# Ramsey, Frank Plumpton

# RAMSEY, FRANK PLUMPTON

(*b*. Cambridge, England, 22 February 1903; *d*. London, 19 January 1930), *philosophy, logic, mathematics, economics, decision theory*.

For the original article on Ramsey see *DSB*, vol. 11.

Admiration of Ramsey’s brilliant career continues to grow. Despite having lived only twenty-six full years, his seminal work in various fields puts him in the top rank of twentieth-century scientists. His contributions to mathematical logic, combinatorics, and economics—which T. A. A. Broadbent describes in his *DSB* article—were quickly appreciated and are still influential. His contributions to decision theory, philosophy of science, semantics, epistemology, and metaphysics—some appearing in posthumous publications as late as 1990—took many years to gain the attention they deserve. They show that he anticipated several currents of thought in twentieth-century philosophy. Ramsey’s ideas have lasting significance for many disciplines. The following summary emphasizes components of his work entering the limelight after Broadbent wrote.

**Logic and the Foundations of Mathematics** Bertrand Russell and Alfred North Whitehead in *Principia Mathematica* sought to derive mathematics from logic. Ramsey simplified their theory of types and dispensed with their axiom of reducibility. His proposals are in “The Foundations of Mathematics” (1925) and “Mathematical Logic” (1926).

**Mathematics** Ramsey’s article “On a Problem of Formal Logic” (1928) treats decision procedures in logic. Along the way, it proves two major theorems in combinatorics. The theorems describe structural patterns appearing in groups. For example, any group of six people contains either three who know each other or three who do not know each other. These theorems generated a thriving branch of mathematics known as Ramsey theory. A typical question asks, what is the minimal number of members that ensures a group’s having a certain structure? Certain Ramsey numbers, as they are called, exist but are unknown.

**Economics** Ramsey’s articles “A Contribution to the Theory of Taxation” (1927) and “A Mathematical Theory of Saving” (1928) lay the foundations for the theory of optimal taxation and the theory of the optimal rate of saving for future generations. Ramsey’s approach to these topics endures in macroeconomics.

**Decision Theory** Ramsey’s groundbreaking paper “Truth and Probability” (1926) presents the main ideas of decision theory. It proves a theorem establishing the existence and uniqueness, given a choice of scale, of probability and utility functions representing preferences among gambles. The demonstration uses simple assumptions about structure and coherence to show that preferences may be represented as agreeing with expected utilities. Ramsey also observed that having degrees of belief conforming to the probability axioms is necessary for avoiding Dutch books, that is, systems of bets that guarantee a loss. Later, he established results about the value for decision making of gathering information.

It took decision theorists some time to grasp the significance of Ramsey’s results. Theorists such as John von Neumann, Oskar Morgenstern, and Leonard Savage obtained similar results during the middle of the twentieth century. They roused enthusiasm for Ramsey’s approach to probability and utility and prompted rediscovery of Ramsey’s pioneering work. Representation theorems such as Ramsey’s are the foundation of Bayesian methods in decision and game theory.

**Philosophy of Science** An interpretation of scientific theories must decide how to understand theoretical entities such as electrons and genes. In “Theories” (1929) Ramsey takes a theory to assert that the roles of its theoretical entities are occupied. The Ramsey sentence for a theory makes this claim explicit. To obtain the Ramsey sentence, begin with a big sentence stating the theory, replace its theoretical terms with variables, then existentially generalize the variables of that open sentence. The resulting sentence asserts that the theory is true under some interpretation of its theoretical entities. Ramsey’s treatment of theoretical terms explains, for example, why their meanings may vary as a theory changes.

To distinguish the causal laws of science from accidentally true generalizations, Ramsey takes the laws as rules of inference in a formalization of complete knowledge of the world. David Lewis later advanced a similar view.

**Semantics and Philosophy of Language** In “Facts and Propositions” (1927) Ramsey expresses pragmatism concerning truth, belief, and meaning. His aim is a simple, naturalistic account of these subjects. Ramsey’s theory of truth observes that an assertion of the truth of a sentence claims no more than the sentence itself. For example, to say that it is true that Caesar was murdered is to say just that Caesar was murdered. His account of acceptance of conditional sentences simplifies their interpretation, too. The Ramsey test says that you accept the conditional, “If *p*, then *q*,” just in case when you add *p* to your beliefs, minimally revising to maintain consistency, you also add *q*. The Ramsey test continues to guide accounts of the semantics and pragmatics of conditionals and also accounts of belief revision.

Analyzing the content of a true belief, Ramsey relied on the belief’s effect on behavior. A belief is true if and only if it generates successful acts. For example, a chicken’s belief that a caterpillar is poisonous is a true belief if and only if the chicken benefits from not eating the caterpillar. A true belief’s content depends on the type of success holding the belief ensures. This principle, called Ramsey’s principle, inspires a field known as “success semantics.” Philosophers such as Fred Dretske, Ruth Millikan, and David Papineau have articulated Ramsey’s naturalistic account of the contents of beliefs.

