The Philosophy of F.P. Ramsey
NilsEric Sahlin
Cambridge University Press 1990
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(this short introduction was written 2001 for this site)
Frank Plumpton Ramsey (19031930), British mathematician and philosopher, best known for his work on the foundations of mathematics. But Ramsey also made remarkable contributions to epistemology, semantics, logic, philosophy of science, mathematics, statistics, probability and decision theory, economics and metaphysics.
1. Brief Biographical Sketch
Frank Plumpton Ramsey was born on February 22, 1903 and died at the age of 26 on January 19, 1930. Ramsey suffered from a chronic and increasingly serious liver complaint, contracted jaundice after an operation and died at Guy’s Hospital in London.
Ramsey came from a distinguished Cambridge family. His father was a mathematician, and the President of Magdalene College. His brother, Michael Ramsey, became the Archbishop of Canterbury. And in 1924, at the age of twentyone, Ramsey himself got a Fellowship at King’s College Cambridge, having graduating the year before as Cambridge’s top mathematics student.
Ramsey was very much a Cambridge thinker and his work is tinted by the work of his friends and colleagues — among them Bertrand Russell, G. E. Moore, J. M. Keynes and Ludwig Wittgenstein.
Few philosophers of the twentieth century have influenced the sciences as much as Ramsey. He did pioneering work in pure mathematics, logic, economics, statistics, probability theory, decision theory and cognitive psychology. He also did groundbreaking work on epistemology, philosophy of science, philosophy of mathematics, metaphysics and semantics. And he accomplished all this before the age of twentyseven.
2. Ramsey’s Pragmatism
Ramsey’s philosophical and scientific work consists of, say, 15 papers. The papers are on disparate subjects but they all contain the same view of philosophy — a method of analysis, Ramsey’s kind of pragmatism. To appreciate his work one has to understand his general view of philosophy.
Ramsey concludes his paper ‘Facts and Propositions’ (1927) by saying:
The essence of pragmatism I take to be this, the meaning of a sentence is to be defined by reference to the actions to which asserting it would lead, or, more vaguely still, by its possible causes and effects.(Philosophical Papers, PP, p. 51.)
In ‘Facts and Propositions’ Ramsey uses his pragmatist philosophy to outline a theory of truth. Ramsey’s theory has been misunderstood in later philosophical literature. The reason for this is that few have clearly comprehended the intimate connection between his theories of truth, partial belief (the subjective theory of probability) and knowledge.
In his paper ‘Truth and Probability’, written in 1926, Ramsey laid the foundations of the modern theory of subjective probability. He showed how people’s beliefs and desires can be measured by use of a traditional betting method. What we want to do is to measure a person’s belief by proposing a bet, and “see what are the lowest odds which he will accept” (PP, p. 68). Ramsey took this method to be ‘fundamentally sound’, but saw that it suffered from “being insufficiently general, and from being necessarily inexact … partly because of the diminishing marginal utility of money, partly because the person may have a special eagerness or reluctance to bet …” (PP, p. 68). To avoid these difficulties he laid the foundations of the modern theory of utility. He then went on to show that if people in their behaviour obey a set of axioms or rules, the measure of our ‘degrees of belief’ will satisfy the laws of probability.
Ramsey was the first one to assert the celebrated Dutch book theorem (if our distribution of degrees of belief follows the rules of probability theory, a book cannot be made against us, we are not bound to lose whatever happens); he had a proof of the value of collecting evidence; he took higher order probabilities seriously; and he had the notion important for Bayesian statistics of ‘exchangeability’. In ‘Truth and Probability’ he also laid the foundations of modern decision theory.
In ‘Facts and Propositions’ Ramsey argues that “if we have analysed judgment we have solved the problem of truth” (PP, p. 39). To carry out such an analysis successfully one has to say what the content of a belief is without falling into a regress by appealing to the meaning of sentences understood as truth conditions.
There is an important paragraph in the paper where Ramsey clearly indicates how such an analysis can be carried out:
…it is, for instance, possible to say that a chicken believes a certain sort of caterpillar to be poisonous, and mean by that merely that it abstains from eating such caterpillars on account of unpleasant experiences connected with them. … An exact analysis of this relation would be very difficult, but it might well be held that in regard to this kind of belief the pragmatist view was correct, i.e. that the relation between the chicken’s behaviour and the objective factors was that the actions were such as to be useful if, and only if, the caterpillars were actually poisonous. Thus any set of actions for whose utility p is a necessary and sufficient condition might be called a belief that p, and so would be true if p, i.e., if they are useful. (PP, p. 40)
In a note Ramsey adds: “It is useful to believe aRb would mean that it is useful to do things which are useful if, and only if, aRb; which is evidently equivalent to aRb.”
This pragmatic theory of truth is something rather different than the redundancy theory of truth credited to Ramsey. If propositions are the carriers of truthvalue, then to say that ‘it is true that Caesar was murdered’ means no more than that Caesar was murdered’. But, Ramsey does not find this a very interesting analysis of truth. (W.E. Johnson and G. Frege had earlier discussed the possibility of getting rid of of the predicate ‘true’.) Far more challenging is to say what it means to have a true belief and to do this without appealing to the meaning of sentences. To succeed in this a pragmatic analysis seems to be the correct way to go.
Ramsey’s theory of truth, like his theory of probability, tells us something about rule obeying. The chicken in Ramsey’s example can be seen as having a decision problem, it has a choice between the two actions: (a) eat the caterpillar; (b) refrain from eating the caterpillar. If the chicken chooses to eat the caterpillar, this choice will lead to one of two consequences, depending on whether the caterpillar is poisonous or edible. If the caterpillar is poisonous, the chicken gets an upset stomach; if it is edible, the chicken gets a good lunch. If, on the other hand, the chicken refrains from eating the caterpillar, this means that it has either avoided an upset stomach or missed its lunch.
This is a welldefined decision problem and Ramsey can therefore use his theories of subjective probability, utility and decision to solve it. ‘Truth and Probability’ tells us that if a chicken does not know whether the caterpillar is poisonous or not, he should “act in the way [he] think[s] most likely to realize the objects of [his] desires” (PP, p. 69); i.e., maximize his subjective expected utility. However, a truthproblem is not one of degrees of belief, but of full belief. We want to make clear what is meant by saying that the chicken believes fully, i.e. believes that the caterpillar is poisonous. What it means is that the chicken refrains from eating the caterpillar: an action that is useful if and only if the caterpillar is poisonous (and the chicken wants to avoid an upset stomach).
This is the gist of Ramsey’s theory of truth. It is an obvious example of a pragmatic theory of truth, but also of what recently has been discussed as success semantics. Having a true belief is having a more or less complicated rule, which if put to use, always leads to success.
In ‘Knowledge’, written in 1929, Ramsey uses his pragmatic theory to give an analysis of what it means to have knowledge. “I have”, he says, “always said that a belief was knowledge if it was (i) true, (ii) certain, (iii) obtained by a reliable process.” (PP, p. 110) On the surface this definition of knowledge looks very much the same as the traditional, truejustifiedbelief theory, but working out the details of this theory one discovers that it diverges significantly from that account of knowledge. Of special interest is his third condition.
