Routledge History of Philosophy Volume IX

Philosophy of the English-Speaking World in the Twentieth Century 1: Science, Logic and Mathematics

S. G. Shanker

• Publisher: Routledge
• Year: 2003
• ISBN:
  9781134393367
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

Volume 9 of the Routledge History of Philosophy surveys ten key topics in the philosophy of science, logic and mathematics in the twentieth century. Each of the essays is written by one of the world's leading experts in that field. Among the topics covered are the philosophy of logic, of mathematics and of Gottlob Frege; Ludwig Wittgenstein's Tractatus; a survey of logical positivism; the philosophy of physics and of science; probability theory, cybernetics and an essay on the mechanist/vitalist debates.
The volume also contains a helpful chronology to the major scientific and philosophical events in the twentieth century. It also provides an extensive glossary of technical terms in the notes on major figures in these fields.

CHAPTER 1
Philosophy of logic

A.D.Irvine

The relationship between evidence and hypothesis is fundamental to the advancement of science. It is this relationship—referred to as the relationship between premisses and conclusion—which lies at the heart of logic. Logic, in this traditional sense, is the study of correct inference. It is the study of formal structures and non-formal relations which hold between evidence and hypothesis, reasons and belief, or premisses and conclusion. It is the study of both conclusive (or monotonic) and inconclusive (non-monotonic or ampliative) inferences or, as it is also commonly described, the study of both entailments and inductions. Specifically, logic involves the detailed study of formal systems designed to exhibit such entailments and inductions. More generally, though, it is the study of those conditions under which evidence rightly can be said to justify, entail, imply, support, corroborate, confirm or falsify a conclusion.

In this broad sense, logic in the twentieth century has come to include, not only theories of formal entailment, but informal logic, probability theory, confirmation theory, decision theory, game theory and theories of computability and epistemic modelling as well. As a result, over the course of the century the study of logic has benefited, not only from advances in traditional fields such as philosophy and mathematics, but also from advances in other fields as diverse as computer science and economics. Through Frege and others late in the nineteenth century, mathematics helped transform logic from a merely formal discipline to a mathematical one as well, making available to it the resources of contemporary mathematics. In turn, logic opened up new avenues of investigation concerning reasoning in mathematics, thereby helping to develop new branches of mathematical research—such as set theory and category theory—relevant to the foundations of mathematics itself. Similarly, much of twentieth-century philosophy—including advances in metaphysics, epistemology, the philosophy of mathematics, the philosophy of science, the philosophy of language and formal semantics—closely parallels this century’s logical developments. These advances have led in turn to a broadening of logic and to a deeper appreciation of its application and extent. Finally, logic has provided many of the underlying theoretical results which have motivated the advent of the computing era, learning as much from the systematic application of these ideas as it has from any other source.

This chapter is divided into four sections. The first, ‘The Close of the Nineteenth Century’, summarizes the logical work of Boole, Frege and others prior to 1900. The second, ‘From Russell to Gödel’, discusses advances made in formal logic from 1901, the year in which Russell discovered his famous paradox, to 1931, the year in which Gödel’s seminal incompleteness results appeared. The third section, ‘From Gödel to Friedman’, discusses developments in formal logic made during the fifty years following Gödel’s remarkable achievement. Finally, the fourth section, ‘The Expansion of Logic’, discusses logic in the broader sense as it has flourished throughout the latter half of the twentieth century.

THE CLOSE OF THE NINETEENTH
CENTURY

‘Logic is an old subject, and since 1879 it has been a great one.’¹ This judgment appears as the opening sentence of W.V.Quine’s 1950 Methods of Logic. The sentence is justly famous—even if it has about it an air of exaggeration—for nothing less than a revolution had occurred in logic by the end of the nineteenth century.

