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c: PROBABILITY AND CERTAINTY
ABSTRACT
I interpret probability as having an output: the subjective degree of belief using Ramseys formula, (Ramsey, F. P., 1926) which is a ratio between loss : gain in an indifferent bet. It also has an input, which is a ratio between the number of trials with the predicted result to the total number of trials. The output misses an important value: the magnitude of the gain minus the loss, in other words the maximum stake you should place on a bet with favourable odds. The input misses an important value: the total number of trials, which I shall call the sample size. I suggest that the larger the sample size, the larger the magnitude of gain minus loss the subject should be willing to bet at favourable odds. In other words the more experience you have, the more certain of your predictions you should be. Certainty so conceived will be a single open ended value measured in units of utility. This accords with natural language use of concepts like guarantees. The sample size can increase indefinitely as can the number series. This means for every degree of certainty there is a higher degree of certainty. There is no absolute certainty.
PROBABILITY AND CERTAINTY
In predicting that a singular event will have a particular property on the basis of a generalisation one can be more or less certain of ones prediction, even though ones prediction is not probabilistic. The generalisation will be that all events of that kind have that property. For example I predict with certainty that Tony Blairs death will occur before he reaches his first millennium on the grounds that all men die before their one thousandth birthday. This prediction of mine is not probabilistic. I dont predict that Blair will probably die before 6.5.2953. In simple terms this means that there are no odds so good that I will bet against Blair dying this side of the thirtieth century. Conversely I will accept any odds as favourable for a bet that he will die sooner. These are output functions for what I mean by saying that I am certain Tony Blair will die before the end of the first millennium after his birth. I also predict with certainty that he will never become Prime Minister again on the grounds that all Prime Ministers who resign from office do not return. But I am less certain of this fact. This is not to say that I find it less probable. The odds I consider fair are the same. No odds for bets that he will, any odds for bets that he wont.
This kind of inductive certainty in prediction is at best obtained when the generalisation from which the prediction is made is true of all recorded events. This is the input function for my certainty. All men have died before a thousand years old in recorded history, and all Prime Ministers, having once resigned have never returned to office. These are the facts that justify my assessment of odds. Put in this way the problem of induction is simply that just because a generalisation is true of all recorded events there is no guarantee that the generalisation will be true of the predicted event. However this absence of a guarantee is a philosophical metaphor. Literally speaking, if the generalisation has been shown through experience to have held in all instances, then there will often be plenty of people who will be willing to guarantee that a prediction made on its basis will be true. I am using the term guarantee in its most literal sense as a promise backed by financial compensation. So by way of illustration I will offer to pay you 10 if Tony Blair once more becomes Prime Minister. This is a kind of guarantee that Tony Blair will not become Prime Minister, it is a guarantee I am prepared to make on inductive grounds. If I am prepared to guarantee a prediction, then it seems fair to say that I am certain of that prediction. A guarantee can be of various amounts. I could have offered to pay you 1 000, but I am not that certain.
Now what seems to be important is not only whether the generalisation has been shown through experience to have held in all instances, but also the extent of the experience. In a rigorous scientific environment the experience can be quantified precisely as the number of trials, or the sample size. This could be a precise figure. But even the most arcane science is informed by background knowledge and typically we cant give a precise quantity to our experience. This said, given that a generalisation holds in all experience, then the greater the experience the greater the certainty of a prediction made on the basis of experience. If I had a lot more experience of politics, then, given that in my experience no one relevantly similar to Tony Blair had ever done anything relevantly similar to returning to an office he had at one time resigned from, I would have been prepared to guarantee you more that Tony Blair will not return. When I say guarantee more, I mean straightforwardly that I would be prepared to offer you more money if Tony Blair returned to office. In this example this may seem trivial, but if we think of pharmaceutical trials and the absence of horrific side effects, the kind of guarantees that scientists give should be proportional to the risks taken by those who may be potentially harmed by acting on their predictions. The same can be said of meat inspectors, aeroplane manufacturers and hedge fund managers. Meat inspectors who declare meat safe to eat should really be prepared to eat the meat themselves. The testimony of experts who say there is no risk associated with smoking should be treated with cynicism if they themselves choose not to smoke.
