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. 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:.
Indeed, he distances himself from that idea. Thus, for Ramsey conditional degree of belief is a synchronic notion, not a diachronic one.
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. 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. That determines the odds on which I would bet on Red given King.
I am not colour blind. This is something he believes. But if he were to learn the antecedent, he would not then believe the consequent! 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 for more details and references. The essay is a pioneering work on subjectivist interpretations of probability also known as personalist interpretations; see the entry on interpretations of probability.
Many of the key ideas and arguments original to the essay have reappeared in later foundational works in the subjectivist tradition e. 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 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. 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. The guiding idea throughout is that:. 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. 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.
To precisify his account, therefore, Ramsey sketches a procedure for the measurement of partial belief, which includes as a part also the measurement of utilities. It can be roughly summarised as follows:. Determine Utilities: Assuming the subject is an expected utility maximiser, use her preferences to determine her numerical utilities. The following paragraphs will briefly discuss these three steps. 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.
Interestingly, Ramsey makes note of a similar problem for an alternative gambling-based measurement procedure a, [FM] , but for unknown reasons does not discuss the worry as it arises for his own proposal. 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. We can precisify matters on his behalf, and assume that for the subject in question,. It begins with the notion of an ethically neutral proposition , which can here be defined as:.
This provides Ramsey with the resources needed to sketch a representation theorem , which forms the centrepiece of his paper.
Ramsey provides his readers with no reasons to believe that even one ethically neutral proposition exists, still less that their existence is a precondition for the consistency of partial beliefs. Having thus defined a way of measuring value we can now derive a way of measuring belief in general. The definition makes sense in light of the background assumptions 6. These are the laws of probability, which we have proved to be necessarily true of any consistent set of degrees of belief. In this passage we find an early version of a representation theorem argument for probabilistic norms on partial belief and for expected utility theory, of the kind later made popular by Savage.
It is dubious, however, that Ramsey really proved that the stated laws must be true of any consistent set of partial beliefs. Amongst the axioms of his representation theorem for utilities are several non-necessary conditions, as well as conditions like those regarding the ethically neutral proposition which are not plausible as conditions of consistency on preferences.
The Philosophy of F.P. Ramsey. cover_ramsey. Nils-Eric Sahlin. Cambridge University Press Buy the book here. (this short introduction was written "Sahlin is to be congratulated for his bold venture sketching an intellectual portrait of Frank P. Ramsey, a man who mastered existing modes of inuirty in many.
He could have a book made against him by a cunning better and would then stand to lose in any event. The reasoning behind the last 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 ; see entry the on Dutch book arguments.
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. He proposes a condition of calibration , or fit with known frequencies:. Let us take a habit of forming opinion in a certain way; e. 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. Something like this condition has reappeared in a number of later works e. It would be hard to understate the importance of the above ideas to the subjectivist tradition. Furthermore, most attempts to characterise degrees of belief in the subjectivist tradition have made central appeal to their connection with preferences.
Indeed, in much of philosophy, economics and psychology 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. Ramsey a, [FM] —53 and thus normal propositions. Thus laws of nature are the axioms and theorems in the simplest axiomatization of all of our knowledge. Rather b, [FM] :.
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 b, [FM] 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 b, [FM] — His non-propositional account of laws, Ramsey b, [FM] , n.
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 [NP] , footnote removed; see also Ramsey , [FM] :. 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.
And Ramsey [NP] stresses in a note on the infinite in mathematics that their use does not entail any ontological commitment:. To investigate the relation of the primary and the secondary system, Ramsey a, [FM] —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. However, Ramsey a, [FM] 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 , 52— This criticism was later rediscovered by Carnap , To avoid these problems, Ramsey a, [FM] suggests a way of applying a theory without defining the secondary 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 a, [FM] ; Bohnert , — While sentences of any order can be Ramseyfied, many technical discussions e. This has been known at least since as the difference between elementary and pseudoelementary classes Hodges , , and is regularly rediscovered in philosophy see Demopoulos , Despite being employed as a mathematician at Cambridge Ramsey only published one paper firmly in that subject, though two ideas he introduced in his more philosophical papers a, b , 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 Ramsey then asserted that for partial belief i.
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 and independently Sherman Lehman 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 A second important mathematical contribution of Ramsey is based on the following brief footnote in his otherwise philosophical paper General Propositions and Causality b, [FM] :.
Starting with Robert Stalnaker this has been mis interpreted see the section on Conditionals as a bridge between logics of indicative conditionals Edgington , Arlo-Costa and logics of belief revision Hansson 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.
This material is uncompromisingly aimed at mathematicians and with apologies its explication herein requires some necessary technicalities. The second theorem gives a version of the first, but for finite sets of sufficiently large size. These two results are now referred to as the Infinite and Finite Ramsey Theorems, respectively. Ramsey then uses the finite theorem in this paper to show that for each sentence, from a relational language with equality, of the form.
An important innovation in his proof was the notion of indiscernibles which was to reappear, apparently independently, three decades later in Model Theory. 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. 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.
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. Other short, posthumously-published works, all of which are reprinted in the collections listed in Subsection A. The SEP editors would like to thank Fraser MacBride for helping us to organize the team of coauthors who contributed sections to this entry.