Ramsey holds that a person X’s belief that p is a case of knowledge only if that belief has been obtained by a reliable process. It is not sufficient that Xhas evidence for believing that p; the way in which we acquire our beliefs should be reliable. The reliability condition thus tells us that the provenance of knowledge is of decisive importance. To have full belief is not enough, not even if the belief is supported by heaps of evidence. Moreover, the future use of those beliefs constituting knowledge is just as important as the provenance of those beliefs. A belief, being a map by which we steer, being a rule to follow, must guide our future actions. A full belief, obtained by a reliable method, is definitely not knowledge if it leads us on the wrong track; to be knowledge it must help us to avoid errors. Thus, knowledge is simply not true justified belief but rather: A belief is knowledge if it is obtained by a reliable process and if it always leads to success.
Theories of evidence and theories of knowledge are intimately linked together. And there are many competing theories of evidence. One way to approach them is by way of looking at the theories of knowledge which are their bedrock.
What might be called the traditional theory of knowledge equates knowledge with true justified belief, i.e. a person is said to have knowledge if a truth condition, a belief condition and a condition of sufficient evidence are satisfied. With this view of knowledge it is natural to argue that the true evidentiary value (of a piece of evidence for a hypothesis) is the probability of the hypothesis given the evidence. It is wellknown that this view of knowledge leads to serious problems. The socalled ‘RussellGettierproblems’show that the ‘traditional’ conditions are not sufficient for knowledge. What Russell and Gettier do is to provide us with counterexamples to the claim that knowledge is but true justified belief. The problems arise because justification is often transitive. If a person’s belief is justified on the basis of another belief that also is justified, but happens to be false, then the true belief will be justified without being an instance of knowledge.
Ramsey’s theory steers clear of the RussellGettierproblems before they were even invented. For Ramsey the beliefgenerating process must be reliable, although it is not assumed that the subject must be aware of that fact. A person might consequently know that p, and not know (or believe) that the process (yielding p) is reliable.
By emphasizing that a reliable process is needed for knowledge (a belief being knowledge if it is obtained by such a process and is true), one sidesteps many of the difficulties of the traditional theory. It is easily noted, for example, that RussellGettier examples are no problem for a theory of knowledge like Ramsey’s. Introducing reliable processes prevents that true beliefs can be justifiably inferred from false premises.
If Ramsey’s account of knowledge is right, then the probability of a hypothesis given the evidence is unsatisfactory as a measure of evidentiary value. It is the reliable processes — what might and have been called the evidentiary mechanisms — that are important. What is needed is a theory of evidence that takes account of these processes.
3. Metaphysics
Ontological questions are at the core of much of Ramsey’s writing, whether it is on numbers, probabilities, the status of theoretical terms or general propositions and causality. One of his most impressive, but underestimated, contributions to philosophy is his analysis of the problem of universals. (See his papers ‘Universals’ and ‘Universals and the “Method of Analysis”’, which are both listed in the Bibliography.)
His paper ‘Universals’ which denies any fundamental distinction between universals and particulars, surmounts serious objections to a realist view of universals and, at the same time, solves several longstanding problems about them, dismissing other venerable enigmas as nonsense. To appreciate Ramsey’s arguments it is important to keep in mind that he believes in facts, believes that the world consists of facts. But that he puts in question a belief strongly held at the time, namely that the logical form of a proposition uniquely can tell us what there is.
There can be various reasons for making the distinction between universals and particulars — psychological, physical and logical. But Ramsey argues that logic justifies no such ontological distinction. Alluding to a grammatical subjectpredicate distinction will not do, since ‘Socrates is wise’, with subject ‘Socrates’ and predicate ‘wise’, “asserts the same fact, and expresses the same proposition” (PP, p. 12) as ‘Wisdom is a characteristic of Socrates’, with subject ‘wisdom’ and predicate ‘Socrates’.
There is, he argues, no essential difference between the (in)completeness of universals and that of particulars. ‘Wise’ can, for example, be used to generate propositions not only of the atomic form ‘Socrates is wise’, but also of the molecular form ‘Neither Socrates nor Plato is wise’. But ‘Socrates’ can also be used to generate propositions of both these forms: e.g. ‘Socrates is wise’ and ‘Socrates is neither wise nor just’. There is thus really a complete symmetry in this respect between individuals and basic properties (qualities). Or, as Ramsey succinctly puts it, “the whole theory of particulars and universals is due to mistaking for a fundamental characteristic of reality what is merely a characteristic of language” (PP, p. 13).
Recently it has been suggested that Ramsey’s argument is built on a simple mistake (see for example, Simon 1991). ‘Socrates is wise’ and ‘Wisdom is a characteristic of Socrates’ imply different propositions. For example ‘Something is wise’ and ‘Something is a characteristic of Socrates’, respectively — thus involving different ontological commitments. The problem with this suggestion is that we cannot evaluate Ramsey’s argument by looking at two arbitrary implications, and from these conclude that the two original sentences are not synonymous. Each proposition implies an (infinite) set of propositions and to be a forceful argument it has to be shown that the two sets of consequences do not contain exactly the same propositions. Hinting at a problem is not too good an argument.
Again, Ramsey argues that there can no more be complex universals (for example, negative, as ‘notwise’; relational, as ‘wiser than’; and compound properties, as ‘grue’) than there can be complex particulars. Suppose that Socrates is to the right of Plato. One could then imagine three propositions: first, that the relation ‘being to the right of’ holds between Socrates and Plato; second, that Socrates has the complex property of ‘being to the right of Plato’; third, that Plato has the complex property which something has if Socrates is to the right of it. Thus if there were complex universals, besides the fact that Socrates is to the right of Plato, there would also be two nonrelational facts, with different constituents. But that is nonsense, the argument goes, there is only one fact, the fact that Socrates is to the right of Plato.
D. H. Mellor (in discussion) has argued that a virtue of Ramsey’s realism is the way it stops the vicious regress started by asking what relates particulars to universals in a fact, e.g. what ties Socrates to wisdom in the fact that Socrates is wise. For Ramsey, universals and particulars are constructions out of facts, not the other way around. He needs no hierarchy of universals to recombine them; they were never separated in the first place. (Other solutions to the regress problem have been suggested; and the problem can be circumvented without accepting Ramsey’s account.)
It is, however, important to keep in mind that the success of Ramsey’s arguments depends on the very existence of facts, and an understanding of what they are; and on the presupposition that the analysed propositions uniquely tell us what there is. There are, of course, contemporary theories consistently maintaining the existence of, for example, complex universals and compound properties. The reason they can do it, in spite of Ramsey’s argument, is that they make other ontological claims. To compare this type of theories, with their new ontological assumptions, and Ramsey’s view would lead us too far away. Important however, evaluating Ramsey’s arguments, is that theories of this kind can be developed and in a consistent manner.
Ramsey’s view of universals also affects much of his other work. Nominalistsfor example reject the socalled ‘Ramsey sentence’ since these involve quantifying over universals, thus expanding our ontological commitments. But given Ramsey’s kind of realism, that is no objection at all.