Predictions of singular events are given probabilities. A good interpretation of probability of singular events is subjectivist. Most simply put, a subjects probability assignment to a singular event is the fair price for a bet that pays a unit if the prediction is correct and nothing otherwise. This can be thought of as an output function. It tells you what to do once you have established the probability of a singular event. Bets shouldnt be construed too narrowly. We can think of all action in life as a kind of bet. When we cross the road, we are betting our lives that we wont get run over for the small prize of getting to the other side. When we apply for a job, we are betting the inconvenience of the application process for the prize of getting the job. Frank Ramsey created a precise measure of degree of belief. (Ramsey, 1926 p. 75). If a subject is indifferent between options: 1. A for certain, or 2. B if p and C if ~p, then the subjects degree of belief can be thought of as the ratio of pay offs:
AC
BC
This is a descriptive definition of a subjects degree of belief given indifference between the options. We can make it prescriptive by saying that given a degree of belief x, one should be indifferent between options 1 and 2 whenever the ratio of pay offs equals x. We can make this more useful and general by thinking of A as the pay off of not acting, B the pay off of acting if p is true and C the pay off for acting if p is false. (A pay off can be either positive of negative and is defined as the relative value of the world to the subject under the circumstances.) In which case the prescription is that if the ratio of pay offs is more than the probability that p then dont act, and if the ratio is less than the probability that p then act. This gives practical meaning to statements of probability. If you were to tell me that there was a probability of 0.01 of getting a particular job, I could use this information to decide whether to bother applying. Roughly speaking, if the utility of getting the job was more than a hundred times the utility of not having to apply then go for it, otherwise dont bother. So well and good, we have a consumer interpretation for probability statements.
The prescription to be certain becomes the prescription to be indifferent between options 1 and 2 whenever A = B, since if one is certain that p the choice of options 1 and 2 becomes a straight choice between A for certain and B for certain. This can be cashed as the prescription to act whenever there is any advantage at all to acting if p is true over not acting regardless of the disadvantage of acting if p is false. It should be noted that C has disappeared out of the equation. For Ramsey this was reason to not apply his formula to certainty. (Ramsey, 1926, p.76) To me it gives a dimension of movement with which to measure in consumer terms the increase in certainty that greater experience gives. The important value is B C, which is the difference in utility between acting if p and acting if ~p. The more certain you are, the larger this value can be before electing not to act. This value can be thought of as how much is at stake in the action.
Another interpretation of probability is the frequentist interpretation. Subjective probability prescribes action given the probability of an event. But how is one to assess the probability of an event? We have got our output theory, or our consumer interpretation. What we need now is a producer interpretation, or an input theory. The most general and simple input theory is that the probability that a predicted event has a particular property is the relative frequency of events in experience with that particular property. If one in three men in your experience has been bald, the probability you should assign that the next man you meet is bald is 1/3. If all the men you have met have a heart, then the probability that the next man you meet will have a heart should be 1. There are many interesting questions as to how to type the predicted event, what to count as positive instances and so on, but the general principle remains the same. We can see that as a general input theory for all epistemological issues it is inadequate for the very simple reason that it gives no extra weight to the sheer size of the sample from which the prediction is made. It also gives no extra weight to the quality of the sample, and the relevance of the sample to the event in question. For example, this input rule would prescribe the same actions whether i) I had only met six men in my life, two of whom were bald; ii) I knew that world wide one in three men are bald; or iii) I knew the next three men I was going to meet, but not in which order, and knew exactly one of them to be bald. Intuitively these three samples are epistemically different, though each would recommend a degree of belief 1/3 that the next man I meet will be bald.
It seems we could do with another input measure that increases by degree as the sample gets larger and more relevant. This could then map onto the maximum value for B C at which one should be indifferent between acting and not acting. The prescription associated with this value would be to not act in situations where too much was as stake. The prescription could be given a precise threshold utility value that is related to the quality and size of the sample. For simplicity I will ignore for the moment all probabilities less than 1 and just talk about certainty, though I have hopes that this measure will generalise. If the input is such that all known events of the predicted type have a particular property, then the probability that the predicted event will have the property will be 1. This means that one should act on the basis that the predicted event will have the property whenever such an action is preferable to not acting given that the prediction is true. However if the cost of acting should the prediction turn out false exceeds a threshold then one should not act. This threshold is a function of the quality and size of the sample.
