Frank Plumpton Ramsey
(1903-1930), 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
Ramsey came
from a distinguished
Ramsey was
very much a
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 ground-breaking work on epistemology, philosophy of science,
philosophy of mathematics, metaphysics and semantics. And he accomplished all
this before the age of twenty-seven.
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 truth-value, 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
well-defined 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 truth-problem 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,
true-justified-belief 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 X has 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 well-known that this view of knowledge leads to
serious problems. The so-called ‘Russell-Gettier problems’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 Russell-Gettier-problems
before they were even invented. For Ramsey the belief-generating 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 Russell-Gettier 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 long-standing 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
subject-predicate 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 ‘not-wise’; 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 non-relational 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. Nominalists
for example reject the so-called ‘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, $y if p is not true, were x > y. The
“traditional method” tells us that the agent’s degree of belief in p is ($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 a
and b is equal to the difference between c and d, and the
difference between c and 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
well-calibrated (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, risk-aversion
(risk-proneness). 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)
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 narrow-minded
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 de dicto-de re problem. To what extent does the theory tell us
anything about human decision-making? Does it simply describe a surface
phenomenon, failing to capture the underlying mechanisms of human
decision-making? 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
Assume
that, after considering it carefully, the subject believes that the probability
that there will be a transit strike in
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
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 and Aumann
and de Finetti, all have a problem with
state-dependent 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 value-neutral. For that, he proposed ethically neutral propositions.
In his theory the outcomes have state-dependent 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’s Tractatus Logico-Philosophicus,
‘the number of all equipossible cases’ is the number
of truth-grounds of a given proposition q; and ‘the number of all
favourable cases’ is the number of truth-grounds of a proposition, p,
which also are truth-grounds of q. Assume that these numbers are m
and k, respectively. Then the conditional probability, the probability
of p given q, is k/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 the Tractatus
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
truth-arguments 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, dated
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
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, given p 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
well-established 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 rule-following, 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 rule-following habits that has to be certified, what has to be
certified is the effectiveness of our rule-following 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 so-called theoretical terms
(for example ‘electron’ and ‘utility’) in favour of so-called
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. 137-8). 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 anti-realist -- 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 often-raised objection is that
proving the theorem assumes second-order 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 nominalist
cannot 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 ‘not-p’ 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 ‘not-p’ 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 q true, p
false and q true, p true and q false,
and disagreement with the remaining possibility p false and q false”
(PP, pp. 45-6). 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 truth-possibilities 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 truth-value,
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 anti-realist (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 truth-values. 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 open-sentence formulas
(because the dictionary does not necessarily contain correspondence rules
for all theoreticals), a
fact which means that the view that theories carry truth-value 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´s later 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’s Entscheidungsproblem. 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 first-order
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 sub-classes of G which have
exactly r
members, or, as we may say, let all r-combinations of the members of G be divided
in any manner into m mutually exclusive classes Ci (i=1, 2, …, m), so that
every r-combination is a member of one and only one
Ci; then, assuming the Axiom of
Selections, G must contain an infinite sub-class D such that
all the r-combinations of the members of D belong to
the same Ci. (F. p. 233)
Ramsey’s second
theorem: Given any r, n, and k such that n + k = r, there is
an m0 such that if m = m0 and the r-combinations of any Gm [a set with m members]
are divided into two mutually exclusive classes C1 and C2, then Gm must contain two mutually exclusive
sub-classes, Gn and Gk such that all the combinations formed by
r members
of Gn + Gk which include at least one member from
Gn belong to the same Ci. (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 well-known 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 Ramsey-number, 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 non-trivial values and upper and lower bounds are
known (and complete and updated lists are published on several
web-pages).
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 measure-theoretic 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, |
|
NPPM |
|
Notes on
Philosophy, Probability and Mathematics, Bibliopolis,
|
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, 1912-1921, January, 1922), 3-5. |
|
|
• |
7#145;The
Douglas Proposal’, The Cambridge
Magazine, 11, no. 1, (January 1922), 74-6. |
|
|
• |
Review of W.
E. Johnson’s Logic Part II, The New Statesman, 19, 29th (July 1922),
469-70. |
|
1923 |
• |
Critical Notice of L. Wittgenstein’s Tractatus Logico-Philosophicus, Mind, 32, no. 128, (October 1923), 465-78. (FM) |
|
1924 |
• |
Review of C.
