PROBABILITY: THE LOGIC OF THE LAW – A RESPONSE
(This is an author-produced electronic version of an article published in Oxford Journal of Legal Studies following peer review. The definitive publisher-authenticated version 1995 14 Oxford Journal of Legal Studies 51-68 is available online here)
In their article ‘Probability: the logic of the law’ (1993), Robertson and Vignaux (henceforth RV) argue that the (quantitative) rules of probability are the ‘uniquely determined set of rules for conducting inference’; and RV make it clear that they intend no qualification to this -- so that they really are saying that rational inference always involves following these rules as closely as possible. I wish to contend that, while compliance with logical rules (including quantitative rules of probability, where they are applicable), is necessary for satisfactory inference-drawing or legal fact-finding, it is very far from sufficient; and that the most important and interesting aspects of satisfactory fact-finding lie elsewhere.
There is a considerable literature on the application of rules of probability to legal inference; but in this article I will take two new approaches. The first is narrow and focused, giving a complete hypothetical case, outlining the kind of reasoning one might expect to find in practice, and considering how such a case could be decided by application of quantitative rules of probability. The second is broad and eclectic, relating the debate to wider philosophical issues concerning the formalization of human rationality and the nature of the human mind. In between, I briefly outline my views on the proper role for quantitative probability in legal decision-making, and I question RV’s approach to combining probabilities.
1 A hypothetical case
Let me begin with my hypothetical case. The facts are fairly simple but realistically messy; and the problem is greatly simplified by the fact that there is only one controversial witness (upon whose demeanour nothing turns), no significant questions of accuracy of recollection or inadvertent reconstruction of events, and no evidence of conversations the precise terms of which could have legal significance. Yet I will be saying that it would be patently absurd to try to decide this case by applying quantitative rules of probability.
In 1982, a de facto couple William Smith and Jane Jones buy a house in Carlton, Melbourne, as joint tenants, for $80,000. They borrow $50,000 from a bank, on the security of a mortgage over the property, which they both sign as borrowers. The balance comes from the sale of jointly owned property. They have no children, they are both employed as high school teachers, they each own a car and have some individual savings, but (so far as the evidence goes) they have no other major assets.
About two years later, William buys a house in Collingwood, Melbourne, for $85,000, borrowing $60,000 from a building society. The building society has a stated policy of not lending to anyone with another house, and in the form applying for the loan William states that he has no interest in any other house or land, that he intends to reside in the Collingwood house, and (in giving the sources of the $25,000 he has to contribute) that a Ms Jane Jones is to provide $8,000 towards the acquisition.
About three months after settlement of the purchase, William goes to live in the Collingwood house, ending the de facto relationship.
At that time, there is still about $50,000 owing on the Carlton house, and the title deed is held by the bank. No lawyers are consulted, and nothing is done to change the legal ownership of this property, and no document evidencing any agreement affecting the ownership is produced to the court.
For the next nine years, Jane lives in the Carlton house, paying all outgoings and mortgage interest, and paying off all but $1,000 of the mortgage principal. William lives in his Collingwood house, paying off his mortgage to the building society. Neither of them forms any other long-term relationship, and they remain on reasonably cordial terms.
In 1993, Jane dies suddenly. Her will appoints her sister (who has lived in Western Australia since 1980) her executrix and sole beneficiary of her estate. Three months later, William goes to the bank, pays the $1,000 owing on the mortgage, obtains the title deed, and claims outright ownership of the Carlton house as surviving joint tenant.
Jane’s sister brings proceedings seeking a declaration that the estate beneficially owns the property; or else has a half interest in it, and/or the benefit of a charge arising out of the payment of $49,000 of the mortgage principal.