The idea of a cut off stake size for certainty is especially pertinent when transmitting warrant through testimony. It is often the case that a more experienced and knowledgeable person will want to transmit the epistemic status of their prediction without having to say exactly what their experience consists in. For this reason there are many words in natural language which give an indication of whether the cut off stake size has been reached. Certainty is the most basic of these. It has often been pointed out that certainty is intuitively incompatible with less than 1 probability. The most famous illustration of this point is the lottery paradox whereby it feels unnatural and wrong to say that it is certain that a ticket will lose before the draw, however unlikely the ticket is to win. Yet certainty is on a sliding scale. There is nothing infelicitous in talking of increasing or decreasing levels of certainty. This paradox also applies to the term sure. Other words in this ball park are knowledge and outright belief. While not so clearly gradable, (Jason Stanley denies that knowledge is gradable at all), both seem at once to denote a probability 1 and yet to be sensitive to non evidential features of the practical situation. Both Williamson and Millikan have noticed this link between probability 1 and variation due to stake size.
"Outright belief still comes in degrees, for one may be willing to use p as a premise in practical reasoning only when the stakes are sufficiently low". (Williamson, T. 2000. pp.99.)
"Unless the stakes are high, we do not require of the person who "knows that p" because he has been told by so-and-so that he be ready to place extremely high stakes on the truth of p," (Millikan, Ruth.1993. pp 253)
The way to look at it is this: if you were in a betting shop and had to decide on whether to bet on p, there are two questions to be asked. At what odds should you bet on p and how much should you bet? These questions can be translated as how probable is p and how certain is p? The first figure will be a ratio that conforms to the axioms of probability calculus. The second will be an integer value with no absolute limit. Both these values have their place in ordinary talk, but in science people only use probability. (Of course other features of the data can be represented in some cases by statistical conventions like standard deviation, but in cases where the probability of an event is 1, standard deviation doesnt have any meaning.) When transmitting warrant for a prediction that a particular event is certain to happen we should expect to find a plethora of ways to indicate the size and quality of the data from which the prediction was made. This is the case. Im certain that p, I am sure that p, p must happen, it is definite that p, it is inevitable that p, Id stake my life that p, Im relatively certain that p, I guarantee that p, I promise you that p, Ill bet the farm that p, Ill stake my reputation that p, Ill bet you a million pounds that p. These last four may sound like bets that could be interpreted as subjective probability claims. But it will be noticed that no odds are mentioned, just a single value to be extracted from the speaker if the prediction is wrong. These phrases can be seen as indicators as to how much the listener should rely on the testimony of the speaker, which can be given a precise value in terms of the maximum stake size at which to act on the basis that the prediction is certain.
Having a separate value that links sample size to stake size is not the mainstream view. The mainstream view I take to be that the probability of the prediction is a conditional probability on the probability that the generalisation is true. The increase in certainty due to a larger sample size on this view is just and increase in probability. In terms of the example at the beginning of the paper, it is less probable that Tony Blair will live to be a thousand than it is that he will once more be Prime Minister, though both eventualities are highly improbable. The increase in the sample size simply increases the extent to which the generalisation is confirmed. As the sample increases in size, Bayesian dynamics will ensure that the probability of the generalisation will tend towards 1 without ever reaching 1. This has the consequence that no predictions are certain since no generalisation is ever certain. Any singular prediction will have a probability conditional on the probability of the generalisation being true. This has the consequence that the probability of a prediction is always lower or equal to the probability that the generalisation is true.
First off this is counterintuitive since a singular prediction is a lesser commitment than a generalisation. I am intuitively a lot more certain that the next person I meet has a heart than I am that all living humans have a heart. Secondly the generalisation is likely to be tailored to the prediction rather than the other way around. If I tell you that I am certain that that pill wont kill you I am not necessarily committing myself to a generalisation that all pills wont kill all people, or even that no pill of that kind ever kills anyone. Some aspirins may kill some people, some pills that find their way into aspirin bottles may kill anyone. What I am guaranteeing is that this aspirin wont kill you.
There is another problem in store for assigning high probabilities to predictions instead of certainty. This is that predictions made on the basis of small samples where all recorded events fit the generalisation are given the same probability as prediction made on the basis of much larger samples where only a high proportion of recorded events fit the generalisation. To illustrate, let us suppose that we want to predict that Jones has X in his blood stream. Here are three sets of data which we can use to make the prediction:
Case 1. We have tested 10 people relevantly similar to Jones for X and all 10 have tested positive. Using Bayesian dynamics and intuitive priors we make a prediction that Jones has a 0.99 probability of have X in his blood.
