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Disquotational theories of truth are often criticised for being too weak to prove interesting generalisations about truth. In this paper we will propose a certain formal theory to serve as a framework for a solution of the generalisation problem. In contrast with Horwich’s original proposal, our framework will eschew psychological notions altogether, replacing them with the epistemic notion of believability. The aim will be to explain why someone who accepts a given disquotational truth theory Th, should also accept various generalisations not provable in Th. The strategy will consist of the development of an axiomatic theory of believability, one permitting us to show how to derive the believability of generalisations from basic axioms that characterise the believability predicate, together with the information that Th is a theory of truth that we accept.

In the section of the Antinomy of pure Reason Kant presents three notions of infinity. By investigating these concepts of infinity, this paper highlights important ‘building blocks’ of the structure of the mathematical antinomies, such as the ability of reason of producing ascending and descending series, as well as the notions of given and givable series. These structural features are discussed in order to clarify Ernst Zermelo’s reading of Kant’s antinomy, according to which the latter is deeply rooted in the tendency of the mind of producing “creative progress” and “inclusive closure”. The aim of this paper is to explain in which sense and why Kant’s treatment of the antinomies attracts the attention of Zermelo in the early 1900s and which aspects of his second axiomatic system have been inspired by Kant’s philosophy. Thus, by reading Kant’s antinomy ‘through Zermelo’s eyes’—with emphasis on the concept of regressive series in indefinitum and on that of regressive series ad infinitum – this paper identifies the echoes of Kant’s work in the making of the ZFC set theory.

The article studies knowledge in multiagent systems where data available to the agents may have small errors. To reason about such uncertain knowledge, a formal semantics is introduced in which indistinguishability relations, commonly used in the semantics for epistemic logic S5, are replaced with metrics to capture how much two epistemic worlds are different from an agent’s point of view. The main result is a logical system sound and complete with respect to the proposed semantics.

There are two prominent viewpoints regarding the nature of rationality and how it should be evaluated in situations of interest: the traditional axiomatic approach and the newer ecological rationality. An obstacle to comparing and evaluating these seemingly opposite approaches is that they employ different language and formalisms, ask different questions, and are at different stages of development. I adapt a formal framework known as SCOP to address this problem by providing a comprehensive common framework in which both approaches may be defined and compared. The main result is that the axiomatic and ecological approaches are in far greater agreement on fundamental issues than has been appreciated; this is supported by a pair of theorems to the effect that they will make accordant rationality judgements when forced to evaluate the same situation. I conclude that ecological rationality has some subtle advantages, but that we should move past the issues currently dominating the discussion of rationality.

In this paper we study the axiomatic system proposed by Bourbaki for the Theory of Sets in the Elements de Mathematique. We begin by examining the role played by the sign $$\uptau $$ in the framework of its formal logical theory and then we show that the system of axioms for set theory is equivalent to Zermelo–Fraenkel system with the axiom of choice but without the axiom of foundation. Moreover, we study Grothendieck’s proposal of adding to Bourbaki’s system the axiom of universes for the purpose of considering the theory of categories. In this regard, we make some historical and epistemological remarks that could explain the conservative attitude of the Group.

The article studies properties of interchangeability of pure, mixed, strict, and strict mixed Nash equilibria. The main result is a sound and complete axiomatic system that describes properties of interchangeability in all four settings. It has been previously shown that the same axiomatic system also describes properties of independence in probability theory, nondeducibility in information flow, and non-interference in concurrency theory.

This paper analyzes Mally’s system of deontic logic, introduced in his The Basic Laws of Ought: Elements of the Logic of Willing (1926). We discuss Mally’s text against the background of some contributions in the literature which show that Mally’s axiomatic system for deontic logic is flawed, in so far as it derives, for an arbitrary A, the theorem “A ought to be the case if and only if A is the case”, which represents a collapse of obligation. We then try to sort out and understand which axioms are responsible for the collapse and consider two ways of amending Mally’s system: (i) by changing its original underlying logical basis, that is classical logic, and (ii) by modifying Mally’s axioms.

Three different ways in which systems of axioms can contribute to the discovery of new notions are presented and they are illustrated by the various ways in which lattices have been introduced in mathematics by Schroder et al. These historical episodes reveal that the axiomatic method is not only a way of systematizing our knowledge, but that it can also be used as a fruitful tool for discovering and introducing new mathematical notions. Looked at it from this perspective, the creative aspect of axiomatics for mathematical practice is brought to the fore.

The modern notion of the axiomatic method developed as a part of the conceptualization of mathematics starting in the nineteenth century. The basic idea of the method is the capture of a class of structures as the models of an axiomatic system. The mathematical study of such classes of structures is not exhausted by the derivation of theorems from the axioms but includes normally the metatheory of the axiom system. This conception of axiomatization satisfies the crucial requirement that the derivation of theorems from axioms does not produce new information in the usual sense of the term called depth information. It can produce new information in a different sense of information called surface information. It is argued in this paper that the derivation should be based on a model-theoretical relation of logical consequence rather than derivability by means of mechanical (recursive) rules. Likewise completeness must be understood by reference to a model-theoretical consequence relation. A correctly understood notion of axiomatization does not apply to purely logical theories. In the latter the only relevant kind of axiomatization amounts to recursive enumeration of logical truths. First-order “axiomatic” set theories are not genuine axiomatizations. The main reason is that their models are structures of particulars, not of sets. Axiomatization cannot usually be motivated epistemologically, but it is related to the idea of explanation.

