Here, I combine the semantics of Mares and Goldblatt [20] and Seki [29, 30] to develop a semantics for quantified modal relevant logics extending ${\bf B}$. The combination requires demonstrating that the Mares–Goldblatt approach is apt for quantified extensions of ${\bf B}$ and other relevant logics, but no significant bridging principles are needed. The result is a single semantic approach for quantified modal relevant logics. Within this framework, I discuss the requirements a quantified modal relevant logic must satisfy to be (...) “sufficiently classical” in its modal fragment, where frame conditions are given that work for positive fragments of logics. The roles of the Barcan formula and its converse are also investigated. (shrink)

In this paper, we investigate neighbourhood semantics for modal extensions of relevant logics. In particular, we combine the neighbourhood interpretation of the relevant implication (and related connectives) with a neighbourhood interpretation of modal operators. We prove completeness for a range of systems and investigate the relations between neighbourhood models and relational models, setting out a range of augmentation conditions for the various relations and operations.

Sedlár and Vigiani [18] have developed an approach to propositional epistemic logics wherein (i) an agent’s beliefs are closed under relevant implication and (ii) the agent is located in a classical possible world (i.e., the non-modal fragment is classical). Here I construct first-order extensions of these logics using the non-Tarskian interpretation of the quantifiers introduced by Mares and Goldblatt [12], and later extended to quantified modal relevant logics by Ferenz [6]. Modular soundness and completeness are proved for constant domain (...) semantics, using non-general frames with Mares–Goldblatt truth conditions. I further detail the relation between the demand that classical possible worlds have Tarskian truth conditions and incompleteness results in quantified relevant logics. (shrink)

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)

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)

Here I show that the one-variable fragment of several first-order relevant logics corresponds to certain S5ish extensions of the underlying propositional relevant logic. In particular, given a fairly standard translation between modal and one-variable languages and a permuting propositional relevant logic L, a formula $$\mathcal {A}$$ A of the one-variable fragment is a theorem of LQ (QL) iff its translation is a theorem of L5 (L.5). The proof is model-theoretic. In one direction, semantics based on the Mares-Goldblatt [15] semantics for (...) quantified L are transformed into ternary (plus two binary) relational semantics for S5-like extensions of L (for a general presentation, see Seki [26, 27]). In the other direction, a valuation is given for the full first-order relevant logic based on L into a model for a suitable S5 extension of L. I also discuss this work’s relation to finding a complete axiomatization of the constant domain, non-general frame ternary relational semantics for which RQ is incomplete [11]. (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)

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)

For relevant logics, the admissibility of the rule of proof $\gamma $ has played a significant historical role in the development of relevant logics. For first-order logics, however, there have been only a handful of $\gamma $ -admissibility proofs for a select few logics. Here we show that, for each logic L of a wide range of propositional relevant logics for which excluded middle is valid (with fusion and the Ackermann truth constant), the first-order extensions QL and LQ admit $\gamma (...) $. Specifically, these are particular “conventionally normal” extensions of the logic $\mathbf {G}^{g,d}$, which is the least propositional relevant logic (with the usual relational semantics) that admits $\gamma $ by the method of normal models. We also note the circumstances in which our results apply to logics without fusion and the Ackermann truth constant. (shrink)

The semantics I develop extend an approach to logic called preservationism. The preservationist approach to logic interprets non-classical consequence relations as preserving something other than truth. I specifically extend a preservationist approach, due to Bryson Brown, which interprets various paraconsistent consequence relations as preserving measures of ambiguity. Relevant logics are constructible by extending one of these logics with an implication connective. I develop a formal semantics which I show to be adequate for interesting relevant logics. I argue that the semantics (...) I develop extend the preservationist approach to relevant logic by showing how the approach treats the implication connective. I conclude by arguing that some of the most pressing objections to the standard semantics for relevant logics do not apply to the ambiguity preservation account. (shrink)

The proof-theoretic results on axiomatic theories oftruth obtained by different authors in recent years are surveyed.In particular, the theories of truth are related to subsystems ofsecond-order analysis. On the basis of these results, thesuitability of axiomatic theories of truth for ontologicalreduction is evaluated.

