"Probability is the very guide of life," in Bishop Butler's famous phrase. He does not mean, of course, that calculations about dice are the guide of life, but that real decision-making involves an essential element of reasoning with uncertainty. Humans have coped with uncertainty without the benefit of advice from mathematicians, both before and after Pascal and Fermat's discovery of the mathematics of probability in 1654. And they have talked and written about how to do so. So, there is a history of probability that concerns neither mathematics nor anticipations of mathematics.

This is a history of rational methods of dealing with uncertainty. It treats, therefore, methods devised in law, science, commerce, philosophy and logic to get at the truth in all cases where certainty is not attainable. It includes the evaluation of evidence by judges and juries, legal presumptions, the balancing of reasons for and against scientific theories, drug trials, and counting shipwrecks to determine insurance rates. It excludes methods like divination or the consulting of oracles, which are substitutes for reasoning about uncertainty.

Three levels of probabilistic reasoning are distinguishable:

• Unconscious inference, the reactions to uncertain situations that the brain delivers automatically at a sub-symbolic level. This system of actions – the cloud out of which, so to speak, talk about uncertainty condensed – can be studied by psychology, but it is not history. A review of what is known about it, and its relation to conscious inference, is given in the Afterword
• Ordinary language reasoning about probabilities. It is this middle level that is the main subject of this book. It may avoid numbers entirely, as in "proof beyond reasonable doubt" in law, or the non-numerical judgements of plausibility that scientists and detectives make in evaluating their hypotheses. Or it may involve rough numerical estimates of probabilities, as in racecourse odds, or guesses about the risks of rare events.
• Formal mathematical reasoning, of the kind found in textbooks of probability and statistics.

The higher levels may be more noble and perfect, but they are so at a cost: they are less widely applicable.

The theme of this book, then, must be the coming to consciousness of uncertain inference. The topic may be compared to, say, the history of perspective. Everyone can see in perspective, but it has been a difficult and long drawn out effort of humankind to become aware of the principles of perspective, in order to take advantage of them and imitate nature. So it is with probability. Everyone can act so as to take a rough account of risk, but understanding the principles of probability and using them to improve performance is an immense task.

There is one further essential distinction to be made. Probability is of two kinds. There is factual or stochastic or aleatory probability, dealing with chance set-ups such as dice-throwing and coin-tossing, which produce characteristic random or patternless sequences. Almost always, in a long sequence of coin tosses, there are about half heads and half tails, but the order of heads and tails does not follow any pattern. On the other hand, there is logical or epistemic probability, or non-deductive logic, concerned with the relation of partial support or confirmation, short of strict entailment, between one proposition and another. A concept of logical probability is employed when one says that, on present evidence, the steady-state theory of the universe is less probable than the big bang theory, or that an accused's guilt is "proved beyond reasonable doubt", though not absolutely certain. How probable a hypothesis is, on given evidence, determines the degree of belief it is rational to have in that hypothesis, if that is all the evidence one has that is relevant to it.

It is a matter of heated philosophical dispute whether one of these notions is reducible to the other, but in any case the surface distinction is clear, and provides an orientation in the history of the subject. In the period covered by the present study, logical probability was the main focus of interest, and the word "probability" was reserved solely for this case. The little study of factual probability there was, concerned with dice and insurance, was not seen as connected with logical probability.

Consequently, the book opens with three chapters on the law of evidence, where there has been the most consistent tradition of dealing explicitly with evidence that falls short of certainty. Conscience, conceived as a kind of internal court of law, could also be in doubt; the rule of "probabilism" concerning it is the subject of the fourth chapter. The fifth chapter describes the (not very successful) attempts by rhetoricians and logicians to give some account of uncertain reasoning. Evidence for scientific theories (understanding "science" widely) is considered in the next two chapters, followed by two chapters on probability in philosophy and religion, dealing largely with inductive arguments and design arguments for the existence of God. The tenth chapter describes commercial and legal thought on the nature of "aleatory contracts" — agreements like insurance, annuities and bets whose fulfilment depends on chance. One aleatory contract has outcomes that can be exactly evaluated mathematically: gaming with dice; it is the subject of the last chapter.

The reader with an average familiarity with received ideas on intellectual history is asked to make a small number of reorientations, at least provisionally.

The first concerns probability specifically. Two points should be made, to avoid perceptions that early writers are indulging merely in confused "anticipations" of later mathematical discoveries. The first is that the process of discovering the principles of uncertain reasoning is far from over. It can sometimes appear that, beginning with Fermat and Pascal's success with dice in 1654, there has been a successful colonisation of all areas of uncertain reasoning by the mathematical theory of probability. As in so many areas, the arrival of the computer has shown that previous knowledge about thinking processes was not nearly as precise as had been thought — not precise enough, in particular, to allow a complete mechanical imitation of them. More will be said about this in the Epilogue; suffice it to say here that there is no agreement on, for example, how to combine evidence for conclusions in computerised expert systems for medical diagnosis. The disagreements are fundamental, and are about quite simple issues that have occupied thinkers about uncertain inference for two thousand years: how to decide the strength with which evidence supports a conclusion, how to combine pieces of evidence that support each other, and what to do when pieces of evidence conflict.

The second point is that while the probability of outcomes of dice throws is essentially numerical, and advances in understanding are measured by the ability to calculate the right answers, it is otherwise with logical probability. Even now, the degree to which evidence supports hypotheses in law or science is not usually quantified, and it is debatable whether it is quantifiable even in principle. Early writers on probability should therefore be regarded as having made advances if they distinguish between conclusive and inconclusive evidence, and if they grade evidence, by understanding that evidence can make a conclusion "almost certain", "more likely than not" and so on. Attempts to give numbers to those grades are not necessarily to be praised. One should not give in to the easy assumption that numbers are good, talk bad.

