## 1. Life and Works



Ramsey was born in Cambridge on 22 February 1903. His father was a
mathematician and Fellow of Magdalene College whilst his mother
campaigned for women's suffrage and other social causes. In 1920
Ramsey matriculated as an undergraduate student at Trinity College,
Cambridge, from which he graduated in 1923 with a First in
Mathematics. The year before he matriculated, whilst still a 17 year
old school boy, Ramsey became friends, through his father's
connections, with C.K. Ogden and I.A. Richards, also a Fellow of
Magdalene. Ramsey had already read Russell's *Introduction to
Mathematical Philosophy* and G.E. Moore's *Ethics*
but because he had become interested in learning German, Ogden gave
Ramsey Mach's *Die Analyse der Empfindungen*
[*Analysis of Sensations*] to read. Ramsey, still in his final
school year, went on to read Louis Couturat, Henri Poincare and
Hermann Weyl. By 1921, Ogden had become so impressed by Ramsey's
philosophical acumen and facility with German that he commissioned
Ramsey, as a second year undergraduate, to translate
Wittgenstein's *Tractatus* into English - in the
face of Moore's scepticism that the *Tractatus* could be
translated into English at all. Back in 1920, Ramsey also discussed
with Ogden and Richards their theory of thought and language, later
famously elaborated in *The Meaning of Meaning* (1923), which
Ramsey reviewed - unfavourably - in *Mind* (1924).
Ramsey nonetheless praised "the excellent appendix on C.S.
Peirce" and this lead him to study the volume of Peirce's
papers, *Chance, Love and Logic* (1923), which Ogden had just
published in his book series for Kegan Paul. During Ramsey's
first year as an undergraduate, Ogden also arranged for him to meet
Russell in London and encouraged Ramsey to attend Moore's
lectures with him and Richards; Ramsey duly went along in his second
term to Moore's lectures on "Incomplete Symbols and
Logical Constructions". Moore was later to recall "In the
early twenties F.P. Ramsey attended at least one course of my
lectures. I had soon come to feel of him, as of Wittgenstein, that he
was much cleverer than I was" (1944, 35).



In his first year as an undergraduate, Ramsey had met and befriended
R.B. Braithwaite, then a mathematics undergraduate a year ahead of
him. Braithwaite introduced Ramsey to J.M. Keynes, the economist.
Ramsey was soon after elected to the Cambridge Apostles, the secret
debating society of which Braithwaite and Keynes were both active
members. Ramsey attended Keynes' economics lectures and they
discussed Keynes' *Treatise on Probability* (1921).
Ramsey's criticisms appeared the following year in a review of
Keynes' book for Ogden's journal, *The Cambridge
Magazine* (1922); Keynes had maintained that probability is an
objective relation between propositions but Ramsey denied we perceive
such relations nor grasp them intellectually. After his finals, Ramsey
turned to writing a critical notice for *Mind* of
Wittgenstein's *Tractatus* (1923). Ogden had written to
Wittgenstein suggesting that he meet with Ramsey and Ramsey duly went
to Puchberg, near Vienna, in September 1923 to spend a fortnight with
Wittgenstein working through the *Tractatus* with him. In 1924
Ramsey spent 6 months in Vienna being psychoanalysed, during which
time he met Moritz Schlick and Hans Hahn, both members of the Vienna
Circle. He returned to Cambridge in October 1924 where he had been
appointed as a Fellow In Mathematics at King's College, where
Keynes, also a Fellow, had pushed for Ramsey's appointment. At a
meeting of the Moral Sciences Club that December, Ramsey met his
future wife, Lettice Baker; they married in 1925 and had two daughters
together.



Once back in Cambridge, Ramsey settled down to teach, including a
course on the foundations of mathematics, and to write the papers in
philosophy, economics and mathematics for which he was to become
famous. He started by completing "The Foundations of
Mathematics", which he had begun in Vienna, and
"Universals", both published in 1925. The former was
intended to rescue logicism by remedying the defects Ramsey found in
*Principia Mathematica*, whilst the latter was a development of
the programme of logical analysis advertised in the
*Tractatus*; they won Ramsey immediate eclat. But it was
the following year in which Ramsey really struck out on his own,
writing "Truth and Probability" (1926a) although he did
not publish it, which started from Ramsey's criticisms of Keynes
and provides a subjective theory of probability and a pragmatic view
of induction, and "Facts and Propositions" (1927a),
initially inspired by his reading of Peirce and Russell's
*Analysis of Mind* (1921), in which Ramsey's philosophy
of mind and language began to have, as he put it, a distinctive
'pragmatist tendency' (1927a, 170). It was also the year
that Ramsey began dedicated work on economics which resulted in the
first of his seminal papers on economics, 'A contribution to the
theory of taxation' (1927b), later followed by his
'Mathematical Theory of Saving' (1928a). Ramsey also wrote
his influential paper, 'On a Problem of Formal Logic'
(1930), which contains the mathematical results now called
'Ramsey's Theorems'. In 1927 Ramsey also began work
on a book manuscript, *On Truth* ([OT]). During the last two
years of his life, Ramsey wrote a number of philosophical notes and
papers, including 'Theories' and 'General
Propositions and Causality', both completed in 1929. They were
published by Braithwaite (in [FM]), along with 'Truth and
Probability', in a posthumous volume of Ramsey's papers,
*The Foundations of Mathematics and other Logical Essays*,
whilst the manuscript on truth did not appear until 1991.



After meeting and falling out with Ramsey when visiting Keynes in
1925, Wittgenstein returned to settle in Cambridge in 1929. He was
admitted as a Ph.D. student, submitting the *Tractatus* as his
thesis, Ramsey was his supervisor; they met regularly during what was
to be the last year of Ramsey's life. Wittgenstein was later to
recall the significance of these conversations for his own
intellectual development in the preface to *Philosophical
Investigations*. Ramsey fell ill at the end of November 1929 and
was diagnosed with jaundice. He had been corresponding with Schlick
and had promised to write a review of Rudolf Carnap's
*Aufbau*. He died on 19 January 1930 in Guy's Hospital in
London. His death was unexpected and its causes remain uncertain. In
an obituary for the *Cambridge Review*, Braithwaite wrote,
'His intellectual subtlety was combined with the most delightful
simplicity and candour. Intolerant of fools, he inspired others by his
mental fertility and integrity. To one at least of his friends his
death is as if a lighthouse were extinguished and we were left to
grope our way in the dark' (1930, 216).



Mellor's BBC radio programme on Ramsey, produced in 1978 (see
Other Internet Resources below), features the first hand testimony of
Braithwaite, Lettice Ramsey, and I.A. Richards. See also Sahlin 1990,
221-9, Mellor 1995 and Taylor 2006. Ramsey's sister,
Margaret Paul, provides a more personal account of Ramsey's
life, drawing upon letters and diaries, in her memoir (2012). Misak
(2020) provides an in-depth intellectual biography of Ramsey.


## 2. The Foundations of Logic and Mathematics



Ramsey, as we saw in the previous section, was still an undergraduate
when, aged 19, he completed a translation of the *Tractatus*
*Logico-Philosophicus* (Wittgenstein 1922). Alas, C. K. Ogden
got all the credit and it has been known since as the 'Ogden
translation'. Ramsey's translation is usually considered
to be superseded by the Pears-McGuinness translation (1961), but one
should not lose sight of the fact that it was carefully scrutinized by
Wittgenstein, who gave it his seal of approval. Ramsey then wrote a
searching review of the *Tractatus* (1923) in which he raised
many serious objections (Methven 2015, chapter 4) (Sullivan 2005). One
such objections is the 'colour-exclusion problem' (1923,
473), against Wittgenstein's claim in 6.3751 that it is
"logically impossible" that a point in the visual field be
both red and blue. This claim was linked to the requirement that
elementary propositions be logically independent (otherwise, the
analysis of the proposition would not be completed), a pillar of the
*Tractatus*. Wittgenstein's recognition in 1929 that he
could not sustain his claim (Wittgenstein 1929), probably under
pressure at that stage from discussions with Ramsey, was to provoke
the downfall of the *Tractatus*.



In 1923 and 1924, Ramsey also visited Wittgenstein, who was teaching
at Puchberg am Schneeberg (Lower Austria). During these visits,
Wittgenstein made changes to the text and translation, that were
incorporated in later editions. (Details are given in (Lewy 1967).) At
first under the spell, Ramsey became disillusioned. In 1923, he wrote
home that he would "try and pump him for ideas for
[*Principia Mathematica*'s] further development which I
shall attempt" (Wittgenstein [LO], 78), but in 1924 he remarked
laconically: "He is no good for my work" (Sahlin 1997,
64).



Ramsey's first major self-assigned task in the foundations of
mathematics was indeed to revise Whitehead and Russell's
*Principia Mathematica*, partly in the hope that his version
would meet some of Wittgenstein's criticisms of its axioms and
treatment of identity, but also because of some further defects he
diagnosed in its treatment of the paradoxes and in its inability
properly to account for the 'extensionality' of
mathematics (1925a, [PP] 177-178). For further details about the
content of the following paragraph, see the entry for *Principia
Mathematica*.



Ramsey drew a distinction between 'logical' and
'semantical' paradoxes, with Russell's paradox
belonging to the former category and Richard's or
Grelling's paradoxes to the latter (1925a, [PP] 183-184).
According to him, the heart of the problem was in the way the theory
of types was devised to deal with the paradoxes, as understood in
terms of the 'vicious-circle principle' ('whatever
involves all of a collection must not be one of the
collection'):



> 
> 
> These contradictions it was proposed to remove by what is called the
> Theory of Types, which consists really of two distinct parts directed
> respectively against two groups of contradictions. These two parts
> were unified by being both deduced in a rather sloppy way from the
> 'vicious-circle principle', but it seems to me essential
> to consider them separately. (1925a, [PP] 187)
> 
> 
> 



Ramsey rightly saw that a 'simple' theory of types
(sometimes called 'ramseyfied') that distinguishes types
of propositional functions by their arguments would deal with
'logical' paradoxes, but in order to deal with the other
paradoxes, Whitehead and Russell had introduced a
'branched' or 'ramified' theory of types with
a hierarchy of levels of functions (and of propositions), that led to
further difficulties, one of which being the inability even to
formulate theorems of classical mathematics such as the least upper
bound theorem, unless one introduced the 'Axiom of
Reducibility' according to which for any function at any level
there exists an equivalent one at the first level (1925a, [PP]
187-192). (Another worry, that one could not formulate
mathematical induction in its full generality turns out to have been
unfounded.) Wittgenstein had pointed out in the *Tractatus*
that it is not a tautology (6.1233), so that it could hardly serve as
an axiom. To circumvent this, Ramsey introduced his own notion of
'predicative function' and showed that if one gets rid of
the 'vicious-circle principle', the levels can be
collapsed and the need for the 'Axiom of Reducibility' is
obviated (1925a, [PP] 195-205). Thus, Ramsey had in effect shown
how to deal with the paradoxes and the 'extensionality' of
mathematics without the need for a 'ramified' theory of
types. (In this he was preceded by Leon Chwistek (Chwistek 1921).)
This, Ramsey concluded, would secure mathematics against the
"Bolshevik menace" of the intuitionist critique of Brouwer
and Weyl (1925a, [PP] 219).



Ramsey also took seriously Wittgenstein's critique
(5.53-5.533) of the definition of identity in *Principia
Mathematica* ([?]13.01) in terms of indiscernibility (with
*F* a predicative function):

 
\[ x\! =\! y \;=\_{df}\; \forall F(Fx \to Fy) \]


Wittgenstein argued that to say that two things are the same,
*x* = *y*, is nonsense, while *x* = *x* is
empty and showed how to get rid of identity with an
'exclusive' reading of the variables (Hintikka 1956),
(Wehmeier 2008), (Wehmeier 2012), with the exception of some
propositions such as the axiom of infinity that cannot be so read, and
are thus to be rejected (5.535). To save the latter (and the axiom of
choice, which was then called the 'Multiplicative Axiom'),
Ramsey introduced 'functions in extension' (including
arbitrary pairings of arguments with values, and indicated here with
the subscript \(e\)), replacing the above with:

 
\[ \forall \phi\_e (\phi\_{e}x \equiv \phi\_{e} y)\]


He then argued that this is a tautology if \(x = y\), and a
contradiction if \(x \neq y\). This is the subject on which he
eventually sparred with Wittgenstein, when the latter wrote a letter
in 1927 (reproduced in [NP] 340-341). Wittgenstein argued that
either way this formula is nonsensical and cannot be substituted for
*x* = *y*. Two drafts of a rejoinder by Ramsey are
preserved ([NP] 342-346), but the matter was dropped
subsequently, although Wittgenstein came back to it repeatedly after
Ramsey's death, for example in (Wittgenstein [BT], SS 113).
(For further discussion see Marion 1995; Sullivan 1995; and Methven
2015, chapter 7.)



It is said that, when Wittgenstein returned to Cambridge in January
1929, it was primarily to discuss philosophy with Ramsey, who he
obviously deemed a worthy interlocutor, and they reportedly met on a
regular basis, until Ramsey's death the following January. They
both left behind manuscripts which show that Ramsey had shifted his
views on a number of topics, while Wittgenstein's thought was
also set in motion towards his later views. In the preface to the
*Philosophical Investigations*, he wrote:



> 
> 
> For since I began to occupy myself with philosophy again, sixteen
> years ago, I could not but recognize grave mistakes in what I set out
> in that first book. I was helped to realize these mistakes - to
> a degree which I myself am hardly able to estimate - by the
> criticism which my ideas encountered from Frank Ramsey, with whom I
> discussed them in innumerable conversations during the last two years
> of his
>  life.[1]
>  (Wittgenstein 1953 [PI, 4])
> 
> 
> 



Although one can at least point out with McGuinness that
"Wittgenstein clearly learnt a lot from Ramsey"
(McGuinness 2006, 25), if one asks who made the larger contribution in
these discussions, there is considerable disagreement, with claims
ranging from the idea that literally none of the later
Wittgenstein's key ideas can be traced to Ramsey (Kienzler 1997,
75-76), to claims of various degrees of a more substantial
influence by Ramsey on Wittgenstein (Glock 2005, Majer 1991; Marion
1998, chapters 4-5; Marion 2012; Methven 2015, chapter 9; Misak
2016a, chapter 7; Misak 2016b; Misak 2018; Sahlin 1990; Sahlin 1997).
It is perhaps better simply to talk of a confluence of ideas, and the
following, being restricted to Ramsey, will show that he reasoned
along lines largely independent from Wittgenstein, and began shifting
his views before 1929.



