Dedekind’s and Frege’s views on logic

A contextual and comparative analysis shows that Dedekind and Frege do not understand the terms “logic” and “arithmetic” in the same way. More specifically the meaning and the scope of the corresponding concepts are essentially different for them. Consequently Dedekind and Frege have different conceptions of the relationship between arithmetic and logic.     

Dedekind和的一个性质

Ｄｅｄｅｋｉｎｄ和的Ｋｎｏｐｐ等式是与Ｈｅｃｋｅ算子有关的一个算术性质，本文不借助ｅｔａ－函数的概念，给予Ｋｎｏｐｐ等式的一个简短的初等证明，同时把Ｋｎｏｐｐ等式拓广到广义Ｄｅｄｅｋｉｎｄ和中。     

Reform of the construction of the number system with reference to Gottlob Frege

Due to missing ontological commitments Frege rejected Hilbert’s Fundamentals of Geometry as well as the construction of the system of real numbers by Dedekind and Cantor. Almost all of school mathematics is ontologically committed. Therefore, H.-G. Steiner considered Frege’s viewpoint of mathematics fundamentals, refined by Tarski’s semantics, as suitable for math education. Frege committed numbers ontologically by using measurement to define numbers. He invented the concept of quantitative domain (Größengebiet), which – it is now known by reconstruction of that concept by the New-Fregean Movement – agrees with the concept of quantity domain (Größenbereich) as established in the German reform of the application-oriented construction of the system of real numbers. Concepts of quantity (ratio-scale) and interval-scale in comparative measurement theory – going beyond Frege – show the way how the negative numbers can be ontologically committed and the operations of addition and multiplication can be included. In this work it is shown how Frege’s viewpoint of mathematics fundamentals, as propagated by H.-G. Steiner, can be better implemented in the current construction of the system of real numbers in school.     

From modules to lattices: Insight into the genesis of Dedekind's Dualgruppen

When Dedekind introduced the notion of a module, he also defined their divisibility and related arithmetical notions (e.g. the LCM of modules). The introduction of notations for these notions allowed Dedekind to state new theorems, now recognized as the modular laws in lattice theory. Observing the dualism displayed by the theorems, Dedekind pursued his investigations on the matter. This led him, 20 years later, to introduce Dualgruppen, equivalent to lattices Dedekind [Über Zerlegungen von Zahlen durch ihre größten gemeinsamen Teiler. In Dedekind (1930–1932), volume II, 1897, 103–147; Über die von drei Moduln erzeugte Dualgruppe. In Dedekind (1930–1932), volume II, 1900, 236–271]. After a brief exposition of the basic elements of Dualgruppe theory, and with the help of his Nachlass, I show how Dedekind gradually built his theory through layers of computations, often repeated in slight variations and attempted generalizations. I study the tools he devised to help and accompany him in his computations. I highlight the crucial conceptual move that consisted in going from investigating operations between modules, to groups of modules closed under these operations. By using Dedekind's drafts, I aim to highlight the concealed yet essential practices anterior to the published text.     

Logic as a Science and Logic as a Theory: Remarks on Frege, Russell and the Logocentric Predicament

Since its publication in 1967, van Heijenoort??s paper, ??Logic as Calculus and Logic as Language?? has become a classic in the historiography of modern logic. According to van Heijenoort, the contrast between the two conceptions of logic provides the key to many philosophical issues underlying the entire classical period of modern logic, the period from Frege??s Begriffsschrift (1879) to the work of Herbrand, G?del and Tarski in the late 1920s and early 1930s. The present paper is a critical reflection on some aspects of van Heijenoort??s thesis. I concentrate on the case of Frege and Russell and the claim that their philosophies of logic are marked through and through by acceptance of the universalist conception of logic, which is an integral part of the view of logic as language. Using the so-called ??Logocentric Predicament?? (Henry M. Sheffer) as an illustration, I shall argue that the universalist conception does not have the consequences drawn from it by the van Heijenoort tradition. The crucial element here is that we draw a distinction between logic as a universal science and logic as a theory. According to both Frege and Russell, logic is first and foremost a universal science, which is concerned with the principles governing inferential transitions between propositions; but this in no way excludes the possibility of studying logic also as a theory, i.e., as an explicit formulation of (some) of these principles. Some aspects of this distinction will be discussed.     

Frege on Identity as a Relation of Names

This essay offers a detailed philosophical criticism of Frege’s popular thesis that identity is a relation of names. I consider Frege’s position as articulated both in ‘On Sense and Reference’, and in the Grundgesetze, where he appears to take an objectual view of identity, arguing that in both cases Frege is clearly committed to the proposition that identity is a relation holding between names, on the grounds that two different things can never be identical. A counterexample to Frege’s thesis is considered, and a positive thesis is developed according to which, in contradistinction to the Fregean position, identity is a reflexive, symmetric, and transitive relation holding only between a thing and itself which can be expressed as a relation between names.     