4. Probability and Utility
In his paper ‘Truth and Probability’ (1926) Ramsey laid the foundations of the modern theory of subjective probability. He showed how, under ideal conditions, people’s beliefs and desires can be measured by use of a betting method, and that given some intuitive principles of rational behaviour are accepted, a measure of our ‘degrees of belief’ will satisfy the laws of probability. He was the first to state the Dutch book theorem and he laid the foundations of modern utility theory and decision theory. In addition, he had a proof of the value of collecting evidence, years before it became known through the independent works of L. J. Savage and I. J. Good; he took higher order probabilities seriously; and, in a derivation of the ‘rule of succession’ he introduced the notion of ‘exchangeability’ (however, not giving it that name.) Ramsey’s decision/probability theory is almost as complete as any such theory can be.
The aim of ‘Truth and Probability’ is to analyse the connection between the subjective degree of belief we have in a proposition and the (subjective) probability it can be given and to find a behavioural way of measuring degrees of belief. More precisely, Ramsey wants to show that: first, we can measure the degree of belief a subject has in a given proposition; and, second, that if the subject is rational his or her degrees of belief will have a measure satisfying the axioms of probability theory, a “subjective” probability. Or, in other words, Ramsey shows that given his method of measuring strength of “partial beliefs” the degrees of belief of an ideally rational agent will obey the laws of probability.
Ramsey argues that people’s beliefs and desires can be measured by use of a traditional betting method. We can measure a subject’s belief simply by proposing a bet, and “see what are the lowest odds which he will accept” (PP, p. 68). The strategy is to offer the agent a bet on the truth value of the proposition p involved in the belief. He took this method to be ‘fundamentally sound,’ but argued that it suffers from “being insufficiently general, and from being necessarily inexact … partly because of the diminishing marginal utility of money, partly because the person may have a special eagerness or reluctance to bet, …” (PP, p. 68). I we for a moment ignore the problem of using money as outcomes, a bet of this type is of the form: $x if p is true, $yif p is not true, were x > y. The “traditional method” tells us that the agent’s degree of belief in p is ($f – $y)/($x – $y), were $f is the greatest amount the agent is willing to pay for the bet. It should be noted that the least amount of money the agent is prepared to pay for the bet coincides with the least amount for which the agent is willing to sell it. If the marginal utility for money is decreasing it is obvious that using money as outcomes does not give correct measures for, for example, bets involving substantial sums of money. As Ramsey says it is “sound” but not completely “general” and not very “exact”.
To have degree of belief ½ in an ethically neutral proposition is by Ramsey defined as being indifferent to two options: a if p is true, b if p is not true; and b if p is true, a if p is not true (a, b, c, …, denoting outcomes). “This comes roughly to defining belief of degree ½ as such a degree of belief as leads to indifference between betting one way and betting the other for the same stakes“ (PP, p. 74). An ethically neutral proposition of degree ½ comes close to something like the very idea of a fair coin.
This gives Ramsey an operational method, a way of measuring value differences. That the value difference between a and b is equal to the difference between c and d, simply means that if ep is an ethically neutral proposition believed to degree ½, the options [a if ep is true, and d if ep is not true] and [b if ep is true, and c if ep is not true] are equally preferable.
He then goes on to prove an important representation theorem. The theorem states that a subject’s preferences can be represented by a utility function determined up to a positive linear transformation. It is the binary preferences that are represented and the very goal of the theorem is to isolate the conditions under which such preferences can be seen as maximizing expected utility. The representation guarantees the existence of a probability function and an unconditional utility function such that the expected utility defined from this probability and utility represents the agent’s preferences. To prove this theorem eight axioms are introduced. The axioms can be divided into three groups: behavioural, ontological, and structural.
A behavioural axiom is a rule that a ‘rational’ person is supposed to satisfy when making a decision. That preferences ought to be transitive is a typical example of a behavioural axiom. One of Ramsey’s axioms states that the subject’s value differences are transitive (if the difference in value between aand b is equal to the difference between c and d, and the difference between cand d is equal to that between e and f, then the difference between a and b is equal to that between e and f).
The ontological and structural axioms tell us what there is and give us the mathematical muscles necessary to prove the representation theorem. Ramsey’s first axiom, for example, states that “[t]here is an ethically neutral proposition p believed to degree ½” (PP, p. 74). And axioms seven and eight are an axiom of continuity and an Archimedean axiom, respectively.
Ramsey’s utility theory is closely related to the theory developed by von Neumann and Morgenstern in Theory of Games and Economic Behavior, about two decades later. Von Neumann and Morgenstern, however, assume ‘objective’ probabilities, prizes and lotteries to derive the utilities. Ramsey avoids these assumptions and thus avoids having to postulate that subjects understand the information contained in a stated probability and that they are wellcalibrated (i.e. that the subjective probabilities mirror the stated objective probabilities). Today it is well known that such a method does not work — that it does not avoid the type of problem of which Ramsey aimed to steer clear. For example, there is robust empirical evidence that shows that subjects have a tendency to overestimate very low (stated) ‘objective’ probabilities but underestimate all other probabilities — a type of behaviour that inevitably will affect the measurement of utilities.
Instead of a traditional betting method Ramsey can now use a refined betting method with differences in utilities rather than with money; in this way avoiding some of the hitches with the traditional method — its being not sufficiently general; necessarily inexact; diminishing marginal utility of money; and, riskaversion (riskproneness). Thus, to avoid the difficulties he identified initially, he laid the foundations of the modern theory of utility.
It is then possible to define the degree of belief in p “by the odds at which the subject would bet on p, the bet being conducted in terms of differences of value as defined” (PP, p. 76). If the subject is indifferent between a with certainty; and, b if p is true (p not necessarily being an ethically neutral proposition, however p’s truth cannot change the relative values of the outcomes) and c if p is not true, the subject’s degree of belief in the proposition is defined as the difference in value between a and c divided by the difference in value between b and c. This can also be expressed as follows:
P(p) = (u(a) u(c))/(u(b) u(c)),
where ‘P(.)’ denotes the subject’s degree of belief function and ‘u(.)’ the subject’s utility function. (But, it has not yet been shown that P(.) is a probability measure.) Ramsey also points out the degree of belief in a proposition given another proposition can be defined along the same lines, using a slightly more complicated pair of bets.
Ramsey then proves that the obtained measure of degree of belief is a probability measure — it obeys the axioms of probability theory. The probability of any proposition is greater than or equal to 0; the probability of a proposition plus the probability of its negation equals 1; and, if two propositions are incompatible, the probability of the disjunction equals the sum of the probability of the disjuncts. Furthermore Ramsey proves the Dutch book theorem, “[h]aving degrees of belief obeying the laws of probability implies a further measure of consistency, namely such a consistency between the odds acceptable on different propositions as shall prevent a book being made against you” (PP, p 79). Having degrees of belief obeying the axioms of probability, having a coherent set of beliefs, is simply a logically necessary and sufficient condition of avoiding a Dutch book. It should be noted that a subject can have, more or less, any degree of belief whatsoever in a proposition provided the set of beliefs to which it belongs is coherent (consistent). It is essentially this feature of Ramsey’s theory that makes the theory subjectivist.
