Clarifying Our Conceptual Scheme: Set Theory and the Role of Explication

2018 ◽  
pp. 114-146
Keyword(s):  
Set Theory ◽  
Conceptual Scheme

Logic and Philosophical Methodology

2016 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Classical Logic ◽  
Proof Theory ◽  
Recursion Theory ◽  
Theory Model ◽  
Philosophical Methodology ◽  
Paraconsistent Logics ◽  
Philosophical Logic

This article explores the role of logic in philosophical methodology, as well as its application in philosophy. The discussion gives a roughly equal coverage to the seven branches of logic: elementary logic, set theory, model theory, recursion theory, proof theory, extraclassical logics, and anticlassical logics. Mathematical logic comprises set theory, model theory, recursion theory, and proof theory. Philosophical logic in the relevant sense is divided into the study of extensions of classical logic, such as modal or temporal or deontic or conditional logics, and the study of alternatives to classical logic, such as intuitionistic or quantum or partial or paraconsistent logics. The nonclassical consists of the extraclassical and the anticlassical, although the distinction is not clearcut.

Global quantification in Zermelo-Fraenkel set theory

1985 ◽  
Vol 50 (2) ◽  
pp. 289-301
Author(s):  
John Mayberry
Keyword(s):  
Set Theory ◽  
Order Theory ◽  
Prima Facie ◽  
Local Theory ◽  
Axiom Schema ◽  
First Order ◽  
First Order Theory ◽  
Naturally Occurring ◽  
Universe Of Sets

My aim here is to investigate the role of global quantifiers—quantifiers ranging over the entire universe of sets—in the formalization of Zermelo-Fraenkel set theory. The use of such quantifiers in the formulas substituted into axiom schemata introduces, at least prima facie, a strong element of impredicativity into the thapry. The axiom schema of replacement provides an example of this. For each instance of that schema enlarges the very domain over which its own global quantifiers vary. The fundamental question at issue is this: How does the employment of these global quantifiers, and the choice of logical principles governing their use, affect the strengths of the axiom schemata in which they occur?I shall attack this question by comparing three quite different formalizations of the intuitive principles which constitute the Zermelo-Fraenkel system. The first of these, local Zermelo-Fraenkel set theory (LZF), is formalized without using global quantifiers. The second, global Zermelo-Fraenkel set theory (GZF), is the extension of the local theory obtained by introducing global quantifiers subject to intuitionistic logical laws, and taking the axiom schema of strong collection (Schema XII, §2) as an additional assumption of the theory. The third system is the conventional formalization of Zermelo-Fraenkel as a classical, first order theory. The local theory, LZF, is already very strong, indeed strong enough to formalize any naturally occurring mathematical argument. I have argued (in [3]) that it is the natural formalization of naive set theory. My intention, therefore, is to use it as a standard against which to measure the strength of each of the other two systems.

scholarly journals COMPLEX MECHANISM OF STATE REGULATION OF LABOR PRODUCTIVITY IN THE HEALTH PROTECTION SECTOR

2018 ◽  
Vol 64 ◽  
Author(s):  
S.H. Kryvusha
Keyword(s):  
Health Care ◽  
Labor Productivity ◽  
Health Sector ◽  
Value Added ◽  
State Regulation ◽  
Health Protection ◽  
Conceptual Scheme ◽  
Complex Mechanism ◽  
Gross Value Added

The article focuses on increasing the role of health care in increasing the productivity of labor employed in the economy, in connection with what is justified the need to develop a mechanism of state influence on its growth. The calculation of the labor productivity of the health sector based on the indicators of gross value added and the number of employed. The conceptual scheme of the mechanism of state influence on labor productivity in the sphere of health protection, which defines its main methods and instruments of state regulation in this sector, is proposed.

scholarly journals The role of qualitative research in management

1987 ◽  
Vol 18 (4) ◽  
pp. 250-254
Author(s):  
C. Orpen
Keyword(s):  
Qualitative Research ◽  
Qualitative Approach ◽  
Quantitative Approach ◽  
Future Research ◽  
Conceptual Scheme ◽  
Organizational Behaviour