This confluence is particularly visible in texts relating to the
foundations of mathematics, where Ramsey became disillusioned with
*Principia Mathematica*, perhaps under the influence of
Wittgenstein, whose criticisms of the axioms of infinity and
reducibility, he had already taken seriously in 'The Foundations
of Mathematics', as was just pointed out. Ramsey initially
adopted Wittgenstein's view of the quantifiers in the
*Tractatus* as 'logical products' and
'sums', with universality being expressed, for predicate
*F*, as an infinite conjunction

 
\[Fa \: \wedge\: Fb \:\wedge\: Fc \:\wedge\: \ldots \]


And existence as an infinite disjunction

 
\[Fa \: \lor\: Fb \:\lor\: Fc \:\lor\: \ldots \]


(1926c, [PP] 240; 1927a, [PP] 48-49). In absence of a rule of
generalization, one has at most that a universality implies any of its
instances:

 
\[\tag{1.1} \forall x Fx \to Fa \]


and the product becomes infinite, and without a rule of existential
elimination all one has is that an instance implies existence:



\[\tag{1.2} Fa \to \exists xFx \]


The disjunction being infinite, it cannot sum up all the
disjuncts.



Wittgenstein had hinted in the *Tractatus* at a logic-free
equational calculus for arithmetic (6.02-6.031 & 6.241), but
Ramsey thought this to be a "ridiculously narrow view of
mathematics" (1925a, [PP] 180), and wanted the convenience
afforded by quantifiers. He assessed two alternatives to Wittgenstein,
Thoralf Skolem's proposal simply to do away with the
quantifiers, in a paper where he introduced primitive recursive
arithmetic (Skolem 1923 [1967]) and Hermann Weyl's suggestion in
'Uber die neue Grundlagenkrise der Mathematik' (Weyl
1921 [1998]) - the very paper in which he joined Brouwer in his
critique of classical mathematics, - that universality should be
understood in analogy with bank drafts, as 'instructions for
judgements', while existence would be a 'judgment
abstract', classical existence being compared to a treasure map
that does not tell us where to find the treasure (Weyl 1921 [1998],
97-98). These explanations happen to justify (1.1) and (1.2)
above. Weyl claimed further that (1.1) and (1.2) cannot be negated
since one could not, say, survey infinitely many disjuncts, so the law
of excluded middle would not hold.



In two of his last papers, 'Principles of Finitist
Mathematics' and 'The Formal Structure of Intuitionist
Mathematics' ([NP] 197-200), Ramsey sided with Weyl. It
appears that he wanted to obtain arithmetic by adding quantifiers,
with Weyl's rules, to Skolem's primitive recursive
arithmetic. But he did not provide a rule of existential elimination
and rules that he proposed in these papers appear to restrict formulas
with quantifiers to those that can be converted to prenex normal form
(with quantifiers prefixed on the left, followed with a
quantifier-free matrix). In intuitionism, the law of the excluded
middle applies to atomic formulas and, if quantifiers are in prenex
normal form, the theory remains constructive. This is what Ramsey
captured, but the result is not the full intuitionistic arithmetic, as
not all intuitionistic formulas are equivalent to prenex formulas. In
'Principles of Finitist Mathematics', he stated that his
rules provide "all the logical modes allowed by the
finitists" ([NP] 201), but if he meant
'intuitionism', the claim was inaccurate.



Ramsey already discussed Weyl earlier, for example in
'Mathematical Logic' (1926c, [PP] 230 & 233), without
approving him, so one needs to explain why he changed his mind in
1928-29. This may be the result of changes in his underlying
conception of theories from his 1928 manuscript *On Truth*
([OT] 33-34) to 'Theories' (1929). According to his
later view, a 'primary system' of true or false
observational statements is entailed, *via* a
'dictionary', by a 'secondary system' of
hypotheses (1929a, [PP] 112-115). This is a view that has roots
in both the introduction to Heinrich Hertz's *The Principles
of Mechanics* (Hertz 1899) and in chapter VI of Norman
Campbell's *Physics. The Elements* (Campbell 1920), but
Ramsey described, as we shall see, hypotheses not as statements or
propositions, but, adapting Weyl's expression, as 'rules
for judging'.



At a deeper level, Ramsey's change of mind can be traced back to
his having read C. S. Peirce's *Chance, Love and Logic*
(Peirce 1923). In 'Fact and Propositions' (1925) he had
already written: "The essence of pragmatism I take to be this,
that 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" (1927a, [PP] 51). This forms
part of what Nils-Eric Sahlin called 'British Pragmatism'
(Sahlin 1997, 65; Marion 2012), and Cheryl Misak 'Cambridge
Pragmatism' (Misak 2016). This causal theory allowed him to tie
logical form with causal properties, for example when he argued that
disbelieving *p* and believing its negation ! *p*
have the same causal properties (1927a, [PP] 44). (There is a link
worth noticing here with his view on 'Universals' (1925),
since this suggestion allows him to avoid 'negative
facts'.) Ramsey suggested at that stage that a causal theory of
this sort could accommodate the quantifiers as understood (as above)
by Wittgenstein: "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" (1927a, [PP] 49). By 1928-1929, however,
his pragmatism finally led him to abandon this suggestion.



In his review of the *Tractatus*, Ramsey thought that
Wittgenstein's suggestion for eliminating the subject (5.542)
was "an important advance" (1923, 469), but he assumed the
opposite in 'Truth and Probability' (1928). In a passage
where he admitted following Peirce (1926a, [PP] 90 n. 2), he argued
that the human mind "works essentially according to general
rules or habits" (1926a, [PP] 90):



> 
> 
> The mind works by general laws; therefore if it infers \(q\) from
> \(p\), this will generally be because \(q\) is an instance of a
> function \(\phi x\) and \(p\) the corresponding instance of a function
> \(\psi x\) such that the mind would always infer \(\phi x\) from
> \(\psi x\). (1926a, [PP] 91)
> 
> 
> 



And, again in pragmatist manner, he argued that one judges habits in
terms of their success: "given a habit of a certain form, we can
praise or blame it accordingly as the degree of belief it produces is
near or far from the actual proportion in which the habit leads to
truth" (1926a, [PP] 92).



In 'General Propositions and Causality' (1929), Ramsey
followed this with Peirce's idea that beliefs "guide our
actions" (Peirce 1992, 114), describing a belief as a "map
by which we steer" (1929b, [PP] 146). It is for this very reason
that he could not countenance the belief *qua* map being
infinite: "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" (1929b, [PP] 146). He then
introduced the notion of 'variable hypotheticals' as
forming the "system with which the speaker meets the
future" and described them in terms strongly reminiscent of
Weyl:



> 
> 
> Variable hypotheticals are not judgments but rules for judging
> 'If I meet a \(\phi\), I shall regard it as a \(\psi\)'.
> This cannot be *negated* but it can be *disagreed* with
> by one who does not adopt it. (1929b, [PP] 149)
> 
> 
> 
> ... when we assert a causal law we are asserting not a fact, not
> an infinite conjunction, nor a connection of universals, but a
> variable hypothetical which is not strictly a proposition at all, but
> a formula from which we derive propositions. (1929b, [PP], 159)
> 
> 
> 



There is indeed a definite connection here with his growing qualms, of
a 'finitist' nature, about the axiom of infinity:
"So too there may be an infinite totality, but what seems to be
propositions about it are again variable hypotheticals and
'infinite collection' is really nonsense" (1929b,
[PP] 160).



Ramsey's last papers thus show that he was moving towards an
original synthesis of Weyl's intuitionism (or finitism) with
pragmatism (Majer 1989; Majer 1991; Marion 2012; Misak 2016a (chapter
6); Misak 2018; Sahlin 1990, chapter 4). As for the confluence of
ideas, it is possible that Wittgenstein's notion of
'hypothesis' had other sources (Marion 2008), but he also
came to reject his own view of the quantifiers in the
*Tractatus* (for example in his Cambridge lectures
(Wittgenstein [LC], 217 & 219)), and his own language bore strong
resemblance to that of Weyl and Ramsey when he spoke of a hypothesis
as a "law for forming expectations" (Wittgenstein [PR],
SS228).


## 3. Ontology



Ramsey is famous for his denial of the particular-universal
distinction. But his ontological vision was even more radical and
austere than this suggests because Ramsey also denied the existence of
complex entities, including complex universals, facts and
propositions. Ramsey's main arguments for this austere ontology
appear in his two landmark papers "Universals" (1925b) and
"Facts and Propositions" (1927a). At a general level we
can think of Ramsey as endeavoring to bring our views of ontology
clear of the shadows of words (MacBride 2005a, 2018 chapter 10).



At this point in his intellectual development, the mid-1920s, Ramsey
was developing the insights he found in Russell and Wittgenstein. But
Ramsey not only took many of their ideas to a new level of detail and
sophistication but pushed boundaries by showing that they had not gone
far enough. "Universals" provides a luminous example of
Ramsey pushing against boundaries that had held Russell back.
According to Russell, "The influence of language on philosophy
has, I believe, been profound and almost unrecognized. If we are not
to be misled by this influence it is necessary to become conscious of
it and to ask how far it is legitimate" (Russell 1924, 330).
Russell was specifically concerned that the subject-predicate form
common to Indo-European languages had misled Western philosophers,
conditioning them to take for granted a substance-attribute ontology
whilst being blind to the existence of relations. Russell proposed to
remedy the situation by introducing alongside the category of
particulars, a broader category of universals which subsumed both
(monadic) attributes and (polyadic) relations. But, in
"Universals", Ramsey argued that Russell had not gone far
enough, because the particular-universal distinction itself was just a
further shadow cast by language. Ramsey wrote, "nearly all
philosophers, including Mr Russell himself have been misled by
language in a far more far-reaching way than that", i.e. more
far-reaching than being led merely to deny the existence of relations,
because "the whole theory of particulars and universals is due
to mistaking for a fundamental characteristic of reality, what is
merely a characteristic of language" (1925b, 405). But Ramsey
wasn't aiming to replace the particular-universal distinction
with an alternative distinction of his own. Instead he aimed to
inculcate in his reader a studied indifference, which meant
withholding judgment, at least a priori, about the forms or categories
of atomic objects, as Wittgenstein had done in the
*Tractatus*.



Ramsey diagnosed that Russell had been led astray by supposing the
subject-predicate distinction to be a logical distinction. So Ramsey
devoted "Universals" to arguing it isn't logical.
Ramsey recognised there wasn't just one candidate for being the
subject-predicate distinction. So "Universals" is divided
into an examination of several candidates. Ramsey argued they reflect
merely accidental features of our language or subjective needs or
biased interests, so none of them provide a basis for an objective
classification of what there is.



Ramsey begins 'Universals' with what was to became a
celebrated argument of early analytic philosophy (see Sahlin 1990,
192-202 and Simons 1991 for helpful accounts of Ramsey's
main lines of argument in "Universals"). To persuade us
there are serious issues to engage here, Ramsey compared


* (3.1)  Socrates is
wise.


and,


* (3.2)  Wisdom is a
characteristic of Socrates.



Whilst they are different sentences, Ramsey invites us to agree that
they say the same about what's out there even though they differ
in their surface forms. He reflected, 'Now of one these
sentences 'Socrates' is the subject, of the other
'wisdom'; and so which of the two is subject, which
predicate, depends upon what particular sentence we use to express our
proposition, and so had nothing to do with the logical nature of
Socrates or wisdom, but is a matter entirely for language'
(1925b, 404). Ramsey hypothesises that the subject-predicate
distinction isn't a semantic or representational distinction at
all, hence not in the business of marking a division of things, but
rather a distinction which we employ to communicate to conversational
partners the focus of our discussion: "If the centre of our
interest is Socrates we say 'Socrates is wise', if we are
discussing wisdom we may say 'wisdom is a characteristic of
Socrates'; but whichever we say we mean the same
thing".



Ramsey's aim was not to conclusively establish this hypothesis
on the basis that (3.1) and (3.2) are used to make the same
assertion-as indeed many of his critics have supposed. What they
miss is that to be fit for purpose, Ramsey's argument only had
to be strong enough to make us seriously entertain the possibility
that the subject-predicate distinction doesn't serve a
representational function, because that's enough to throw doubt
upon the procedure of classifying things on the basis of the
subject-predicate distinction, hence strong enough to make it evident
that "the question requires a new examination" (1925b,
405).



(Anscombe [1959, 108], Geach [1950, 474-5; 1975, 143-4]
and Dummett [1973, 63-4] have all criticised Ramsey's
argument here on the grounds that whilst a predicate may be negated a
name may not - although see Anscombe's
'Retractation' [1965]. Coming from a different direction,
Simons [1991, 159], Dokic and Engel [2002, 40-1] and Lowe [2004,
307] argue that Ramsey was mistaken to think that ontological issues
could ever be settled in logico-linguistic terms. MacBride [2005a,b]
argues that the former criticisms miss their intended mark because
Ramsey was committed to denying that negative predicates correspond to
the constituents of atomic facts; whereas the latter criticisms miss
their mark because Ramsey was in fact arguing that logical reflection
fails to determine the nature of the atomic facts. Hochberg [2004,
197-9], Mulligan [2000], Simons [1992, 151-2] raise
further difficulties for Ramsey's initial argument that builds
upon the equivalence of (3.1) and (3.2) but MacBride [2018,
207-10] argues that these criticisms rest upon a failure to
appreciate the radical character of Ramsey's challenge.)



Ramsey begins his new examination by considering whether molecular
statements can be analysed into subjects and predicates -
specifically predicates which stand for complex universals. He argues,
*reductio ad absurdum*, that they don't. He assumes that
if two analyses of a proposition invoke different constituents then
they are analyses of different propositions. Absurdity results when
this assumption is combined with the claim that complex predicates are
terms for complex universals. This is because more than one complex
predicate may be isolated in a sentence of the form
'*aRb*' - predicates that may be represented
by '*xRb*', '*aRy*'
and'*xRy*'. If these predicates are terms for
different universals, then the sentence '*aRb*'
expresses no less than three propositions. The propositions are
distinct because they correspond to the three different collections of
constituents: (i) *xRb*, *a*, (ii) *aRx*,
*b*, and (iii) *xRy*, *a*, *b*. But this
is absurd because '*aRb*' says only one thing: that
*aRb*. Ramey concluded, "so the theory of complex
universals is responsible for an incomprehensible trinity, as
senseless as that of theology" (1925b, 406). Ramsey must also
have thought but doesn't spell out, that it's a
consequence of this argument that complex particulars should be
dismissed too - otherwise particulars would be distinguished
from universals in terms of complexity. (For discussion of
Ramsey's argument against complex universals see Oliver 1992;
Mellor 1992; and MacBride 2005a, 86-94; 2018,
210-220.)



