We introduce a generalization of MV algebras motivated by the investigations into the structure of quantum logical gates. After laying down the foundations of the structure theory for such quasi-MV algebras, we show that every quasi-MV algebra is embeddable into the direct product of an MV algebra and a “flat” quasi-MV algebra, and prove a completeness result w.r.t. a standard quasi-MV algebra over the complex numbers.
Recent decades have witnessed tremendous progress in artificial intelligence and in the development of autonomous systems that rely on artificial intelligence. Critics, however, have pointed to the difficulty of allocating responsibility for the actions of an autonomous system, especially when the autonomous system causes harm or damage. The highly autonomous behavior of such systems, for which neither the programmer, the manufacturer, nor the operator seems to be responsible, has been suspected to generate responsibility gaps. This has been the cause of (...) much concern. In this article, I propose a more optimistic view on artificial intelligence, raising two challenges for responsibility gap pessimists. First, proponents of responsibility gaps must say more about when responsibility gaps occur. Once we accept a difficult-to-reject plausibility constraint on the emergence of such gaps, it becomes apparent that the situations in which responsibility gaps occur are unclear. Second, assuming that responsibility gaps occur, more must be said about why we should be concerned about such gaps in the first place. I proceed by defusing what I take to be the two most important concerns about responsibility gaps, one relating to the consequences of responsibility gaps and the other relating to violations of jus in bello. (shrink)
Subjectivism about wellbeing rests on the idea that what is good for a person must ‘fit’ her, ‘resonate’ with her, not be ‘alien’ to her, etc. This idea has been called the ‘beating heart’ of subjectivism. In this article, I present the No-Beating-Heart Challenge for subjectivism, which holds that there is no satisfactory statement of this idea. I proceed by first identifying three criteria that any statement of the idea must meet if it is to provide support for subjectivism: Distinctness, (...) Exclusiveness, and Explicitness. I then argue that no statement of this idea meets these criteria. (shrink)
Schlusslogische Letztbegründung is a collection of essays in honor of Kurt Walter Zeidler. Mr. Zeidler is a distinguished Kant- and Neo-Kantian-scholar who has reconstructed Kant's concept of transcendental logic in connection with the logic of the concept of Hegel and the logic of symbolization of Peirce. (cf. Zeidler: Grundriss der transzendentalen Logik, 3rd ed., Wien 2017) He has most notably inquired intensively into the relation of transcendental logic to philosophy of science (cf. Zeidler: Prolegomena zur Wissenschaftstheorie, Wien 2000) and to (...) phenomenology (cf. Zeidler: Vermittlungen. Zum antiken und neueren Idealismus, Wien 2016). He has also published several studies on Neo-Kantianism (cf. Zeidler: Provokationen. Zu Problemen des Neukantianismus, Wien 2018). This is refelected in the collection of essays by distinguished scholars who discuss and critically examine Zeidler's work. It includes contributions by Steinar Mathisen (Oslo), Wolfdietrich Schmied-Kowarzik (Vienna), Werner Flach (Lichtenau), Thomas Knoppe (Straberg), Geert Edel (Wyk/Föhr), Martin Bunte (Münster), Reinhard Hiltscher (Dresden), Walter Tydecks (Bensheim), Christian Krijnen (Amsterdam), Hartwig Wiedebach (Zürich), Max Gottschlich (Linz), Thomas Sören Hoffmann (Hagen), Rudolf Meer (Kaliningrad), Hans-Jürgen Müller (Frankfurt am Main), Robert König (Vienna), Ulrich Blau (Marburg), Karen Gloy (Luzern/Munich), Reinhold Breil (Aachen), Erhard Oeser (Vienna), Hans-Dieter Klein (Vienna), Hans Martin Dober (Tübingen), Kurt Walter Zeidler (Vienna) and Lois Marie Rendl (Vienna). (shrink)
We develop measure theory in the context of subsystems of second order arithmetic with restricted induction. We introduce a combinatorial principleWWKL (weak-weak König's lemma) and prove that it is strictly weaker thanWKL (weak König's lemma). We show thatWWKL is equivalent to a formal version of the statement that Lebesgue measure is countably additive on open sets. We also show thatWWKL is equivalent to a formal version of the statement that any Borel measure on a compact metric space is countably additive (...) on open sets. (shrink)
In his posthumous book from 1914, "New foundations of logic, arithmetic and set theory", Julius Konig develops his philosophy of mathematics. In a previous contribution, we attracted attention on the positive part (his truth and falsehood predicates being excluded) of his "pure logic": his "isology" being assimilated to mutual implication, it constitutes a genuine formalization of positive intuitionistic logic. Konig's intention was to rebuild logic in such a way that the excluded third's principle could no longer be logical. (...) However, his treatment of truth and falsehood (boiling down to negation) is purely classical. We explain here this discrepancy by the choice of the alleged more primitive notions to which the questioned notions of truth and falsehood have been reduced. Finaly, it turns out that the disjunctive and conjunctive forms of the principles of the excluded third and of contradiction have effectively been excluded, but none of their implicative forms. (shrink)
