Definitional and axiomatic theories of truth -- Objects of truth -- Tarski -- Truth and set theory -- Technical preliminaries -- Comparing axiomatic theories of truth -- Disquotation -- Classical compositional truth -- Hierarchies -- Typed and type-free theories of truth -- Reasons against typing -- Axioms and rules -- Axioms for type-free truth -- Classical symmetric truth -- Kripke-Feferman -- Axiomatizing Kripke's theory in partial logic -- Grounded truth -- Alternative evaluation schemata -- Disquotation -- Classical logic -- Deflationism (...) -- Reflection -- Ontological reduction -- Applying theories of truth. (shrink)

We investigate axiomatizations of Kripke's theory of truth based on the Strong Kleene evaluation scheme for treating sentences lacking a truth value. Feferman's axiomatization KF formulated in classical logic is an indirect approach, because it is not sound with respect to Kripke's semantics in the straightforward sense: only the sentences that can be proved to be true in KF are valid in Kripke's partial models. Reinhardt proposed to focus just on the sentences that can be proved to be true in (...) KF and conjectured that the detour through classical logic in KF is dispensable. We refute Reinhardt's Conjecture, and provide a direct axiomatization PKF of Kripke's theory in partial logic. We argue that any natural axiomatization of Kripke's theory in Strong Kleene logic has the same proof-theoretic strength as PKF, namely the strength of the system RA< ωω ramified analysis or a system of Tarskian ramified truth up to ωω. Thus any such axiomatization is much weaker than Feferman's axiomatization KF in classical logic, which is equivalent to the system RA<ε₀ of ramified analysis up to ε₀. (shrink)

According to the disquotationalist theory of truth, the Tarskian equivalences, conceived as axioms, yield all there is to say about truth. Several authors have claimed that the expression of infinite conjunctions and disjunctions is the only purpose of the disquotationalist truth predicate. The way in which infinite conjunctions can be expressed by an axiomatized truth predicate is explored and it is considered whether the disquotationalist truth predicate is adequate for this purpose.

At the centre of the traditional discussion of truth is the question of how truth is defined. Recent research, especially with the development of deflationist accounts of truth, has tended to take truth as an undefined primitive notion governed by axioms, while the liar paradox and cognate paradoxes pose problems for certain seemingly natural axioms for truth. In this book, Volker Halbach examines the most important axiomatizations of truth, explores their properties and shows how the logical results impinge on (...) the philosophical topics related to truth. In particular, he shows that the discussion on topics such as deflationism about truth depends on the solution of the paradoxes. His book is an invaluable survey of the logical background to the philosophical discussion of truth, and will be indispensable reading for any graduate or professional philosopher in theories of truth. (shrink)

If □ is conceived as an operator, i.e., an expression that gives applied to a formula another formula, the expressive power of the language is severely restricted when compared to a language where □ is conceived as a predicate, i.e., an expression that yields a formula if it is applied to a term. This consideration favours the predicate approach. The predicate view, however, is threatened mainly by two problems: Some obvious predicate systems are inconsistent, and possible-worlds semantics for predicates of (...) sentences has not been developed very far. By introducing possible-worlds semantics for the language of arithmetic plus the unary predicate □, we tackle both problems. Given a frame (W, R) consisting of a set W of worlds and a binary relation R on W, we investigate whether we can interpret □ at every world in such a way that □ $\ulcorner A \ulcorner$ holds at a world ᵆ ∊ W if and only if A holds at every world $\upsilon$ ∊ W such that ᵆR $\upsilon$ . The arithmetical vocabulary is interpreted by the standard model at every world. Several 'paradoxes' (like Montague's Theorem, Gödel's Second Incompleteness Theorem, McGee's Theorem on the ω-inconsistency of certain truth theories, etc.) show that many frames, e.g., reflexive frames, do not allow for such an interpretation. We present sufficient and necessary conditions for the existence of a suitable interpretation of □ at any world. Sound and complete semi-formal systems, corresponding to the modal systems K and K4, for the class of all possible-worlds models for predicates and all transitive possible-worlds models are presented. We apply our account also to nonstandard models of arithmetic and other languages than the language of arithmetic. (shrink)