Ramsey next looks to see if a logical distinction between subject and
predicate applies to atomic statements. Russell had argued that
predicates can be distinguished from subjects by the fact that the
former exhibit a distinctive incompleteness because they carry the
form of the statements in which occur. Ramsey agreed that it's
natural to feel that an adjective like 'wise' is
incomplete in a manner that 'Socrates' is not. But, Ramsey
argued, predicates only appear to be especially incomplete because as
ordinary language users we overlook relevant data: we routinely
distinguish wide from narrow scope occurrences of predicates but we
don't do the same for subjects even though the same distinction
can be drawn for them. Ramsey cites Russell's theory of
descriptions as the inspiration for the wide-narrow scope distinction,
which Ramsey elaborates in terms of a distinction between more and
less encompassing ranges of propositions in which an expression
figures. Ramsey concludes that to be a substantive (or adjective)
"is not an objective but a subjective property, in the sense
that it depends not indeed on any one mind but on the common elements
in all men's minds and purposes" (1925b, 413).



Since the difference between 'Socrates' and
'wise' turns out to be subjective in this way, it tells us
nothing about the objective classification of the constituents of
atomic facts. Moreover, Ramsey continued, once we are explicit about
the wide and narrow scope occurrences of atomic expressions, there is
no reason to think that any of them are especially incomplete. Ramsey
held that it is only the 'biassed interest' of
mathematicians, i.e. their extensionalism, that make things appear
otherwise (1925b, 416). They neglect to distinguish between complex
predicates which really do require variables for their depiction
(\(xRa \lor xSb\)) and atomic expressions which don't
('\(f\)' in '\(fa\)'), because they are only
interested in the extensional matter whether these expressions are
true or false of the same things. But to avoid 'useless
complication' mathematicians treat the latter terms as requiring
variables for their representation too ('\(fx\)'), hence
producing the appearance that there is some class of atomic terms
which are incomplete because they require variables (1925b,
414-5). But, Ramsey continued, if it were not for the biased
interests of the mathematician, we would allow '\(f\)' to
stand alone, so removing the appearance of an especially incomplete
class of atomic terms.



Having completed his examination, Ramsey concluded that we can know
nothing whatever about the classification of the constituents of
atomic facts. But when he came to reprise his argument a year later in
"Universals and 'the Method of Analysis'"
(1926b), Ramsey adopted the more modest conclusion that whilst we
cannot know *a priori* the forms of the atomic facts we may yet
know them *a posteriori*: "When I wrote my article I was
sure that it was impossible to discover atomic propositions by actual
analysis. Of this I am now very doubtful, and I cannot be sure that
they may not be discovered to be all of one or another of a series of
forms which can be expressed by \(R\_1(x)\), \(R\_2(x,y)\),
\(R\_3(x,y,z)\).... This I admit may be found to be the case, but
no one can as yet be certain what atomic propositions there are, it
cannot be positively asserted; and there is no strong presumption in
its favour, for I think that the argument of my article establishes
that nothing of the sort can be known *a priori*" (1926b,
31) (see Methven 2018 for discussion of the relationship between
Ramsey 1925b and 1926b).



Ramsey's less well-known arguments against the ontology of facts
and propositions appear in the context of Ramsey's discussion of
the nature of truth and belief. In "Facts and
Propositions" (1927a) he argued against the existence of facts
on the grounds that the expression 'the fact that...'
cannot be completed to make either a name or a description of a fact
(1927a, 155-7). Indeed, he claimed, 'the fact
that...' disappears upon paraphrase of the contexts in
which it occurs. As Ramsey put the point (1927a, 159):


* (3.3)  the fact that
\(a\) *has* \(R\) *to* \(b\) exists



is no different from


* (3.4)  \(a\) has \(R\)
to \(b\)



By 'is no different from' (his words) Ramsey meant
something like "semantically equivalent". So his argument
is that (3.3) is just a long-winded way of saying (3.4) and because
(3.4) has no constituent expression that refers to a fact, it follows
that (3.3) has no constituent expression that refers to a fact
either.



We can imagine Ramsey unpacking this argument further. Statements like
(3.3) commit us to the existence of facts if any statements do; they
constitute our best effort to explicitly affirm the existence of
facts. Since even statements like (3.3) don't commit us to
facts, because they're just paraphrases of statements like
(3.4), we cannot have reason to believe in the existence of facts at
all. To transpose a remark of Ramsey's from another context, if
you can't say that facts exist, you can't whistle it
either. We could call this Ramsey's 'redundancy theory of
facts', emphasising another parallel, this time with his famous
redundancy theory of truth (according to which '\(a\) has \)(R\)
to \(b\) is true' says the same as '\(a\) has \(R\) to
\(b\)').



Ramsey renounced propositions conceived as the single objects of
judgment, which may be true or false, for the reasons that had already
led Russell to give up propositions so conceived, namely 'the
incredibility of the existence of objects as "that Caesar died
in his bed" which could be described as objective falsehoods,
and the mysterious nature of the difference, on this theory, between
truth and falsehood' (1927a, 153-4). Ramsey, like Russell,
didn't spell this out, but we can think of Ramsey as pointing,
*inter alia*, to the difficulty of explaining the unity of a
proposition \(aRb\) in terms of \(R\)'s holdings between \(a\)
and \(b\), which is difficult to understand if the proposition is
false since then \(R\) doesn't hold between \(a\) and \(b\).
Ramsey continued to find objective falsehoods incredible in his later
manuscript 'On Truth': 'it is only the hardiest
verbalists who can persuade themselves 'that the earth is
flat' is the name of something real' ([OT] 85).



Ramsey concluded with Russell, 'that a judgment had no single
object, but is a multiple relation of the mind or mental factors to
many objects, those, namely, which we should ordinarily call
constituents of the propositions judged' (1927a, 154). So we can
think of Ramsey offering two arguments for renouncing propositions
conceived as single objects: a negative argument, that they're
incredible and mysterious, but also a positive argument arguing that
propositions so conceived are dispensable based upon Ramsey's
own views about truth and belief which don't require them. In
'Facts and Propositions' Ramsey put forward a version of
Wittgenstein's picture theory according to which a subject
'will believe that \(aRb\) by having names for \(a\), \(R\), and
\(b\) connected in his mind and accompanied by a feeling of
belief' (1927a, 160) but Ramsey was already pushing in more
radical, pragmatist direction which emphasised the significance of
action for understanding the notion of belief, hence shifting
attention further away from an ontology of facts and propositions.
(See the section on Belief and Truth.)



It is Ramsey's ontological views on the particular-universal
distinction that have attracted the most attention. On the historical
front, Ruth Barcan Marcus (1993) argues that Ramsey's
"Universals" bore a significant influence on
Russell's subsequent intellectual development; see MacBride 2004
and Hochberg 2004 for further discussion. MacBride (2018) puts
Ramsey's "Universals" in historical context, arguing
that "Universals" should be interpreted as the final stage
in an extended conversation about particulars and universals,
involving G.E. Moore, Russell, Wittgenstein and Whitehead. Misak
(2020) explores the connection between "Universals" and
Ramsey's later pragmatist writings. Braithwaite (1926) and
Strawson (1959, chapter 5) argued, contra Ramsey, that the
particular-universal distinction may be understood in spatio-temporal
terms. See MacBride 2005c and Lowe 2006 (chapter 7) for further
discussion of Ramsey's views on universals from a contemporary
point of view.


## 4. Belief and Truth


### 4.1 Belief and Assertion



Ramsey's primary analysis of belief and assertion appears in his
papers 'Truth and Probability' (1926a), 'Facts and
Propositions' (1927a) and 'The Nature of Truth'
(1929c), with some significant remarks on general beliefs made in
'Universals of Law and of Fact' (1928b) and 'General
Propositions and Causality' (1929b).



In 'Truth and Probability' Ramsey presents his view on the
nature of partial belief, and this paper is considered the inaugural
work of the family of approaches to probability known as
*subjectivism*. Since the present section focuses on his
semantic analysis of belief and assertion, and not on subjective
probabilities, the analysis of partial belief developed in
'Truth and Probability' will be left aside.



Ramsey was a pragmatist about belief (Hookway 2005, 182ff; Dokic and
Engel 2002, 2, 8, 55), although not about truth. He explicitly
declares his pragmatism in 'Truth and Probability' (1926a)
and in 'Fact and Propositions' (1927a). In 'Truth
and Probability', he explains his approach to rationality as
'a kind of pragmatism: we judge mental habits by whether they
work' (1926a, [PP] 93-94). In 'Facts and
Propositions', in contrast, the topic is the notion of meaning:
'The essence of pragmatism I take to be this, that the meaning
of a sentence is to be determined by reference to the actions to which
asserting it would lead' (1927a, [PP] 51). He attributes his
pragmatism about meaning to the influence of Bertrand Russell. In
fact, in *The Analysis of Mind* (1921), Russell explained the
intentional properties of words through their causal history. As Acero
(2005, 16) stresses, Russell at that time identified the meaning of a
word with its use, and it seems to be this aspect of Russell's
philosophy of mind that Ramsey acknowledges as the source of his
pragmatism about meaning in 'Facts and Propositions' (cf.
Dokic and Engel 2002, 22). Sahlin (1990, 70) and Hookway (2005,
185-187) point out that Peirce must also have had an effect on
Ramsey's pragmatist views since many themes in Peirce's
early works are detectable in Ramsey's writings (Hookway 2005,
185-6). More recently, Misak (2016a) has tracked Peirce's
influence on Ramsey in full detail, using the entries on Peirce that
Ramsey inserted into his diary. It is now clear that Ramsey read
Peirce extensively and that the American philosopher deeply influenced
Ramsey's views on meaning, truth and probability (Misak 2016a,
157).



As a pragmatist, Ramsey individuates beliefs by attending to their
causes and effects. Thus, attitudes towards two sentences prompted by
the same causes and producing the same effects express the same
belief, no matter the linguistic constitution of the sentences used to
express it. 'Belief', Ramsey acknowledges, is an ambiguous
term (1927a, [PP] 40). Both human beings and non-linguistic animals
can be said to possess beliefs. In non-linguistic animals, beliefs are
dispositions to act; in linguistic animals, beliefs are mental states
with a particular kind of content that involves entertaining words.
Chickens that avoid eating a particular kind of caterpillar, due to
systematically connecting it with disagreeable past experiences,
*believe* that caterpillars of this kind are poisonous. This is
Ramsey's causal theory of belief, which Dokic and Engel (2002,
24) extend to beliefs of any kind and attribute to Peirce's
influence on Ramsey. The beliefs of linguistic animals are
specifically defined in terms of attitudes towards complexes of signs
with logical structure (Hookway 2005, 185). Both kinds of belief,
linguistic and non-linguistic, are analysed in pragmatist terms, in
which 'the relations between belief and action do take centre
stage in some of [Ramsey's] most influential writings on this
topic' (Hookway 2005, 186).



Even if the main targets of Ramsey's analysis are those beliefs
expressed in words and 'consciously asserted and denied'
(1927a, [PP] 40), it is still illuminating to consider his treatment
of animal beliefs, in which the pragmatist connection between belief
and action is yet more perspicuous.



Ramsey distinguishes mental and objective factors in his analysis of
belief. In the case of the chicken, the simplest one, the mental
factor is part of the animal's external behaviour (1927a, 159),
i.e. its behaviour of refraining from eating caterpillars on the basis
of disagreeable past experiences. The objective factor in this case is
a conglomerate of relevant aspects of the world, i.e. the caterpillar
and the property of being poisonous, with which the mental factor
stands in some kind of relation. In beliefs expressed in words, on the
other hand, the mental factor is partly constituted by those words or
symbols 'spoken aloud or merely imagined, connected
together'. Beliefs expressed in words can be classified into
three different categories: (i) those whose content is represented by
predicates or relations together with their arguments, i.e. by atomic
sentences with simple structures such as '*aRb*',
(ii) those whose content includes logical connectives
('not', 'and', 'or'), and (iii)
general beliefs. In (i), the simplest linguistic case, an agent having
a belief that *aRb* means that she connects the words
'*a*', '*R*' and
'*b*' in her mind in a certain way. A further
ingredient of the mental factor in linguistic creatures is a sort of
feeling that accompanies the words in one's mind. By
'feeling' Ramsey does not mean a private sentiment, but
rather some attitude of the agent that might be expressed in less
subjective terms, such as '"specific quality" or
"act of assertion" and "act of denial"'
(loc. cit., n. 2).



Purely private feelings towards contents, Ramsey argues, play no role
in the analysis of belief and are useless for measuring degrees of
belief, as he explains in 'Truth and Probability':



> 
> As soon as we regard belief quantitatively, this seems to me the only
> view we can take of it. It could well be held that the difference
> between believing and not believing lies in the presence or absence of
> introspective feelings. But when we seek to know what is the
> difference between believing more firmly and believing less firmly, we
> can no longer regard it as consisting in having more or less of
> certain observable feelings; at least I personally cannot recognize
> any such feelings. (1926a, [PP] 65).
> 



There are basically two reasons why he takes private feelings to be
irrelevant for the analysis of belief. The first of these concerns the
difficulty of assigning objective measurements, but undoubtedly the
definitive reason rests on the fact that our most unshakable beliefs
come without any particular sensation (Dokic and Engel 2002, 7). As
Ramsey puts it: '[N]o one feels strongly about things he takes
for granted' (loc. cit.). At this point, the similarities with
the view that Wittgenstein held in his pragmatist period are evident
(for the mutual influence between Ramsey and Wittgenstein, see SS2
above).



Linguistic beliefs of the second category, i.e. those that also
include logical terms such as 'not', 'and' and
'or' are more complex. Concerning them, Ramsey heavily
relies on the Tractarian view of logical constants. Like Wittgenstein,
Ramsey thinks that logical words are not names, i.e. they do not
represent. The meanings of 'not' and 'or' are
determined by the agents' attitudes towards the sentences that
include them. Believing that not-*p* and disbelieving that
*p* are 'equivalent occurrences' (1927a, [PP] 43),
i.e. they express the same attitude because their practical effects
are indistinguishable (1927a, [PP] 44; Dokic and Engel 2002, 24).