It is important to emphasize that Ramsey was far from being the narrowminded subjectivist/Bayesian as others have often presented him. He did not, for example, believe that ‘probabilities do not exist,’ meaning objective probabilities; rather he saw that some types of probability are a matter for physics and not for logic. Ramsey would probably have argued that some probability assessments are not all that rational. If a subject has a degree of beliefs not reflecting the chances given by an accepted theory, then the subjective probabilities are clearly not well calibrated. Ramsey does discuss these matters.
Thus Ramsey’s decision/probability theory is close to being as complete as any such theory can be. That is, complete in the sense that he deals with and gives answers to the fundamental questions we have. But its assumptions can be put in doubt; and the validity of its applicability can be questioned.
Ramsey’s theory is a descriptive theory. It is not a normative theory. Its primary purpose is not to tell people what they ought to do (though it has normative ramifications.) It portrays the ideal decision maker. There is a dedictode re problem. To what extent does the theory tell us anything about human decisionmaking? Does it simply describe a surface phenomenon, failing to capture the underlying mechanisms of human decisionmaking? Are the concepts introduced the appropriate concepts for a theory of human decision making? Not to mention the axioms that from a psychological point of view are somewhat unrealistic.
Theories like Ramsey’s, Savage’s (and similar) aim at representing a subject’s state of belief by a unique probability measure, and to any degree of precision. There is a class of examples that show that this aim leads to far too strong assumptions (axioms).
Assume, for example, that the subject is offered two lotteries and the task is to choose the one considered to be most preferable. The first lottery gives 100 dollars if a white ball is drawn from an urn containing 30 white balls and 70 black balls; otherwise nothing. The second lottery gives 100 dollars if there is a transit strike in Verona, Italy, next week; otherwise nothing.
Assume that, after considering it carefully, the subject believes that the probability that there will be a transit strike in Verona next week is 0.30. Thus, provided dollars and utilities are exchangeable, the (subjective) expected utility of this lottery is 30 dollars. The second lottery has the same expected value as the first one. But although the two gambles have the same expected utility, the subject will trade the first gamble for the second. And this preference conflicts with the recommendations of the theories.
The reason why the subject prefers the first lottery to the second is a feeling of knowing more about the urn than about Italian wages, working conditions and other important factors that may provoke a transit strike in Verona. There are situations in which there is an important difference in degree between our knowledge about the various factors underlying our decisions, a difference in ignorance that cannot be mirrored by a unique probability measure.
The probabilities given by Ramsey’s theory (and akin theories, for example Savage’s) are the result of the subject’s inability to express fully strength of preference. Ramsey’s theory does not take this unreliability into account. The clarity of perception of uncertainty (in part caused by the quantity and quality of information the subject has) is not introduced.
In a recent paper (1990), Schervish et al. show that classical theories, such as those developed by Savage, von Neumann and Morgenstern, Anscombe andAumann and de Finetti, all have a problem with statedependent utilities. Theories using (horse) lotteries and prizes to derive probabilities cannot guarantee the existence of unique probabilities. The problem is that the utility of a prize is the utility of that prize given that a particular state of nature obtains. And even ‘constant’ prizes might have different values in different states of nature — meaning that the subject’s preferences can be represented by far to many utility functions. As a consequence there is no unique subjective probability distribution over states of nature. Ramsey saw that his method of using preferences among bets to quantify value differences required that the states defining the bets were themselves valueneutral. For that, he proposed ethically neutral propositions. In his theory the outcomes have statedependent utilities, which can be measured through bets involving an ethically neutral proposition. The question, however, is whether the concept of an ethically neutral proposition can be understood and put to use without making use of lotteries. If not Ramsey faces the same problems as did the progenies of his theory.
Ramsey begins ‘Truth and Probability’ by discussing the frequency theory (for example, as it was developed by R. L. Ellis and later J. Venn) and Keynes’ theory of logical probability. It has some value to compare Ramsey’s own theories with one or two of the theories discussed at the time when he wrote his paper.
What has been called Wittgenstein’s theory of probability is, as a theory of probability, somewhat incomplete. The classical definition of probability says that probability is the ratio of the number of favourable cases to the number of all equipossible cases. Transformed into the world of Wittgenstein’sTractatus LogicoPhilosophicus, ‘the number of all equipossible cases’ is the number of truthgrounds of a given proposition q; and ‘the number of all favourable cases’ is the number of truthgrounds of a proposition, p, which also are truthgrounds of q. Assume that these numbers are m and k, respectively. Then the conditional probability, the probability of p given q, isk/m, ‘the degree of probability that the proposition [‘q’] gives to the proposition [‘p’]’ (5.15).
G. H. von Wright (1982, p. 241) reminds us that when Wittgenstein wrote theTractatus he obviously believed that logically independent propositions give one another the probability ½. The classical view of probability needs a set of exclusive and exhaustive alternatives. Wittgenstein must have thought that his system of propositions was sufficiently rigorous to allow a straightforward transformation of the classical definition. But it all hangs on the notion of independence. If two propositions are logically independent, if they have no trutharguments in common (5.152), (i.e. if they have no elementary propositions in common), they do not necessarily give one another the probability ½. Von Wright also tells us that Wittgenstein must have realized his mistake and in the second edition we are told that two elementary propositions give one another the probability ½.
It is most likely that it was Ramsey who taught him this in Puchberg. According to C. Lewy (1967), the relevant correction was made in the German text and is in Wittgenstein’s handwriting. In a letter to Wittgenstein, datedSeptember 15, 1924, Ramsey talks about “… a lot of corrections we made to the translation.”
What we then have is a set of exclusive, exhaustive and equally probable elementary propositions, i.e. a hidden principle of indifference. An interesting hypothesis is that we here have the seed of Wittgenstein’s abandonment of the completeness idea. However, it would take us too far away to discuss this surmise and I thus have to leave it for the time being.
For Wittgenstein, probability is a logical relation between propositions. Keynes (1921) advocates a similar idea. (Keynes began his work on probability in 1906 and it was almost completed in 1911; however, it was not published until 1921.) In the preface Keynes says that he has “been much influenced by W. E. Johnson, G. E. Moore, and Bertrand Russell, that is to say by Cambridge.” It is known that Johnson already around 1907 entertained ideas similar to Keynes’, but they were not published until after his death. Thus, when Wittgenstein came to Cambridge in the autumn of 1911, probability theory must have been on the philosophical agenda. It seems therefore safe to say that Wittgenstein was as much influenced by Cambridge as Keynes. However, as a logical theory of probability Wittgenstein’s attempt in the Tractatus is but a pale and incomplete counterpart of the theories of Keynes and Johnson.
At the bottom of Keynes’ theory we have a primitive logical (probability) relation. When this relation is measurable it tells us how strong an inference is from one proposition to another. The conclusive inferences of deductive logic are in Keynes’ theory replaced by objective inconclusive inferences. ‘Objective’ means that for any two propositions one and only one probability relation holds. If this relation has the degree k/m, it is irrational to have any other degree of belief in the derived conclusion.
Ramsey’s key argument against Keynes is that this probability relation does not seem to exist. Ramsey says that he cannot perceive it and that he shrewdly suspects that no one else can. What is, he asks, the probability relation between ‘That is red’ and ‘That is blue’? (PP, p. 58). Is it 1, 1/4, 1/9, 1/25, 1/64, 1/169, or …? One could, of course, give up the idea of a primitive logical probability relation and instead make probability depend on ordinary logical relations. But can a logical relation justify a unique degree of belief? What logical relations justify what degrees of belief?