It is argued that there are currently two separate research approaches to behaviour in organizations; the qualitative approach and the quantitative approach, each with their own framework, criteria of proof, and accepted procedures. Reasons for the dominance of the quantitative approach and the renewed interest in the qualitative approach are presented. Implications for future research in organizational behaviour are developed. There is a need for more multi-approach research efforts and for a conceptual scheme that will locate the two approaches in relation to one another.

scholarly journals Evaluating the Sustainability of Microfinance Institutions Considering Macro-Environmental Factors: A Cross-Country Study

2019 ◽  
Vol 11 (21) ◽  
pp. 5947 ◽  
Author(s):  
Wei Xu ◽  
Hongyong Fu ◽  
Huanpeng Liu
Keyword(s):  
Environmental Factors ◽  
Set Theory ◽  
Poverty Reduction ◽  
Soft Set ◽  
Ideal Solution ◽  
Microfinance Institutions ◽  
Order Preference ◽  
Cross Country ◽  
Types Of Information

Microfinance institutions (MFIs) have attracted great attention, due to their significant role in poverty reduction. Given the features of MFIs, this paper proposes a novel hybrid model of soft set theory, and an improved order preference by similarity to ideal solution (HMSIT) to evaluate the sustainability of MFIs, considering accounting ratios, corporate governance factors, and macro-environmental factors, from a cross-country perspective. This setting enables the examination of the role of macro-environmental factors in the sustainability of MFIs. For this purpose, soft set theory is adopted to select optimal criteria. An improved order preference by similarity to ideal solution method, in which the weight of each criterion is determined by soft set theory, is proposed to rank the sustainability of MFIs. This algorithm enables HMSIT to make full use of various types of information. The case study uses cross-country samples. Results indicate that macro-environmental factors are significant in evaluating the sustainability of MFIs from a cross-country perspective. Particularly, they can play a key role in distinguishing MFIs with low sustainability. The results also indicate that HMSIT has strong robustness. Ranked results, produced from the proposed HMSIT are reliable enough to provide some managerial suggestions for MFIs and help stakeholders make decisions.

Alfred Tarski's work in set theory

1988 ◽  
Vol 53 (1) ◽  
pp. 2-6 ◽  
Author(s):  
Azriel Levy
Keyword(s):  
Set Theory ◽  
Axiom System ◽  
Large Cardinals ◽  
Inaccessible Cardinal ◽  
Promised Land ◽  
And Mathematics ◽  
The University ◽  
Axiomatic Set Theory ◽  
The Way

Alfred Tarski started contributing to set theory at a time when the Zermelo-Fraenkel axiom system was not yet fully formulated and as simple a concept as that of the inaccessible cardinal was not yet fully defined. At the end of Tarski's career the basic concepts of the three major areas and tools of modern axiomatic set theory, namely constructibility, large cardinals and forcing, were already clearly defined and were in the midst of a rapid successful development. The role of Tarski in this development was somewhat similar to the role of Moses showing his people the way to the Promised Land and leading them along the way, while the actual entry of the Promised Land was done mostly by the next generation. The theory of large cardinals was started mostly by Tarski, and developed mostly by his school. The mathematical logicians of Tarski's school contributed much to the development of forcing, after its discovery by Paul Cohen, and to a lesser extent also to the development of the theory of constructibility, discovered by Kurt Gödel. As in other areas of logic and mathematics Tarski's contribution to set theory cannot be measured by his own results only; Tarski was a source of energy and inspiration to his pupils and collaborators, of which I was fortunate to be one, always confronting them with new problems and pushing them to gain new ground.Tarski's interest in set theory was probably aroused by the general emphasis on set theory in Poland after the First World War, and by the influence of Wactaw Sierpinski, who was one of Tarski's teachers at the University of Warsaw. The very first paper published by Tarski, [21], was a paper in set theory.

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