Marian Boykan Pour-El and Hilary Putnam. Recursively enumerable classes and their application to recursive sequences of formal theories. Archiv für mathematische Logik und Grundlagenforschung, vol. 8 no. 3–4 (1965), pp. 104–121. - Marian Boykan Pour-El and William A. Howard. A structural criterion for recursive enumeration without repetition. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 10 (1964), pp. 105–114. - A. H. Lachlan. On recursive enumeration without repetition. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 11 (1965), pp. 209–220. - A. H. Lachlan. On recursive enumeration without repetition: a correction. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 13 (1967), pp. 99–100.

As in Rogers [3], we treat the partial degrees as notational variants of the enumeration degrees (that is, the partial degree of a function is identified with the enumeration degree of its graph). We showed in [1] that there are no minimal partial degrees. The purpose of this paper is to show that the partial degrees below 0′ (that is, the partial degrees of the Σ2 partial functions) are dense. From this we see that the Σ2 sets play an analagous role within the enumeration degrees to that played by the recursively enumerable sets within the Turing degrees. The techniques, of course, are very different to those required to prove the Sacks Density Theorem (see [4, p. 20]) for the recursively enumerable Turing degrees.Notation and terminology are similar to those of [1]. In particular, We, Dx, 〈m, n〉, ψe are, respectively, notations for the e th r.e. set in a given standard listing of the r.e. sets, the finite set whose canonical index is x, the recursive code for (m, n) and the e th enumeration operator (derived from We). Recursive approximations etc. are also defined as in [1].Theorem 1. If B and C are Σ2sets of numbers, and B ≰e C, then there is an e-operator Θ withProof. We enumerate an e-operator Θ so as to satisfy the list of conditions:Let {Bs ∣ s ≥ 0}, {Cs ∣ s ≥ 0} be recursive sequences of approximations to B, C respectively, for which, for each х, х ∈ B ⇔ (∃s*)(∀s ≥ s*)(х ∈ Bs) and х ∈ C ⇔ (∃s*)(∀s ≥ s*)(х ∈ Cs).

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Hilary Putnam

10.1017/cbo9780511614187 ◽

2005 ◽

Cited By ~ 7

Keyword(s):

Hilary Putnam

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Narrow Content

10.1093/oso/9780198785965.001.0001 ◽

2018 ◽

Cited By ~ 3

Author(s):

Juhani Yli-Vakkuri ◽

John Hawthorne

Keyword(s):

Philosophy Of Mind ◽

Mental Content ◽

Thought Experiments ◽

Hilary Putnam ◽

Central Topic ◽

Tyler Burge ◽

Narrow Content

Narrow mental content, if there is such a thing, is content that is entirely determined by the goings-on inside the head of the thinker. A central topic in the philosophy of mind since the mid-1970s has been whether there is a kind of mental content that is narrow in this sense. It is widely conceded, thanks to famous thought experiments by Hilary Putnam and Tyler Burge, that there is a kind of mental content that is not narrow. But it is often maintained that there is also a kind of mental content that is narrow, and that such content can play various key explanatory roles relating, inter alia, to epistemology and the explanation of action. This book argues that this is a forlorn hope. It carefully distinguishes a variety of conceptions of narrow content and a variety of explanatory roles that might be assigned to narrow content. It then argues that, once we pay sufficient attention to the details, there is no promising theory of narrow content in the offing.

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Introduction

10.1093/oso/9780198785965.003.0001 ◽

2018 ◽

Author(s):

Juhani Yli-Vakkuri ◽

John Hawthorne

Keyword(s):

Thought Experiments ◽

Hilary Putnam ◽

Tyler Burge ◽

Narrow Content ◽

History Of ◽

To Come

The Introduction outlines the history of the narrow content debate. It introduces the famous thought experiments by Hilary Putnam and Tyler Burge, discusses why the debate only came to prominence in the 1970s, and outlines what is to come.

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Is Rationality Normative?

10.1093/oso/9780198802693.003.0002 ◽

2017 ◽

Cited By ~ 1

Author(s):

Ralph Wedgwood

Keyword(s):

Rational Choice ◽

Rational Belief ◽

False Beliefs ◽

Ought Implies Can ◽

Normative Concept ◽

Original Meaning ◽

Formal Theories

In its original meaning, the word ‘rational’ referred to the faculty of reason—the capacity for reasoning. It is undeniable that the word later came also to express a normative concept—the concept of the proper use of this faculty. Does it express a normative concept when it is used in formal theories of rational belief or rational choice? Reasons are given for concluding that it does express a normative concept in these contexts. But this conclusion seems to imply that we ought always to think rationally. Four objections can be raised. (1) What about cases where thinking rationally has disastrous consequences? (2) What about cases where we have rational false beliefs about what we ought to do? (3) ‘Ought’ implies ‘can’—but is it true that we can always think rationally? (4) Rationality requires nothing more than coherence—but why does coherence matter?

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Introduction

10.1093/oso/9780198803430.003.0001 ◽

2017 ◽

Author(s):

Simon Kirchin

Keyword(s):

Hilary Putnam ◽

Bernard Williams ◽

General Methodology ◽

Thick Concepts ◽

History Of ◽

Philippa Foot ◽

The Difference

This chapter introduces the distinction between thin and thick concepts and then performs a number of functions. First, two major accounts of thick concepts—separationism and nonseparationism—are introduced and, in doing so, a novel account of evaluation is indicated. Second, each chapter is outlined as is the general methodology, followed, third, by a brief history of the discussion of thick concepts, referencing Philippa Foot, Hilary Putnam, Gilbert Ryle, and Bernard Williams among others. Fourth, a number of relevant contrasts are introduced, such as the fact–value distinction and the difference between concepts, properties, and terms. Lastly, some interesting and relevant questions are raised that, unfortunately, have to be left aside.

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