The other requested reorientations concern two features of the history of ideas generally. They will seem strange to anyone even slightly familiar with the usual portrayals of the rise of modern science. The template "Antiquity — Medieval Decline — Renaissance — Scientific Revolution" does not fit the history of probability; certainly not the history of logical probability. In particular, it is not possible to read the story with the medieval scholastics as "them" and the men of the seventeenth century as "us". The scholastics made many advances in the clarification and deepening of concepts, such as is necessary to understand probability. And contrary to the myths put about by their many enemies, they explained themselves perfectly clearly.

Finally, the reader is asked to regard it as normal to find many ideas developing in legal contexts. Like the scholastics, lawyers are often thought of as pursuing esoteric interests of little consequence for the outside world, and as, by and large, enemies of scientific progress. It will be argued that the prominence of both scholastics and lawyers is not unique to probability, but that their contributions to the development of modern ideas generally have been substantially underrated. A brief overview of their wider importance in the history of ideas is given in the Afterword, in order to situate the development of probability in its appropriate context.

It will be useful to keep a few questions in mind while reading the detailed history. Previous researchers in the field have wondered why the development of probability theory was so slow — especially, why the apparently quite simple mathematical theory of dice-throwing did not appear until the 1650s. The main part of the answer lies in appreciating just how difficult it is to make concepts precise, especially when mathematical precision is asked for in an area which seems at first glance to be imprecise by nature. Mathematical modelling is always difficult, as is evident in contemporary parallel cases such as the mathematization of continuity that led to the calculus of Newton and Leibniz. The very idea of a "geometry of chance", as Pascal put it, is revolutionary. An evaluation of this and alternative explanations of the slowness of the rise of probability will be given in the Afterword. It will be suggested that, nevertheless, some mystery remains.

The book has an unusually high proportion of quotation. It is in the nature of the material that, once a small amount of context has been supplied, the authors can be allowed to speak for themselves. Paraphrase is pointless. The book is to be read in only one place at a time: the footnotes contain references only, the purpose of which is solely to increase the reader's degree of belief in the statements referenced. It is written in only one language, except for occasional words from the original language of texts, included to indicate that there is no overinterpretation through tendentious translation.

The purpose of history may not be to teach us lessons, but the story told here does have a certain contemporary relevance, even though it ends in 1660. The last century of the old millennium saw a gradual waning of faith in the objectivity of the relation of uncertain evidence to conclusion. In the philosophy of science, Popper, Kuhn and their schools denied that observational evidence could make scientific theories more probable, and attention in the field moved to sociological and other non-evidential influences. Postmodernism, presuming rather than arguing for the absence of objective methods of evaluating theories, offered a number of other reasons — or rather causes — of actual beliefs, such as the demands of "power".

The situation is not so bad in law, which has largely retained a commitment to the objectivity of evidence, but even there, theory is not as robust as practice. The past is a counterweight to these febrile inanities of pygmies who stand on the shoulders of giants only to mock their size. Just as one who feels battered by the relentless enfant terriblisme of "modern" art or music can revive his spirit by communion with Vermeer or Mozart, so the friend of Reason can draw comfort from the achievements left by the like-minded of the past. The story of the discovery of rational methods of evaluating evidence can serve as a point of reference and can supply material for the defences of rationality that will have to be undertaken.

Acknowledgements: I owe a great debt to David Stove, who introduced me to the subject of logical probability, and provided constant encouragement as well as reading and commenting on the manuscript.

For advice on various matters, I thank James Brundage, David Burr, Hilary Carey, Michael Cowling, Anthony Garrett, Paul Gross, Grahame Harrison, Michael Hasofer, Jamie Kassler, Roger Kimball, Joe McCarthy, Br Michael Naughtin, Carlos Pimenta, Margaret Sampson, Sandy Stewart, Jenny Teichman, E.H. Thompson, Stephen Voss, John O. Ward, George Winterton, and the Institute of Medieval Canon Law, Berkeley.

For financial support, I am grateful to Enid Jenkins and the University of Sydney's Thyne Reid Memorial Fellowship.

1. J. Butler, The Analogy of Religion, Introduction, §4 (in The Works of Joseph Butler, ed. W.E. Gladstone, Oxford, 1896, vol I p. 5).
2. Introductions to this distinction in I. Hacking, The Emergence of Probability (Cambridge, 1975) ch 2; D.C. Stove, Probability and Hume's Inductive Scepticism (Oxford, 1973) ch 1.
3. J.M. Keynes, Treatise on Probability (London, 1921) (all probability is logical); J. Neyman, `Outline of a theory of statistical estimation based on the classical theory of probability', Philosophical Transactions of the Royal Society of London A 236 (1937): 333-80 (all probability is factual); R. Carnap, Logical Foundations of Probability (London, 1950) (neither factual nor logical probability is reducible to the other).
4. D.C. Stove, Popper and After: Four Modern Irrationalists (Oxford, 1982), repr. as Anything Goes (Sydney, 1998).
5. For the situation in historiography see K. Windschuttle, The Killing of History (N.Y., 1997), especially ch 7.
6. W. Twining, Rethinking Evidence (Oxford, 1990), especially ch 4; symposium in Hastings Law Journal 49 (2-4) (1998).

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