No Class

2014 ◽  
Author(s):  
Scott Soames
Keyword(s):  
Philosophy Of Mathematics ◽  
Philosophical Logic ◽  
Substitutional Quantification ◽  
Class Theory ◽  
Contextual Definition ◽  
Definition Of

This chapter explores Russell’s “no class theory,” originally expressed by his contextual definition of classes in Principia Mathematica. In recent years, some Russell scholars have trumpeted the virtues of the interpretation of Russell’s quantification as substitutional, among which is the sense it makes of the “no-class theory.” Such an interpretation does make some sense of Russell’s philosophical remarks about that theory, about the significance of his logicist reduction, and about the ability of the reduction to serve as a model for similar reductions outside the philosophy of mathematics. However this substitutional interpretation is not sufficient, since it is inconsistent with important aspects of Russell’s philosophical logic and is technically inadequate to support his logicist reduction. In short, if substitutional quantification is the source of the “no class theory,” then the theory is not vindicated, but refuted.

The anatytic conception of truth and the foundations of arithmetic

2000 ◽  
Vol 65 (1) ◽  
pp. 33-102 ◽  
Author(s):  
Peter Apostoli
Keyword(s):  
Philosophy Of Mathematics ◽  
Logical Truth ◽  
Numerical Identity ◽  
Hume’S Principle ◽  
Contextual Definition ◽  
Definition Of ◽  
Axiomatic Set Theory ◽  
In Virtue Of ◽  
Second Order Logic ◽  
Foundations Of Arithmetic

Until very recently, it was thought that there couldn't be any current interest in logicism as a philosophy of mathematics. Indeed, there is an old argument one often finds that logicism is a simple nonstarter just in virtue of the fact that if it were a logical truth that there are infinitely many natural numbers, then this would be in conflict with the existence of finite models. It is certainly true that from the perspective of model theory, arithmetic cannot be a part of logic. However, it is equally true that model theory's reliance on a background of axiomatic set theory renders it unable to match Frege's Theorem, the derivation within second order logic of the infinity of the number series from the contextual “definition” of the cardinality operator. Called “Hume's Principle” by Boolos, the contextual definition of the cardinality operator is presented in Section 63 of Grundlagen, as the statement that, for any concepts F and G,the number of F s = the number of G sif, and only if,F is equinumerous with G.The philosophical interest in Frege's Theorem derives from the thesis, defended for example by Crispin Wright, that Hume's principle expresses our pre-analytic conception of assertions of numerical identity. However, Boolos cites the very fact that Hume's principle has only infinite models as grounds for denying that it is logically true: For Boolos, Hume's principle is simply a disguised axiom of infinity.

scholarly journals Carnap’s Structuralist Thesis

2020 ◽  
pp. 383-420
Author(s):  
Georg Schiemer
Keyword(s):  
Present Article ◽  
Structural Properties ◽  
Philosophy Of Mathematics ◽  
Mathematical Structure ◽  
Mathematical Structuralism ◽  
Definition Of ◽  
Philosophical Debates ◽  
Implicit Definitions

This chapter investigates Carnap’s structuralism in his philosophy of mathematics of the 1920s and early 1930s. His approach to mathematics is based on a genuinely structuralist thesis, namely that axiomatic theories describe abstract structures or the structural properties of their objects. The aim in the present article is twofold: first, to show that Carnap, in his contributions to mathematics from the time, proposed three different (but interrelated) ways to characterize the notion of mathematical structure, namely in terms of (i) implicit definitions, (ii) logical constructions, and (iii) definitions by abstraction. The second aim is to re-evaluate Carnap’s early contributions to the philosophy of mathematics in light of contemporary mathematical structuralism. Specifically, the chapter discusses two connections between his structuralist thesis and current philosophical debates on structural abstraction and the on the definition of structural properties.

Extending Gödel's negative interpretation to ZF

1975 ◽  
Vol 40 (2) ◽  
pp. 221-229 ◽  
Author(s):  
William C. Powell
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Stable Sets ◽  
Class Theory ◽  
Heyting Arithmetic ◽  
Inner Model ◽  
Order Arithmetic ◽  
Definition Of ◽  
Excluded Middle ◽  
Negative Sense

In [5] Gödel interpreted Peano arithmetic in Heyting arithmetic. In [8, p. 153], and [7, p. 344, (iii)], Kreisel observed that Gödel's interpretation extended to second order arithmetic. In [11] (see [4, p. 92] for a correction) and [10] Myhill extended the interpretation to type theory. We will show that Gödel's negative interpretation can be extended to Zermelo-Fraenkel set theory. We consider a set theory T formulated in the minimal predicate calculus, which in the presence of the full law of excluded middle is the same as the classical theory of Zermelo and Fraenkel. Then, following Myhill, we define an inner model S in which the axioms of Zermelo-Fraenkel set theory are true.More generally we show that any class X that is (i) transitive in the negative sense, ∀x ∈ X∀y ∈ x ¬ ¬ x ∈ X, (ii) contained in the class St = {x: ∀u(¬ ¬ u ∈ x→ u ∈ x)} of stable sets, and (iii) closed in the sense that ∀x(x ⊆ X ∼ ∼ x ∈ X), is a standard model of Zermelo-Fraenkel set theory. The class S is simply the ⊆-least such class, and, hence, could be defined by S = ⋂{X: ∀x(x ⊆ ∼ ∼ X→ ∼ ∼ x ∈ X)}. However, since we can only conservatively extend T to a class theory with Δ01-comprehension, but not with Δ11-comprehension, we will give a Δ01-definition of S within T.