Case 2. We have tested 1 000 000 people for X and discover that only 990 000 have X in the blood. We change the hypothesis so that it states that 99 % of people have X in their blood. We conclude that there is a 0.99 probability that Jones has X in his blood. (Let us suppose that the test for X is infallible).
Case 3. We have tested 1 000 000 for X and discover that all 1 000 000 have X in their blood. Using Bayesian dynamics we conclude that the hypothesis that there is X in the blood of all humans has a probability of 0.9999999. As before we use this to predict that Jones has a probability of 0.9999999 of having X in his blood.
What is wrong here is that Case 1 yields the same prediction as Case 2, yet the data is very different. Intuitively Case 1 should be at least neutral between the other two cases, if not positively favouring Case 3. To make vivid this intuition let us suppose that Jones has been injected in the arm with substance Y. If Jones does not have X in his blood the Y will kill him within half an hour. The only way to save him would be to amputate his arm. If Jones does have X in his blood, then he is in no danger. Would a reasonable man advise Jones to amputate his arm given either of the three cases above? Lets suppose that Jones would much prefer to live without his arm than to die, but would equally prefer not amputating his arm and living over having his arm amputated needlessly. I would only advise him to amputate given Case 2. Given Case 1 and 3, I would tell him he was safe. I would be a lot less certain in Case 1 than in Case 3. In Case 2, I would tell Jones that he had a 1 in a hundred chance of dying if he didnt amputate his arm. If I told him this in Case 1 I would feel that I was misleading him. The best way to transmit the warrant given by Case 1 would be to say that I am certain that he has X is in his blood, but not that certain. If he doesnt have X in his blood then he will die unless he amputates his arm. To say more than this will be false precision and going beyond the evidence. If Jones pressed me further, what more can I do than tell him that 10 out of 10 people who matched his profile would be perfectly safe and no one who matched his profile would have died in his situation?
Another argument against probabilism is that Bayesians are committed to the idea that no amount of confirmation can give us a degree of belief 1. Also that degree of belief 1 is monotonic, which means that any proposition believed to degree 1 cannot through Bayesian dynamics come to be believed to degree less than 1. This has the consequence that either we should never be certain of anything, or that once certain, no new evidence can undermine our certainty. Neither disjunct is very attractive. We are certain of many things and occasionally when we are certain we are also wrong.
A third possibility is that the term certainty as used by the folk is to be interpreted as a Bayesian degree of belief over some threshold less than 1. But this is also problematic for the Bayesian who considers the degree to which evidence confirms a theory (T) to be the probability that the theory is true conditional on the evidence. This has the consequence that the results of a scientific experiment have a probability 1. If the result of a scientific experiment was (e), and the prior probability of (T | e) = n, then the new probability of T should be n. This has the implicit assumption that if (e) is scientific evidence, then the probability (e) = 1. A reason to reject probabilism is that it would seem peculiar if the results of scientific experiments had a higher probability than certainty since even scientific observations are fallible. So if the results of scientific experiments have a probability 1, then propositions of which we are certain must have a probability of at least 1. But there can be no probability higher than 1. Further more if we allow observations to have a probability of 1, and yet the threshold of certainty to be less than 1, we get an alarming failure of modus ponens.
Suppose for the sake of reductio that a proposition is certain if it has a probability of at least 0.99. Suppose that (h|e) is certain and (e) is certain. Then the probability (h) could be less than probability (e). But this is absurd, since it has the consequence that (e) could be certain and the probability of (h|e) also certain and yet (h) be less than certain. For example let (h) be that Jones is mortal and (e) be that Jones is dead. Let us say that (e) is certain and (h|e) is certain and this means that they both have probability 0.99. Suppose also that it is certain that Jones is dead. It is possible, with the right prior probabilities, that the posterior probability that Jones is mortal should be less than 0.99, and therefore not be certain. The absurd consequence is that the certainty that Jones is dead does not entail certainty that he is mortal even on the assumption that it is certain that Jones is mortal given that he is dead. These arguments I think are conclusive that if certainty implies any probability at all it implies probability 1. Probabilism therefore cannot explain varying degrees of certainty. This leaves variation due to stake size as the only account available.
