*Amartya Sen*

- Published in print:
- 1973
- Published Online:
- November 2003
- ISBN:
- 9780198281931
- eISBN:
- 9780191715815
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198281935.003.0003
- Subject:
- Economics and Finance, Public and Welfare

It is argued that the difficulty of using the positive and normative measures of inequality described in the previous chapter arises from the fact that they are ‘complete’ measures. Each of these ...
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It is argued that the difficulty of using the positive and normative measures of inequality described in the previous chapter arises from the fact that they are ‘complete’ measures. Each of these measures may give absurd results, because they aim to give a complete‐ordering representation to a concept that is essentially one of partial ranking. Hence, a weakening of the inequality measures to a mixture of partly descriptive and partly normative considerations is proposed. A number of reasons for taking inequality rankings as quasi‐orderings rather than complete orderings are suggested.Less

It is argued that the difficulty of using the positive and normative measures of inequality described in the previous chapter arises from the fact that they are ‘complete’ measures. Each of these measures may give absurd results, because they aim to give a complete‐ordering representation to a concept that is essentially one of partial ranking. Hence, a weakening of the inequality measures to a mixture of partly descriptive and partly normative considerations is proposed. A number of reasons for taking inequality rankings as quasi‐orderings rather than complete orderings are suggested.

*Luc Bovens and Stephan Hartmann*

- Published in print:
- 2004
- Published Online:
- January 2005
- ISBN:
- 9780199269754
- eISBN:
- 9780191601705
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199269750.003.0003
- Subject:
- Philosophy, Metaphysics/Epistemology

Shows how to construct a coherence quasi-ordering that respects the claim that the more coherent a set of propositions is, the greater the degree of confidence ought to be in its content, ceteris ...
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Shows how to construct a coherence quasi-ordering that respects the claim that the more coherent a set of propositions is, the greater the degree of confidence ought to be in its content, ceteris paribus. Applies this result to the problem of scientific-theory choice.Less

Shows how to construct a coherence quasi-ordering that respects the claim that the more coherent a set of propositions is, the greater the degree of confidence ought to be in its content, *ceteris paribus*. Applies this result to the problem of scientific-theory choice.

*James Oxley*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198566946
- eISBN:
- 9780191774904
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566946.003.0015
- Subject:
- Mathematics, Educational Mathematics

This chapter considers recent progress on the Well-Quasi-Ordering Conjecture for matroids and on Rota's Conjecture. It also discusses some related work on matroids representable over finite fields. ...
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This chapter considers recent progress on the Well-Quasi-Ordering Conjecture for matroids and on Rota's Conjecture. It also discusses some related work on matroids representable over finite fields. One constant theme throughout the chapter is the many contrasts between the classes of GF(q)-representable and ℝ-representable matroids.Less

This chapter considers recent progress on the Well-Quasi-Ordering Conjecture for matroids and on Rota's Conjecture. It also discusses some related work on matroids representable over finite fields. One constant theme throughout the chapter is the many contrasts between the classes of *GF*(*q*)-representable and ℝ-representable matroids.

*Marco Giunti*

- Published in print:
- 1997
- Published Online:
- November 2020
- ISBN:
- 9780195090093
- eISBN:
- 9780197560600
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195090093.003.0005
- Subject:
- Computer Science, Mathematical Theory of Computation

The main thesis of this chapter is that a dynamical viewpoint allows us to better understand some important foundational issues of computation theory. Effective procedures are traditionally studied ...
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The main thesis of this chapter is that a dynamical viewpoint allows us to better understand some important foundational issues of computation theory. Effective procedures are traditionally studied from two different but complementary points of view. The first approach is concerned with individuating those numeric functions that are effectively calculable. This approach reached its systematization with the theory of the recursive functions (Gödel, Church Kleene).This theory is not directly concerned with computing devices or computations. Rather, the effective calculability of a recursive function is guaranteed by the algorithmic nature of its definition. In contrast, the second approach focuses on a family of abstract mechanisms, which are then typically used to compute or recognize numeric functions, sets of numbers, or numbers. These devices can be divided into two broad categories: automata or machines (Turing and Post), and systems of rules for symbol manipulation (Post). The mechanisms that have been studied include: a. Automata or Machines 1. gate-nets and McCulloch-Pitts nets 2. finite automata (Mealy and Moore machines) 3. push-down automata 4. stack automata 5. Turing machines 6. register machines 7. wang machines 8. cellular automata b. Systems of Rules 9. monogenic production systems in general 10. monogenic Post canonical systems 11. monogenic Post normal systems 12. tag systems. I call any device studied by computation theory a computational system. Computation theory is traditionally interested in studying the relations between each type of computational system and the others, and in establishing what class of numeric functions each type can compute. Accordingly one proves two kinds of theorem: (1) that systems of a given type emulate systems of another type (examples: Turing machines emulate register machines and cellular automata; cellular automata emulate Turing machines, etc.), and (2) that a certain type of system is complete relative to the class of the (partial) recursive functions or, in other words, that this type of system can compute all and only the (partial) recursive functions (examples of complete systems: Turing machines, register machines, cellular automata, tag systems, etc.). All different types of computational systems have much in common. Nevertheless, it is not at all clear exactly which properties these mechanisms share.
Less

The main thesis of this chapter is that a dynamical viewpoint allows us to better understand some important foundational issues of computation theory. Effective procedures are traditionally studied from two different but complementary points of view. The first approach is concerned with individuating those numeric functions that are effectively calculable. This approach reached its systematization with the theory of the recursive functions (Gödel, Church Kleene).This theory is not directly concerned with computing devices or computations. Rather, the effective calculability of a recursive function is guaranteed by the algorithmic nature of its definition. In contrast, the second approach focuses on a family of abstract mechanisms, which are then typically used to compute or recognize numeric functions, sets of numbers, or numbers. These devices can be divided into two broad categories: automata or machines (Turing and Post), and systems of rules for symbol manipulation (Post). The mechanisms that have been studied include: a. Automata or Machines 1. gate-nets and McCulloch-Pitts nets 2. finite automata (Mealy and Moore machines) 3. push-down automata 4. stack automata 5. Turing machines 6. register machines 7. wang machines 8. cellular automata b. Systems of Rules 9. monogenic production systems in general 10. monogenic Post canonical systems 11. monogenic Post normal systems 12. tag systems. I call any device studied by computation theory a computational system. Computation theory is traditionally interested in studying the relations between each type of computational system and the others, and in establishing what class of numeric functions each type can compute. Accordingly one proves two kinds of theorem: (1) that systems of a given type emulate systems of another type (examples: Turing machines emulate register machines and cellular automata; cellular automata emulate Turing machines, etc.), and (2) that a certain type of system is complete relative to the class of the (partial) recursive functions or, in other words, that this type of system can compute all and only the (partial) recursive functions (examples of complete systems: Turing machines, register machines, cellular automata, tag systems, etc.). All different types of computational systems have much in common. Nevertheless, it is not at all clear exactly which properties these mechanisms share.