This paper aims to provide a comprehensive explanation for the likely failure in the decoupling of economic growth from environmental degradation, and also intends to offer perspectives on the new role of competition in a steady state or a degrowth economy. The analysis is based on five different scenarios, and uses the European Union as an example. It is concluded that we must prepare ourselves for a potential incompatibility between sustainability and economic growth. In this respect one can say that (...) the current EU situation is in some ways already quite close to an economic system without growth, although far from sustainable as yet. Two of the four perspectives developed, regarding the new role of competition in an economy without growth, indicate an increase of direct and indirect competition over resources. The other two perspectives point out that regulatory public intervention as well as financial intervention will increase in order to ensure that - despite the revised role of competition - the poorest are still given a chance to develop (e.g. through basic minimum incomes), even in a degrowth economy. (shrink)

Solutions to semantic paradoxes often involve restrictions of classical logic for semantic vocabulary. In the paper we investigate the costs of these restrictions in a model case. In particular, we fix two systems of truth capturing the same conception of truth: of the system KF of Feferman formulated in classical logic, and the system PKF of Halbach and Horsten, formulated in basic De Morgan logic. The classical system is known to be much stronger than the nonclassical one. We assess the (...) reasons for this asymmetry by showing that the truth theoretic principles of PKF cannot be blamed: PKF with induction restricted to non-semantic vocabulary coincides in fact with what the restricted version of KF proves true. (shrink)

Over the past decades, we have witnessed calls for greater transdisciplinary engagement between scientific and societal actors to develop more robust answers to complex societal challenges. Although there seems to be agreement that these approaches might nurture innovations of a new kind, we know little regarding the research practices, their potential, and the limitations. To fill this gap, this article investigates a funding scheme in the area of transdisciplinary sustainability research. It offers a detailed analysis of the imaginaries and expectations (...) on which the funding scheme rests and how researchers actually practice transdisciplinarity within the respective projects. Identifying three ideal typical models of science–society relations at work, attention is paid to how, where, and when societal and scientific arenas get entangled. This article discusses the tensions between classical academic values and efforts to open research to society, the prevailing power structures that make societal participation challenging, the importance of place and technopolitical cultures, and how temporal project structures impede more radical openings to new ways of knowledge production. We finally emphasize that transdisciplinary knowledge production can only become a serious option for addressing societal challenges if broader changes are made to the knowledge regimes in place. (shrink)

This article critically engages a recent essay Jeffrey Alexander has published on ‘societalization’, whose conceptualization it finds problematic; first, because in contrast to the impression conveyed by the essay, the term itself is anything but new (as shown in a summary of six theories of societalization which precede Alexander’s by decades, in two cases, by more than a century), and, second, because the way Alexander employs the term is highly aporetic, while also being emblematic of much deeper problems that afflict (...) the whole discipline. Following a reconstruction of the term’s morphology and the transmutations it underwent during its gradual incorporation into the English language, the article identifies an undertheorized concept of society as the root cause behind the difficulties into which Alexander maneuvers himself. It concludes with a brief sketch of an alternative that can contribute to overcoming these difficulties. (shrink)

To the axioms of Peano arithmetic formulated in a language with an additional unary predicate symbol T we add the rules of necessitation and conecessitation T and axioms stating that T commutes with the logical connectives and quantifiers. By a result of McGee this theory is -inconsistent, but it can be approximated by models obtained by a kind of rule-of-revision semantics. Furthermore we prove that FS is equivalent to a system already studied by Friedman and Sheard and give an analysis (...) of its proof theory. (shrink)

The purpose of this article is to explore the potential contribution of Axel Honneth's critical theory of recognition to empirical and normative debates on global justice. I first present, very briefly, an overview of recent theories of global distributive justice. I argue that theorists of distributive justice do not pay enough attention to sources of self-respect and conditions for identity formation, and that they are blind toward the danger of harming people's sense of self even by well-intentioned redistributive policies. Honneth's (...) theory suffers from complementary shortcomings; it is anti-technocratic but largely oblivious to the global nature of many contemporary justice claims. Given this situation, I seek to broaden the theory's scope by outlining transnational extensions of the recognition principles of love, rights and solidarity identified by Honneth. In conclusion, I show how utilizing a broadened conceptualization of the struggle for recognition allows us to better understand the changing logic of justice-oriented foreign policies. (shrink)