A specific difficulty in analyzing beliefs of this second category
derives from the need to identify attitudes held towards several
atomic sentences at the same time, while when atomic sentences are
considered one by one only two mutually exclusive feelings must be
taken into account: full belief and full disbelief. The
'intermediate attitudes of partial belief', which he
proposes to 'leave to the theory of probability' (1927a,
[PP] 45), are discussed in 'Truth and Probability'. To
deal with attitudes held towards various atomic sentences at once,
Ramsey borrows from the former Wittgenstein the notion of
truth-possibility. When *n* atomic sentences are at play, there
are 2*n* possible combinations of fully believing
and fully disbelieving. Thus, 'the meaning of a sentence is
agreement and disagreement with such and such
truth-possibilities' (op. cit., 46), and a proposition is
'a disjunction of the truth-possibilities with which [it]
agrees' (op. cit., 47).



The third category of linguistic beliefs is general beliefs, i.e.
those whose expression involves the words 'all' and
'some'. Concerning general propositions, Ramsey
successively held two different views. The first of these, explained
in 'Facts and Propositions' (1927a) and 'Universals
of Law and Fact' (1928b), is essentially the Tractarian view
that quantified sentences are conjunctions and disjunctions of atomic
propositions (Sahlin 1990, 76; Dokic and Engel 2002, 30). The only
difference he found between conjunctions and disjunctions, on the one
hand, and universal and particular sentences, on the other, was that
the arguments of genuine conjunctions and disjunctions can be
enumerated, whereas the arguments of quantified sentences are given by
means of propositional functions. These propositional functions show
their usefulness in those cases in which either we do not know their
particular instances, or there are so many of them that enumerating
them is impossible. When such enumeration is logically possible we
have universals of fact; when it is not, we have universals of law.
One might think that the details that distinguish conjunctions,
disjunctions, and universals of fact from genuine general propositions
would vanish from the perspective of an omniscient subject who could
entertain the totality of all application instances. Ramsey
nevertheless rejects this idea because the role of general
propositions is pragmatic, as an aid in the task of systematizing
'our knowledge as a deductive system' (1928b, [PP] 143),
rather than semantic.



The second view appears in 'General Propositions and
Causality' (1929b), and this is a fully pragmatist account
according to which general beliefs do not have a truth-value (Dokic
and Engel 2002, 30; Sahlin 1990, 78). Here Ramsey presents his famous
account of atomic beliefs as maps: 'A belief of the primary
sort', i.e. one without quantifiers, 'is a map of
neighbouring space by which we steer' (1929b, [PP] 146). If
beliefs are maps of the neighbourhood, general beliefs -
understood as codifying general inferences - should be maps of
infinite extension, except that an infinite map is 'no longer a
map; we cannot take it in or steer by it. Our journey is over before
we need its remoter parts' (loc. cit.). Quantified sentences are
thus '*rules for action* or *attitudes*'
(Dokic and Engel 2002, 31), and express 'an inference we are at
any time prepared to make' (1929b, [PP] 146). Dokic and Engel
(2002) also stress the similarities between Ramsey's account of
general beliefs and Wittgenstein's view of laws as rules of
inference.



General beliefs with general contents, 'variable
hypotheticals' as Ramsey calls them, are not propositions at
all. With respect to the role of general beliefs, Ramsey is prepared
to explicitly endorse a central thesis of any pragmatist approach to
meaning: '[m]any sentences express cognitive attitudes without
being propositions' (1929b, [PP] 147). The cognitive attitude
they express is still belief, which in the case of general beliefs
Ramsey characterizes as composed of (i) a general enunciation, i.e.
the endorsement of a sentence that expresses it - 'all men
are mortal', for instance - and (ii) 'a habit of
singular belief' (1929b, [PP] 148). These two steps are
connected by the psychological law that gives the meaning of
'all' (1929b, [PP] 149).


### 4.2 Truth



The posthumously-published paper 'The Nature of Truth'
(1929c) is specifically devoted to the analysis of truth, and there
Ramsey develops the few lines about truth included in 'Facts and
Propositions' (1927a). Even if that earlier paper mostly deals
with belief, the brief remarks it contains on truth are among
Ramsey's most influential contributions to philosophy. What he
has to say in there about truth is stated almost as 'an obvious
fact' (Sahlin 1990, 56), hence his famous remark that
'there is really no separate problem of truth but merely a
linguistic muddle' (1927a, [PP] 38).



In 'The Nature of Truth' Ramsey explains that beliefs, in
the sense that is relevant here, are necessarily beliefs that
'something or other is so-and-so' (1929c, 7ff). He calls
this 'so-and-so' the 'propositional reference'
(loc. cit.) and uses it as the criterion of propositional
individuation, in the sense that if 'two men both believe that
the earth is flat we say they have the same belief' (loc. cit.).
Individuating beliefs by their propositional references might seem at
odds with the pragmatist method given in terms of causes and effects
that Ramsey used in 'Facts and Propositions'. But this
superficial tension is defused if propositional references, which are
not linguistic but conceptual, are individuated by their consequences
(which include actions). Other mental states - doubts, desires,
fears, regrets, etc. - have propositional references too, but
only beliefs have 'affirmative or assertive character'
(1929c, 8). Ramsey proposes to extend the ordinary meaning of the term
'belief' to cover mental states with any degree of
assertive force - from mere conjecture to full certainty -
and propositional content. The contents of beliefs, in this extended
sense, are the bearers of truth: 'Truth and falsity are ascribed
primarily to propositions', he says (1927a, [PP] 38). Later, he
substitutes 'proposition' with 'propositional
reference' (1929c, 7), but this rewording owes more to
philosophical precaution than any substantial change in view.



Truth adds nothing substantive to the content of beliefs, according to
Ramsey. He claims that the meaning of truth is 'perfectly
obvious, that anyone can *see* what it is and that difficulty
only arises when we try to *say* what it is, because it is
something which ordinary language is rather ill-adapted to
express' (1929c, 9). Because of this, Ramsey's view is
generally interpreted as a redundancy theory of truth (Sahlin 1990,
56-59; Dokic and Engel 2002, 18; Hookway 2005, 56-58), a
kind of minimalism or deflationism about truth similar to the views
put forward by Aristotle and Frege (cf. Sahlin 1990, 56), and later on
by Tarski, Wittgenstein, Quine (cf. Hookway 2005, 57), Field and
Horwich (cf. Koslow 2005, 106).



Nevertheless, there is more to Ramsey's view than what the label
'redundancy' suggests. The bearers of truth, i.e.
propositions, can be referred to directly, as in 'that Caesar
was murdered', or identified by description, as in 'what
she said'. Correspondingly, truth can be attached to the direct
expression of a proposition, as in 'It is true that Caesar was
murdered', or to a description, as in 'what she says is
true'. When truth applies to expressions of the first kind,
Ramsey characterizes its role as eliminable: 'it is evident that
"It is true that Caesar was murdered" means no more than
that Caesar was murdered, and "It is false that Caesar was
murdered" means that Caesar was not murdered' (1927a,
[PP], 38-9). But when the truth predicate applies to expressions
of the second kind, Ramsey explicitly claims that it cannot be
dispensed with: 'In the second case in which the proposition is
described and not given explicitly we have perhaps more of a problem,
for we get statements from which we cannot in ordinary language
eliminate the words "true" and "false"'
(loc. cit.).



Let us consider the following three examples of truth ascriptions in
English:


* (4.1)  All Joan's
thoughts are true.
* (4.2)  What follows
from a true premise is true.
* (4.3)  Evolutionism is
true.



To *say* what truth consists in, and deal with any kind of
truth ascription, Ramsey enriches ordinary English into English\* by
the addition of single propositional variables -
'*p*', '*q*', etc., ranging over
the set of declarative sentences - and propositional
quantifiers. In English\* the truth predicate vanishes and its role is
taken over by propositional variables and quantifiers. The semi-formal
sentences (4.1\*)-(4.3\*) are translations of (4.1)-(4.3)
into English\*:


* (4.1\*)  For all
*p*, if Joan thinks that *p*, *p*.
* (4.2\*)  For all
*p* and *q*, if *q* follows from *p*, and
*p*, then *q*.
* (4.3\*)  For all
*p*, if *p* follows from the theory of evolution, then
*p*.



Note that the translation shows that the truth apparatus is redundant
in English\*, but *not* that it is redundant in English.



The definition of truth Ramsey provides in 'The Nature of
Truth' is: 'A belief is true if it is a belief that
*p*, and *p*' (1929c, [OT] 9), where the first
instance of the propositional variable '*p*'
represents the propositional reference that individuates the belief at
issue (1929c, [OT] 7), and the second instance represents the
objective factor. This definition is expressed in English\*, since
ordinary English does not have the necessary resources:



As we claim to have defined truth we ought to be able to substitute
our definition for the word "true" wherever it occurs. But
the difficulty we have mentioned renders this impossible in ordinary
language which treats what should really be called
*pro-sentences* as if they were *pro-nouns*. [...]
"that" and "what" even when functioning as
short for sentences always require to be supplied with a verb: this
verb is often "is true" and this peculiarity of language
gives rise to artificial problems as to the nature of truth, which
disappear at once when they are expressed in logical symbolism, in
which we can render "what he believed is true" by
"if *p* was what he believed, *p*" (1929c,
[OT] 10, his emphasis).



This text is possibly the first unequivocal expression of a
prosentential theory of truth (Dokic and Engel 2002, 19;
Frapolli 2005a, 113-138; 2013, 61ff).



Ramsey (1929c, [OT] 9) anticipated a possible reaction to his use of
propositional variables and to the translation of (4.1)-(4.3) as
(4.1\*)-(4.3\*). The objection is the following:
(4.1\*)-(4.3\*) cannot be English\* counterparts of English
sentences because they are not well-formed. Free-standing instances,
the objection goes, of the variables '*p*' and
'*q*' cannot fill the argument-slots of
truth-functions, such as conjunction and material implication, which
need complete sentences as arguments. Free-standing uses of the
variables in 'and *p*' and 'then
*q*' would lack a verb to complete these clauses into
genuine instances of the grammatical category of sentences.
Ramsey's reply insists on the sentential status of the variables
'*p*' and '*q*', which stand for
items that already contain a verb and express a truth-bearer (loc.
cit.).



What Ramsey did not anticipate was the classification of his view
either as a redundancy or as a pro-sentential theory. On the contrary,
he thought that it would be interpreted as a correspondence theory of
truth, and although he did not accept the metaphysical commitments of
the many versions of correspondentism, he did not reject the label
either. In fact, he acknowledges that truth is objective
correspondence to the facts (see for instance 1929c, [OT] 11; 1929d,
36). Dokic and Engel (2002, 25) point out that Ramsey was a realist
about truth conditions and also about truth, but this realism was
modulated by his pragmatism: 'his kind of realism is distinct
from the kind of realism which underlies a full-blown correspondence
theory of truth. It is a form of pragmatic realism, which associates
truth to beliefs-habits and to actions.' Thus, Ramsey's
take on truth has a pragmatist background, but it is not a pragmatist
view of truth. It is a kind of sophisticated redundancy theory that is
realist about truth conditions, and also includes some of the
technical apparatus that prosentential theories developed later on
(see Williams 1976, Grover 1992, Brandom 1994, Frapolli
2013).


## 5. Conditionals



In "General Propositions and Causality" (1929b) Ramsey
made some remarks about conditional judgements, which have influenced
thinking about conditionals since the 1960s. Most famous is this
footnote:



> 
> If two people are arguing 'If \(p\), will \(q\)?' and are
> both in doubt about \(p\), they are adding \(p\) hypothetically to
> their stock of knowledge and arguing on that basis about \(q\);
> ... they are fixing their degrees of belief in \(q\) given \(p\).
> (1929b, [FM] 247)
> 



To understand what Ramsey means by "degree of belief in \(q\)
given \(p\)" we need to turn to his earlier essay "Truth
and Probability" (1926a), where he develops an interpretation of
probability as the logic of partial belief ([FM] 157). (Note, Ramsey
did not take this to be the only notion of probability: "The
conclusions we shall come to as to the meaning of probability in Logic
must not ... be taken as prejudging its meaning in physics"
(ibid.). The editor adds that a final chapter, on probability in
science, was designed but not written.) Here he introduces a
"very useful new idea - 'the degree of belief in
\(q\) given \(p\)'" (1926a, [FM] 180). One of his basic
laws is:



> 
> Degree of belief in (\(p\) and \(q\)) = degree of belief in \(p\)
> x degree of belief in \(q\) given \(p\) (1926a, [FM] 181)
> 



As a law of probability, this was not new. It was in use since the
eighteenth century. It is needed to answer a compulsory question: how
likely is it that two events will happen? Answer: it is the
probability of the first, multiplied by the probability of the second
on the supposition that the first happens. What was novel in
Ramsey's essay was the interpretation of probability as degree
of belief; and what was novel in the 1929 paper (1929b) was the
linking of "degree of belief in \(q\) given \(p\)" with
our ordinary, typically uncertain, conditional judgements.



Explaining "degree of belief in \(q\) given \(p\)", Ramsey
says "It roughly expresses the odds at which [the subject] would
now bet on \(q\), the bet only to be valid if \(p\) is true"
(1926a, [FM] 180). Ramsey was aware that the link between degrees of
belief and betting behaviour is imperfect (1926a, [FM] 170). Here the
message is that a *conditional* degree of belief is to a
*conditional* bet, as an unconditional degree of belief is to
an unconditional bet. For a bet on \(q\), you ask yourself, which is
more likely and by how much, \(q\) or \(\neg q\)? For a bet on \(q\)
given \(p\), you ask yourself, which is more likely, and by how much,
\(p \:\&\: q\), or \(p \:\&\: \neg q\)? If you judge \(p
\:\&\: q\) much more likely than \(p \:\&\: \neg q\), you have
a high degree of belief in \(q\) given \(p\). Equivalently, under the
hypothesis or supposition that \(p\), you judge it to be likely that
\(q\).



Ramsey then warns against some possible misunderstandings of
"degree of belief in \(q\) given \(p\)", which will be
discussed below.