The only controversial oral evidence in the case (which lasts one day) is given by William. He says he purchased the Collingwood house mainly as an investment but also as somewhere to live if the relationship with Jane should break down; that although at one stage he intended to borrow $8,000 from Jane, he was in the event able to find the whole $25,000 himself, with the help of a personal loan of $8,000 from another financier; that Jane gave him no financial assistance with the purchase, and there was no agreement or discussion between them concerning his interest in the Carlton property or his liability under the joint mortgage to the bank. He says that, although he told lies to the building society to obtain the loan, he regards giving evidence on oath in court as a very serious matter, and he is telling the truth.
The only bank records still available from the time of the purchase of the Collingwood property are some of William’s bank statements: no records are available concerning his alleged personal loan, or of Jane’s bank accounts at the time. William gives evidence about the sources of the other $17,000 of the purchase price (savings, earnings, sale of car), but cross-examination shows he does not account for at least $4,000 of this $17,000.
The judge gives the following reasons for her decision in the case.
I first consider how much reliance can be placed on William’s sworn evidence. (1) He was vigorously cross-examined, but maintained his version of events, and nothing in his demeanour assists me one way or the other. (2) If his evidence is true, he was prepared deliberately to misrepresent important matters to the Building Society to gain a financial advantage, so he may be willing now to give false evidence to gain a financial advantage. Further, (3) it seems unlikely that, in a reasonably cordial parting between these two people, there should be no agreement or even discussion concerning the house they had lived in or the joint mortgage liability in respect of it, particularly if another house was being purchased as part of the separation process. (If William had admitted to a discussion, he would have been cross-examined about its terms, and his answers tested against a range of circumstances). It also seems unlikely (4) that the Collingwood property was purchased primarily as an investment, and not as part of the process of separation, having regard to the means of the parties and the fact that they parted soon afterwards, or (5) that Jane made no contribution at all to the price of the Collingwood property, having regard to the intention stated to the Building Society and William’s failure to give any explanation of the source of $4,000 of the price. (6) While I could give William the benefit of the doubt (that his evidence was false, and deliberately so) on one or two of these matters (3) to (5), the three of them added to (2) lead me to decide that he is probably willing to lie on oath in order to advance his case, and probably has done so.
I now have to consider what, if any, positive inferences can be drawn from the other facts and evidence, and in particular whether I can infer that there was an agreement by William to give up his interest in the Carlton house in return for some appropriate consideration. (7) Such an agreement has some support from William’s statements to the Building Society which, if there was such an agreement, could be considered true in substance if not strictly true in form; although (8) I must allow for the fact that they were made by a person I believe is probably willing to lie on oath. (9) I think it is quite probable that Jane did provide the $8,000 which William told the Building Society she would provide, since I have no reason to doubt that her intention was as William said it was in his application, and (10) I have no confidence in William’s evidence about a personal loan from a financier; and (11) I think it is also quite probable that, in one way or another, Jane provided the $4,000 which William did not account for. (12) I would expect that the consideration, in any such agreement, for William’s interest in the Carlton house would approximate the value of a half interest, that is about $15,000 or perhaps a little more, and there is positive support in the evidence for payments by Jane to William of no more than about $12,000; but the evidence does not rule out other adjustments between the parties. (13) The parties failed over nine years to give proper legal effect to any such agreement, but this by no means precludes the existence of such an agreement, because no lawyers were involved in the separation, the parties remained on cordial terms, and Jane would have got possession of the title deed when she had finished paying off the mortgage. (14) Indeed the fact that for as long as nine years William was living in and paying the mortgage on his house, while Jane was living in and paying the mortgage on the Carlton house, supports the existence of an agreement of that general kind. (15) I am more confident about drawing an inference against William, because whereas Jane is unable to tell her story, and there is no suggestion that there is any other person in a position to know the facts, William has probably given deliberately false evidence, suggesting that the truth is probably against his interests; and all in all, I am satisfied, on the balance of probabilities, that such an agreement was made.
The judge goes on to decide that the lack of any evidence of documentation of the agreement is overcome by part performance, and decides that Jane’s estate is entitled to the Carlton house (subject to reimbursement of $1,000 to William). In case an appeal court might take a different view, she also considers whether or not a case was made out for severance of the joint tenancy, and/or for an equitable charge in favour of Jane’s estate; but I will not set out her reasons on these matters. The judge gives her decision the morning after the hearing, and then goes on to another case, no less resistant to decision by application of quantitative rules of probability.