A fifth fairly well known reason to reject probabilism is that the generalisations themselves must be counterfactual supporting since we can make predictions of counterfactual events of which we are certain. This has to be the case otherwise we could not use predictions in order to avoid doing actions with dangerous or fatal consequences. If I shoot myself in the head then I am certain that I will die therefore I will not shoot myself in the head. Because generalisations have to be counterfactual supporting, the population over which they are quantified is infinite. Since any data set is going to be finite, the subjective probability that the generalisation is true can never raise above 0 since the generalization can never be confirmed. Also, if the rationale behind the increase of probability of the generalisation due to increase in sample size is that the bigger the sample, the more likely it is to accurately represent the total population, then the rationale fails for infinite populations. (Strawson, 1956 p253 -255) This is because the constitution of any sample of any size has a zero probability of matching the constitution of the total population since any number divided by infinity is zero.
So from the input side there is plenty of reason to endorse an independent measure of certainty which is a function of the size and quality of the sample. It is not hard to find criticisms of subjective probability on the output side as well, and these can easily be tackled by having an independent certainty scale. The focus of the criticisms is that subjective probability is insensitive to the subjects attitude to risk and to the diminishing marginal utility of money. The most famous criticism is from Allais problem. (Allais, 1956), who showed that intuitively non gambles are more valuable than gambles even when the expected utility is the same.
My strategy is to use B C, as a separate measure of certainty. B is the value of an action conditional on p and C is the value of the same action conditional on ~p. Think of B C as representing what is at stake in assuming p with regard to a specific action On my view there is no need to assume that the rational degree of belief will remain the same as B - C varies in magnitude. The assumption that the magnitude B - C has no affect on the rational degree of belief is demonstratably false since if B - C is 0 then the truth or otherwise of p becomes irrelevant to the choice of whether to act or not and the subjects degree of belief that p becomes undefined by her choice of actions. This is a purely mathematical result since whenever the denominator is 0, the value of the fraction is undefined. The Allais problem exploits this and presents a series of options one of which has B - C as zero. It seems intuitively rational to favour the option where B - C is zero, especially when B and C are themselves of great positive utility. (Anecdotally Savage found himself preferring the sure gain over the higher expected utility, going against his own theory. However, instead of amending his theory, he amended his intuitions.) Seeing this mechanism it is easy to come up with Allais type problems for subjective probability. Given the choice of either
Option 1: WIN 1 million if a coin lands heads and WIN 1 million if it does not land heads, or
Option 2: WIN 50 million pounds if 2 sixes are thrown on a fair die but LOSE 100 000 if two sixes are not thrown.
It would be a reckless person who chose option 2 even though the expected utility is much higher. Please ask yourself if, on a one offer of this kind, you would really view it as more rational to go for a 1/36 chance of winning 50 million and a 35/36 chance of losing 100 000, over a certain million?
Also if one could bet as much as one liked on the single toss of a coin at odds of 11:20 in one's favour, how much should one bet? Surely not every penny to one's name. Commentators are split on these issues. Ramsey, (1926 p.72) Friedman and Savage (1948), and Jeffreys (1970, p.161 -2) were persuaded that these problems can be solved by recourse to a robust utility theory. Ramsey added the value A to counterbalance the subject's attitude to bets. He presumed that his utility theory would do the rest. Others think these problems are decisive and that subjective probability theory (Kaplan 1996, p172) (Allais, 1953) should be seriously modified or abandoned. These latter tend to ignore the solution inherent in adding the counterbalancing value A for not accepting the bet. I am suggesting a middle ground. This is to drop the assumption that the rational bettor should accept the same range of odds whatever the magnitude of the difference between the loss and the gain. This assumption is counterintuitive, as the Allais problem shows. It is also demonstratably false since when B - C = 0 then the bettor is rational to accept any odds. Furthermore it is not argued for by a Dutch book argument since there is no way a cunning bettor can have simultaneous bets at different stake sizes, since this would be in effect to have a single bet with the sum of the various stake sizes. (If I bet you 10 that it is raining and 100 that it is raining, I am not making two bets, I am making one bet of 110 that it is raining.)
To conclude: a subjective probability theory could be augmented by an independent scale of certainty that would transmit the warrant given by greater experience into actions of greater risk. This value has its place in Ramsey's measure for degrees of belief already as the magnitude of the denominator B - C. This amounts to the difference in value of an action based on a prediction given that the prediction is correct and given that the prediction is incorrect. On the input side the sample size dictates the maximum value for B C by assessing the cost of attaining the sample. If the cost of a false prediction exceeds the cost of the experiences that warrant the prediction then one shouldnt act on the prediction. This has a straightforward rationale because if the cost of being wrong is more than the cost of doing more research, then it seems straightforward that one should do more research. If on the other hand the cost of being wrong is less than the cost of doing more research, then there is no reason not to proceed with certainty. It also must be noted that whenever one acts on a prediction, one increases ones knowledge. I have presented a possible addition to subjective probability theory that links features of the input: sample size and relevance, to features of the output, stake size and certainty.