scholarly journals Some Problems in Erik Olin Wright’s Theory of Class

2014 ◽  
Vol 33 ◽  
pp. 20-40 ◽  
Author(s):  
Jacek Tittenbrun
Keyword(s):  
Social Class ◽  
Social Science ◽  
Critical Analysis ◽  
Social Differentiation ◽  
Class Theory ◽  
Definition Of

Nowadays, Erik Olin Wright‘s class theory is one of the most influential approaches to social differentiation, although in the mainstream social science the popularity of the so-called EGP class scheme is perhaps greater, Wright‘s framework has virtually no rivals amongst Marxists and neo-Marxists. Both those considerations add to the relevance of the present paper, which is a critical analysis of some of the most salient issues present in the aforementioned framework. Indeed, it turns out that Wright‘s flaws are more often than not shared by the remaining theorists dealing with social class or stratification. Nomen omen, one of those problems consists in failing to adequately distinguish between those two axes of differentiation. Perhaps the pivotal problem plaguing Wright‘s framework concerns his flawed understanding of ownership that cannot but reflect on his definition of class and exploitation.

Unleashing the Latent Potential of Young People in Africa: The Example of Dalumuzi Happy Mhlanga and Salathiel Ntakirutimana

2020 ◽  
Vol 16 (2) ◽  
Author(s):  
Josephine Olufunmilayo Alexander
Keyword(s):  
Young People ◽  
National Development ◽  
Youth Empowerment ◽  
Servant Leaders ◽  
Social Entrepreneurs ◽  
Personal Interaction ◽  
Contextual Definition ◽  
Latent Potential ◽  
Definition Of ◽  
Insight Into

This paper tells the story of two young people, Dalumuzi Happy Mhlanga from Zimbabwe and Salathiel Ntakirutimana from Burundi, to show how they have defied the lack of structured opportunities to impact on the development of their home countries and to make a mark globally. The intention is to highlight the potential of young people and to show how this might be unleashed when they are allowed to innovate and flourish. The paper begins by providing a contextual definition of youth from global and African perspectives, followed by an insight into youth participation. Their stories are then told, based on my personal interaction with them during their two years at Waterford Kamhlaba, United World College of Southern Africa in Swaziland, their activities in school, university and in their home communities, their postings on social media and interviews. The discussion identifies Dalumuzi and Salathiel as social entrepreneurs and servant leaders with an enlightened vision of community development and the empowerment of young people. They demonstrate the interrelationship between youth empowerment and sustainable national development. The paper concludes with a message for African leaders and institutions around the world that it is essential to invest creatively in young people as they can be powerful catalysts for African development.

The Problem of Definition of Legal Concepts with “Open Texture”

2020 ◽  
Vol 2 (1) ◽  
pp. 150-161
Author(s):  
V. V. Ogleznev ◽  
Keyword(s):  
Theoretical Basis ◽  
Special Kind ◽  
Analytical Philosophy ◽  
Clear Cut ◽  
The Core ◽  
Open Texture ◽  
Contextual Definition ◽  
Definition Of ◽  
Legal Concepts ◽  
Core Meaning

Introduction: The article discusses the problems associated with the definition of legal concepts which have the feature of “open texture”. The introduction presents the nature and meaning of “open texture”, which is understood as a special kind of indeterminacy. Such concepts are considered in the form in which they were postulated in the works of the Austrian linguistic philosopher Friedrich Waismann and the British legal philosopher Herbert Hart. Theoretical Basis. Methods. It is contested that, in Hart’s interpretation, “open texture” appears in legal concepts in borderline cases, when the meaning of the term of “concept” becomes indeterminate, unclear, uncertain, and we do not know whether or not it should be applied. Such cases should be distinguished from clear-cut cases where such doubt does not arise. The methodological basis of the study is Hart’s thesis stating that legal concepts have “core” and “penumbra” of meaning. The “core” meaning indicates a set of certain conditions, in which the use of the term “concept” is clear, while a “penumbra” meaning refers to conditions in which the its use becomes less clear. “Open texture” in this case, is an irreducible feature of legal concepts. Results. The main result of the study is the assertion that “open texture” as an irreducible feature of legal concepts, can be disproved by changing its definition. It is shown that the most appropriate kind of definition of open-textured legal concepts is the definition or contextual definition, widely used in analytical philosophy.

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