The uniform reflection principle for the theory of uniform T-sentences is added to PA. The resulting system is justified on the basis of a disquotationalist theory of truth where the provability predicate is conceived as a special kind of analyticity. The system is equivalent to the system ACA of arithmetical comprehension. If the truth predicate is also allowed to occur in the sentences that are inserted in the T-sentences, yet not in the scope of negation, the system with the reflection (...) schema for these T-sentences assumes the strength of the Kripke-Feferman theory KF, and thus of ramified analysis up to go. (shrink)

It is shown that David Hilbert's formalistic approach to axiomaticis accompanied by a certain pragmatism that is compatible with aphilosophical, or, so to say, external foundation of mathematics.Hilbert's foundational programme can thus be seen as areconciliation of Pragmatism and Apriorism. This interpretation iselaborated by discussing two recent positions in the philosophy ofmathematics which are or can be related to Hilbert's axiomaticalprogramme and his formalism. In a first step it is argued that thepragmatism of Hilbert's axiomatic contradicts the opinion thatHilbert style (...) axiomatical systems are closed systems, a reproachposed by Carlo Cellucci. In the second section the question isdiscussed whether Hilbert's pragmatism in foundational issuescomes close to an a-philosophical ``naturalism in mathematics'' assuggested by Penelope Maddy. The answer is ``no'', because forHilbert philosophy had its specific tasks in the general projectto found mathematics. This is illuminated in the concludingsection giving further evidence for Hilbert's foundationalapriorism by discussing his ``axiom of the existence of mind'' andrelating it to the ``one and only axiom'' of the German algebraistof logic, Ernst Schröder, postulating the inherence of signs onthe paper. (shrink)

On the one hand, the concept of truth is a major research subject in analytic philosophy. On the other hand, mathematical logicians have developed sophisticated logical theories of truth and the paradoxes. Recent developments in logical theories of the semantical paradoxes are highly relevant for philosophical research on the notion of truth. And conversely, philosophical guidance is necessary for the development of logical theories of truth and the paradoxes. From this perspective, this volume intends to reflect and promote deeper interaction (...) and collaboration between philosophers and logicians investigating the concept of truth than has existed so far.Aside from an extended introductory overview of recent work in the theory of truth, the volume consists of articles by leading philosophers and logicians on subjects and debates that are situated on the interface between logical and philosophical theories of truth. The volume is intended for graduate students in philosophy and in logic who want an introduction to contemporary research in this area, as well as for professional philosophers and logicians. (shrink)

In this sequel toSelf-reference in arithmetic Iwe continue our discussion of the question: What does it mean for a sentence of arithmetic to ascribe to itself a property? We investigate how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed point for the expressing formulae are obtained. In this second part we look at some further examples. In particular, we study sentences apparently expressing their Rosser-provability, their own${\rm{\Sigma (...) }}_n^0$-truth or their own${\rm{\Pi }}_n^0$-truth. Finally we offer an assessment of the results of both papers. (shrink)

Wittgenstein’s Whewell’s Court Lectures contains previously unpublished notes from lectures given by Ludwig Wittgenstein between 1938 and 1941. The volume offers new insight into the development of Wittgenstein’s thought and includes some of the finest examples of Wittgenstein’s lectures in regard to both content and reliability.

Every educational system requires some kind of management and control. This is particularly true in the case of occupational education and dual apprenticeship. Nevertheless, the didactical value and the overall success of such kind of educational systems can vary considerably. Any educational system, any system of apprenticeship requires either some kind of coordination or some kind of co-operation. In Germany, despite the fact that there is a lot of rumors about the importance of cooperation for over a decade by now, (...) the well-known dual system of vocational education is firmly grounded on the management principle of co-ordination. The control of the system ‘by plans’ apparently has proved to be the most adequate approach to its organization and management structures. Starting from the idea of a duality which is characterized by its didactical dimension, the set of qualities needed for this purpose can be illustrated by following the logic of different didactical approaches in vocational education. (shrink)