The main business of "General Propositions and Causality"
is to discuss the nature of causal generalisations, also called
variable hypotheticals - and naturally this leads him to discuss
their singular conditional instances. He argues that "variable
hypotheticals are not judgements but rules for judging 'If I
meet a \(\phi\) I shall treat it as a \(\psi\)'. This cannot be
*negated*, but it can be *disagreed* with by one who
does not adopt it" (1929b, [FM] 241, italics original). There is
emphasis throughout on the idea that these conditional judgements are
not propositions. After an example in which \(A\) says "If I eat
the pie I will get a stomach-ache" and \(B\) disagrees, Ramsey
writes:



> 
> Many sentences express cognitive attitudes without being propositions;
> and the difference between saying yes and no to them is not the
> difference between saying yes or no to a proposition. This is even
> true of the ordinary hypothetical (as can be seen from the last
> example, it asserts something for the case in which the [antecedent]
> is true: we apply the Law of Excluded Middle not to the whole thing
> but to the [consequent] only). (1929b, [FM] 239-240)
> 



Later, with the same example, "it is not that he believes [a
proposition] \(p\), we \(\overline{p}\); but he has a different degree
of belief in \(q\) given \(p\) from ours; and we can obviously try to
convert him to our view" (1929b, [FM] 247, followed by the
famous footnote). Here is another passage:



> 
> 
> When we deliberate about a possible action, we ask ourselves what will
> happen if we do this or that. If we give a definite answer of the form
> 'If I do \(p\), \(q\) will result', this can be properly
> regarded as a material implication or disjunction 'Either not
> \(p\), or \(q\)'. But it differs, of course from any ordinary
> disjunction in that one of its members is not something of which we
> are trying to discover the truth, but something within our power to
> make true or false. ...
> 
> 
> 
> Besides definite answers 'If \(p\), \(q\) will result', we
> often get ones 'If \(p\), \(q\) might result' or
> '\(q\) would probably result'. Here the degree of
> probability is not a degree of belief in 'Not-\(p\) or
> \(q\)', but a degree of belief in \(q\) given \(p\), which it is
> evidently possible to have without a definite degree of belief in
> \(p\), \(p\) not being an intellectual problem. And our conduct is
> largely determined by these degrees of hypothetical belief. (1929b,
> [FM] 246)
> 
> 
> 



Two observations are in order: first, in the case of certainty that if
\(A\), \(B\), provided that \(A\) does not have probability 0, degree
of belief in the material conditional and the conditional degree of
belief in \(B\) given \(A\) coincide. Writing "\(\bel\)"
for degree of belief: if \(\bel(A \amp \neg B\)) = 0 but \(\bel(A)
\neq 0\), both \(\bel(A \supset B) = 1\) and \(\bel(B \given A) = 1.\)
The only other case in which they coincide is if \(\bel(A) = 1.\)
Otherwise, \(\bel(A \supset B) \gt \bel(B \given A),\) radically so
when \(\bel(B \given A)\) is low but \(\bel(\neg A)\) is high. Second,
it is clear that Ramsey does not reductively define conditional degree
of belief in terms of the ratio: \(\bel(A \amp B)/\bel(A),\) for the
conditional degree of belief can be assessed even without any specific
value for \(\bel(A).\) Indeed, in the fundamental law stated above,
the right-hand side is much more use as a route to the probability of
a conjunction, rather than the other way round.



Ramsey does not treat counterfactual conditionals in terms of
conditional degrees of belief. The famous footnote continues:



> 
> If \(p\) turns out false, these degrees of belief are rendered
> *void*. If either party believes !\(p\) for certain, the
> question ceases to mean anything to him except as a question about
> what follows from certain laws or hypotheses.
> 



That is, a degree of belief in \(q\) given \(p\) exists only when the
degree of belief in \(p\) is non-zero. Ramsey does make a few remarks
about counterfactuals, noticing the 'subtle syntactic
variations'. For instance, "If he was there, he must have
voted for it (for it was passed unanimously), but if he had been
there, he would have voted against it (such being his nature)"
(1929b, [FM] 249). (Here we have an early precursor of the
Oswald-Kennedy pair: if Oswald didn't do it, someone else did;
but if Oswald hadn't done it, no one else would have; see Adams
1970.) There are a few remarks about counterfactuals in Ramsey's
paper but no settled view of them is presented.



Bruno de Finetti (1937, based on lectures of 1935), shortly after
Ramsey and independently, also developed an account of probability as
degree of belief. (Unlike Ramsey, de Finetti argued that this was the
only kind of probability.) In a separate lecture (1936) he gave a
theory of "conditional events" or "tri-events"
as he called them. Like Ramsey, he provided an account of conditional
judgements in terms of conditional probabilities, and gave a
three-valued account of such a conditional "if \(p\),
\(q\)": true if \(p \:\&\: q\), false if \(p \:\&\: \neg
q\), void if \(\neg p\). He also connects this with conditional bets:
won if \(p \:\&\: q\), lost if \(p \:\&\: \neg q\), called off
if \(\neg p\). He gives three-valued truth tables for the other
connectives when applied to conditionals, and claims that the
probability of a "tri-event" is the probability of its
truth given that it has a truth value - which, in the case of
"if \(p\), \(q\)" is just the probability of \(q\) given
\(p\). The results for more complex sentences are controversial, but
there has been a fair amount of work in this tradition, much of it
summarised by Peter Milne (1997). It is interesting that this second
exposition of probability as degree of belief also connected
conditional degree of belief to ordinary conditional judgements, and
also recognised that the latter could not then be treated as ordinary
propositions.



Ernest Adams wrote two papers (1965, 1966), and later a book (1975)
developing a logic for conditionals assessed as conditional
probabilities. Like Ramsey, he was sceptical of the idea of
conditionals as propositions with truth conditions, and developed a
logic based on probability. He showed that valid arguments have an
interesting probability-preserving property: the uncertainty (i.e. 1
[?] the probability) of the conclusion cannot exceed the sum of
the uncertainties of the premises. He applied this criterion to
arguments with conditionals, and an interesting logic emerged. The
paradoxes of material implication are invalid. Specifically, with
material implication, \(\neg A\) guarantees the truth of \(A \to B\).
But here, the probability of \(\neg A\) can be high while the
probability of \(B\) given \(A\) is low. (\(B\) given \(A\) is the
analog of \(A \to B\).) More surprisingly, there are counterexamples
to transitivity, contraposition, and strengthening of the antecedent
(see Section 3.2 of the entry on
 indicative conditionals,
 for examples and more details).



Robert Stalnaker's first paper on the subject, "A Theory
of Conditionals", appeared in 1968. Stalnaker refers to Ramsey.
He also mentions Adams (1966) in a footnote, and indicates that he
will devote another paper to the probabilistic aspects of the theory.
This is his "Probability and Conditionals" (1970), and its
abstract announces that "conditional propositions are introduced
as propositions whose absolute probability is equal to the conditional
probability of the consequent given the antecedent". Then David
Lewis (1976) proved that there is no proposition the probability of
whose truth can be systematically equated with the conditional
probability of \(B\) given \(A\) - a result which Ramsey and de
Finetti appear to have foreseen.



Probability is not discussed in the 1968 paper, but Adams's
logic is in the background as a guide: it turns out that
Stalnaker's logic coincides with Adams over their common domain
(Adams puts aside sentences in which conditionals are embedded in
other connectives). Thus, for example, Stalnaker also has the
counterexamples to transitivity, contraposition and strengthening of
the antecedent (1968, 48-9).



Stalnaker puts Ramsey's suggestion thus: "add the
antecedent (hypothetically) to your stock of knowledge (or beliefs),
and then consider whether the consequent is true. Your belief about
the conditional should be the same as your hypothetical belief, under
this condition, in the consequent" (1968, 43).



Stalnaker was interested in a theory which covered both indicative
conditionals and counterfactual conditionals. He held that something
similar happened with the latter:



> 
> 
> First, add the antecedent (hypothetically) to your stock of beliefs;
> second, make whatever adjustments are required to maintain consistency
> (without modifying the hypothetical belief in the antecedent);
> finally, consider whether the consequent is then true.
> 
> 
> 
> ... Now we have found an answer to the question, "How do we
> decide whether or not to believe a conditional statement?" the
> problem is to make the transition from belief conditions to truth
> conditions; that is, to find a set of truth conditions for statements
> having conditional form which explains why we use the method we do use
> to evaluate them. The concept of a *possible world* is just
> what we need to make this transition, since a possible world is the
> analogue of a stock of hypothetical beliefs. The following set of
> truth conditions, using this notion, is a first approximation to the
> account that I shall propose:
> 
> 
> 
> Consider a possible world in which \(A\) is true, and which otherwise
> differs minimally from the actual world. "*If A, then
> B*" *is true* (*false*) *just in case B is
> true* (*false*) *in that possible world*.
> (44-45, italics original)
> 
> 
> 



And close-possible-world semantics for conditionals was born.



A consequence of Stalnaker's setting the probabilistic
considerations aside in his influential 1968 paper is that
Ramsey's idea came to be taken differently: it was applied in
the context of full, all-or-nothing belief; it was applied to
counterfactuals as well as indicatives; and consequently, there was
much emphasis on belief *revision*. The phrase "the
Ramsey Test" was coined by William Harper (1975, 1976), and it
caught on. He explains it thus: "Accept 'if \(A\),
\(B\)' if and only if the minimum revision of your system of
beliefs needed to accept \(A\) also requires accepting \(B\)"
(1975, 245; 1976, 118). Peter Gardenfors (1986, 81) says the
same. Sven Ove Hansson (1992) says "In modern studies of belief
dynamics, the Ramsey test is taken to 'presume ... some
method of revising states of belief' [quoting Gardenfors].
Thus 'If \(p\) then \(q\)' is taken to be believed if and
only if \(q\) would be believed in the belief state that would result
from revising the present belief state by \(p\)" (1992, 522).
Stephen Read says "Ramsey's idea was simple, but
appealing. One should believe a conditional 'If \(A\),
\(B\)' if one would come to believe \(B\) if one were to add
\(A\) to one's stock of beliefs" (1995, 47).



This emphasis on belief *revision* is not found in
Ramsey's work. Indeed, he distances himself from that idea. On
the page where he introduces "degree of belief in \(q\) given
\(p\)", Ramsey writes:



> 
> This does not mean the degree of belief which the subject would have
> in \(q\) is he knew \(p\), or that which he ought to have. ...
> [And again] This is not the same as the degree he would believe \(q\),
> if he believed \(p\) for certain; for knowledge of \(p\) might for
> psychological reasons profoundly alter his whole system of beliefs.
> (1926a, [FM] 180)
> 



Thus, for Ramsey conditional degree of belief is a synchronic notion,
not a diachronic one. It is a feature of one's present
belief-state, not to be defined in terms of how one's
belief-state would change on acquiring more information.



Why did Ramsey issue these warnings? Here is one example which may be
what he had in mind. Mary is an atheist. However, if she were to learn
that she had a fatal disease, she would acquire a belief in God. But
she doesn't now believe "If I get a fatal disease, there
is a God"! Another example: I am about to pick a card and look
at it. I know there are three kings in the pile, two red and one
black. My conditional degree of belief that I pick a red card, given
that I pick a king, is 2/3. That determines the odds on which I would
bet on Red given King. But if I do pick a king, I won't have a
degree of belief 2/3 that it is red. I am not colour blind. Then there
examples like Lewis's "If Reagan was in the pay of the
KGB, we'll never find out". This is something he believes.
But if he were to learn the antecedent, he would not then believe the
consequent! As Lewis says, "It might be no easy thing to judge
what would be learned if [\(A\)] were learned, in view of the variety
of ways that something might be learned" (1986, 156). He
concludes that a conditional degree of belief is a good but fallible
guide to the degree of belief you would have in \(B\) if you learned
\(A\). Edgington's (1995) reply to Read, SS1, discusses the
disparity between what Ramsey himself said, and much that goes under
the name "the Ramsey Test". In the famous footnote, Ramsey
speaks of people "adding \(p\) *hypothetically* to their
stock of knowledge" - that is, supposing that \(p\). The
consequences of doing so for one's conditional beliefs, is not
invariably equivalent to the consequences of coming to believe \(p\),
for one's unconditional beliefs.



Many philosophers have followed the path along which Ramsey took the
first steps, and Adams developed - of understanding our
typically uncertain conditional judgements in terms of conditional
degrees of belief (see Edgington 2014 for more details and
references).


## 6. Partial Belief and Subjective Probability



In 'Truth and Probability' (1926a), Ramsey sets out an
influential account of the nature, measurement, and norms of partial
belief. The essay is a pioneering work on *subjectivist*
interpretations of probability (also known as *personalist*
interpretations; see the entry on
 interpretations of probability).
 According to subjectivism, probabilities can be interpreted as
numerical representations of an individual's subjective degrees
of confidence. First written in 1926 and still incomplete at the time
of its publication in 1931 (a year after Ramsey's death),
'Truth and Probability' was not very widely discussed
until after the publication of Savage's *The Foundations of
Statistics* (1954), at which time subjectivism came to prominence.
Many of the key ideas and arguments original to the essay have
reappeared in later foundational works in the subjectivist tradition
(e.g., de Finetti 1937, Savage 1954, Jeffrey 1965). This section
outlines some of Ramsey's major contributions to this tradition,
focusing primarily on the procedure he suggests for measuring
subjective probability.



Ramsey's central goal in 'Truth and Probability' is
to show that the laws of probability provide us with a 'logic of
partial belief'. That is, the laws specify necessary conditions
on any consistent set of partial beliefs, in a manner analogous to how
the laws of classical logic might be taken to generate necessary
conditions on any consistent set of full beliefs. The argument for
this claim is grounded in Ramsey's subjectivist approach to
probability. The essay begins, however, with a discussion of the two
major approaches to understanding probabilities at the time in
Cambridge: frequentism, and a version of the logical interpretation
put forward by the economist John Maynard Keynes in *A Treatise on
Probability* (1921).



Regarding frequentism - according to which the probability of an
event is the relative frequency with which that type of event occurs,
or would occur, over repeated trials - Ramsey adopts a
conciliatory tone. For many cases (e.g., coin flips and rolls of a
die), frequencies provide a natural interpretation of the probability
calculus, but they are insufficiently general for Ramsey's
purposes and relate not to the consistency of partial beliefs but to
their reasonableness. (We'll return to the role of frequencies
in Ramsey's theory below.) On the other hand, Ramsey presents a
detailed critique of Keynes' theory. According to Keynes,
probabilities are an objective and quantifiable relation between
propositions - roughly, the probability of an hypothesis \(h\),
given evidence \(e\), is the degree to which \(h\) is logically
implied by \(e\). Importantly, Keynes' assumed that this
relation could be perceived through intuition - to which Ramsey
objects:



> 
> 
> [T]here really do not seem to be any such things as the probability
> relations [Keynes] describes. He supposes that, at any rate in certain
> cases, they can be perceived; but speaking for myself I feel confident
> that this is not true. I do not perceive them... moreover I
> shrewdly suspect that others do not perceive them either, because they
> are able to come to so very little agreement as to which of them
> relates any two given propositions. (1926a, [FM] 161)
> 
> 
> 



For example, for even very simple pairs of propositions (such as
'This is red' and 'That is blue'), where one
might have expected objective relations to be more readily accessible,
there is very little agreement as to what probability relation might
connect them. Ramsey also notes that, while most people will agree
that the probability of a fair coin landing heads is \(\frac{1}{2}\),
'we can none of us say exactly what is the evidence which forms
the other term for the probability relation about which we are then
judging' (1926a, [FM] 162).