Informational Semantics and Frege Cases

One of the most important objections to information-based semantic theories is that they are incapable of explaining Frege cases. The worry is that if a concept’s intentional content is a function of its informational content, as such theories propose, then it would appear that coreferring expressions have to be synonymous, and if this is true, it’s difficult to see how an agent could believe that a is F without believing that b is F whenever a and b are identical. I argue that this appearance is deceptive. If we heed the distinction between the analog and digital contents of a signal, it is actually possible to reconstruct something akin to Frege’s sense/reference distinction in purely information-theoretic terms. This allows informational semanticists to treat coreferring expressions as semantically distinct and to solve Frege cases in the same way that Frege did—namely, by appealing to the different contents of coreferring expressions.     

In Defense of Logical Universalism: Taking Issue with Jean van Heijenoort

Van Heijenoort??s main contribution to history and philosophy of modern logic was his distinction between two basic views of logic, first, the absolutist, or universalist, view of the founding fathers, Frege, Peano, and Russell, which dominated the first, classical period of history of modern logic, and, second, the relativist, or model-theoretic, view, inherited from Boole, Schr?der, and L?wenheim, which has dominated the second, contemporary period of that history. In my paper, I present the man Jean van Heijenoort (Sect. 1); then I describe his way of arguing for the second view (Sect. 2); and finally I come down in favor of the first view (Sect. 3). There, I specify the version of universalism for which I am prepared to argue (Sect. 3, introduction). Choosing ZFC to play the part of universal, logical (in a nowadays forgotten sense) system, I show, through an example, how the usual model theory can be naturally given its proper place, from the universalist point of view, in the logical framework of ZFC; I outline another, not rival but complementary, semantics for admissible extensions of ZFC in the very same logical framework; I propose a way to get universalism out of the predicaments in which universalists themselves believed it to be (Sect. 3.1). Thus, if universalists of the classical period did not, in fact, construct these semantics, it was not that their universalism forbade them, in principle, to do so. The historical defeat of universalism was not technical in character. Neither was it philosophical. Indeed, it was hardly more than the victory of technicism over the very possibility of a philosophical dispute (Sect. 3.2).     

Dedekind和及第一类Chebyshev多项式

本文利用分析方法、Dedekind和及第一类Chebyshev多项式的算术性质,研究了一类关于Dedekind和及第一类Chebyshev多项式混合均值的渐近估计问题,并得到了一个较强的渐近公式.     

The Death of the Expert

The texts of OR are littered, explicitly and implicitly, with myths about the ‘expert’ that are taken for self-evident truths. We would like to challenge these. This paper presents arguments following a postmodern route which views the world as text, where all phenomena and events can be regarded as text and, as such, subject to narrative analysis. Narrative analysis explodes and disperses text to reveal forms and codes according to which meanings are possible. The paper will introduce a case study drawing on our experiences in community OR, which we aim to use to demonstrate this approach.     

The Concept of Relevance and the Logic Diagram Tradition

What is logical relevance? Anderson and Belnap say that the “modern classical tradition [,] stemming from Frege and Whitehead-Russell, gave no consideration whatsoever to the classical notion of relevance.” But just what is this classical notion? I argue that the relevance tradition is implicitly most deeply concerned with the containment of truth-grounds, less deeply with the containment of classes, and least of all with variable sharing in the Anderson–Belnap manner. Thus modern classical logicians such as Peirce, Frege, Russell, Wittgenstein, and Quine are implicit relevantists on the deepest level. In showing this, I reunite two fields of logic which, strangely from the traditional point of view, have become basically separated from each other: relevance logic and diagram logic. I argue that there are two main concepts of relevance, intensional and extensional. The first is that of the relevantists, who overlook the presence of the second in modern classical logic. The second is the concept of truth-ground containment as following from in Wittgenstein’s Tractatus. I show that this second concept belongs to the diagram tradition of showing that the premisses contain the conclusion by the fact that the conclusion is diagrammed in the very act of diagramming the premisses. I argue that the extensional concept is primary, with at least five usable modern classical filters or constraints and indefinitely many secondary intensional filters or constraints. For the extensional concept is the genus of deductive relevance, and the filters define species. Also following the Tractatus, deductive relevance, or full truth-ground containment, is the limit of inductive relevance, or partial truth-ground containment. Purely extensional inductive or partial relevance has its filters or species too. Thus extensional relevance is more properly a universal concept of relevance or summum genus with modern classical deductive logic, relevantist deductive logic, and inductive logic as its three main domains.     

Insights into Dedekind and Weber’s edition of Riemann’s <Emphasis Type="Italic">Gesammelte Werke</Emphasis>

In this paper, I present and analyse Dedekind’s and Weber’s editorial work which led to the edition of Riemann’s Gesammelte Werke in 1876. With several examples, I suggest that this editorial work is to be understood as a mathematical activity in and of itself and provide evidence for it.     

Axiomatic systems for rough sets and fuzzy rough sets

Rough set theory is an important tool for approximate reasoning about data. Axiomatic systems of rough sets are significant for using rough set theory in logical reasoning systems. In this paper, outer product method are used in rough set study for the first time. By this approach, we propose a unified lower approximation axiomatic system for Pawlak’s rough sets and fuzzy rough sets. As the dual of axiomatic systems for lower approximation, a unified upper approximation axiomatic characterization of rough sets and fuzzy rough sets without any restriction on the cardinality of universe is also given. These rough set axiomatic systems will help to understand the structural feature of various approximate operators.     