We study operator equations within the Turing machine based framework for computability in analysis. Is there an algorithm that maps pairs to solutions of Tx = u ? Here we consider the case when T is a bounded linear mapping between Hilbert spaces. We are in particular interested in computing the generalized inverse T†u, which is the standard concept of solution in the theory of inverse problems. Typically, T† is discontinuous and hence no computable mapping. However, we will use effective (...) versions of theorems from the theory of regularization to show that the mapping ↦ T†u is computable. We then go on to study the computability of average-case solutions with respect to Gaussian measures which have been considered in information based complexity. Here, T† is considered as an element of an L2-space. We define suitable representations for such spaces and use the results from the first part of the paper to show that ↦ T† is computable. (shrink)

The general notions of object- and metalanguage are discussed and as a special case of this relation an arbitrary first order language with an infinite model is expanded by a predicate symbol T0 which is interpreted as truth predicate for . Then the expanded language is again augmented by a new truth predicate T1 for the whole language plus T0. This process is iterated into the transfinite to obtain the Tarskian hierarchy of languages. It is shown that there are natural (...) points for stopping this process. The sets which become definable in suitable hierarchies are investigated, so that the relevance of the Tarskian hierarchy to some subjects of philosophy of mathematics are clarified. (shrink)

A theory of the transfinite Tarskian hierarchy of languages is outlined and compared to a notion of partial truth by Kripke. It is shown that the hierarchy can be embedded into Kripke's minimal fixed point model. From this results on the expressive power of both approaches are obtained.

Against competing political theories of the integration of immigrants, we propose to reframe the relationship between the populations of host countries and arriving refugees in terms of a neo-Maussian theory of gift exchange. Using the example of the European refugee crisis of 2015 and the welcoming attitude of significant parts of German civil society, we argue that this particular situation should be understood as epitomizing the trend toward internal transnationalism. Increasingly, the “international” is becoming part and parcel of the “domestic” (...) sphere. Since Marcel Mauss was concerned with the question of how separate, culturally different communities can establish ties of solidarity and cooperation between each other, we use his work to answer key questions about the relations between international refugees and native citizens in their home countries: What are the expectations underlying gift-giving in the context of welcoming refugees? Should refugees feel obliged to repay the arrival gifts? How sh... (shrink)

The book provides a thematic account of the changing political thought of critical theorists from Adorno to Habermas and Honneth. Its purpose is to establish the relevance of this tradition for contemporary political theory and philosophy.

Zermelos Zeit in Göttingen (1897?1910) kann als wissenschaftlich fruchtbarste Periode in seiner Karriere angesehen werden. Gleichwohl stehen bisher Untersuchungen aus. die eine Einbettung von Zermelos Werk in den biographischen und sozialen Kontext ermöglichen Die vorliegende Studie will diese Lücke unter Konzentration auf zwei Gegenstandsbereiche teileweise ausfüllen: (1) den historischen Entstehungskontext von Zermelos ersten Arbeiten über die Grundlagen der Mengenlehre; (2) die Vorgeschichte und näheren Umstände des 1907 an Zermelo verliehenen Lehrauftrages für mathematische Logik und verwandte Gegenstände. mit dem ein erster (...) Schritt zur Institutionalisierung dieses Faches als mathematischer Teildisziplin gemacht wurde. Beides wird in enger Verbindung zur ersten Phase des Hilbertschen Programms zur Grundlegung der Mathematik gesehen. Es wird aber auch gezeigt, daß für die Erteilung des Lehrauftrages neben diesen systematischen und forschungspolitischen Motiven auch persönliche und institutspolitische Rücksichten ausschlaggebend waren. Im Anhang werden Teile eines Briefwechsels zwischen Ernst Zermelo und dem Göttinger Philosophen Leonard Nelson ediert sowie die Anträge der Direktoren des Göttinger mathematisch-physikalischen Seminars auf Erteilung des Lehrauftrags und auf Ernennung Zermelos zum außerordentlichen Professor. Zermelo's Göttingen period (1897?1910) can be regarded as the most fruitful period in his scientific career. Nevertheless, up till now there have been no investigations which enable us to embed his work in its biographical and social contexts. This study tries to partially fill the gap, concentrating on two major themes: (I) the historical context of the development of Zermelo's early work on the foundations of set theory; (2) the prehistory and particulars of his lectureship for mathematical logic and related topics, this lectureship, which he was awarded in 1907, being the first step taken in Germany towards the establishment of this subject as a separate mathematical discipline. Both will be seen in close connection with the first phase of Hilbert's programme of founding mathematics. It is shown, however, that this lectureship was not only created to further the aims of Hilbert's research programme but that personal and institutional motives also played a role. In the appendices, parts of a correspondence between Ernst Zermelo and the Göttingen Philosopher Leonard Nelson are edited, as well as the applications of the directors of the Göttingen Mathematisch-physikalisches Seminar for awarding the lectureship to Zermelo in 1907, and for appointing Zermelo to an extraordinary professorship in 1910. (shrink)