Having dismissed Keynes' theory, Ramsey moves on to his
alternative, subjectivist view. The guiding idea throughout is
that:



> 
> 
> [T]he degree of a belief is a causal property of it, which we can
> express vaguely as the extent to which we are prepared to act on it.
> (1926a, [FM] 169)
> 
> 
> 



That is, partial beliefs are connected to choice and action, in that
the more confidence one has in a proposition \(p\), the more willing
one will be to choose those options that lead to favourable outcomes
under the assumption that \(p\) is true. Although he does not say much
about the nature of this connection in 'Truth and
Probability,' in 'Facts and Propositions' (1927a,
170) Ramsey expresses sympathy for the pragmatist view according to
which a full belief that \(p\) just *is* a set of actions that
tends to lead to favourable outcomes at worlds where \(p\) is
true:



> 
> 
> 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. The mental factors in such a belief
> would be parts of the chicken's behaviour... Thus any set
> of actions for whose utility \(p\) is a necessary and sufficient
> condition might be called a belief that \(p\)... (1927a, [FM]
> 144)
> 
> 
> 



It is plausible that Ramsey intended a similar account for partial
beliefs - that partial beliefs either *are* patterns of
behaviour, or perhaps behavioural dispositions, or that they're
otherwise definable primarily in terms of such things. In any case,
Ramsey recognised that, if this vague connection between belief and
choice could be suitably precisified, then it could be used to build a
definition of degrees of belief in terms of choices.



Like other quantities throughout the sciences, Ramsey argues,
'the degree of a belief... has no precise meaning unless we
specify more exactly how it is to be measured' (1926a, [FM]
167). To precisify his account, therefore, Ramsey sketches a procedure
for the measurement of partial belief, which includes as an initial
part also the measurement of desires (or 'utilities'). The
procedure takes as input the subject's preferences over


* *worlds*, or more accurately propositions that are
maximally specific with respect to matters the subject cares about,
and
* *gambles*, including especially binary gambles of the form
'world \(\omega\_1\) if \(p\), and world \(\omega\_2\)
otherwise', and ternary gambles of the form '\(\omega\_1\)
if \(p \amp q\), \(\omega\_2\) if \(p \amp \neg{q}\), and \(\omega\_3\)
otherwise'.



With those as inputs, the procedure supplies precise numerical
representations of her partial beliefs and utilities provided the
agents' preferences satisfy a number of 'coherence'
requirements. The procedure can be roughly summarised as follows.
First, we elicit the subject's preferences by offering her a
sequence of choices between pairs of worlds and/or gambles. Then, on
the assumption that the subject is an expected utility maximiser, we
can use her preferences to determine her numerical utilities. Given
her utilities, we can finally define the subject's degree of
belief towards the proposition \(p\) in terms of her utilities for
gambles conditional on \(p\). The following paragraphs will briefly
discuss these three steps in turn. More thorough treatments of
Ramsey's procedure can be found in Sahlin 1990, Bradley 2001,
and Elliott 2017; see also Neth 2019 for an extension of the Ramseyan
procedure beyond expected utility theory.



Regarding the first step, which requires us to elicit the
subject's preferences, Ramsey says:



> 
> 
> [Suppose] that our subject has certain beliefs about everything; then
> he will act so that what he believes to be the total consequences of
> his action will be the best possible. If then we had the power of the
> Almighty, and could persuade our subject of our power, we could, by
> offering him options, discover how he placed in order of merit all
> possible courses of the world... Suppose next that the subject is
> capable of doubt; then we could test his degree of belief in different
> propositions by making him offers of the following kind. Would you
> rather have world [\(\omega\_1\)] in any event; or world [\(\omega\_2\)]
> if \(p\) is true, and world [\(\omega\_3\)] if \(p\) is false? (1926a,
> [FM] 177)
> 
> 
> 



A common objection to this early stage of the procedure is that if the
experimenter were to convince their subject that they 'had the
power of the Almighty', then this would radically alter the
beliefs supposedly being measured (e.g., Jeffrey 1983, 158-60;
cf. Sobel 1998, 255-6; Bradley 2001, SS3.2; Eriksson &
Rabinowicz 2013). A measurement procedure can be considered accurate
only if it doesn't itself significantly alter the measurand.
Measurement procedures usually involve *some* unavoidable
change to the quantity being measured - placing a cold
thermometer into a hot liquid will slightly cool the liquid, for
example, but for most purposes this effect is negligible. The
objection at hand is therefore that merely having the subject *come
to believe* that they have some choice over these worlds and
gambles will generally involve a *substantial* change to their
beliefs - enough to undermine the accuracy of Ramsey's
procedure. For example, the process might alter the subject's
beliefs regarding the claim 'Experimenters have the capacity to
make it the case that \(\omega\_1\) obtains if \(p\) is true, and
\(\omega\_2\) obtains otherwise', with potentially many ripple on
effects through to her other beliefs. Interestingly, Ramsey makes note
of a similar problem for an alternative procedure (1926a, [FM] 170),
though does not discuss the worry as it arises for his own
proposal.



The second step of Ramsey's procedure takes us from preferences
to utilities. This step is the most complicated, and it requires a
suite of background assumptions about the partial beliefs, utilities,
and preferences of the subject. In his words,



> 
> 
> I propose to take as a basis a general psychological theory, which is
> now universally discarded, but nevertheless comes, I think, fairly
> close to the truth in the sort of cases with which we are most
> concerned. I mean the theory that we act in the way we think most
> likely to realize the objects of our desires, so that a person's
> actions are completely determined by his desires and opinions. (1926a,
> [FM] 174)
> 
> 
> 



Ramsey's point here is that, given some background psychological
assumptions, it will be possible - at least sometimes - to
specify *exactly* what a subject's degrees of belief and
utilities are once we have enough information about her preferences.
However, Ramsey leaves the details of this 'general
psychological theory' almost entirely implicit. We can clarify
matters on his behalf by breaking the theory into three parts:


* First, there exists a function \(\varphi\) that associates gambles
with real numbers, such that the subject weakly prefers a gamble
\(\gamma\_1\) to another gamble \(\gamma\_2\) just in case
\(\varphi(\gamma\_1)\geq\varphi(\gamma\_2)\)
* Second, this function \(\varphi\) 'decomposes' into
two further real-valued functions \(\bel\) and \(des,\) in the sense
that if \(\gamma\) is the binary gamble 
'\(\omega\_1\) if
\(p\), \(\omega\_2\)
if \(\neg{p}\)' then



> \(\varphi(\gamma) =
> bel(p){\cdot}des(\omega\_1\&{}p) +
> (1-bel(p)){\cdot}des(\omega\_2\&\neg{}p)\)


(and likewise for ternary gambles, *mutatis
 mutandis*)
* Third, \(bel\) and \(des\) represent the subject's
partial beliefs and utilities respectively; for instance,
\(bel(p)\geq\bel(q)\) just in case the subject's degree of
belief towards \(p\) is at least as great as their degree of belief
towards \(q\), and likewise \(des(p)\geq{}des(q)\) just in case the
subject desires that \(p\) at least as much as they desire that
\(q\)



In other words, Ramsey assumes that our preferences conform to the
rule of *expected utility maximisation*, where the expected
utility of a gamble is given by the weighted average of the utilities
of its outcomes, the weights being provided by the subject's
degrees of belief. While Ramsey recognised that these assumptions
involve some idealisation, he justifies them by noting that the
development of any measurement process 'cannot be accomplished
without introducing a certain amount of hypothesis or fiction'
(1926a, [FM] 168), and that the assumptions come close enough to the
truth to render them still useful (1926a, [FM] 173).



The reasoning by which we go from preferences to a numerical
representation of utilities is complicated, and in Ramsey's
paper mostly left unstated. It begins with the notion of an
*ethically neutral proposition*, which can here be defined
as:



**Ethical Neutrality:** \(p\) is *ethically
neutral* for a subject iff, for all worlds \(\omega\) consistent
with \(p\) and \(\neg{p}\), the subject is indifferent between
\(\omega\), \(\omega \amp p\), and \(\omega \amp \neg{p}\).



That is, a proposition \(p\) is ethically neutral if the truth or
falsity of \(p\) is a matter of indifference to the subject,
regardless of the wider context in which its truth or falsity obtains.
Having characterised ethical neutrality in terms of preferences,
Ramsey uses it to construct a definition (in terms of preferences over
gambles) for when the *difference in utility* between two
worlds \(\omega\_1\) and \(\omega\_2\) *is equal to* the
difference in utility between \(\omega\_3\) and
 \(\omega\_4\).[2]
 This *equal differences in utility* definition provides Ramsey
with the resources needed to sketch a *representation theorem*,
which constitutes the centrepiece of his paper. According to this
theorem, the subject's preferences over worlds and gambles will
satisfy eight relatively simple conditions (or 'axioms')
only if there exists at least one function, \(des\), which represents
the subject's preferences over worlds in the sense that
\(\omega\_1\) is weakly preferred to \(\omega\_2\) if and only if
\(des(\omega\_1) \geq des(\omega\_2)\). Moreover, this function will be 
unique up to positive linear transformations - that is, for any
other function \(des^\*\) with the same property, \(des^\*(x) =
des(x)r+c \) (for some positive real number \(r\) and constant \(c\)).
As such, \(des\) is an *interval scale* representation:a numerical scale in which intervals of differences are
meaningful. For instance, the difference in utility between
\(\omega\_1\) and \(\omega\_2\) is at least as great as that between
\(\omega\_3\) and \(\omega\_4\) just in case \(des(\omega\_1) -
des(\omega\_2) \geq des(\omega\_3) - des(\omega\_4) \). (Other examples
of interval scales include temperatures as measured in degrees Celsius
or Fahrenheit, and yearly dates as measured in the A.D. and Buddhist
or Hindu systems.) Given Ramsey's background 'general
psychological theory', the implication is that the
subject's preferences will satisfy Ramsey's eight axioms
only if this function \(des\) represents her utilities on an interval
scale.



Included among Ramsey's eight axioms are some obvious coherence
conditions, such as that preferences ought to be transitive. But the
axioms also include some less obvious conditions, for instance that
for any pair of worlds \(\omega\_1\), \(\omega\_2\), there's a
further world \(\omega\_3\) whose utility is exactly halfway between
that of \(\omega\_1\) and \(\omega\_2\). There has been very little
empirical investigation into Ramsey's axioms, primarily due to
the widespread opinion that his representation result has been
superseded by the results of Savage (1954) and later decision
theorists (see, e.g., Fishburn 1981). However, Ramsey's
assumption that there exists an ethically neutral proposition (needed
for the definition of equal differences in utility to make sense) has
attracted substantial critical discussion. Ramsey does not discuss any
reasons to believe that even one ethically neutral proposition exists,
still less that their existence is a precondition for the
*consistency* of partial beliefs. For more discussion,
including on why ethically neutral propositions are problematic for
Ramsey's proposal and whether they're necessary for it,
see Sobel 1998, Bradley 2001, Eriksson & Hajek 2007, and
Elliott 2017. A further question concerns whether this stage of
Ramsey's procedure presupposes the controversial
'deliberation crowds out prediction' thesis, according to
which an agent deliberating about an action cannot have meaningful
degrees of belief regarding whether she will perform that action; see
Liu & Price (2020) for discussion and Rabinowicz (2002) for a
critique of the thesis.



The third and final stage of Ramsey's measurement procedure
takes us from the subject's utilities to a definition of her
degrees of belief:



> 
> 
> Having thus defined a way of measuring value we can now derive a way
> of measuring belief in general. If the option of [\(\omega\_2\)] for
> certain is indifferent with that of ['\(\omega\_1\) if \(p\),
> \(\omega\_3\) otherwise'], we can define the subject's
> degree of belief in \(p\) as the ratio of the difference between
> [\(\omega\_2\)] and [\(\omega\_3\)] to that between [\(\omega\_1\)] and
> [\(\omega\_3\)]. (1926a, [FM] 179)
> 
> 
> 



In a footnote, Ramsey adds that \(\omega\_1\) must imply \(p\), and
\(\omega\_3\) must imply \(\neg{p}\). The definition follows from the
background assumption of the 'general psychological
theory' mentioned above: the world \(\omega\_2\) has the same
utility as the gamble '\(\omega\_1\) if \(p\), \(\omega\_3\)
otherwise' just in case

 
\[des(\omega\_2) = bel(p)des(\omega\_1) + (1 - bel(p))des(\omega\_3)\]


Where \(des(\omega\_1) \neq des(\omega\_3)\), this equality can be
rearranged to give a definition of \(bel(p)\)as follows:



\[bel(p) = \frac{des(\omega\_2) - des(\omega\_3)}{des(\omega\_1) - des(\omega\_3)}\]


As Ramsey notes,



> 
> 
> This amounts roughly to defining 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. (1926a, [FM]
> 179-80)
> 
> 
> 



Ramsey's definition of conditional probability (1926a, [FM] 180)
follows an essentially similar strategy. Having thus outlined his
strategy for defining degrees of belief, Ramsey proves that from his
preference conditions and subsequent definitions, the following
'laws of probability' follow:

 
\[\begin{align}
&bel(p) + bel(\neg p) = 1\\
 &bel(p \,|\, q) + \bel(\neg p \,|\, q) = 1\\
 &bel(p \amp q) = bel(p)bel(q \,|\, p)\\
 &bel(p \amp q) + bel(p \amp \neg q) = bel(p)\\
 &bel(p \,\vee\, q) = bel(p) + bel(q), \text{where }p\amp q\text{ is impossible} 
\end{align}\]


Ramsey goes on to say:



> 
> 
> These are the laws of probability, which we have proved to be
> necessarily true of any consistent set of degrees of belief. Any
> definite set of degrees of belief which broke them would be
> inconsistent in the sense that it violated the laws of preference
> between options... (1926a, [FM] 182)
> 
> 
> 



In this passage we find an early version of what has come to be known
as a *representation theorem argument* for probabilistic norms
on partial belief and for expected utility theory, of the kind later
made popular by Savage. (See also Skyrms 1987, Hajek 2008, and the
entry on
 normative theories of rational choice: expected utility,
 SS2.2, for discussion). It is dubious, however, that Ramsey
'proved' that the stated laws must be true of any
consistent set of partial beliefs, even where
'consistency' is understood as being tied to the
'laws of preference' in the manner that Ramsey suggests.
Among the axioms of his representation theorem are some (like those
regarding the ethically neutral proposition) which are not plausible
as conditions of the *consistency* on those preferences.
Furthermore, Ramsey's 'proof' requires substantive
psychological assumptions about the connection between partial beliefs
and preferences, which Ramsey admits are idealizations. Even if those
assumptions happen to be true as a matter of fact, though, or close to
the truth, they are presupposed in the derivation and require
independent justification.