Throughout more than two millennia philosophers adhered massively to ideal standards of scientific rationality going back ultimately to Aristotle's Analytica posteriora. These standards got progressively shaped by and adapted to new scientific needs and tendencies. Nevertheless, a core of conditions capturing the fundamentals of what a proper science should look like remained remarkably constant all along. Call this cluster of conditions the Classical Model of Science. In this paper we will do two things. First of all, we will propose a general and systematized account of the Classical Model of Science. Secondly, we will offer an analysis of the philosophical significance of this model at different historical junctures by giving an overview of the connections it has had with a number of important topics. The latter include the analytic-synthetic distinction, the axiomatic method, the hierarchical order of sciences and the status of logic as a science. Our claim is that particularly fruitful insights are gained by seeing themes such as these against the background of the Classical Model of Science. In an appendix we deal with the historiographical background of this model by considering the systematizations of Aristotle's theory of science offered by Heinrich Scholz, and in his footsteps by Evert W. Beth. © The Author(s) 2008.

Axiomatic characterization results in social choice theory are usually compared either regarding the normative plausibility or regarding the logical strength of the axioms involved. Here, instead, we propose to compare axiomatizations according to the language used for expressing the axioms. In order to carry out such a comparison, we suggest a formalist approach to axiomatization results which uses a restricted formal logical language to express axioms. Axiomatic characterization results in social choice theory then turn into definability results of formal logic. The advantages of this approach include the possibility of non-axiomatizability results, a distinction between absolute and relative axiomatizations, and the possibility to ask how rich a language needs to be to express certain axioms. We argue for formal minimalism, i.e., for favoring axiomatizations in the weakest language possible.

We discuss three problems concerning the use of formal languages in theoretical physics: (i) the definability of time and spacetime in classical physical theories; (ii) how to cope with indistinguishable elementary particles in quantum mechanics without labeling them; and (iii) how to get a formal picture of quantum states jumping.

After a brief flirtation with logicism in 1917-1920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more comprehensive areas of mathematics and finitistic proofs of consistency of these systems. Early advances in these areas were made by Hilbert (and Bernays) in a series of lecture courses at the University of Goettingen between 1917 and 1923, and notably in Ackermann's dissertation of 1924. The main innovation was the invention of the epsilon-calculus, on which Hilbert's axiom systems were based, and the development of the epsilon-substitution method as a basis for consistency proofs. The paper traces the development of the "simultaneous development of logic and mathematics" through the epsilon-notation and provides an analysis of Ackermann's consistency proofs for primitive recursive arithmetic and for the first comprehensive mathematical system, the latter using the substitution method. It is striking that these proofs use transfinite induction not dissimilar to that used in Gentzen's later consistency proof as well as non-primitive recursive definitions, and that these methods were accepted as finitistic at the time., 44 pp

In this paper I discuss Husserl's solution of the problem of imaginary elements in mathematics as presented in the drafts for two lectures hegave in Gottingen in 1901 and other related texts of the same period,a problem that had occupied Husserl since the beginning of 1890, whenhe was planning a never published sequel to Philosophie der Arithmetik(1891). In order to solve the problem of imaginary entities Husserl introduced,independently of Hilbert, two notions of completeness (definiteness in Husserl'sterminology) for a formal axiomatic system. I present and discuss these notionshere, establishing also parallels between Husserl's and Hilbert's notions ofcompleteness.

This paper develops — within an axiomatic theory of properties, relations, and propositions which accords them well-defined existence and identity conditions — a sententialist-functionalist account of belief as a symbolically mediated relation to a special kind of propositional entity, theproxy-encoding abstract proposition. It is then shown how, in terms of this account, the truth conditions of English belief reports may be captured in a formally precise and empirically adequate way that accords genuinely semantic status to familiar opacity data.

I first encountered Keith Lehrer's work on consensus during the summer of 1977,l and was immediately intrigued by the possibility of developing a formal account of his model of rational group decision making. I adopted as a model for this enterprise the axiomatic method of social choice theory and was not surprised to discover that the question of how to aggregate probabilities was as complex and prob lematic as the question of how to aggregate preferences and utilities. The former issue has only recently received the kind of attention which has been directed at the latter for several decades, and is finally beginning to be as vigorously debated, as attested to by the essays in this volume. I am grateful to Barry Loe wer for organizing this forum on Rational Consensus in Science and Society. My replies to the preceding critical essays follow.

This special issue of Synthese ‘The Classical Model of Science II: The axiomatic method, the order of concepts and the hierarchy of sciences’ follows up on the previous ‘The ClassicalModel of Science I: Amillennia-oldmodel of scientific rationality’. Both issues centre on the role, the significance and the impact of the axiomatic ideal of scientific knowledge in the history of philosophy. The first issue focuses on the relation between axiomatics and a number of issues in the development of logic, mathematics, and methodology and philosophy of science in Aristotle, Proclus, the seventeenth century, Kant, Bolzano, Frege and Leśniewski. The papers collected in this second issue on the one hand continue to investigate that relation in Kant and Bolzano, stretching it further on to our days, via mathematicians such as Schroder, Dedekind, and Birkhoff, and on the other hand they extend that investigation to related and current issues concerning the empirical sciences, in a systematic evaluation of modern (formal) axiomatic conceptions of science. The contributions in both issues take their cue from the axiomatic ideal in question as captured in the ‘Classical Model (or Ideal) of Science’ (de Jong and Betti 2008)