Mathematical logic One of the
theorems proved by Ramsey in his 1928 paper
On a Problem of Formal Logic now bears his name (
Ramsey's theorem). While this theorem is the work Ramsey is probably best remembered for, he proved it only in passing, as a minor
lemma along the way to his true goal in the paper, solving a special case of the
decision problem for first-order logic, namely the
decidability of what is now called the
Bernays–Schönfinkel–Ramsey class of
first-order logic, as well as a characterisation of the spectrum of sentences in this fragment of logic.
Alonzo Church would go on to show that the general case of the decision problem for first-order logic is
unsolvable and that first-order logic is undecidable (see
Church's theorem). A great amount of later work in mathematics was fruitfully developed out of the ostensibly minor lemma used by Ramsey in his decidability proof: this lemma turned out to be an important early result in
combinatorics, supporting the idea that within some sufficiently large systems, however disordered, there must be some order. So fruitful, in fact, was Ramsey's theorem that today there is an entire branch of mathematics, known as
Ramsey theory, which is dedicated to studying similar results. In 1926, Ramsey proposed a simplification of the
Theory of Types developed by
Bertrand Russell and
Alfred North Whitehead in their
Principia Mathematica. The resulting theory is known today as
Theory of Simple Type (TST) or Simple Type Theory. Ramsey observed that a hierarchy of types was sufficient to deal with mathematical
paradoxes, so removed Russell's and Whitehead's ramified hierarchy, which was meant to elude semantic paradoxes. Ramsey's version of the theory is the one considered by
Kurt Gödel in the original proof of his
first incompleteness theorem. Ramsey's Theory of Simple Types was further simplified by
Willard van Orman Quine in his
New Foundations for
set theory, in which any explicit reference to types is eliminated from the language of the theory.
Philosophy His main philosophical works included
Universals (1925),
Facts and Propositions (1927) (which proposed a
redundancy theory of truth),
Universals of Law and of Fact (1928),
Knowledge (1929),
Theories (1929),
On Truth (1929),
Causal Qualities (1929), and
General Propositions and Causality (1929). Ramsey was perhaps the first to propose a
reliabilist theory of knowledge. He also produced what philosopher Alan Hájek has described as an "enormously influential version of the subjective interpretation of probability". His thought in this area was outlined in the paper
Truth and Probability (discussed below) which was written in 1926 but first published posthumously in 1931. Ramsey's economic views were
socialist.
Truth and Probability In
A Treatise on Probability (1921), Keynes argued against the subjective approach in
epistemic probabilities. For Keynes, the subjectivity of probabilities does not matter as much, as for him there is an objective relationship between knowledge and probabilities, as knowledge is disembodied and not personal. Ramsey disagreed with this approach. In his article "Truth and Probability" (1926), he argued that there is a difference between the notions of
probability in
physics and in
logic. For Ramsey, probability is not related to a disembodied body of knowledge but is related to the knowledge that each individual possesses alone. Thus personal beliefs that are formulated by this individual knowledge govern probabilities, leading to the notions of
subjective probability and
Bayesian probability. Consequently, subjective probabilities can be inferred by observing actions that reflect individuals' personal beliefs. Ramsey argued that the degree of probability that an individual attaches to a particular outcome can be measured by finding what
odds the individual would accept when
betting on that outcome. Ramsey suggested a way of deriving a consistent theory of choice under uncertainty that could isolate beliefs from preferences while still maintaining subjective probabilities, although Ramsey later noted that "taking the whole field of chance events no generalizations about them are possible (consider e.g. infectious diseases, dactyls in hexameters, deaths from horse kicks, births of great men)." Despite the fact that Ramsey's work on probabilities was of great importance, no one paid any attention to it until the publication of
Theory of Games and Economic Behavior by
John von Neumann and
Oskar Morgenstern in 1944 (1947 2nd ed.), although after Ramsey's death, an approach to probability similar to his was developed independently by the Italian
mathematician Bruno de Finetti.
A Contribution to the Theory of Taxation This paper, first published in 1927 has been described by
Joseph E. Stiglitz as "a landmark in the economics of public finance" Ramsey poses the question that is to be solved at the beginning of the article: "A given revenue is to be raised by proportionate taxes on some or all uses of income, the taxes on different uses being possibly at different rates; how much should these rates be adjusted in order that the decrement of utility may be a minimum?" It employed, as
Paul Samuelson described it, "a strategically beautiful application of the
calculus of variations" The Ramsey model is today acknowledged as the starting point for
optimal accumulation theory although its importance was not recognised until many years after its first publication. The main contributions of the model were firstly the initial question Ramsey posed on how much savings should be and secondly the method of analysis, the intertemporal maximisation (optimisation) of collective or individual utility by applying techniques of dynamic optimisation.
Tjalling C. Koopmans and
David Cass modified the Ramsey model incorporating the dynamic features of
population growth at a steady rate and of Harrod-neutral technical progress again at a steady rate, giving birth to a model named the
Ramsey–Cass–Koopmans model where the objective now is to maximise household's
utility function. ==Legacy==