Immediately following, Ramsey also states that:



> 
> 
> If anyone's mental condition violated these laws, his choice
> would depend on the precise form in which the options were offered
> him, which would be absurd. He could have a book made against him by a
> cunning better and would then stand to lose in any event. (1926a, [FM]
> 182)
> 
> 
> 



The reasoning behind this claim is never made explicit, though it is
evident that Ramsey was putting forward what has come to be known as a
Dutch Book Argument, later made more precise by de Finetti (1937; see
the entry on
 Dutch book arguments
 for details).



Having established what he takes to be the conditions of
*consistency* for partial beliefs, Ramsey concludes his paper
with a lengthy discussion on what (in addition) might make a set of
partial beliefs *reasonable* (as opposed to just
*consistent*). He proposes a condition of *calibration*,
or fit with known frequencies:



> 
> 
> Let us take a habit of forming opinion in a certain way; e.g. the
> habit of proceeding from the opinion that a toadstool is yellow to the
> opinion that it is unwholesome. Then we can accept the fact that the
> person has a habit of this sort, and ask merely what degree of opinion
> that the toadstool is unwholesome it would be best for him to
> entertain when he sees it... And the answer is that it will in
> general be best for his degree of belief that a yellow toadstool is
> unwholesome to be equal to the proportion of yellow toadstools which
> are in fact unwholesome. (This follows from the meaning of degree of
> belief.) (1926a, [FM] 195)
> 
> 
> 



So, for example, if 1 in 100 yellow toadstools is unwholesome, then
*ceteris paribus* one should believe that *this*
toadstool is unwholesome, given that it is yellow, to degree
\(0.01\).Something like this condition has reappeared in a number of
later works (e.g., Shimony 1988; Lewis 1980); see the entry on
 Interpretations of Probability,
 SS3.3.4, for more discussion.



It would be hard to understate the importance of the above ideas to
the subjectivist tradition. It is a major testament to the originality
of Ramsey's essay that it contains not only the first
appearances of two of the main contemporary arguments for
probabilistic norms on partial belief, but also an influential case
for a normative link between partial beliefs and known frequencies.
Furthermore, most attempts to characterise degrees of belief in the
subjectivist tradition have made central appeal to their connection
with preferences. (See de Finetti 1937, Savage 1954, Anscombe &
Aumann 1963, Maher 1993, Elliott 2021; for criticisms of the approach,
see Joyce 1999, SS1.3; Eriksson & Hajek 2007; Meacham
& Weisberg 2011; Stefansson 2017.) Indeed, in much of
philosophy, economics and psychology still today, the default or
orthodox way to operationalise degrees of belief is in terms of
choices, along essentially the same lines that Ramsey put forward.


## 7. Laws and Theories



Ramsey's main contributions to the philosophy of science stem
from three posthumously published papers written in 1928 and 1929, one
on theoretical terms in theories and two on the distinction between
(in modern terminology) laws of nature and accidental
generalizations.


### 7.1 Laws



In both papers on laws, Ramsey (1928b, SS2; 1929b, [FM]
237-38) considers accidental generalization to be conjunctions
([NP], n. 40; cf. Ramsey 1927a, [FM] 152-53) and thus normal
propositions (for the relation to Wittgenstein, see
 SS2).
 In "Universals of Law and Fact", Ramsey (1928b, SS12)
considers laws of nature to be propositions with a special status:



> 
> [E]ven if we knew everything, we should still want to systematize our
> knowledge as a deductive system, and the general axioms in that system
> would be the fundamental laws of nature. The choice of axioms is bound
> to some extent to be arbitrary, but what is less likely to be
> arbitrary if any simplicity is to be preserved is a body of
> fundamental generalizations, some to be taken as axioms and others
> deduced.
> 



Thus laws of nature are the axioms and theorems in the simplest
axiomatization of all of our knowledge. And while we currently do not
know everything, we can still find the simplest axiomatization for
what we do know and what we expect to come to know (SS16).
Following Lewis (1973 [1986], SS3.3), a similar view of laws has
been developed and discussed as the "Best System" account
of laws, which, however, does not demand that we find the simplest
axiomatization of *all* our knowledge, but rather of as much of
our knowledge as is possible while retaining a high enough level of
simplicity (see also Carroll 2016, SS2).



"General Propositions and Causality", written in the
summer of 1929 (Braithwaite 1931, xiii) and with additional notes from
September 1929 ([NP] note 74), explicitly retracts the view that laws
(here called 'variable hypotheticals') are propositions
(1929b, [FM] 238; for the relation to Weyl, see
 SS2).
 Rather (1929b, [FM] 241):



> 
> Variable hypotheticals are not judgments but rules for judging
> 'If I meet a \(\phi\), I shall regard it as a \(\psi\)'.
> This cannot be *negated* but it can be *disagreed* with
> by one who does not adopt it.
> 



Thus variable hypotheticals "form the system with which the
speaker meets the future" in that "we trust [a conjunction
of their instances] to guide us in a new instance" exactly
because the conjunction is evidence for the variable hypothetical.
Simplicity may still play a role, but not as constitutive of laws (as
in the earlier, Best System account), rather only as epistemic
justification for adopting a rule of judgment (1929b, [FM] 242). This
allows again for the non-arbitrariness if not uniqueness of laws.
Unknown laws are those rules of judgment whose adoption is justified
by unknown facts (1929b, [FM] 243-45).



In both papers, 'if ... then ...' statements,
called 'conditionals', are taken to differ from material
implications in that they can be inferred from the system of laws
(1928b, SS15; 1929b, [FM] 248). Ramsey (1929b, [FM] 249) takes
conditionals to encompass both the indicative and subjunctive mood
(see SS5 on Conditionals) and he identifies causal laws with
conditionals in which the event described in the antecedent takes
place before the event described in the consequent.



Ramsey's earlier, Best System account of laws fits well with
Humean constraints of metaphysics (Carroll 2016, SS2), and Ramsey
(1929b, [FM] 252-53; 254-55, n. 4) is explicit that his
later, non-propositional account of laws avoids "real
connections of universals" and is in the spirit of Hume. At
first Ramsey (1928b, SS15) even states that the material
implication is all we practically need, while later acknowledging that
conditionals are necessary for assigning praise and blame (1929b, [FM]
245-46).


### 7.2 Theories



His non-propositional account of laws, Ramsey (1929b, [FM] 254, n. 2)
notes, renders theoretical laws twice removed from propositions:



> 
> [Laws] are used in the same way in a theoretical system as in a
> primary system; cause, too, if the theoretical system is temporal. Of
> course the theoretical system is all like a variable hypothetical in
> being there just to be deduced from; and a law in the theoretical
> system is at two removes of deduction.
> 



In his paper "Theories" (1929a), written in the summer of
1929 (Braithwaite 1931, xiii), Ramsey calls the observational language
'primary system'. The non-observational (theoretical)
language is called 'secondary system' (see also
 SS2).
 The paper is thoroughly instrumentalistic, since the factual content
of theories consists only of their primary implications. This is made
clear in a comment on the paper, written in August 1929 ([NP] 229,
footnote removed; see also Ramsey (1931, [FM] 260):



> 
> 
> The essence of a theory is that we make our assertions in a form
> containing a lot of parameters, which have to be eliminated in order
> to get our real meaning.
> 
> 
> 
> We mean to assert everything that can be deduced and does not contain
> a parameter.
> 
> 
> 



The "parameters" here are the secondary terms. And Ramsey
([NP] 236) stresses in a note on the infinite in mathematics that
their use does not entail any ontological commitment:



> 
> It is obvious that mathematics does not require the existence of an
> infinite number of things. We say at once that imaginary things will
> do, i.e. theoretical secondary terms. But there are no imaginary
> things, they are just words, and mathematicians and physicists who use
> the infinite are just manipulating symbols with some analogy to
> propositions.
> 



To investigate the relation of the primary and the secondary system,
Ramsey (1929a, [FM] 215-19) provides a primary system for an
agent who, in discrete time intervals, can move forwards and
backwards, open and close their eyes, and see blue, red, or nothing.
In the secondary system, Ramsey formulates a toy theory that describes
the movement of the agent among three locations and the sometimes
changing color of the locations. A dictionary defines the primary
terms in the secondary system so that, for instance, the agent sees
blue if and only if situated at a blue location with open eyes.



Ramsey (1929a, [FM] 217-19) shows that all the theory's
implications in the primary system can be listed without the use of
the secondary system and, contrary to "Russell, Whitehead, Nicod
and Carnap" (220), that it is virtually impossible to define the
secondary terms in the primary system:



> 
> [T]here is neither in this case nor in general any simple way of
> inverting the dictionary so as to get either a unique or an obviously
> preeminent solution which will also satisfy the axioms, the reason for
> this lying partly in difficulties of detail in the solution of the
> equations, partly in the fact that the secondary system has a higher
> multiplicity, i.e. more degrees of freedom, than the primary. (222)
> 



Carnap (1923, 100) had already noted the additional degrees of freedom
in the secondary system, but without appreciating the implication
(obvious for Ramsey) regarding the inversion of the dictionary.
Ramsey's qualification that there is no *simple*
inversion is needed to exclude the following case: Assume that all the
models of the theory and the dictionary are finite and all objects
have names in the primary system. Then in each model \(M\_i\), each
term \(T\_j\) of the secondary system is co-extensional with a formula
\(\phi\_{i,j}\) in the primary system. If furthermore each model is
named by a sentential constant \(C\_i\) of the primary system, one can
define each term \(T\_jx\_1\ldots x\_n\) as the conjunction of the
material implications \(C\_i \to \phi\_{i,j}(x\_1,\ldots ,x\_n)\) for all
models \(M\_i\) (220-21). However, Ramsey (1929a, [FM] 230)
argues, apart from being cumbersome, definitions for secondary terms
actually hinder scientific development because they require conceptual
decisions based on fiat rather than further empirical discoveries (cf.
Braithwaite 1953, 52-76). This criticism was later rediscovered
by Carnap (1936, 449).



To avoid these problems, Ramsey (1929a, [FM] 231) suggests a way of
applying a theory without defining the secondary terms:



> 
> The best way to write our theory seems to be this:
>   
> 
> \((\exists \alpha, \beta, \gamma)\): *dictionary* \(\cdot\)
> *axioms*.
> 



\(\alpha\), \(\beta\), and \(\gamma\) are variables for the terms of
the secondary system and are interpreted purely extensionally, so that
the theory states "There are extensions \(\alpha\), \(\beta\),
and \(\gamma\) that satisfy the sentences of the dictionary and the
axioms". In contemporary presentations, the axioms are usually
denoted \(T\) (for 'theory'), the dictionary is denoted
\(C\) (for 'correspondence rules'), and the secondary
terms \(T\_1,\ldots ,T\_n\) are treated as parameters in \(T\) and
\(C\), so that the conjunction of axioms and dictionary is written as
\(TC(T\_1,\ldots ,T\_n)\). What is now called the 'Ramsey
sentence' \({TC}^R\) is then written as

 
\[ {TC}^R = \exists X\_1,\ldots ,X\_nTC(X\_1,\ldots ,X\_n), \]


that is, in the conjunction of all axioms and sentences of the
dictionary, the theoretical terms \(T\_1,\ldots ,T\_n\) are replaced by
variables \(X\_1,\ldots ,X\_n\) and bound by existential quantifiers.
This procedure is now known as 'Ramseyfication' or
'Ramsification'. If some of the theoretical terms are of
the highest order (predicates or functions of \(n^{\mathrm{th}}\)
order in \(n^{\mathrm{th}}\) order logic), then Ramseyfication
increases the order of the theory by one. While higher-order
existential quantification was commonly used to express consistency
when Ramsey wrote "Theories", he was the first to quantify
on only some of the terms (Demopoulos 2003, 256). Crucially, the
Ramsey sentence contains no theoretical terms while still entailing
the same primary statements as \(TC\) (for a simple proof, see Bohnert
1967, 342-43). Since all of \(TC\)'s sentences are in the
scope of the existential quantifiers introduced during Ramseyfication,
\({TC}^R\) does not assign meaning to individual secondary sentences
or terms. However, we can still reason with them by deriving primary
propositions from them within the scope of the quantifiers, just as we
can derive propositions from variable hypotheses (1929a, [FM] 232;
Bohnert 1967, 344-47). Hence secondary universal sentences are
"at two removes" from propositions, and secondary
sentences and terms are only meaningful in the context of the theory
to which they belong (1931, 260).



As an example, consider a simplification of Ramsey's toy theory
that states that being at a blue location with open eyes is followed
by being at a red location with open eyes: \(\forall x(Bx \wedge Ox
\to Rx' \wedge Ox')\), where \(x'\) is the successor of \(x\).
'Blue' (\(B\)), 'red' (\(R\)), and 'open
eyes' (\(O\)) are theoretical terms. The correspondence rules
define the observational terms 'perceiving blue' as
'being at a blue location with open eyes', \(\forall x(Px
\leftrightarrow Bx \wedge Ox)\), and correspondingly 'perceiving
red' \(\forall x(Qx \leftrightarrow Rx \wedge Ox)\). The Ramsey
sentence of \(TC\) is then:

 
\[\begin{align}
\exists X\_1,X\_2,X\_3[& \forall x(Px \leftrightarrow X\_1x \wedge X\_2x) \;\wedge \\
 & \forall x(Qx \leftrightarrow X\_3x \wedge X\_2x) \;\wedge \\
 & \forall x(X\_1x \wedge X\_2x \to X\_3x' \wedge X\_2x')] 
\end{align}\]


This is equivalent to the observational sentence \(\forall x(Px \to
Qx')\), which states that perceiving blue is followed by perceiving
red. For any specific location \(a\), this variable hypothetical then
becomes the proposition \(Pa \to Qa'\).