In a paper written in the autumn of 1922 and read at one of the meetings of the Apostles (October 20, 1923), Ramsey briefly touches upon Wittgenstein’s theory of probability. In the first part of the paper he attacks Keynes’ view of induction. He then gives a simple example of the Tractarian view of probability. If p and q are elementary propositions, the probability of p, givenp or q, is 2/3. If the disjunction is to be true, the propositions cannot both e false. Thus it is easily seen that p is true in 2 out of 3 cases.
Ramsey finds two objections to this theory. First, it is of almost no practical use. How do we know “the logical forms of the complicated relations of every day life”? (p. 300 in Notes on Philosophy, Probability and Mathematics, NPPM.) Second, it definitely does not justify induction. The theory does not tell us how to make an inference from one set of facts to another distinct set of facts. And he might have added, third, it does not allow us to learn by experience. Following Wittgenstein, the probability of an elementary proposition, given any conjunction of elementary propositions (from which it is not entailed), is 1/2.
Ramsey’s own theory of probability avoids these difficulties. The basic idea is that probability is to be interpreted as degree of belief, i.e., to give the notion of probability a subjective interpretation. This requires the measurement of partial beliefs. He therefore showed how people’s beliefs and desires can be measured by use of a betting method and also gave a joint axiomatisation of probability and utility. Given some intuitive rules of rational behaviour, a system of preferences among options, he could prove that the measure of our ‘degrees of belief’ satisfies the laws of probability.
A classical blemish that Ramsey succeeds in getting rid of is the principle of indifference. Keynes thought it possible to base it on purely logical conditions, but did not succeed in doing so. A careless reading of the Tractatus seems to suggest that Wittgenstein could do without the principle. But this is not true. He simply hid it in the fabric of elementary propositions. Ramsey’s probability theory also gives a justification for the axioms of the calculus; Keynes’ theory has to assume the existence of a probability relation. In Ramsey’s theory probable knowledge is effectively accommodated. In Keynes’ theory my rational degree of belief is given by the probability relation (between the hypothesis and what is known for certain, the evidence) and in Wittgenstein’s theory by some logical relation (between sets of elementary propositions).
Von Wright (op. cit.) tells us that there are two poles in Wittgenstein’s thinking about probability. The one pole is the logical theory of probability as it is sketched in the Tractatus. The other pole is the epistemological view of probability as it is briefly outlined in Philosophical Remarks and Philosophical Grammar. von Wright emphasizes that Wittgenstein’s later theory of probability is linked to the notions of imperfect knowledge and incomplete descriptions. But so is Ramsey’s theory! A subjective theory of probability handles probable knowledge and incomplete descriptions: that is the whole point. A subjectivist does not need any logical relations to guide his or her probability assessments.
We want our subjective probabilities to stem from as complete and accurate knowledge as possible. They should be well calibrated. The best way to calibrate them is to take account of wellestablished frequencies and objective probabilities. To bet 1 to 1 on the toss of an American penny is not to be well calibrated. Similarly it seems rather stupid not to follow the probabilities given by accepted physical theories. The same ideas I think we can find in the later Wittgenstein’s writings on probability. To give an account of the relationship between frequencies and probabilities, as von Wright puts it, you have to expand the bulk of knowledge by various hypotheses.
Wittgenstein says that the logic of probability is only concerned with the state of expectation in the sense in which logic is concerned with thinking. This could well be Wittgenstein’s understanding of what Ramsey was doing.
Wittgenstein’s upheaval of the Tractarian view also forced him to reconsider his view on probabilities. von Wright says that the bridge between the two poles in Wittgenstein’s thinking on probability is the idea of a probability which is relative to the bulk of our knowledge. One could say that some of the reasons for, and the drawings and material for this bridge come from Ramsey.
In the Tractatus, Wittgenstein argues that induction consists in accepting as true the simplest law that harmonizes with our experience (6.363). This procedure, however, has no logical justification, only a psychological one (6.3631). Ramsey, however, thought it would ‘be a pity, out of deference to authority, to give up trying to say anything useful about induction’ (PP, p. 87).
Ramsey concludes the paper on induction that he read to the Apostles in 1923 by saying “a type of inference is reasonable or unreasonable according to the relative frequencies with which it leads to truth and falsehood. Induction is reasonable because it produces predictions which are generally verified, not because of any logical relation between its premiss and conclusion. On this view we should establish by induction that induction was reasonable, …” (NPPM, p. 301). In ‘Truth and Probability’ we find the same idea again, but this time more fully developed. He says: ‘We are all convinced by inductive arguments, and our conviction is reasonable because the world is so constituted that inductive arguments lead on the whole to true opinions. We are not, therefore, able to help trusting induction, nor if we could help it do we see any reason why we should, because we believe it to be a reliable process’ (PP, p. 93). That is, our conviction is justified because the world houses reliable processes; inductive arguments on the whole lead to success.
Hume’s problem is a problem of justification or validity. The premises of an inductive argument do not logically entail its conclusion. But what is it that has to be certified? The truth of the belief? Of course not! General beliefs carry no truth value; they “are not judgments but rules for judging ‘If I meet a, I shall regard it as a ’“ (PP, p. 149). D. H. Mellor (1988) has strongly argued that what has to be certified is the effectiveness of our general inferential habits or beliefs. The only way in which this can be done adequately is by assuming the existence of underlying reliable processes.
Instead of accepting the Tractarian view Ramsey showed why some type of rulefollowing, some beliefs, are better habits qua basis for action than others. It is not because they are backed up by more evidence; because they have proved successful in the past. It is because there are underlying reliable processes or mechanisms accounting for our habits. Our habit of acting as if all men are mortal is successful simply because there is an underlying biological mechanism which more or less rapidly breaks down our minds and bodies. We do not need to assume that we can account for the underlying mechanisms or the reliable processes. No one has a clue to the enigma of aging. But this fact does not make our habit less successful; it is successful because there is an underlying mechanism.
Ramsey’s and the later Wittgenstein’s views on induction merit discussion. In the Philosophical Investigations, Wittgenstein rejects the traditional demand for a justification of our expectations. Induction tells us that if we drop a book it will fall to the floor. According to Wittgenstein, “we don’t need any grounds for this certainty … ,” because “[w]hat could justify the certainty better than success?” (324) Wittgenstein also tells us that “[j]ustification by experience comes to an end”. “If it did not it would not be justification.”
To me some of Wittgenstein’s renowned remarks are but echoes of Ramsey’s. Induction does not need the type of justification for which we have traditionally been looking. Wittgenstein is not talking about underlying mechanisms. He saw that it is not the truth of our rulefollowing habits that has to be certified, what has to be certified is the effectiveness of our rulefollowing habits, and nothing can do this better than success.
5. Philosophy of Science
Ramsey argued that the best way to understand how the theoretical entities of a theory function is to picture them as existentially bound variables. If the entities of our theory are , , and , the “best way to write our theory” according to Ramsey is: ,,(dictionary & axioms) (PP, p. 131). This is the theory’s ‘Ramsey sentence’. The existentially bound variables are the carriers of ontological commitment; if the Ramsey sentence is true, they tell us what there is.
