## Ramsey, Frank Plumpton (1903–1930)

## Ramsey, Frank Plumpton (1903–1930)

# RAMSEY, FRANK PLUMPTON

*(1903–1930)*

Frank Plumpton Ramsey, the Cambridge mathematician and philosopher, was one of the most brilliant men of his generation; his highly original papers on the foundations of mathematics, the nature of scientific theory, probability, and epistemology are still widely studied. He also wrote two studies in economics, the second of which was described by J. M. Keynes as "one of the most remarkable contributions to mathematical economics ever made." Ramsey's earlier work led to radical criticisms of A. N. Whitehead and Bertrand Russell's *Principia Mathematica*, some of which were incorporated in the second edition of the *Principia*. Ramsey was one of the first to expound the early teachings of Ludwig Wittgenstein, by whom he was greatly influenced. In his last papers he was moving toward a modified and sophisticated pragmatism.

## The Foundations of Mathematics

A stumbling block in the reduction of mathematics to logic attempted in *Principia Mathematica* has long been its appeal to the so-called ramified theory of types, introduced in order to cope with the paradoxes discovered by Russell and others. The excessive restrictions demanded by the theory of types were mitigated by introducing an ad hoc axiom of reducibility, which Ramsey, following Wittgenstein, held to be at best contingently true. Ramsey was one of the first to argue, following Giuseppe Peano, that many of the notorious paradoxes depended on the use of equivocal semantic notions having no place in mathematics. By introducing the notion of "predicative functions"—roughly speaking, truth-functions permitting infinitely many arguments—Ramsey was able to show that the paradoxes could be avoided without appeal to an axiom of reducibility. In order to improve what he regarded as an unsatisfactory conception of identity in *Principia Mathematica*, Ramsey proposed the wider concept of "propositional functions in extension," considered as correlations, not necessarily definable, between individuals and associated propositions. Fully elaborated, this view would seem to lead to a markedly nonconstructivistic set theory, which most contemporaries would find unacceptable. Ramsey's distinction between semantic and logical paradoxes and his rejection of that part of the theory of types that subdivides types into "orders" has been almost universally accepted by his successors.

## Philosophy of Science

In a striking paper, "Theories," Ramsey developed a novel method for eliminating overt reference to theoretical entities in the formal statement of scientific theory. The method consists of replacing, in the axioms of the formal system expressing the scientific theory in question, every constant designating a theoretical entity with an appropriate variable and then applying universal quantification over the propositional matrices thus obtained. Ramsey was able to show that the conjunction of the universally quantified statements thus derived from the original axioms would have the same observational consequences as the original axiom system. This technique is of interest to philosophers concerned with the ontological implications or commitments of scientific theory.

## Probability

Ramsey sketched a theory of probability considered as measuring a degree of "partial belief," thereby providing a stimulus to what are sometimes called "subjective" or "personalistic" analyses of probability. His most important idea was an operational test for degree of belief. Suppose somebody, *P*, has no preference between the following options: (1) to receive *m* _{1} for certain, and (2) to receive *m* _{2} if *p* is true but *m* _{3} if *p* is false, where *p* is some definite proposition and *m* _{1}, *m* _{2}, and *m* _{3} are monetary or other suitable measures of utility for *P*. Then *P* 's degree of belief in *p* is proposed to be measured by the ratio (*m* _{1} − *m* _{3})/(*m* _{2} − *m* _{3})—roughly speaking, therefore, by the betting odds that *P* will accept in favor of *p* 's being true, given the relative values to him of the possible outcomes.

## General Philosophy

Ramsey's most suggestive idea in general philosophy was that of treating a general proposition, say of the form "all *A* 's are *B*," as a "variable hypothetical," considered not as a truth-function (as it had been in his earlier papers) but rather as a rule for judging that if something is found to be an *A* it will be judged to be a *B* —that is, as a formula for deriving propositions in certain ways rather than as an authentic proposition having truth-value. This idea is connected with Ramsey's unfortunately fragmentary explorations into the connections between belief, habit, and behavior. Ramsey's papers on facts, propositions, and universals also have not outlived their usefulness.

** See also ** Keynes, John Maynard; Logical Paradoxes; Mathematics, Foundations of; Peano, Giuseppe; Philosophy of Science; Pragmatism; Probability; Russell, Bertrand Arthur William; Scientific Theories; Type Theory; Whitehead, Alfred North; Wittgenstein, Ludwig Josef Johann.

## Bibliography

A collection of Ramsey's work, including previously unpublished papers, was published posthumously as *The Foundations of Mathematics and Other Logical Essays*, edited by Richard B. Braithwaite (London: K. Paul, Trench, Trubner, 1931). This collection has a preface by G. E. Moore, a useful editor's introduction, and a complete bibliography. For the definitions of "predicative functions" and "functions in extension," see especially pp. 39–42, 52–53; Ramsey's discussion of theories is mainly on pp. 212–236; the generalized betting definition of degree of belief occurs on p. 179.

For discussions of Ramsey's work, see Israel Scheffler, *The Anatomy of Inquiry* (New York: Knopf, 1963), pp. 203–222, which contains a critical exposition of Ramsey's procedure for eliminating theoretical terms; Herbert Gaylord Bohnert, *The Interpretation of Theory* (PhD diss., University of Pennsylvania, 1961), further elaboration of Ramsey's work on the nature of scientific theory; Leonard J. Savage, *The Foundations of Statistics* (New York: Wiley, 1954), which acknowledges indebtedness to Ramsey's definition of partial belief; and Gilbert Ryle, "'If,' 'So,' and 'Because,'" in *Philosophical Analysis*, edited by Max Black (Ithaca, NY: Cornell University Press, 1950; reprinted, New York, 1963), which is a discussion of hypothetical statements as "inference licenses."

*Max Black (1967)*