The article focuses on re-evaluating Kant’s Transcendental Dialectic by initially highlighting its seemingly negative function within the Critique of Pure Reason as a mere regulative form for cognition and experience. The Dialectic, however, does not only have such a negative-regulative function but also its very own positive and founding character for cognition that even is present in the supposedly most immediate forms of intuition. In exploring this positive side of the Transcendental Dialectic it becomes clear that it manifests itself as (...) a bridge between the so-called theoretical and practical reason inasmuch as it fills in their gap within Kant’s philosophy. From the practical side, the Dialectic is manifest as an action full of purposiveness, maxims, and imperatives within cognition, from a theoretical side it assumes the form of syllogistic inference, which is the adequate and acting theoretical form of practical reason. Therefore, the unity of reason is shown in presenting its inner gap as a dialectical misunderstanding that Kant not only highlights in the Transcendental Dialectic but also tends to leave unsolved mostly. Nevertheless, the Dialectic can be shown as the a priori synthetic act of unifying reason, if investigated in the context of Kant’s complete critical endeavour. (shrink)
Zusammenfassung Bei der Erforschung der politischen Kommunikation in der hellenistischen Welt haben die sportlichen Wettkämpfe bislang nicht die gebührende Aufmerksamkeit erfahren. Hier setzt der Aufsatz an, der aufzuzeigen versucht, wie Könige und Poleis die Agonistik als Kommunikationsraum nutzten, um Sieghaftigkeit zu demonstrieren, Loyalität zu bekunden und Status zu verhandeln. Konkret werden drei Phänomene in den Blick genommen: die Teilnahme der Könige an den Pferde- und Wagenrennen, die Partizipation der Könige am agonistischen Ruhm anderer und die Konstituierung und Ausrichtung von Agonen (...) durch oder mit Bezug auf Könige. Dabei werden sowohl allgemeine Muster als auch spezifische Strategien einzelner Könige und Dynastien sichtbar. (shrink)
König, D. [1926. ‘Sur les correspondances multivoques des ensembles’, Fundamenta Mathematica, 8, 114–34] includes a result subsequently called König's Infinity Lemma. Konig, D. [1927. ‘Über eine Schlussweise aus dem Endlichen ins Unendliche’, Acta Litterarum ac Scientiarum, Szeged, 3, 121–30] includes a graph theoretic formulation: an infinite, locally finite and connected graph includes an infinite path. Contemporary applications of the infinity lemma in logic frequently refer to a consequence of the infinity lemma: an infinite, locally finite tree with a root (...) has a infinite branch. This tree lemma can be traced to [Beth, E. W. 1955. ‘Semantic entailment and formal derivability’, Mededelingen der Kon. Ned. Akad. v. Wet., new series 18, 13, 309–42]. It is argued that Beth independently discovered the tree lemma in the early 1950s and that it was later recognized among logicians that Beth's result was a consequence of the infinity lemma. The equivalence of these lemmas is an easy consequence of a well known result in graph theory: every connected, locally finite graph has among its partial subgraphs a spanning tree. (shrink)
In his posthumous book from 1914, “New foundations of logic, arithmetic andset theory”, Julius König develops his philosophy of mathematics. In a previous contribution, we attracted attention on the positive part of his “pure logic”: his “isology” being assimilated to mutual implication, it constitutes a genuine formalization of positive intuitionistic logic. König’s intention was to rebuild logic in such a way that the excluded third’s principle could no longer be logical. However, his treatment of truth and falsehood is purely classical. (...) We explain here this discrepancy by the choice of the alleged more primitive notions to which the questioned notions of truth and falsehood have been reduced. Finaly, it turns out that the disjunctive and conjunctive forms of the principles of the excluded third and of contradiction have effectively been excluded, but none of their implicative forms. (shrink)
Various theorems for the preservation of set-theoretic axioms under forcing are proved, regarding both forcing axioms and axioms true in the Lévy collapse. These show in particular that certain applications of forcing axioms require to add generic countable sequences high up in the set-theoretic hierarchy even before collapsing everything down to א₁. Later we give applications, among them the consistency of MM with אω not being Jónsson which answers a question raised in the set theory meeting at Oberwolfach in 2005.
Let CKDT be the assertion that for every countably infinite bipartite graph G, there exist a vertex covering C of G and a matching M in G such that C consists of exactly one vertex from each edge in M. (This is a theorem of Podewski and Steffens .) Let ATR0 be the subsystem of second-order arithmetic with arithmetical transfinite recursion and restricted induction. Let RCA0 be the subsystem of second-order arithmetic with recursive comprehension and restricted induction. We show that (...) CKDT is provable in ART0. Combining this with a result of Aharoni, Magidor, and Shore , we see that CKDT is logically equivalent to the axioms of ATR0, the equivalence being provable in RCA0. (shrink)
We show that large fragments of MM, e. g. the tree property and stationary reflection, are preserved by strongly -game-closed forcings. PFA can be destroyed by a strongly -game-closed forcing but not by an ω2-closed.