While sentences of any order can be Ramseyfied, many technical
discussions (e.g., the overview by van Benthem 1978) treat the special
case in which axioms and dictionary are of first order and the Ramsey
sentence is of second order. In this case, the class of models of the
observational first-order implications of \(TC\) can be a proper
superset of the class of models of \({TC}^R\). This has been known at
least since 1941 as the difference between elementary and
pseudoelementary classes (Hodges 1993, 207, 260) and is regularly
rediscovered in philosophy (see Demopoulos 2011, 179). It entails that
\({TC}^R\) can be false in a structure even though all the first order
implications of \(TC\) are true. Specifically, \(TC\) may only have
tautological observational implications even though the Ramsey
sentence is not logically true; this situation is one version of the
distinction between syntactic and semantic conservativeness known from
definition theory (Gupta 2015, SS2.3; Przelecki 1969,
52-53).



Carnap (1963, SS24.D) used the Ramsey sentence for distinguishing
between the analytic and synthetic components of higher-order theories
(Uebel 2016, SS3.5), and Lewis (1970) significantly modified the
Ramsey sentence to arrive at explicit definitions of secondary terms
(Andreas 2017, SS4.4). This Ramsey-Lewis sentence has become
central in the philosophy of mind (Levin 2017, SS4.1) and
philosophical methodology (Papineau 2016, SS2.3).



In current terminology, the semantics of the Ramsey sentence is the
following: \({TC}^R\) is true in a structure \(S\) for the primary
terms if and only if it is possible to add interpretations for the
secondary terms to \(S\) such that \(TC\) is true (Przelecki
1969, ch. 6; Andreas 2017, SS4.2). Sneed (1971, 106) modified the
semantics of the Ramsey sentence for the use in structuralism (Schmidt
2014, SS2.2.1; Stegmuller 1976, 88-89). Maxwell (1970,
186-88) identified the Ramsey sentence with its semantics to
argue that the Ramsey sentence expresses structural realism (see also
Ladyman 2016, SS3.2). Dewar (2019) argues that this is
misguided.


## 8. Contributions to Mathematics



Despite being employed as a mathematician at Cambridge Ramsey only
published one paper (1930) firmly in that subject, though two ideas he
introduced in his more philosophical papers (1926a, 1929b), also led
eventually to new lines of mathematical research. Each of these three
contributions has proved seminal and they are now rightly associated
with his name.



Of these three Ramsey is best known within philosophy for his somewhat
sketchy description of what later became known as the Dutch Book
argument or theorem, see Vineberg 2016. In his unpublished paper
"Truth and Probability" (1926a), Ramsey proposed a set of
axioms and assumptions, his *Laws of Preference*, which enable
a subject to correlate the value of options \(\alpha, \beta, \gamma\)
and the preference relation between them with real numbers under the
standard ordering. Using \(\alpha\) etc. as Ramsey does to denote both
the option and its correlated real number allows a subject in turn to
equate their degree of belief in a proposition \(p\) with the
ratio

 
\[\bel(p)= \frac{\alpha- \gamma}{\beta - \gamma}\]


when, as far as the subject is concerned, the option

 
\[ \alpha \text{ (for certain)} \]


and the option

 
\[ \beta \text{ if } p \text{ is true, } \gamma \text{ if } p \text{ is false} \]


are equally preferable.



Ramsey then asserted that for partial belief (i.e. not certain belief)
"This amounts roughly to defining 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". In
making this rough identification then Ramsey draws on the already
"old-established" idea of measuring one's degree of
belief in a proposition \(p\) in terms of ones willingness to bet on
its being true (1926a, [FM] 172). While he does not elaborate further
on this connection it is implicit in the derivations that follow that
Ramsey concludes that there is a number \(x\) such that for any option
\(\xi\) and any stake \(t\) (positive or negative), \(\xi\) for
certain is equivalent to:

 
\[\begin{align}
\xi + \text{receive } (1-x)t, &\text{ if } p \text{ is true} \\
 \xi + \text{receive } -xt, &\text{ if } p \text{ is false} 
\end{align}\]


As above this gives \(\bel(p)=x\), from which we might surmise that
Ramsey intends \(\bel(p)\) to be that number for which the
options:

 
\[\begin{align}
\text{receive } (1-\bel(p))t, &\text{ if } p \text{ is true} \\
 \text{receive } {-}\bel(p)t, &\text{ if } p \text{ is false} 
\end{align}\]


are all equally acceptable to the subject. (There is an assumption
here that the stake \(t\) is not so great as to incur
"diminishing marginal utility".)



With this definition of the function \(\bel(p)\) and its further
extension \(\bel(p\mid q)\) to conditional belief in \(p\) given
\(q\), for which Ramsey claims credit, he argues that if a
subject's degrees of belief are consistent in the sense of
obeying these Laws of Preference (alternatively, as Ramsey simply
states, do not allow a [Dutch] book to be made against him by which he
would lose in any event) then \(Bel\) must satisfy the
"fundamental laws of probable belief (degrees of belief between
0 and 1)", which he gives as:

 
\[\begin{align}
&\bel(p) + \bel(\neg p) = 1, \\
 &\bel(p\mid q) + \bel(\neg p\mid q) = 1, \\
 &\bel(p \wedge q) = \bel(q\mid p) \cdot \bel(p), \\
 &\bel(p \wedge q) + \bel(p \wedge \neg q) = \bel(p). 
\end{align}\]


Ramsey also states, again without proof, the converse (1926a, [FM],
183):



> 
> 
> Having 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.
> 
> 
> 



Nowadays, this result is commonly attributed to John Kemeny (1955) and
independently Sherman Lehman (1955).



Such casual allusions to important new mathematical results without
further explanation seems to have been a feature of Ramsey's
writing.



Unaware of Ramsey's result, Bruno de Finetti published (1931) a
detailed direct proof that if a subject's (unconditional)
degrees of belief \(\bel(p)\) do not allow such a book, he calls such
a subject *coherent*, then \(\bel\) must satisfy what would
later become know as Kolmogorov's Axioms (1933) for a finitely
additive probability function and conversely. In consequence of this
fuller explication the forward direction of the Dutch Book argument
for belief as probability is commonly attributed to de Finetti
alone.



In recent times Dutch Book arguments have become an established topic
in philosophy and mathematics in terms of analysing and extending the
original argument to other logics, see Vineberg 2016.



A second important mathematical contribution of Ramsey is based on the
following brief footnote in his otherwise philosophical paper
*General Propositions and Causality* (1929b, [FM] 247):



> 
> 
> If two people are arguing 'If \(p\) will \(q\)?' and are
> both in doubt as to \(p\), they are adding \(p\) hypothetically to
> their stock of knowledge and arguing on that basis about \(q\); so
> that in a sense 'If \(p\), \(q\)' and 'If \(p\), not
> \(q\)' are contradictories.
> 
> 
> 



Starting with Robert Stalnaker (1968) this has been (mis)interpreted
(see the section on Conditionals) as a bridge between logics of
(indicative) conditionals (Edgington 2014, Arlo-Costa 2016) and logics
of belief revision (Hansson 2017). In the former the interest is in
formalising the properties of conditional assertions such as
*if* \(p\) *then* \(q\), denoted \(p>q\), their
semantics and the rules and constraints that, arguably, they should
obey. For the latter the focus is on capturing the rules and
constraints that, again arguably, should apply when an agent's
set of beliefs \(K\), is to be revised, to \(K\*p\) say, on receipt of
new information \(p\) which, in particular, may not even be consistent
with the original \(K\).



The above footnote has been formalised within this notation as the so
called *Ramsey's Test*:



An agent with belief set \(K\) should accept \(p \gt q\) iff \(q \in
K\*p\)



Through this recipe then any logic of conditionals generates a logic
of belief revision, and conversely. In this way it can act as a
*test* of the appropriateness of a logic of conditionals by
opening up the corresponding logic of belief revision to scrutiny, and
again conversely.



Not surprisingly then Ramsey's Test, in this or in its many
amended forms, has been hugely influential in the formal development
of the logics of conditionals and belief revision in recent years, in
philosophy and equally in artificial intelligence.



Despite its important contribution however Ramsey's Test has in
certain guises fallen foul of so called *Triviality Theorems*,
starting with Gardenfors (1986), which show that under certain
conditions and apparently reasonable constraints on the belief
revision Ramsey's Test can only apply to a very limited family
of belief sets \(K\). That is, agents who satisfied Ramsey's
Test could only consistently subscribe to very limited and trivial
belief sets. In particular this can happen if we allow \(>\) to
freely appear in the logical language from which the sentences \(p,q\)
etc. are formed since then it would seem that the Ramsey Test would
equally amount to:



**Ramsey Test (RT):**
  

For an agent with belief set \(K\), \((p>q) \in K\) iff \(q\in K \*
p\)



From this it follows that \(\*\) must be monotonic in the sense that if
\(K \subseteq K'\) then \(K\*p \subseteq K'\*p\), since if \(q \in K\*p\)
then \((p>q) \in K\) by (RT) so \(K \subseteq K'\) gives \((p>q)
\in K'\) and hence by (RT) again \(q \in K'\*p\). But this is a
conclusion which goes against intuition, and unless the allowed belief
sets \(K\) are severely limited, contradicts ostensibly reasonable
assumptions about \(\*\), including in particular that if \(p\) is
consistent with \(K\) then \(K\*p\) should be simply the logical
consequences of \(K \cup\{p\}\).



Ramsey's third contribution, and the one most directly important
for mathematics, consists of two closely related theorems proved in
the paper *On a Problem of Formal Logic* (1930) which was read
to the London Mathematical Society on the 13th December 1928. This
material is uncompromisingly aimed at mathematicians and with
apologies its explication herein requires some necessary
technicalities.



The first of these theorems says that if \(n,r\) are finite positive
numbers and \(X\) is an infinite set and we partition the set of
subsets of \(X\) of size \(n\) into \(r\) parts then there is an
infinite subset \(Y\) of \(X\) such that every subset of \(Y\) of size
\(n\) is in the same part. The second theorem gives a version of the
first, but for finite sets of sufficiently large size. It says that,
for any finite positive numbers \(m,n,r\) there is a finite positive
\(k\) such that if \(X\) has at least \(k\) elements, and we partition
the sets of subsets of \(X\) of size \(n\) into \(r\) parts, then
there is a subset \(Y\) of \(X\) with at least \(m\) elements such
that every subset of \(Y\) of size \(n\) is in the same part of the
partition. Ramsey gave an upper bound on such a \(k\) in terms of
\(m,n,r\) but even today the actual value of the least such \(k\),
known as the *Ramsey Number* \(R\_n(m,r)\), is unknown in all
but a handful of cases for small \(n,m,r\). These two results are now
referred to as the Infinite and Finite Ramsey Theorems, respectively.
They can be viewed as generalizing Dirichlet's much earlier
*Schubfachprinzip* or *Pigeon Hole Principle*
(1832).



Ramsey then uses the finite theorem in this paper to show that for
each sentence, from a relational language with equality, of the
form

 
\[\exists z\_1, \ldots, z\_m \forall x\_1, \ldots, x\_n \, \theta(z\_1, \ldots, z\_m,x\_1, \ldots, x\_n),\]


where \(\theta\) is quantifier free, one could effectively decide if
it was satisfiable in some structure of cardinality \(\kappa\). Ramsey
achieved this by showing that one can write down a natural number
\(k\) such that for any cardinal \(\kappa \geq k\) this sentence can
be satisfied in some structure of cardinality \(\kappa\) just if it
can be satisfied in some structure of cardinality \(k\), and to
determine this latter just requires checking its satisfiability in one
of the finitely many such structures. An important innovation in his
proof was the notion of indiscernibles which was to reappear,
apparently independently, three decades later in Model Theory. It was
subsequently employed in the proof of Michael Morley's seminal
Categoricity Theorem (1965) and later in the proof of the
Paris-Harrington Theorem (Paris & Harrington 1977).



This result of Ramsey's improved on earlier joint work of Paul
Bernays and Moses Schonfinkel (1928) and solves the prospective
*Spectrum Problem* (Scholz 1952), for sentences of the above
form, now referred to as the Bernays-Schonfinkel-Ramsey Class.
That is, for such sentences it enables us to determine the class of
cardinalities of structures in which that sentence can be satisfied.
As a corollary it gives that one can effectively decide if a sentence
in this class is satisfiable in *some* structure. However it
seems clear from the accompanying discussion that Ramsey was aware
that there was also a direct and simple proof of this fact.



This corollary would seem to justify Ramsey's stated aim in this
paper to provide a step towards a positive solution to Hilbert's
*Entscheidungsproblem* though it is possible that the aim he
actually had in mind was the related Spectrum Problem. Hilbert's
goal was later shown to be not achievable by the undecidability
results of Alonzo Church (1936) and independently Alan Turing (1937).
In fact Ramsey's positive contribution turned out to be the best
possible in the sense that decidability fails once the order of the
blocks of existential and universal quantifiers is reversed, see
Kalmar & Suranyi 1950, and Mortimer 1975.



Ramsey's Theorems were later popularised in an influential paper
of Paul Erdos and George Szekeres (1935) and initiated in the
early 1970s the establishment of *Ramsey Theory* as a major
subdiscipline of Combinatorics. The basic aim in this subdiscipline is
to show that mathematical structures of a certain type and size (and
generally non-uniform) must have within them uniform substructures of
a certain type and size, see Katz & Reimann 2018,
Nesetril & Rodl 1990, and Spencer & Graham
1990.



Ramsey's Theorems themselves have yielded numerous important
applications, generalizations and concepts in other branches of
mathematics. For example the notion of a *Ramsey Cardinal*,
\(\kappa\), where we require that for a set \(X\) of cardinality
\(\kappa\) and any partition of the finite subsets of \(\kappa\) into
two sets there is a subset \(Y\) of \(X\), also of cardinality
\(\kappa\), such that any two finite subsets of \(Y\) of the same size
are in the same part of the partition. Ramsey Cardinals were one of
the first examples of *large cardinals*, a central area of
research in Set Theory. One feature of such cardinals is that they are
so large that their existence cannot be proved within ZFC.



In addition to these contributions to mathematics Frank Ramsey (1925a)
devoted much effort to addressing what were by then seen as problems
in Whitehead & Russell's attempt in their *Principia
Mathematica* (1910-1913), to base mathematics on formal
logic, an enterprise which Ramsey initially strongly supported though
he ultimately became disappointed with his efforts to correct its
shortcomings, see Paul 2012 (232). Since then this approach to the
foundations of mathematics has been very largely overshadowed amongst
mathematicians by the practically simpler device of assuming as a
foundation Zermelo-Fraenkel Set Theory.