Ramsey sentences have been used in the attempt to eliminate socalled theoretical terms (for example ‘electron’ and ‘utility’) in favour of socalled observational terms. But Ramsey’s aim was not to do away with the theoretical terms. In fact he had an argument showing that, depending on the type of dictionary (containing definitions connecting theoretical expressions and observational ones) used, the type of definitions introduced (for example explicit definitions) such a strategy leads to static theories. Ramsey’s goal was to explain the function of theoretical terms. To do this he does not, for example, use the dictionary to ‘define’ the theoretical terms in terms of the observational terms, instead he does the opposite, he uses the dictionary to define the observational terms of the ‘primary language’ (the observation language) in terms of the theoretical terms of the ‘secondary language’ (the theoretical language). This gives us an understanding of how the two type of terms work together in a theory.
Ramsey’s view of theories has several advantages. First, Ramsey sentences help us understand the dynamics of scientific theories and scientific growth. Second, they explain the phenomenon of ‘incommensurability.’ We note, for example, that no proposition of a theory ‘can be understood apart from the whole theory to which it belongs. If a man says ‘Zeus hurls thunderbolts’, that is not nonsense because Zeus does not appear in my theory, and is not definable in terms of my theory. I have to consider it as part of a theory and attend to its consequences, e.g. that sacrifices will bring the thunderbolts to an end’ (PP, p. 1378). Thus, the ‘adherents of two such theories could quite well dispute, although neither affirmed anything the other denied’ (PP, p. 133). And analogous for terms like ‘mass’ and ‘utility.’ As Ramsey explains, any additions to our theory, be it a new axiom, a “particular assertion” or a new definition, have to be made within the scope of a theories quantifiers. And making additions we have to consider, what else we want to add in the future, and whether the addition or its negation is the one best suited for upcoming revisions. We can dispute between theories and how to expand them, but our reasoning, taking place within a theory, within the scope of the quantifiers, is not affected. Ramsey was an ontological antirealist — theoretical terms acquire their meaning by their function in the theory.
Do the Ramsey sentence and the original theory have the same empirical content? It has been shown that they do. That each observational consequence of the Ramsey sentence is also a consequence of the original theory follows from the fact that the former is an existential generalization of the latter and thus implied by it. The reverse implication is not that straightforward (and was first proved by H. G. Bohnert). An oftenraised objection is that proving the theorem assumes secondorder logic; quantification over properties and sets. The Ramsey sentences force us to make stronger ontological claims than we may wish to go along with. Assumptions that, for example, a nominalistcannot accept. But for Ramsey this was no problem. In ‘Universals’ (1925) he argues that there is no intrinsic difference between universals and particulars — in his view there is no essential difference between a predicate’s and a subject’s incompleteness. Thus the Ramsey sentences do not lead to a wanton expansion of our ontology. (See the section on Metaphysics above).
6. General Propositions
Ramsey argued that the logical form of a belief determined its causal properties. The difference between the belief ‘notp’ and the belief ‘p’ lies in their causal properties. Thus disbelieving ‘p’ and believing its negation have the same causal properties. They express, as Ramsey puts it, really the same attitude: “It seems to me that the equivalence between believing ‘notp’ and disbelieving ‘p’ is to be defined in terms of causation, the two occurrences having in common many of their causes and many of their effects” (PP, p. 44). One of the advantages that Ramsey found in this theory is how it avoids the ontological proliferation of Russell’s theory; negative facts, for example, are not needed. (See the section on Metaphysics above.)
A causal property theory of this kind also has to handle more complex beliefs. What precise differences are there between the various logical forms of a belief and its causes and effects? Disjunctive beliefs engender no problems. To “believe p or q is to express agreement with the possibilities p true and qtrue, p false and q true, p true and q false, and disagreement with the remaining possibility p false and q false” (PP, pp. 456). However, quantification introduces a set of problems which are not that easily handled.
In ‘Facts and Propositions’, Ramsey follows W. E. Johnson and L. Wittgenstein and sees general propositions as the logical products and the logical sums of atomic propositions. ‘For all x, Fx’ is to be interpreted as: a is F, b is F, c is F… and ‘There is an x such that Fx’ is consequently equivalent to the logical sum of the values of ‘Fx.’ If all propositions are truth functions of elementary propositions, traditional quantification leads to truth functions of an infinite number of arguments. With this analysis the causal property theory is easily extended to cover also the case of general propositions: “Thus general propositions, just like molecular ones, express agreement and disagreement with the truthpossibilities of atomic propositions, but they do this in a different and more complicated way. Feeling belief towards ‘For all x, Fx’ has certain causal properties which we call its expressing agreement only with the possibility that all the values of Fx are true” (PP, p. 49).
In ‘General Propositions and Causality’ (1929), Ramsey no longer found this a defensible analysis. He has four arguments against analysing ‘For all x, Fx’ as a conjunction. First, ‘For all x, Fx’ cannot be written out as a conjunction. Second, it is never used as a conjunction. The statements are different as a basis for action. Third, ‘For all x, Fx’ exceeds by far what we know or of what we have knowledge. What we know are, at most, a few instances of this generalization: “belief of the primary sort is a map of neighbouring space by which we steer. It remains such a map however much we complicate it or fill in details. But if we professedly extend it to infinity, it is no longer a map; we cannot take it in or steer by it. Our journey is over before we need its remoter parts.” (PP, p. 146)
Fourth and finally, he argues that what we can be certain about is the particular case, or a finite set of particular cases. Of an infinite set of particular cases we could not be certain at all. Thus, ‘For all x, Fx’ expresses, as Ramsey puts it, an inference we are at any time prepared to make, not a belief of the primary sort.
But, if general propositions are not conjunctions and thus not propositions, and assuming that general facts do not exist, how then are we to look upon sentences of this type? What status do they have? In what way can they be right or wrong? Ramsey gives a pragmatic answer to this question. That general propositions are neither true nor false, that they carry no truthvalue, does not imply that they are meaningless. This type of sentence is the very foundation of the expectations that steer our actions. If I accept that for all x,Fx, this means that when I have an x, I act as if it is F. As Ramsey puts it, a general proposition is not a judgment but a rule for judging: it cannot be negated but it can be disagreed with.
In the previous sections Ramsey’s theory of (scientific) theories was discussed and it was noted that he was an ontological antirealist (or instrumentalist). We now note that his analysis of general proposition means that he is indirectly advocating a form of theoretical instrumentalism. General propositions do not have truthvalues. Scientific laws and hence theories constitute the system (or instrument if you like) by which we meet the future, they are not judgments but rules for judging. This means that scientific theories cannot be negated, they cannot be proved true or false, but they can be disagreed with. On this point Ramsey was influenced by C. S. Peirce(although their views differs considerably) but also by his new conception of mathematics. It is important to keep in mind that Ramsey’s problem was not the realist’s. The problem that theories are not statements but opensentence formulas (because the dictionary does not necessarily contain correspondence rules for all theoreticals), a fact which means that the view that theories carry truthvalue does not hold. The sentences are not statements since they do not fulfil the grammatical conditions for statements.
