Jonny Blamey

Probability can be used to measure degree of belief in two ways: objectively and subjectively. The objective measure is a measure of the rational degree of belief in a proposition given a set of evidential propositions. The subjective measure is the measure of a particular subject’s dispositions to decide between options. In both measures, certainty is a degree of belief 1. I will show, however, that there can be cases where one belief is stronger than another yet both beliefs are plausibly measurable as objectively and subjectively certain. In ordinary language, we can say that while both beliefs are certain, one belief is more certain than the other. I will then propose second, non probabilistic dimension of measurement, which tracks this variation in certainty in such cases where the probability is 1. A general principle of rationality is that one’s subjective degree of belief should match the rational degree of belief given the evidence available. In this paper I hope to show that it is also a rational principle that the maximum stake size at which one should remain certain should match the rational weight of certainty given the evidence available. Neither objective nor subjective measures of certainty conform to the axioms of probability, but instead are measured in utility. This has the consequence that, although it is often rational to be certain to some degree, there is no such thing as absolute certainty.

Abstract Argument is seen as central to philosophy, especially epistemology. It is often said that philosophy teaches you to argue for any position. Arguments are used to justify beliefs and many people suppose that for a belief to be counted for knowledge it must be justified. In science, scientific theories must be backed by the evidence and it has been proposed that the relationship of evidence to theory is that of argument to conclusion. But is argument really so important? Arguments in practice are hardly ever persuasive, and a deductive argument adds no new information. Knowledge and wisdom of the philosophical kind can be garnered and created through observation, exploration, experiment, collaboration, narrative and sharing. None of these ways of knowing necessarily involve argument, and can actually in some cases be harmed by argument. It is fairly uncontroversial to say that argument has a central role in contemporary English speaking philosophy. But if argument is a form of combat then this emphasis on argument could be a source of gender bias in philosophy and in certain sciences. Men are physiologically attracted to combat through the mechanism of testosterone fluctuations in response to competitive situations. But apart from being attractive to some men, arguments do not create good philosophy. Good philosophy aims at the love of wisdom and knowledge. Two principle goods in philosophy are the creation of knowledge and the spreading of knowledge. Arguments neither create knowledge nor are persuasive. On the contrary, arguments create divisions and cognitive biases. As such philosophy may be better with less of an emphasis on argument, and more emphasis on non-violence, sharing and love.

In this essay I hope to establish that truth is not a necessary condition for knowledge. This is not to go so far as that it is possible to know falsehoods, since not everything that is not true is therefore false. Rather the aim is to show that knowledge is that in which we are fully confident, where our confidence is supported by conclusive evidence. If these two conditions are met, then there is no further condition, truth, that needs to be met in order for us to be in possession of knowledge.

The value of knowledge can vary in that knowledge of important facts is more valuable than knowledge of trivialities. This variation in the value of knowledge is mirrored by a variation in evidential standards. Matters of greater importance require greater evidential support. But all knowledge, however trivial, needs to be evidentially certain. So on one hand we have a variable evidential standard that depends on the value of the knowledge, and on the other, we have the invariant standard of evidential certainty. This paradox in the concept of knowledge runs deep in the history of philosophy. We approach this paradox by proposing a bet settlement theory of knowledge. Degrees of belief can be measured by the expected value of a bet divided by stake size, with the highest degree of belief being probability 1, or certainty. Evidence sufficient to settle the bet makes the expectation equal to the stake size and therefore has evidential probability 1. This gives us the invariant evidential certainty standard for knowledge. The value of knowledge relative to a bet is given by the stake size. We propose that evidential probability can vary with stake size, so that evidential certainty at low stakes does not entail evidential certainty at high stakes. This solves the paradox by allowing that certainty is necessary for knowledge at any stakes, but that the evidential standards for knowledge vary according to what is at stake. We give a Stake Size Variation Principle that calculates evidential probability from the value of evidence and the stakes. Stake size variant degrees of belief are probabilistically coherent and explain a greater range of preferences than orthodox expected utility theory, namely the Ellsberg and Allais preferences. The resulting theory of knowledge gives an empirically adequate, rationally grounded, unified account of evidence, value and probability.

Abstract The Preface Paradox, first introduced by David Makinson (1961), presents a plausible scenario where an agent is evidentially certain of each of a set of propositions without being evidentially certain of the conjunction of the set of propositions. Given reasonable assumptions about the nature of evidential certainty, this appears to be a straightforward contradiction. We solve the paradox by appeal to stake size sensitivity, which is the claim that evidential probability is sensitive to stake size. The argument is that because the informational content in the conjunction is greater than the sum of the informational content of the conjuncts, the stake size in the conjunction is higher than the sum of the stake sizes in the conjuncts. We present a theory of evidential probability that identifies knowledge with value and allows for coherent stake sensitive beliefs. An agent’s beliefs are represented two dimensionally as a bid – ask spread, which gives a bid price and an ask price for bets at each stake size. The bid ask spread gets wider when there is less valuable evidence relative to the stake size, and narrower when there is more valuable evidence according to a simple formula. The bid-ask spread can represent the uncertainty in the first order probabilistic judgement. According to the theory it can be coherent to be evidentially certain at low stakes, but less than certain at high stakes, and therefore there is no contradiction in the Preface. The theory not only solves the paradox, but also gives a good model of decisions under risk that overcomes many of the problems associated with classic expected utility theory.