The important difference between degrees of certainty and probability is there is no argument available to show that the likelihood increase due to the increase in sample size conforms to the axioms of probability calculus so there is no reason why degrees of certainty should be expressed as a ratio between 0 and 1. We can now explain sceptical problems as stemming from probabilism. Sceptical problems are premised on the assumption that where there exists the possibility of doubt there cannot be certainty. Certainty is supposed to be probability 1. We then get the conclusion that where there exists the possibility of doubt, there exist a probability of less than 1. Sceptical hypotheses simply show that anything can be doubted. A survey of the philosophy of logic reveals that even logical tautologies can be doubted. Consequently nothing is certain. If however certainty is express by a single integer, then certainty can increase infinitely. This has the consequence that there can be no highest level of certainty. The observation that any proposition is capable of being doubted then has the trivial consequence that for any degree of certainty, there exists a higher degree of certainty. For any prediction based on a sample size there exists the possibility of a more certain prediction based on a greater sample size.
I predict with certainty that Tony Blair will never again be Prime Minister. If you measure my degree of belief in this prediction it is 1, since I will accept any odds on this bet and no odds on a bet against. However, there are predictions that I can make that are more certain than this prediction and predictions I can make that are less certain. We can measure this non probabilistic scale of certainty in terms of degrees of certainty. This can be given an input function which is that ones certainty should increase with the breadth, depth, relevance and size of ones sample. It can be also given an output function, which is the difference between the value of being correct and the value of being incorrect. The output function can be expressed naturally as the amount you would guarantee that your prediction is correct.
Bibliography
Bibliography.
Allais, Maurice. 1953. Le comportment de lhomme rationnel devant le risqu: critique des postulats et axioms de lecole Americaine, Econometrica 21: 503 46.
Descartes, Rene. 1984/85. The Philosophical Writings of Descartes, 2 vols. John Cottingham, Robert Stoothoff and Dugald Murdoch (eds. and trans.), CUP. Vol 2
DeRose, Keith. 2002. Assertion, Knowledge, and Context Philosophical Review, Vol. 111, No. 2. Apr.,. pp. 167-203
Friedman, M and Savage, L. J. (1948) The Utility Analysis of Choices Involving Risk Journal of Political Economy 56, pp 279 -304.
Gettier, E.L. 1963. Is Justified True Belief Knowledge In Analysis 23 pp. 121-123
Hawthorne, James and Bovens, Luc. 1999. The Preface, the Lottery and the Logic of Belief Mind, Apr., Vol.108, pp. 430.
Jeffrey, Richard C. 1970. Dracula meets Wolfman: Acceptance VS. Partial belief. In Swaine, M. ed. Induction Acceptance and Rational Belief. D. Reidel Publishing Company, Dordrecht Holland.
Kaplan, Mark. 1996. Decision Theory as Philosophy. MIT
Kaplan, Mark. 2006. Coming to terms with our human fallibility Epistemic Value Conference and Pre-Conference Workshop August 18th-20th, Stirling Management Centre
Kyburg, Henry, Jr. 1961. Probability and the Logic of Rational Belief. Wesleyan University Press.
Makinson, David C. 1965. The Paradox of the Preface Analysis, 25, pp. 205-7.
Millikan, Ruth. 1993. White Queen Psychology MIT.
Quine, W. V.. 1977. Natural Kinds. In Naming Necessity and Natural Kinds ed. Stephen P. Schwartz. Cornell university Press
Ramsey, F. P. 1926. Truth and Probability. In Philosophical Papers, ed. D.H.Mellor. 1990. C.U.P.
Stalnaker, 1984. Inquiry. MIT.
Jason Stanley Knowledge and Practical Interests OUP 2005
Strawson, P. F.1956. Introduction to Logical Theory Methuen
Williamson, T. 2000. Knowledge and its limits. OUP.
This was from a talk he gave in St Andrews and London in 2006 called Knowledge and Certainty.
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