A substitutional account of logical validity for formal first‐order languages is developed and defended against competing accounts such as the model‐theoretic definition of validity. Roughly, a substitution instance of a sentence is defined as the result of uniformly substituting nonlogical expressions in the sentence with expressions of the same grammatical category and possibly relativizing quantifiers. In particular, predicate symbols can be replaced with formulae possibly containing additional free variables. A sentence is defined to be logically true iff all its substitution (...) instances are satisfied by all variable assignments. Logical consequence is defined analogously. Satisfaction is taken to be a primitive notion and axiomatized.For every set‐theoretic model in the sense of model theory there exists a corresponding substitutional interpretation in a sense to be specified. Conversely, however, there are substitutional interpretations – in particular the ‘intended’ interpretation – that lack a model‐theoretic counterpart. The substitutional definition of logical validity overcomes the weaknesses of more restrictive accounts of substitutional validity; unlike model‐theoretic logical consequence, the substitutional notion is trivially and provably truth preserving. In Kreisel's squeezing argument the formal notion of substitutional validity naturally slots into the place of intuitive validity. (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)

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)

A transcription of Richard Routley's manuscript, "Semantic Analysis of Entailment and Relevant Implication: I".

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.

This chapter discusses the complex conditions for the emergence of 19th-century symbolic logic. The main scope will be on the mathematical motives leading to the interest in logic; the philosophical context will be dealt with only in passing. The main object of study will be the algebra of logic in its British and German versions. Special emphasis will be laid on the systems of George Boole and above all of his German follower Ernst Schröder.

) 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)

Some axiomatic theories of truth and related subsystems of second-order arithmetic are surveyed and shown to be conservative over their respective base theory. In particular, it is shown by purely finitistically means that the theory PA ÷ "there is a satisfaction class" and the theory FS of [2] are conservative over PA.

This article suggests reading Theodor Adorno not as a notoriously pessimistic sociologist but as a committed public educator. Partly drawing on still unpublished transcripts of lectures, public talks and radio broadcasts from the 1950s and ’60s, the article offers an account of Adorno’s concept and practice of a ‘democratic pedagogy’. The key question is how we should understand the difference between Adorno the social philosopher, on the one hand, and Adorno the educator, on the other. It is argued that Adorno’s (...) pedagogical interventions are not a footnote to his social theory, but a key to understanding his entire oeuvre. (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)

Historically, blatantly untrue and defamatory conspiracy theories had disastrous consequences for those who were portrayed in them as evil-doers. At the same time, conspiratorial agreements at the expense of the common good between powerful groups in society do exist and have occasionally been uncovered. Against this background, the article describes different ways in which critical theory has looked at conspiracies. First, an attempt is made to show that Max Horkheimer's notes on `rackets' are an ambitious but flawed attempt to theorize (...) conspiracy. It is argued that Horkheimer's theory is imbued by the very conspiracy thinking that he proposed to criticize. Second, the author suggests recovering Franz Neumann's concept of `political alienation' as a more appropriate starting point to think critically about the ethical and epistemological questions raised by conspiracy theories. (shrink)

This article—mainly referring to the situation in Germany—consists of three parts. In a first section the current presence of neurosciences in the public discourse will be described in order to illuminate the background which is relevant for contemporary educational thinking. The prefix ‘neuro-’ is ubiquitous today and therefore concepts like ‘neuropedagogy’ or ‘neurodidactics’ seem to be in the mainstream of modern thinking. In the second part of the article the perspective changes from the public discourse to the disciplinary discourse; a (...) brief excursus into developmental psychiatry, neuropsychology and modern psychoanalysis will be made in order to demonstrate how the results of neuroscientific research are integrated in their theoretical frameworks. These three disciplines have no difficulty in integrating neuroscientific findings because each of them possesses a systematic core composed of ‘native concepts’. In contrast to them, educational theory has much more difficulty with such integration, as will be shown in the third part of the essay. On the one hand, neuroscientific thinking seems to be able to dominate education rather easily and without great resistance, especially in the fields of early childhood education, instruction and learning—mainly by simplifying educational processes and by reducing the complexity of the educational task to a mere ‘relationship problem’. On the other hand, this attraction of neuroscience in education might be understood as the reflection of a theoretical deficit in educational theory itself, with the significance of affect and emotion not receiving proper attention. (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 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)

The distinction between the contexts of discovery and justification has left a turbulent wake in the philosophy of science. This book recognizes the need to re-open the debate about the nature, development, and significance of the context distinction, about its merits and flaws. The discussion clears the ground for the productive and fruitful integration of these new developments into philosophy of science.