**Epistemology** In a short note titled “Knowledge” (1929), Ramsey presents an account of knowledge according to which it is defined as true, full belief acquired by a reliable process. Epistemologists, such as Alvin Goldman, who hold similar views, note the advantages of making knowledge depend on a belief’s source rather than on evidence supporting the belief. Reliabilism explains, for instance, why a child may acquire knowledge through perception. The child may learn that a ball is red by perceiving its color. No review of the evidence concerning its color is necessary.

**Metaphysics** In “Universals” (1925) Ramsey denies the metaphysical significance of the grammatical distinction between subject and predicate. The sentence “Socrates is wise” is equivalent to the sentence “Wisdom is a characteristic of Socrates.” Hence, it is arbitrary to take Socrates as a particular and wisdom as a universal.

Besides pursuing lines of thought that Ramsey initiated, scholars have also shown their esteem for Ramsey by naming awards after him. The Decision Analysis Society awards its Ramsey Medal to outstanding decision theorists. The journal *Macroeconomic Dynamics* awards its Ramsey Prize to eminent economists.

## SUPPLEMENTARY BIBLIOGRAPHY

### WORKS BY RAMSEY

*Foundations: Essays in Philosophy, Logic, Mathematics, and Economics*. Edited by D. H. Mellor. London: Routledge & Kegan Paul, 1978.

*Philosophical Papers*. Edited by D. H. Mellor. Cambridge, U.K.: Cambridge University Press, 1990.

*On Truth: Original Manuscript Materials (1927–1929) from the Ramsey Collection at the University of Pittsburgh*. Edited by Nicholas Rescher and Ulrich Majer. Dordrecht, Netherlands: Kluwer Academic, 1991.

*Notes on Philosophy, Probability, and Mathematics*. Edited by Maria Carla Galavotti. Naples, Italy: Bibliopolis, 1991.

### OTHER SOURCES

Dokic, Jérôme, and Pascal Engel. *Frank Ramsey: Truth and Success*. London: Routledge, 2002. Treats belief, knowledge, truth, probability, decision, and universals.

Frápolli, María J., ed. *F. P. Ramsey: Critical Reassessments*. London: Thoemmes Continuum, 2005. Treats economics, logic, truth, pragmatism, reliabilism, Ramsey sentences, and universals.

Galavotti, Maria Carla, ed. “The Philosophy of F. P. Ramse *y*.” *Theoria* 57 (1991). A special issue of the journal that treats probability and universals.

———, ed. *Cambridge and Vienna: Frank P. Ramsey and the Vienna Circle*. New York: Springer, 2006. Treats logic, belief, the value of knowledge, and Ramsey sentences.

Jeffrey, Richard. “Ramsey.” In the *Cambridge Dictionary of Philosophy*. 2nd ed., edited by Robert Audi. Cambridge, U.K.: Cambridge University Press, 1999.

Lillehammer, Hallvard, and D. H. Mellor, eds. *Ramsey’s Legacy*. Oxford: Clarendon Press, 2005. Treats logic, metaphysics, and philosophy of mind.

Mellor, D. H., ed. *Prospects for Pragmatism: Essays in Memory of F. P. Ramsey*. Cambridge, U.K.: Cambridge University Press, 1980. Treats belief, probability, and pragmatism.

Sahlin, Nils-Eric. *The Philosophy of F. P. Ramsey*. Cambridge, U.K.: Cambridge University Press, 1990. A comprehensive account of Ramsey’s philosophical and scientific research.

*Paul Weirich*

# Ramsey, Frank Plumpton

# RAMSEY, FRANK PLUMPTON

(*b*. Cambridge, England, 22 February 1903; *d*. Cambridge, 19 January 1930), *mathematical logic*.

Ramsey, the elder son of A. S. Ramsey, president of Magdalene College, Cambridge, was educated at Winchester and Trinity colleges, Cambridge. After graduation in 1923 he was elected a fellow of King’s College, Cambridge, where he spent the rest of his short life. His lectures on the foundations of mathematics impressed young students by their remarkable clarity and enthusiasm, and his untimely death deprived Cambridge of one of its most brilliant thinkers.

Whitehead and Russell, in their system of mathematical logic, *Principia Mathematica* (1910–1913), had avoided the antinomies (paradoxes) by creating both a theory of types, which dealt with the nature of propositional functions, and an axiom of reducibility. Ramsey accepted the Whitehead-Russell view of mathematics as part of logic but said of the axiom of reducibility that it is “certainly not self-evident and there is no reason to suppose it true.” In his first paper (1926) he took up Wittgenstein’s work on tautologies and truth functions, reinterpreting the concept of propositional functions and thus obviating the need for the axiom of reducibility. He also drew an important distinction between the logical antinomies, for example, that concerning the class of all classes not members of themselves, and those antinomies that cannot be stated in logical terms alone, for example, “I am lying.” For the first class, he accepted Russell’s solution; for the second, which are not formal but involve meaning, he applied his reinterpretation of the *Principia Mathematica*.

In a paper “On a Problem of Formal Logic” (1930) Ramsey discussed the *Entscheidungsproblem*—the search for a general method for determining the consistency of a logical formula—using some ingenious combinatorial theorems. To the Oxford meeting of the British Association in 1926, he described the development of mathematical logic subsequent to the publication of *Principia Mathematica*; this address was printed in the *Mathematical Gazette* (1926).