Our knowledge of quantifiers and quantification theory has grown rapidly over the last fifty years. It might be argued that Ramsey’s epistemological arguments miss some of the logical alternatives, and that his pragmatic theory doesn’t answer the logical and ontological problems with which it started off. It might also be argued that Ramsey sidesteps the original problem rather than solves it.
It is worth noting that Ramsey’s questions are very important for the development of logic and the philosophy of mathematics. Wittgenstein, for one, worked on these fundamental questions of logic. In (1982), von Wright recalls that in one of the first conversations he and Wittgenstein had in 1939, Wittgenstein said, “the biggest mistake he made in the Tractatus was that he had identified general propositions with infinite conjunctions and disjunctions of singular propositions” (p. 151). It is well known that the later Wittgenstein took a different view of quantification, a view which in fact is very similar to the one adopted by Ramsey. Wittgenstein came to endorse the view that propositions containing quantifiers are not ‘genuine’ propositions (se von Wright, op. cit. p. 151). We cannot say with certainty whether Wittgenstein got these ideas from Ramsey, or via some other source. It would not be too bold a conjecture to say that the discussions they had in 1929 must have made Wittgenstein see his earlier view of quantification as the biggest mistake he made in the Tractatus.
In 1928, the year before he wrote “General Propositions and Causality”, Ramsey puts forward a completely different theory on law and causality. Taking off from the works of W. E. Johnson and R. Braithwaite Ramsey discusses the difference between laws and accidental generalizations. Johnson’s, or rather J. S. Mill’s, idea is that an accidental generalization state but what is a fact. A law, on the other hand, takes us beyond what we know, states something about the possible. Braithwaite argues that the difference between laws and accidental generalizations is that laws are accepted on grounds that are evidently correct. In “Universals of Law and of Fact” (1928) Ramsey rejects both views. Johnson’s view, he argues, would mean that laws would apply over a wider range than accidental generalizations, which means that they would apply over a wider range than everything. Braithwaite’s view is wrong since in fact we believe some accidental generalizations on grounds that are not demonstrably correct; we do not believe certain laws of nature simply because we do not know them; and we believe some laws on demonstrable grounds.
Following Ramsey we should not look for a distinction grounded in space and time. Likewise, our beliefs are not of any importance to the distinction sought. The difference between laws and accidental generalizations, he argues, lies quite simply in the fact that a true law would exist even though we knew everything. If we were omniscient and organise our knowledge into a simple deductive system, true laws would remain unchanged. Laws are those sentences that would be the logical consequences of the simplest and most powerful account of the world we could device if we knew everything. Ramsey’s theory has many advantages. It explains why the choice of scientific laws is not only a question of generality or universality. It explains the difference between the statements we regard as laws and true laws. Ramsey’s account of laws has recently become popular and it has been developed and defended by, for example, D. Lewis.
As we have already seen Ramsey himself abandoned the idea. In “General Propositions and Causality” he writes: “it is impossible to know everything and organise it in a deductive system” (PP, p. 150).
7. Philosophy of Mathematics and The Ramsey Theory
Ramsey is probably best known for his work on the foundations of mathematics. R. B. Braithwaite (1931, Introduction) notes that ‘The Foundations of Mathematics’ (1925) is an “attempt to reconstruct the system of Principia Mathematica so that the blemishes may be avoided but its excellencies retained.” What Ramsey did was to streamline and strengthen Russell’s and Whitehead’s system. In Principia, a mathematical proposition is defined as a proposition that is completely general. Ramsey saw that this was too loose a definition and gives the following counterexample: “Any two things differ in at least thirty ways” — a completely general proposition but clearly not a mathematical truth. Instead, mathematical propositions are such that “their content must be completely generalized and their form tautological” (PP, p. 167); as, for example, “Any two things together with any other two things make four things.”
To avoid the classical paradoxes, Russell and Whitehead introduced a theory of types and with it a number of axioms. One of these axioms, The Axiom of Reducibility, asserts the reducibility of functions to predicative functions (that the quantifiers are not needed). Without it the attempted reduction of mathematics to logic cannot be carried out within their framework. But, as Ramsey points out there is no reason to suppose this axiom to be a tautology and he therefore constructs an alternative theory of types to evade it. With the Ramseyfied theory of types, Principia Mathematica gets a new and far more solid foundation than it originally had.
Doing this he also shows that it is only the logical axioms that are problematic for the theory. The semantical paradoxes can be dealt with separately. For example, he shows how The Liar can be handled by the introduction of a hierarchy of ‘naming’ relations, and to some extent, this anticipated Tarski´slater hierarchy of ‘languages’ or truth predicates.
At the end of his life Ramsey gives up logicism for a finitistic view of mathematics and in 1929 he takes steps in the direction towards intuitionism. For example, he strongly argues against The Axiom of Infinity (the assumption that there are an infinite number of individuals). And his paper ‘Theories’ (1929) can be read both as a theory of scientific theories and as an essay on the foundations of mathematics.
In ‘On a Problem of Formal Logic’, Ramsey attempts to solve Hilbert’sEntscheidungsproblem. That is he tries to find a procedure of “determining the truth or falsity of any given logical formula.” He succeeds in solving the problem for a segment of firstorder predicate calculus. Today we know that the problem cannot be solved, there is no general mechanical method. But trying to find a solution to the problem Ramsey proves two theorems that have given rise to a thriving branch of mathematics: the Ramsey theory.
For historical reasons, if for nothing else, I will quote Ramsey’s own formulation of the theorems:
Ramsey’s first theorem: Let G be an infinite class, and m and r positive integers; and let all those subclasses of G which have exactly r members, or, as we may say, let all rcombinations of the members of G be divided in any manner into m mutually exclusive classes C_{i} (i=1, 2, …, m), so that every rcombination is a member of one and only one C_{i}; then, assuming the Axiom of Selections, G must contain an infinite subclass D such that all the rcombinations of the members of D belong to the same C_{i}. (F. p. 233)
Ramsey’s second theorem: Given any r, n, and k such that n + k = r, there is an m_{0} such that if m = m_{0} and the rcombinations of any G_{m} [a set with mmembers] are divided into two mutually exclusive classes C_{1} and C_{2}, then G_{m}must contain two mutually exclusive subclasses, G_{n} and G_{k} such that all the combinations formed by r members of G_{n} + G_{k} which include at least one member from G_{n} belong to the same C_{i}. (F. p. 237)
A classical example is that in any group of six people there are either three mutual friends or three mutual strangers. Another wellknown example runs as follows. Take six points in the plane. The points are connected in pairs by edges. The edges being coloured in one of two colours. Then we know that there are three points such that the edges connecting them have one and the same colour. The Ramseynumber, R(3,3,2), is equal to 6. Since five points in the plane can be coloured in pairs leaving no unicoloured triangle, we know that R(3,3,2)>5. A lot of efforts have been made to determine the Ramsey numbers. Only a few nontrivial values and upper and lower bounds are known (and complete and updated lists are published on several webpages).