The judge’s reasoning is not conclusive, and some may disagree with her decision; but no one could deny that the reasons are rational and quite persuasive. I believe most people would prefer to have their cases decided on the basis of reasons of that kind, than to have them decided wholly by a mathematical computation of probabilities, which almost no one could understand, so that a case like my example might be decided by a computed probability of 0.51 in favour of one result or another - even if such a computation were possible. More to the point, I contend it is not generally possible (as well as being wildly impractical).
B.2 The application of quantitative probabilities
How would RV’s ideal judge tackle this case?
According to them (p462), it should be decided by application of Bayes’ theorem, which is an important theorem of probability theory, devised in the eighteenth century by the Revd Thomas Bayes. It states:
P(H|E) = P(H) x P(E|H) / [P(H) x P(E|H) + P(not-H) x P(E|not-H)]
P(H|E) is the probability of the hypothesis, given the truth of the evidence (posterior probability of the hypothesis);
P(H) is the probability of the hypothesis, before considering the evidence (prior probability of the hypothesis);
P(E|H) is the probability of the evidence, given the truth of the hypothesis;
P(not-H) is the (prior) probability of the falsity of the hypothesis;
P(E|not-H) is the probability of the evidence, given the falsity of the hypothesis.
To apply this theorem, real numbers should be used to represent degrees of belief, with 1 representing certainty of truth and 0 representing certainty of falsity. Intermediate degrees of (reasonable) belief should be represented by appropriate intermediate numbers, with 0.5 representing equal likelihood of truth and falsity, and so on. RV recommend that inference proceed by assigning such numbers to the probabilities of hypotheses, given certain evidence, and vice versa. These probabilities should then be manipulated in accordance with the uniquely determined rules of probability (including Bayes’ theorem) to find the probability for the ultimate question before the court.
In attempting to apply this method to my hypothetical case, an initial problem is that every one of the judge’s statements (2) to (15) are judgements of probability entirely unsupported by any applications of rules of probability. They are judgements or opinions, based on nothing more than common sense, experience of the world, and beliefs as to how people behave (folk psychology). If it is accepted, as I believe it must be, that the case has to be decided through consideration of statements of this general kind (not necessarily the same ones as the judge used), then the postulated ideal judge would either have to accept them unsupported by rules of probability, or somehow justify them by applying rules of probability.
I would be intrigued to see an attempt to justify a numerical probability for any of these statements by applying rules of quantitative probability to certain or reliable information - for example, the statement in (2) to the effect that a person who has lied in a substantial way to a building society, to get a favourable loan, may be willing to give false evidence in court many years later, to win a case. But I don’t believe I will see such an attempt, because (I think) it would be an absurdity. And the same applies to all the other statements.
If this is accepted, then the application of the rules of probability has to begin after statements of that general kind are accepted. This is already a huge inroad into the suggestion that the rules of (quantitative) probability are the rules for conducting inference; because some very important steps in conducting the inference in this case (perhaps the most important) must first be taken without regard to those rules.
Overlooking this, RV’s ideal judge now tries to reach his conclusion from statements of this kind by applying the rules of probability to them. First, he will need to represent the probability involved in each statement by a number. From the previous discussion, the number cannot be calculated. The judge just has to use his common sense, experience of the world, and folk psychology, to hit upon a number which he thinks fairly represents the probability in question. If he is anything like me, the best he could do would be to give a range - say, from 0.05 to 0.1 to things which seem really unlikely; 0.2 to 0.3 to things which seem rather unlikely; 0.4 to 0.6, say, for things which seem moderately likely; and 0.7 to 0.9, say, for things which seem really likely.