K. Ogden and I. A. Richards’ The Meaning of
Meaning, Mind, 33, no. 129, (January
1924), 108-9. |
|
1925 |
• |
‘The New Principia’ (review of A. N. Whitehead and B. Russell’s
Principia Mathematica, volume I, second
edition), Nature, 116, no. 2908, (25th
July 1925), 127-8. |
|
|
• |
Review of the
same book in Mind, 34, no. 136, (October
1925), 506-7. |
|
|
• |
‘Universals’,
Mind, 34, no. 136, (October 1925),
401-17. (FM, F, and PP) |
|
|
• |
‘The Foundations of Mathematics’, Proceedings of the London Mathematical Society, ser. 2, 25, part 5, (read 12th November 1925), 338-84. (FM, F, and PP) |
|
1926 |
• |
‘Mathematical
Logic’, The Encyclopædia
Britannica, supplementary volumes constituting thirteenth
edition, 2, (1926), 830-32. |
|
|
• |
‘Universals
and the “Method of analysis”’, Aristotelian
Society Supplementary, 6, (July 1926), 17-26.[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), 185-94. [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), 47-61. (F) |
|
|
• |
‘Facts and Propositions’, Aristotelian Society Supplementary Volume VII, (July 1927), 153-70. [Symposium with G. E. Moore]. (FM, F, and PP) |
|
1928 |
• |
‘A Mathematical Theory of Saving’, The Economic Journal, 38, no. 192, (December 1928), 543-49. (F) |
|
|
• |
‘On a Problem of Formal Logic’, Proceedings of the London Mathematical Society, ser. 2, 30, (read 13 December 1928), 338-84. (FM and in F as ‘Ramsey’s theorem’) |
|
1929 |
• |
‘Foundations
of Mathematics’, The Encyclopædia
Britannica, 14th edition, 15,
(1929), 82-4. |
|
|
• |
‘Bertrand
Arthur William Russell’ (in part), The Encyclopædia
Britannica, 14th edition, 19,(1929), 678. |
Posthumously Published Papers
|
1931 |
• |
Epilogue
(1925), in FM. |
|
|
• |
Truth and
probability (1926), in FM, F and PP. |
|
|
• |
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), 357-59. Ed. N. Rescher. |
|
|
• |
Principles of
finitist mathematics,
History of PhilosophyQuarterly, 6, (1989),
255-58, appendix to U. Majer, Ramsey’s conception of theories:
An intuitionistic
approach, 233-58. |
|
1990 |
• |
Weight or the value of knowledge, The British Journal for the Philosophy of Science, 41, (1990), 1-3. Ed. N.-E. Sahlin. |
|
1991 |
• |
Notes on
Philosophy, Probability and Mathematics, Bibliopolis,
|
Secondary Sources
· Dokic, J., and Engel, P., Vérité et succeès, Presses Universitaires de France,, Paris 2001.
· -------, Frank Ramsey : Truth and Success, Routledge, London 2002.
·
Galavotti, Maria
Carla (ed.), Special Issue on the Philosophy of F.
P. Ramsey, Theoria, 57, (1991). (With contributions
by Rosaria Egidi,
·
Keynes, J. M., A Treatise on Probability,
·
Lewis, D., Counterfactuals, Blackwell, Oxford 1973.
·
Lewy, C., ‘A Note on the
Text of the Tractatus’, Mind, 76, 1967, pp.
417-23.
·
Mellor, D.H., ‘The Eponymous F. P.
Ramsey’, Journal of Graph Theory, 7, (1983),
9-13.
·
------, The Warrant of Induction, Cambridge University
Press, Cambridge 1988.
·
·
Graham, R.,
Rothschild, B., and Spencer, J. H., Ramsey Theory,
·
Sahlin, N.-E., The Philosophy of F. P. Ramsey,
·
------, Review of F.
P. Ramsey, ‘Notes on Philosophy, Probability and Mathematics’, ed. by
·
------, ‘F. P.
Ramsey’, in A Companion to
Metaphysics. Ed. J. Kim and E. Sosa, Basil Blackwell, Oxford 1994,
429-30.
· ------, ‘On the Philosophical Relations Between Ramsey and Wittgenstein’, in The British Tradition in the 20th Century Philosophy , Ed. J. Hintikka and K. Puhl. Hölder-Pischler-Tempsky, Wien 1995, 150-63.
·
-------, ‘“"He is no
good for my work”: On the Philosophical Relations Between Ramsey and Wittgenstein’, in Knowledge and Inquiry: Essays on Jaakko Hintikkas Epistemology and Philosophy of
Science. Ed. M. Sintonen.
·
Simon, P., ‘Ramsey,
particulars, and universals’, in Special Issue on
the Philosophy of F. P. Ramsey, Theoria, ed. by
·
Schervish, M., Seidenfeld J.,
and Kadane, T.,
‘State-dependent utilities’, Journal of the
American Statistical Association, 1990, Vol. 85, No. 411, pp. 840-7.
·
von Neumann, J. and
Morgenstern, O., Theory of Games and Economic
Behavior, Princeton;
Princeton University Press, 1944
·
von Wright, G.H., Wittgenstein,
·
Wittgenstein, L., Tractatus Logico-Philosophicus,
·
------, Philosophical Investigations, trans. by E. Anscombe, Oxford: Blackwell,
1958.
Other Internet Resources
·
Mellor, D.H., Frank Ramsey: A Portrait,
·
O’Connor, J., and
Robertson, E., Frank Plumpton Ramsey, The MacTutor History of Mathematics Archive,
University of
·
"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
·
Ramsey MSS, Cambridge DSpace repository: http://www.dspace.cam.ac.uk/handle/1810/192853
Copyright ©
2001
by
Nils-Eric.Sahlin@fil.lu.se