400 years after Bruno's death by fire in Rome the question arises whether Bruno did in fact die as martyr of truth. If we consider his philosophical arguments we can understand his cosmological concept of the infinite universe as essentially contradictory to some of Aristotle's ideas of philosophy of nature. Alongside his belief in the omnipresence of divinity in nature he takes the view of the universe as a living organism, an area where he was influenced by the hermetic tradition. (...) Above all Bruno's cosmology also includes ethical elements. So if we ask for the actual reason of Bruno's conviction, some theological consequences of his cosmological ideas did presumably influence the judgement of the inquisition. In contrast to Galileo Bruno's condemnation has not so far been withdrawn. However such reconsideration is possible and counting on the church's spirit of reconciliation should come to pass. (shrink)

It is a commonplace that in the development of modern logic towards its actual shape at least two directions or traditions have to be distinguished. These traditions may be called, following the mo...

This paper gives a survey of David Hilbert's (1862–1943) changing attitudes towards logic. The logical theory of the Göttingen mathematician is presented as intimately linked to his studies on the foundation of mathematics. Hilbert developed his logical theory in three stages: (1) in his early axiomatic programme until 1903 Hilbert proposed to use the traditional theory of logical inferences to prove the consistency of his set of axioms for arithmetic. (2) After the publication of the logical and set-theoretical paradoxes by (...) Gottlob Frege and Bertrand Russell it was due to Hilbert and his closest collaborator Ernst Zermelo that mathematical logic became one of the topics taught in courses for Göttingen mathematics students. The axiomatization of logic and set-theory became part of the axiomatic programme, and they tried to create their own consistent logical calculi as tools for proving consistency of axiomatic systems. (3) In his struggle with intuitionism, represented by L. E. J. Brouwer and his advocate Hermann Weyl, Hilbert, assisted by Paul Bemays, created the distinction between proper mathematics and meta-mathematics, the latter using only finite means. He considerably revised the logical calculus of thePrincipia Mathematica of Alfred North Whitehead and Bertrand Russell by introducing the ε-axiom which should serve for avoiding infinite operations in logic. (shrink)

) Modern theories of formal truth have traditionally been used to analyse semantic paradoxes, but their field of application goes well beyond this field to include ontological issues, Godel′s incompleteness phenomena, and the relationship between object language, meta language and reduction. All these fields have had new light sched upon them by studies on the theories of truth. In providing a first summary of the various approaches in this field the author documents their respective advantages and areas of application. The (...) emphasis here is on the axiomatic theories of truth developed by Cantini, Feferman, Friedman and the autor himself, which over the past ten years have increasingly gained ground over the semantic theories of Kripke et al. The book describes the way in which many of the aciomatic proposals are built up and expand on semantic theories. This ist the first time that the results in this field have been made accessible to readers in their larger context. The book deals in detail with the technical aspects and so is ideal as a point of entry into the vast array of literature on the metamathematics of this field. A key objective of the book is to extablish a connection between the latest theories of truth and the older constructions elaborated by Tarski. He made a number of important findings about the formal properties of truth back in the 1930s, but some of his constructions are now considered to be obsolete. Tarski′s views appear in a new light in this book since some of his ideas have only been formulated in recent years. (shrink)