Several important factors led to this revolution, but without doubt the most important of these concerned the mathematization of logic. Since the time of Aristotle, logic had taken as its subject matter formal patterns of inference, both inside and outside mathematics. Aristotle’s Organon had been intended as nothing less than a tool or canon governing correct inference. However, it was not until the mid-nineteenth century that logic came to be viewed as a subject which could be developed mathematically, alongside other branches of mathematics. The leaders in this movement—George Boole (1815–64), Augustus DeMorgan (1806– 71), William Stanley Jevons (1835–82), Ernst Schröder (1841–1902), and Charles Sanders Peirce (1839–1914)—all saw the potential for developing what was to be called an ‘algebra of logic’, a mathematical means of modelling the abstract laws governing formal inference. However, it was not until the appearance, in 1847, of a small pamphlet entitled The Mathematical Analysis of Logic, that Boole’s calculus of classes—later extended by Schröder and Peirce to form a calculus of relations—successfully achieved this end.

Boole had been prompted to write The Mathematical Analysis of Logic by a public dispute between DeMorgan and the philosopher William Hamilton (1788–1856) over the quantification of the predicate. As a result, Boole’s landmark pamphlet was the first successful, systematic application of the methods of algebra to the subject of logic. So impressed was DeMorgan that two years later, in 1849, despite Boole’s lack of university education, he was appointed Professor of Mathematics at Queen’s College, Cork, Ireland, largely on DeMorgan’s recommendation. Five years following his appointment, Boole’s next work, An Investigation of the Laws of Thoughts, expanded many of the ideas introduced in his earlier pamphlet. In the Laws of Thought, Boole developed more thoroughly the formal analogy between the operations of logic and mathematics which would help revolutionize logic. Specifically, his algebra of logic showed how recognizably algebraic formulas could be used to express and manipulate logical relations.

Boole’s calculus, which is known today as the theory of Boolean algebras, can be viewed as a formal system consisting of a set, S, over which three operations,

(or ×, representing intersection),

(or +, representing union), and ‘ (or -, representing complementation) are defined, such that for all a, b, and c that are members of S, the following axioms hold:

(1) Commutativity:
a

b=b

a, and a

b=b

a

(2) Associativity:
a

(b

c)=(a

b)

c, and a

(b

c)=(a

b)

c

(3) Distributivity:
a

(b

c)=(a

b)

(a

c), and a

(b

c)=(a

b)

(a

c)

(4) Identity:
There exist two elements, 0 and 1, of S such that,a

0=a, and a

1=a

(5) Complementation:
For each element a in S, there is an element a’ such that a

a’=1, and a

a’=0.

The logical utility of the system arises once it is realized that many logical relations are successfully formalizable in it. For example, by letting a and b represent variables for statements or propositions,

represent the truth-functional connective ‘and’, and

represent the truth-functional connective ‘or’, the commutativity axioms assert that statements of the form ‘a and b’ are equivalent to statements of the form ‘b and a’, and that statemen...

Table of contents

• COVER PAGE
• TITLE PAGE
• COPYRIGHT PAGE
• GENERAL EDITORS’ PREFACE
• NOTES ON CONTRIBUTORS
• ACKNOWLEDGEMENTS
• CHRONOLOGY
• INTRODUCTION
• CHAPTER 1: PHILOSOPHY OF LOGIC
• CHAPTER 2: PHILOSOPHY OF MATHEMATICS IN THE TWENTIETH CENTURY
• CHAPTER 3: FREGE
• CHAPTER 4: WITTGENSTEIN’S TRACTATUS
• CHAPTER 5: LOGICAL POSITIVISM
• CHAPTER 6: THE PHILOSOPHY OF PHYSICS
• CHAPTER 7: THE PHILOSOPHY OF SCIENCE TODAY
• CHAPTER 8: CHANCE, CAUSE AND CONDUCT: PROBABILITY THEORY AND THE EXPLANATION OF HUMAN ACTION
• CHAPTER 9: CYBERNETICS
• CHAPTER 10: DESCARTES’ LEGACY: THE MECHANIST/VITALIST DEBATES
• GLOSSARY