The attempts to mitigate the unprecedented health, economic, and social disruptions caused by the COVID-19 pandemic are largely dependent on establishing compliance to behavioral guidelines and rules that reduce the risk of infection. Here, by conducting an online survey that tested participants’ knowledge about the disease and measured demographic, attitudinal, and cognitive variables, we identify predictors of self-reported social distancing and hygiene behavior. To investigate the cognitive processes underlying health-prevention behavior in the pandemic, we co-opted the dual-process model of thinking (...) to measure participants’ propensities for automatic and intuitive thinking vs. controlled and reflective thinking. Self-reports of 17 precautionary behaviors, including regular hand washing, social distancing, and wearing a face mask, served as a dependent measure. The results of hierarchical regressions showed that age, risk-taking propensity, and concern about the pandemic predicted adoption of precautionary behavior. Variance in cognitive processes also predicted precautionary behavior: participants with higher scores for controlled thinking (measured with the Cognitive Reflection Test) reported less adherence to specific guidelines, as did respondents with a poor understanding of the infection and transmission mechanism of the COVID-19 virus. The predictive power of this model was comparable to an approach (Theory of Planned Behavior) based on attitudes to health behavior. Given these results, we propose the inclusion of measures of cognitive reflection and mental model variables in predictive models of compliance, and future studies of precautionary behavior to establish how cognitive variables are linked with people’s information processing and social norms. (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 Mares-Goldblatt semantics for quantified relevant logics have been developed for first-order extensions of R, and a range of other relevant logics and modal extensions thereof. All such work has taken place in the the ternary relation semantic framework, most famously developed by Sylvan and Meyer. In this paper, the Mares-Goldblatt technique for the interpretation of quantifiers is adapted to the more general neighbourhood semantic framework, developed by Sylvan, Meyer, and, more recently, Goble. This more algebraic semantics allows one to (...) characterise a still wider range of logics, and provides the grist for some new results. To showcase this, we show, using some non-augmented models, that some quantified relevant logics are not conservatively extended by connectives the addition of which do conservatively extend the associated propositional logics, namely fusion and the dual implication. We close by proposing some further uses to which the neighbourhood Mares-Goldblatt semantics may be put. (shrink)

The writings of Volker Gerhardt have had substantial influence on recent Nietzsche research. This volume collects a variety of his influential essays and makes them available to scholars and causal readers alike. The present collection takes stock of Nietzsche’s most prominent ideas. Gerhardt investigates which of these ideas still offer solutions for the problems of today. In the process, it becomes clear what sort of inspiration a future-oriented philosophizing can draw from Nietzsche.

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.

The article addresses the following question: if an extensive period of globalization and also democratization after the fall of the Berlin Wall has been followed by populism, does this mean that there is something wrong with liberalism itself? Must liberalism be substituted by alternative economic and political concepts? The article presents three alternatives to liberalism that are supposed to counter populism: a new communitarianism, a renewal of the democratic project as much as novel conceptions of social justice. However, it takes (...) also into account positions that address the current crisis from within the liberal framework itself. (shrink)

Summary The non-statement view of scientific theories contains a new conception of theoreticity: A function is „T-theoretical if T must be presupposed for its calculation. On the basis of this conception some philosophers came to the conclusion that scientific theories are not empirically testable because they contain T-theoretical functions. It is claimed that the attempt to test them ends in a circularity: The test of T presupposes T itself.

The history of modern logic is usually written as the history of mathematical or, more general, symbolic logic. As such it was created by mathematicians. Not regarding its anticipations in Scholastic logic and in the rationalistic era, its continuous development began with George Boole's The Mathematical Analysis of Logic of 1847, and it became a mathematical subdiscipline in the early 20th century. This style of presentation cuts off one eminent line of development, the philosophical development of logic, although logic is (...) evidently one of the basic disciplines of philosophy. One needs only to recall some of the standard 19th century definitions of logic as, e.g., the art and science of reasoning (Whateley) or as giving the normative rules of correct reasoning (Herbart). In the paper the relationship between the philosophical and the mathematical development of logic will be discussed. Answers to the following questions will be provided: 1. What were the reasons for the philosophers' lack of interest in formal logic? 2. What were the reasons for the mathematicians' interest in logic? 3. What did "logic reform" mean in the 19th century? Were the systems of mathematical logic initially regarded as contributions to a reform of logic? 4. Was mathematical logic regarded as art, as science or as both? (shrink)

This article presents a critical reading of Albert O. Hirschman’s typology of exit, voice and loyalty as a heuristic for understanding the changing meanings of exile in the 20th and early 21st centuries. It is argued that Hirschman’s experiences as well as the theory he distilled from them are highly relevant for researchers of forced migration and exile. After first defending the usefulness of Hirschman’s analytical framework for exile and diaspora studies, the article then highlights the need to revise and (...) complicate his approach. Hirschman could not foresee the emerging global possibilities of cultivating ‘the art of voice’, new forms of internal and self-exile as a result of post-fascist versions of authoritarianism, and the growing difficulties faced by refugees including, refugee scholars and writers, to exit their countries and find a safe haven somewhere else. The gaps in Hirschman’s theory are addressed by drawing on insights from the writings of Judith Shklar. (shrink)