One problem with this is that mechanical application of the rules of probability to such rubbery figures seems hardly likely to give a result which will decide the case. A more precise (and associated) problem is that in fact the processes of inference from such statements, which are likely to occur in any realistic case, are not mere manipulations of numerical probabilities, but do themselves involve common-sense judgements of likelihood of same type as those made in reaching the basic statements. For example, step (6) in the hypothetical case, by which a conclusion was drawn from statements (2) to (5), involved among other things a common-sense view as to the unlikelihood of an innocent explanation for three, as opposed to two or one, probable falsehoods. And step (15) involved a common-sense view as to the probability of the truth being unfavourable to William.
As an exercise, I have written a judgement for the hypothetical case, which applies Bayes’ theorem; and set it out in a postscript (Section B.5). It required two assumptions of prior probabilities of hypotheses, and twelve Bayesian steps, each involving two assumptions of numerical probabilities of evidence, given the truth or falsity of hypotheses: twenty-six guesses in all. In all twenty-six, I found I had virtually no confidence in the numbers I initially selected (in some cases partly because of unsureness of exactly what question I was asking, as well as because I just had to guess the answer); and I felt I had to check the numbers against the plausibility of the results, and then adjust (and re-adjust) the numbers, in order to arrive at numbers in which I had very slightly more confidence. That is, I had to cheat. Such little confidence as I ended up with depended very heavily on my common-sense assessment of the plausibility of the intermediate results and the conclusion. The exercise strongly suggests that in realistic situations Bayes’ theorem can fairly be regarded as a procedure for checking the consistency of one’s intuitions as to probability - and not as anything more than this.
I think my hypothetical case shows that, for ordinary contested cases, it is fanciful to envisage a process by which a court manipulates probabilities fixed upon for certain basic statements (premisses) to arrive at a decision of the case (conclusion). In all steps from the premisses to the conclusion, a judge will generally have in the forefront of her mind the actual particular circumstance of the case, and will be making common-sense judgements of (non-quantitative) probability in making these steps (as well as in determining upon the premisses). Indeed, the ultimate decision on the facts will generally itself be a common-sense judgement of non-quantitative probability concerning the overall situation, of very much the same kind as gave rise to the premisses - and very often the judge will (rightly) be more confident of reaching a correct overall conclusion ‘on the balance of probabilities’ than of assigning even approximate numerical probabilities to the premisses.
One may call RV’s model of fact-finding an actuarial model. I would suggest a better model for most judicial fact-finding would be a recognition model, assimilating it to a decision as to whether a person now in one’s presence is a particular person with whom one was acquainted many years ago, but has not seen since. In arriving at this decision in a doubtful case, one will pay close attention to component features and characteristics - eyes, nose, mouth, facial shape, hair, height, manner, etc. - having regard to the changes likely to have resulted from the passage of years. However, one does not assess numerical probabilities concerning each of these characteristics (e.g. that this is X’s nose, etc.) and compute from that a probability of identity. Rather, one will make an overall judgement, in which the contributions of the component characteristics cannot be clearly separated out. This is a process of inference, to which quantitative rules make no explicit contribution.
The superiority of the recognition model will be even more pronounced in most court cases involving disputed questions of fact, than in my hypothetical case; because most of such cases depend even more heavily on common-sense reasoning of the type I have outlined, because (inter alia):
(1) Generally there will be different versions of the facts given by different witnesses.
(2) There will often be difficult questions, not just as to honesty, but also as to the accuracy of recollection and the possibility of reconstruction of events partially recalled.
(3) The vexed issue of demeanour is sometimes important, in which case it needs to be considered in connection with the nature of the evidence being given from time to time by the witness in question.
(4) In property and contract cases, very often the precise terms of imperfectly remembered conversations have important legal consequences.
In their article, RV discuss various objections to their thesis, including some a little like mine: that evidence must be interpreted, that people actually compare hypotheses, that the complexity of evidence precludes the application of quantitative rules of probability, and that decisions are made in a holistic way rather than by dissection of elements. Their answers are abstract: I think the full impossibility of their position is best demonstrated by a realistic example.