Two papers on the mathematical theory of economics appeared in the *Economic Journal*. In a biographical essay on Ramsey, J. M. Keynes described Ramsey’s interest in economics, the importance of his two papers, and the way in which Cambridge economists made use of his critical powers to test their theories. An interesting, if not altogether convincing, study of the bases of probability theory was published posthumously in a collection of Ramsey’s essays.

## BIBLIOGRAPHY

I. Original Works. Ramsey’s works include “The Foundations of Mathematics,” in *Proceedings of the London Mathematical Society*, 2nd ser., **25** (1926), 338–384; “Mathematical Logic,” in *Mathematical Gazette*, **13** (1926), 185–194; “A Contribution to the Theory of Taxation,” in *Economic Journal* (Mar. 1927); “A Mathematical Theory of Saving,” *ibid.* (Dec. 1928); and “On the Problem of Formal Logic,” in *Proceedings of the London Mathematical Society*, 2nd ser., **30** (1930), 264–286.

A convenient source is the posthumous volume *Foundations of Mathematics and Other Essays*, R. B. Braithwaite, ed. (London, 1931); this work contains Ramsey’s published papers, excluding the two on economics, and a number of items from his unpublished manuscripts.

II. Secondary Literature. An obituary notice by Braithwaite is in *Journal of the London Mathematical Society*, **6** (1931), 75–78. The essay on Ramsey by J. M. Keynes, in *Essays in Biography* (London, 1933), is valuable for Ramsey’s work in economics and in philosophy.

T. A. A. Broadbent

# Frank Plumpton Ramsey

# Frank Plumpton Ramsey

The English mathematician and philosopher Frank Plumpton Ramsey (1903-1930) was recognized as an authority in mathematical logic.

Frank Ramsey was born on Feb. 22, 1903. His father, Arthur Ramsey, was president of Magdalen College. Ramsey's excellent work at Winchester College won him a scholarship to Trinity College, Cambridge. He was Allen University scholar in 1924 and in the same year was elected a fellow of King's College and appointed lecturer in mathematics at the university.

Ramsey's precocious talents were legendary at Cambridge. From about his sixteenth year he was consulted by theorists in mathematics and other subjects in which mathematics is largely used. The economist John Maynard Keynes reported, "Economists living in Cambridge have been accustomed from his undergraduate days to try their theories on the keen edge of his critical and logical faculties." And indeed in his brief life Ramsey made two important contributions to economic theory: "A Mathematical Theory of Saving" and "A Contribution to the Theory of Taxation." Of the first of these, Keynes wrote that it is "one of the most remarkable contributions to mathematical economics ever made."

But Ramsey's contributions to the subject that taxed the best abstract theorists of the day, the foundations of mathematics, were even more impressive. At the age of 22 he presented a brilliant defense of the mathematical theories of Bertrand Russell and Alfred North Whitehead against Continental critics. Using the *Tractatus* of Ludwig Wittgenstein, which he was among the first to appreciate, Ramsay succeeded in removing some of the most serious objections to the logicist theory. He showed how the ad hoc axiom of reducibility, one of the most vulnerable parts of *Principia Mathematica,* by Russell and Whitehead, could be eliminated, and he offered ways of improving the concept of identity used in that work.

Ramsey also made important contributions to the philosophy of science. In an effort to clarify the role played by theory in science, he introduced the important idea that scientific laws could be regarded as "inference licenses," a theme that was developed further by Gilbert Ryle and S. E. Toulmin. Taking up the work of the American philosopher C. S. Peirce on inductive logic, Ramsey sought to provide sharper criteria for the acceptability of beliefs.

On the question of whether there are important truths inaccessible to language, Ramsey went still further than his friend and colleague Wittgenstein. He gave an answer that was repeated by a generation of Cambridge students: "What we can't say, we can't say and we can't whistle it either."

Ramsey was widely regarded as having no equal in his generation for sheer power and quality of mind and in the originality and promise of his work. A man of large, "Johnsonian" build, he was straightforward and blunt in conversation and modest about his exceptional gifts. He died after an operation at the age of 26 on Jan. 19, 1930, survived by his wife and two daughters.

## Further Reading

Ramsey's most important essays were published posthumously by R. B. Braithwaite, *The Foundations of Mathematics and Other Logical Essays* (1931), which also contains a eulogy by G. E. Moore and a bibliography of the remaining works. Further background is in John Maynard Keynes, *Essays in Biography* (1933; rev. ed. 1951). □

# Frank Plumpton Ramsey

# Frank Plumpton Ramsey

**1903-1930**

English mathematician best known for his criticisms of Alfred North Whitehead and Bertrand Russell's *Principia Mathematica.* Ramsey showed how to eliminate the axiom of reducibility, which they had introduced to deal with paradoxes arising from their theory of types. Ramsey also published two studies in economics, the last of which John Maynard Keynes described as "one of the most remarkable contributions to mathematical economics ever made." Ramsey's contributions to philosophy were small but significant, including work on universals and scientific theories.

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