Theories and Theories of Truth

Formal theories, as in logic and mathematics, are sets of sentences closed under logical consequence. Philosophical theories, like scientific theories, are often far less formal. There are many axiomatic theories of the truth predicate for certain formal languages; on analogy with these, some philosophers (most notably Paul Horwich) have proposed axiomatic theories of the property of truth. Though in many ways similar to logical theories, axiomatic theories of truth must be different in several nontrivial ways. I explore what an axiomatic theory of truth would look like. Because Horwich’s is the most prominent, I examine his theory and argue that it fails as a theory of truth. Such a theory is adequate if, given a suitable base theory, every fact about truth is a consequence of the axioms of the theory. I show, using an argument analogous to Gödel’s incompleteness proofs, that no axiomatic theory of truth could ever be adequate. I also argue that a certain class of generalizations cannot be consequences of the theory.     

Cynefin: uncertainty,small worlds and scenarios

Uncertainty, its modelling and analysis have been discussed across many literatures including statistics and operational research, knowledge management and philosophy: (i) adherents to Bayesian approaches have usually argued that uncertainty should either be modelled by probabilities or resolved by discussion that clarifies meaning; (ii) some have followed Knight in distinguishing between contexts of risk and of uncertainty: the former admitting modelling and analysis through probability; the latter not; (iii) there are also host of approaches in the literatures stemming from Zadeh’s concept of a fuzzy set; (iv) theories of sense-making in the philosophy and management literatures see knowledge and uncertainty as opposite extremes of human understanding and discuss the resolution of uncertainty accordingly. Here I provide a personal perspective, taking a Bayesian stance. However, I adopt a softer position than conventional and recognise the concerns in other approaches. In particular, I use the Cynefin framework of decision contexts to reflect on processes of modelling and analysis in statistical, risk and decision analysis. The approach builds on several recent strands of discussion that argue for a convergence of qualitative scenario planning ideas and more quantitative approaches to analysis. I discuss how these suggestions and discussions relate to some earlier thinking on the methodology of modelling and, in particular, the concept of a ‘small world’ articulated by Savage.     

Relevance and Relationalism

This paper will provide support for relationalism; the claim that the identity of objects is constituted by the totality of their relations to other things in the world. I will consider how Kit Fine’s criticisms of essentialism within modal logic not only highlight the inability of modal logic to account for essential properties but also arouse suspicion surrounding the possibility of nonrelational properties. I will claim that Fine’s criticisms, together with concerns surrounding Hempel’s paradox, show that it is not possible to provide a satisfactory account of certain properties in abstraction from their place within a wider context. Next, we will shift attention to natural kinds and consider the notion that relevance plays in metaphysical accounts of identity, by examining Peter Geach’s notion of relative identity. I will argue that the intensional relation between subject and object must be included in a satisfactory account of metaphysical identity.     

Relative arithmetic

In nonstandard mathematics, the predicate ‘x is standard’ is fundamental. Recently, ‘relative’ or ‘stratified’ nonstandard theories have been developed in which this predicate is replaced with ‘x is y ‐standard’. Thus, objects are not (non)standard in an absolute sense, but (non)standard relative to other objects and there is a whole stratified universe of ‘levels’ or ‘degrees’ of standardness. Here, we study stratified nonstandard arithmetic and the related transfer principle. Using the latter, we obtain the ‘reduction theorem’ which states that arithmetical formulas can be reduced to equivalent bounded formulas. Surprisingly, the reduction theorem is also equivalent to the transfer principle. As applications, we obtain a truth definition for arithmetical sentences and we formalize Nelson's notion of impredicativity (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

AN ARITHMETICAL CHARACTERIZATION OF FINITE ELEMENTARY 2-GROUPS

Beginning with Carlitz's well known characterization of algebraic number rings with class number two, arithmetical characterizations of divisor class groups have been a topic of interest in the literature. In this note we develop a characterization of Dedekind domains with finite elementary 2-group class group. The characterization is in terms of the asymptotic behavior of the number of distinct factorizations of powers of an irreducible element.     

The main sources for the Arte Mayor in sixteenth century Spain

One of the main changes in European Renaissance mathematics was the progressive development of algebra from practical arithmetic, in which equations and operations began to be written with abbreviations and symbols, rather than in the rhetorical way found in earlier arithmetical texts. In Spain, the introduction of algebraic procedures was mainly achieved through certain commercial or arithmetical texts, in which a section was devoted to algebra or the ‘Arte Mayor’. This paper deals with the contents of the first arithmetical texts containing sections on algebra. These allow us to determine how algebraic ideas were introduced into Spain and what their main sources were. The first printed arithmetical Spanish text containing algebra was the Libro primero de Arithmetica Algebratica (1552) by Marco Aurel. Therefore, the aim of this paper is to analyse the possible sources of this book and show the major influence of the German text Coss (1525) by Christoff Rudolff, on Aurel's work.