, Gandy analyzed in his paper Church’s Thesis and Principles for Mechanisms (...) what can be done by “discrete deterministic mechanical devices” and appealed to physical considerations to motivate restrictive principles. These devices include, crucially, ones that carry out parallel processes, e.g., cellular automata. Following Turing’s methodological ways and guided by Conway’s “Game of Life” as a paradigm, Gandy formulated four restrictive principles and proved that devices satisfying them are computationally equivalent to Turing machines. (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)

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the fifteenth publication in the Lecture Notes in Logic series, collects papers presented at the symposium 'Reflections on the Foundations of Mathematics' held in celebration of Solomon Feferman's 70th birthday (The 'Feferfest') at Stanford University, California in 1988. (...) Feferman has shaped the field of foundational research for nearly half a century. These papers reflect his broad interests as well as his approach to foundational research, which emphasizes the solution of mathematical and philosophical problems. There are four sections, covering proof theoretic analysis, logic and computation, applicative and self-applicative theories, and philosophy of modern mathematical and logic thought. (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)

In this rapid commentary, a mini-review is given of the present state-of-knowledge regarding the etiology and epidemiology of the new coronavirus 2019-nCoV and the risks for developing Acute respiratory distress syndrome. The available knowledge on the viral genomics, molecular biology and pathogenicity of viruses of the Coronaviridae family and other Nidovirales, forms a helpful template for understanding the present pandemic outbreak. However, important questions remain unanswered about the underlying mechanism causing the very high case fatality ratios and mechanisms regarding severe (...) reactions like ARDS, fatal cardiac and renal failures, associated with a number of important comorbidity factors. Immunological reactions to lung alveoles in particular in late phase ARDS in SARS-like CoV diseases, so far may not have received enough attention. Finally a shortlist of questions for high priority further research is suggested. (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)

Using the concept of notations for infinitary derivations we give an explanation of Takeuti's reduction steps on finite derivations (used in his consistency proof for Π1 1-CA) in terms of the more perspicious infinitary approach from [BS88].

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)

In this article we prove that a certain intuitionistic version of the well-known fixed point theory \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\widehat{\rm ID}_1$\end{document} is conservative over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\mbox{\sf HA}$\end{document} for almost negative formulas.

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)

The Federation of Belgian companies is a confederation which groups 36 branch associations covering 30. 000 individual firms. About 85 % are small and medium-sizedfirms occupying less than 50 employees. The VBO/FEB is the most important exponent of the high degree of affiliation on the employers' side. The process of decision making reflects the structure of the federation. The mandate is defined in the board of the VBO/FEB, composed by a representative ofeach branch association. This representative holds his mandate on (...) the basis of the deliberations held in the elected board of his association. The guiding principle is consensus, which is not synonymous of unanimity. When a consistent and representative majority emerges from the debate, the members holding another opinion join that majority in a spirit ofemployers' solidarity. The decision making process aims in this way to combine efficiency and democracy. This is particularly important for the VBO/FEB, as that organisation not only acts as a lobby but also as a social partner able to conclude agreements. In the Belgian system, social partners are consulted by the authorities, because of their representativeness. It must however be clear that they may not interfere with the political decisions which in our democratic system rest with government and parliament. (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)

[1] Textteil : Grundlagen, Prozeduren, Evaluation -- [2] Anhang : Interviewleitfaden, Tonbandtranskript und Auswertungsbeispiele.

There is a tension between the idea of popular sovereignty and our understanding that basic constitutional rights and liberties have a normative authority which is independent from the results of democratic decision‐making procedures. On the one hand there is the claim that the content of political justice, at least as far as the basic liberties are concerned, is to be fixed solely by substantive moral and political argument, while on the other there is the claim that it is the people (...) who determine the specific scheme of basic liberties that they live under. The apparent tension between the two claims can be resolved by introducing, firstly, a distinction between the concept of justice and the concept of legitimacy and, secondly, between proto‐rights and constitutional rights. Unlike constitutional rights, proto‐rights are moral, not legal, rights, and they are less specific than constitutional rights: one set of proto‐rights allows for various schemes of legal constitutional rights. Proto‐rights are part of the principles of justice and can be established only by substantive argument while constitutional rights have to be established by the procedures of democratic legitimization, and, therefore, they are a result of the exercise of popular sovereignty. The correlated distinctions between justice and legitimacy, and between proto‐rights and constitutional rights, allow us to specify the scope of the two claims and, hence, to affirm both simultaneously. The philosophical and legal relevance of the distinctions made is elucidated by a brief discussion of the constitutional rights of freedom of speech in the US and Germany. (shrink)