Proving the theorem Ramsey makes explicit use of the axioms of choice. Today it is known that for countable sets the axiom is not needed. It is also known that the axiom of choice implies Ramsey’s theorem, but is not implied by it. The theorem is important in the theory of infinitary combinatorics (large cardinals); large cardinal versions also have interesting measuretheoretic consequences for the theory of conditional probability of finitely additive measures; and it has been used to prove theorems in, for example, plane geometry.
Ramsey’s theorem is one of the first examples of a Gödel sentence (see Paris and Harrington 1977) — in (first order) Peano arithmetic an extension of the finite version of the theorem can be shown to be true but unprovable. However, the theorem can be proved using an extended, second order, notion of natural numbers.
Ramsey Theory is the result of an unsuccessful attempt to solve a classical, today known as unsolvable, problem of logic, and to answer a fundamental philosophical question. In R. C. Jeffrey succinct words: “Trying to solve the unsolvable, Ramsey proved the unprovable”
Bibliography
Primary Sources
FM  Foundations of Mathematics and Other Logical Essays, 1931, R. B. Braithwaite (ed.), …  
F  Foundations: Essays in Philosophy, Logic, Mathematics and Economics, D. H. Mellor (ed.), 1978, …  
PP  Philosophical Papers, D. H. Mellor (ed.), 1990, …  
•  On Truth, N. Rescher and U. Majer (eds.), Kluwer, Dordrecht 1991.  
NPPM  Notes on Philosophy, Probability and Mathematics, Bibliopolis, Napoli1991, edited by Maria Carla Galavotti 
Chronological Catalog of Ramsey’s Work
Items marked FM, F or PP are reprinted or published for the first time in The Foundations of Mathematics and Other Logical Essays; Foundations: Essays in Philosophy, Logic, Mathematics and Economics; and Philosophical Papers, respectively.
1922  •  ‘Mr Keynes on Probability’, The Cambridge Magazine , 11, no. 1 (Decennial number, 19121921, January, 1922), 35. 
•  7#145;The Douglas Proposal’, The Cambridge Magazine, 11, no. 1, (January 1922), 746.  
•  Review of W. E. Johnson’s Logic Part II, The New Statesman, 19, 29th (July 1922), 46970.  
1923  •  Critical Notice of L. Wittgenstein’s Tractatus LogicoPhilosophicus,Mind, 32, no. 128, (October 1923), 46578. (FM) 
1924  •  Review of C. K. Ogden and I. A. Richards’ The Meaning of Meaning,Mind, 33, no. 129, (January 1924), 1089. 
1925  •  ‘The New Principia’ (review of A. N. Whitehead and B. Russell’sPrincipia Mathematica, volume I, second edition), Nature, 116, no. 2908, (25th July 1925), 1278. 
•  Review of the same book in Mind, 34, no. 136, (October 1925), 5067.  
•  ‘Universals’, Mind, 34, no. 136, (October 1925), 40117. (FM, F, and PP)  
•  ‘The Foundations of Mathematics’, Proceedings of the London Mathematical Society, ser. 2, 25, part 5, (read 12th November 1925), 33884. (FM, F, and PP)  
1926  •  ‘Mathematical Logic’, The Encyclopædia Britannica, supplementary volumes constituting thirteenth edition, 2, (1926), 83032. 
•  ‘Universals and the “Method of analysis”’, Aristotelian Society Supplementary, 6, (July 1926), 1726.[Symposium with H. W. B. Joseph and R. B. Braithwaite]. (A few pages of this paper is reprinted in FM and PP, see ‘Note on the preceding paper’.)  
•  ‘Mathematical Logic’, The Mathematical Gazette , 13, no. 184, (October 1926), 18594. [A paper read before the British Association, section A, Oxford, August, 1926]. (FM, F,and PP)  
1927  •  ‘A Contribution to the Theory of Taxation’, The Economic Journal, 37, no. 145, (March 1927), 4761. (F) 
•  ‘Facts and Propositions’, Aristotelian Society Supplementary Volume VII, (July 1927), 15370. [Symposium with G. E. Moore]. (FM, F, and PP)  
1928  •  ‘A Mathematical Theory of Saving’, The Economic Journal, 38, no. 192, (December 1928), 54349. (F) 
•  ‘On a Problem of Formal Logic’, Proceedings of the London Mathematical Society, ser. 2, 30, (read 13 December 1928), 33884.(FM and in F as ‘Ramsey’s theorem’)  
1929  •  ‘Foundations of Mathematics’, The Encyclopædia Britannica, 14^{th}edition, 15, (1929), 824. 
•  ‘Bertrand Arthur William Russell’ (in part), The EncyclopædiaBritannica, 14th edition, 19,(1929), 678. 
Posthumously Published Papers
1931  •  Epilogue (1925), in FM. [A paper read to a Cambridge discussion society in 1925.] 
•  Truth and probability (1926), in FM, F and PP. [Parts of this paper was read to the Moral Science Club at Cambridge.] 

•  Reasonable degree of belief (1928), in FM and PP.  
•  Statistics, (1928) in FM and PP.  
•  Chance, (1928) in FM and PP.  
•  Theories, (1929) in FM, F and PP.  
•  General propositions and causality (1929), in FM, F and PP.  
•  Probability and partial belief (1929), in FMand PP.  
•  Knowledge (1929), in FM, F and PP.  
•  Causal qualities (1929), in FM and PP.  
•  Philosophy (1929), in FM and PP.  
1978  •  Universals of law and of fact (1928), in F and PP. 
1987  •  The ‘long’ and ‘short’ of it or a failure of logic, American Philosophical Quarterly, 24,no. 4, (October 1987), 35759. Ed. N. Rescher. 
•  Principles of finitist mathematics, History of PhilosophyQuarterly, 6, (1989), 25558, appendix to U. Majer, Ramsey’s conception of theories: An intuitionistic approach, 23358.  
1990  •  Weight or the value of knowledge, The British Journal for the Philosophy of Science, 41, (1990), 13. Ed. N.E. Sahlin. 
1991  •  Notes on Philosophy, Probability and Mathematics, Bibliopolis, Napoli1991, Ed. Maria Carla Galavotti 
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 ——, Frank Ramsey : Truth and Success, Routledge, London 2002.
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 Keynes, J. M., A Treatise on Probability, London: Macmillan, 1921, reprinted 1957
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 Lewy, C., ‘A Note on the Text of the Tractatus’, Mind, 76, 1967, pp. 41723.
 Mellor, D.H., ‘The Eponymous F. P. Ramsey’, Journal of Graph Theory, 7, (1983), 913.
 ——, The Warrant of Induction, Cambridge University Press, Cambridge 1988.
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 ——, Philosophical Investigations, trans. by E. Anscombe, Oxford: Blackwell, 1958.
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 ”Better than the stars”, a radio portrait of Frank Ramsey, broadcast on BBC Radio 3 on February 27, 1978, produced by Fraser Steel (with contributions by A. J. Ayer, Richard Braithwaite, Dick Jeffrey, Lord Ramsey, Mrs Lettice Ramsey and I. A. Richards, and excerpts from Ramsey’s writings are read by Hugh Dickson and from Maynard Keynes’s writings by Gabriel Woolf).
 Frank Ramsey Appreciation Society
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Other Internet Resources
Copyright © 2001 by
NilsEric Sahlin
University of Lund
nilseric.sahlin@telia.se