*Jon Williamson*

- Published in print:
- 2017
- Published Online:
- March 2017
- ISBN:
- 9780199666478
- eISBN:
- 9780191749292
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199666478.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

Inductive logic (also known as confirmation theory) seeks to determine the extent to which the premisses of an argument entail its conclusion. This book offers an introduction to the field of ...
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Inductive logic (also known as confirmation theory) seeks to determine the extent to which the premisses of an argument entail its conclusion. This book offers an introduction to the field of inductive logic and develops a new Bayesian inductive logic. Chapter 1 introduces perhaps the simplest and most natural account of inductive logic, classical inductive logic, which is attributable to Ludwig Wittgenstein. Classical inductive logic is seen to fail in a crucial way, so there is a need to develop more sophisticated inductive logics. Chapter 2 presents enough logic and probability theory for the reader to begin to study inductive logic, while Chapter 3 introduces the ways in which logic and probability can be combined in an inductive logic. Chapter 4 analyses the most influential approach to inductive logic, due to W.E. Johnson and Rudolf Carnap. Again, this logic is seen to be inadequate. Chapter 5 shows how an alternative approach to inductive logic follows naturally from the philosophical theory of objective Bayesian epistemology. This approach preserves the inferences that classical inductive logic gets right (Chapter 6). On the other hand, it also offers a way out of the problems that beset classical inductive logic (Chapter 7). Chapter 8 defends the approach by tackling several key criticisms that are often levelled at inductive logic. Chapter 9 presents a formal justification of the version of objective Bayesianism which underpins the approach. Chapter 10 explains what has been achieved and poses some open questions.Less

Inductive logic (also known as confirmation theory) seeks to determine the extent to which the premisses of an argument entail its conclusion. This book offers an introduction to the field of inductive logic and develops a new Bayesian inductive logic. Chapter 1 introduces perhaps the simplest and most natural account of inductive logic, classical inductive logic, which is attributable to Ludwig Wittgenstein. Classical inductive logic is seen to fail in a crucial way, so there is a need to develop more sophisticated inductive logics. Chapter 2 presents enough logic and probability theory for the reader to begin to study inductive logic, while Chapter 3 introduces the ways in which logic and probability can be combined in an inductive logic. Chapter 4 analyses the most influential approach to inductive logic, due to W.E. Johnson and Rudolf Carnap. Again, this logic is seen to be inadequate. Chapter 5 shows how an alternative approach to inductive logic follows naturally from the philosophical theory of objective Bayesian epistemology. This approach preserves the inferences that classical inductive logic gets right (Chapter 6). On the other hand, it also offers a way out of the problems that beset classical inductive logic (Chapter 7). Chapter 8 defends the approach by tackling several key criticisms that are often levelled at inductive logic. Chapter 9 presents a formal justification of the version of objective Bayesianism which underpins the approach. Chapter 10 explains what has been achieved and poses some open questions.

*Joseph Mazur*

- Published in print:
- 2016
- Published Online:
- January 2018
- ISBN:
- 9780691173375
- eISBN:
- 9781400850112
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691173375.003.0023
- Subject:
- Mathematics, History of Mathematics

This chapter considers the mental pictures of thought and images in relation to algebraic symbols. According to Ludwig Wittgenstein, “We make to ourselves pictures of facts.” For Wittgenstein, the ...
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This chapter considers the mental pictures of thought and images in relation to algebraic symbols. According to Ludwig Wittgenstein, “We make to ourselves pictures of facts.” For Wittgenstein, the picture is a model of what we take to be real. The geneticist Francis Galton claimed that his thoughts almost never suggested words, and when those rare moments did suggest words, they were nonsense words like “the notes of a song might accompany thought.” As for words, the French mathematician Jacques Hadamard suggested that words are neither followed by thoughts, nor thoughts by words. He goes on to say that this is also the case when he is thinking about algebraic symbols. More revealing is Hadamard's presentation of his mental pictures of the steps in a proof that there are an unlimited number of prime numbers. The chapter also discusses thought without verbal language and in relation to proofs.Less

This chapter considers the mental pictures of thought and images in relation to algebraic symbols. According to Ludwig Wittgenstein, “We make to ourselves pictures of facts.” For Wittgenstein, the picture is a model of what we take to be real. The geneticist Francis Galton claimed that his thoughts almost never suggested words, and when those rare moments did suggest words, they were nonsense words like “the notes of a song might accompany thought.” As for words, the French mathematician Jacques Hadamard suggested that words are neither followed by thoughts, nor thoughts by words. He goes on to say that this is also the case when he is thinking about algebraic symbols. More revealing is Hadamard's presentation of his mental pictures of the steps in a proof that there are an unlimited number of prime numbers. The chapter also discusses thought without verbal language and in relation to proofs.

*Jon Williamson*

- Published in print:
- 2017
- Published Online:
- March 2017
- ISBN:
- 9780199666478
- eISBN:
- 9780191749292
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199666478.003.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

Chapter 1 introduces perhaps the simplest and most natural account of inductive logic, classical inductive logic, which is attributable to Ludwig Wittgenstein. The key concept in inductive logic is ...
More

Chapter 1 introduces perhaps the simplest and most natural account of inductive logic, classical inductive logic, which is attributable to Ludwig Wittgenstein. The key concept in inductive logic is that of the degree to which given premisses partially entail (or confirm) a given conclusion. An inductive logic needs to capture two kinds of partial entailment. The first kind, logical entailment, quantifies the extent to which premisses entail a conclusion by virtue of logical connections between them. The second kind, inductive entailment, quantifies the extent to which a sample of past observations entails the next outcome to be sampled. This chapter shows that classical inductive logic fails to adequately capture inductive entailment. Thus, there is a need to develop more sophisticated inductive logics. This chapter also explores the reasons for studying inductive logic.Less

Chapter 1 introduces perhaps the simplest and most natural account of inductive logic, classical inductive logic, which is attributable to Ludwig Wittgenstein. The key concept in inductive logic is that of the degree to which given premisses *partially entail* (or *confirm*) a given conclusion. An inductive logic needs to capture two kinds of partial entailment. The first kind, logical entailment, quantifies the extent to which premisses entail a conclusion by virtue of logical connections between them. The second kind, inductive entailment, quantifies the extent to which a sample of past observations entails the next outcome to be sampled. This chapter shows that classical inductive logic fails to adequately capture inductive entailment. Thus, there is a need to develop more sophisticated inductive logics. This chapter also explores the reasons for studying inductive logic.