3 The role of probability
It is not my position that quantitative rules of probability are unimportant in legal fact-finding - although I do say they are less important than RV imagine. And I am certainly not suggesting disregard of quantitative rules of probability. What I am saying is that logical rules of all kinds, including quantitative rules of probability, should be a guide, adjunct, and corrective for plausible common-sense reasoning, but not a substitute for it.
And there is much in RV’s article I agree with. I accept their characterisation of (quantitative) probability as a numerical measure of the strength of belief based on rational consideration of available evidence (p462) (although, as I will note later, I do not agree with everything they say about ‘the mind projection fallacy’). I think they make out a powerful case for the view that, for those inferences where a quantitative analysis of probability is appropriate, their quantitative rules of probability (p463-8) are the uniquely determined set of rules for conducting inference; and I agree that the application of such rules is by no means limited to frequentist probabilities (p469) - so I agree with their rejection of any valid distinction between different kinds of mathematical probability. I also agree that application of quantitative rules of probability can help avoid fallacies detected in untutored thinking (p460), and I will return to this.
(a) Non-quantitative probability
However, as argued above, I say that much legal inference is necessarily non-quantitative. And I suggest that the probability which the law requires for a finding for the plaintiff in a civil case, as well as for a finding of guilt in a criminal case, is not primarily a quantitative probability at all.
In criminal cases, juries are never directed in terms which refer to a quantitative probability (such as 0.9 or 0.95); but rather are directed in terms requiring a finding ‘beyond reasonable doubt’, with a minimum of elaboration on these hallowed words. Quantitative ideas are suggested by the civil standard of proof of ‘the balance of probabilities’, and civil juries are sometimes directed in terms that it is enough to find for plaintiffs that the scales of justice be weighed down, by ever so little, in their favour; but even in civil cases, I think the better view is that what is required is reasonable satisfaction, for which a mere quantitative preponderance of probability may be insufficient.
This view was strongly put by Dixon J of the High Court of Australia in Briginshaw v Briginshaw (1938) 60 CLR 336 at 361-2:
... when the law requires the proof of any fact, the tribunal must feel an actual persuasion of its occurrence or existence before it can be found. It cannot be found as a result of a mere mechanical comparison of probabilities independently of any belief in its reality. No doubt an opinion that a state of facts exists may be held according to indefinite gradations of certainty ... [A]t common law ... it is enough that the affirmative of an allegation is made out to the reasonable satisfaction of the tribunal. But reasonable satisfaction is not a state of mind that is attained or established independently of the nature and consequences of the fact or facts to be proved. The seriousness of an allegation made, the inherent unlikelihood of an occurrence of a given description or the gravity of the consequences flowing from a particular finding are considerations which must affect the answer to the question whether the issue has been proved to the reasonable satisfaction of the tribunal.
I suggest there are two main reasons why legal standards of proof are not primarily matters of mathematical probability, reasons which themselves suggest guidelines for the proper role of mathematical probability in legal fact-finding.
First, as argued above, in the general run of cases, decision-making is not a matter of mathematical computation, but is rather a matter of coming to a decision by the exercise of judgement, to the best of one’s ability and in accordance with common sense. Common sense, experience of the world, and folk psychology are generally far more important to fact-finding than quantitative rules of probability; and it is consistent with this that the ultimate question for the court should generally be defined in common-sense rather than purely quantitative terms.
Secondly, mathematical probabilities can be based on the slightest and most general of information, whereas reasonable fact-finding in particular cases requires, so far as reasonably possible, that there be evidence directed to the particular facts. Accordingly, courts should decline to give effect to mere numerical probabilities, in the absence of evidence directed to the particular facts, particularly where a reasonable endeavour by the party with the onus of proof should have produced such evidence.
(b) The place of quantitative probabilities
The two reasons given above suggest that the role for quantitative probabilities should be greatest where their common-sense probative force in particular cases is greatest, and/or where the party with the onus of proof cannot reasonably be expected to produce evidence bearing more directly on the particular case.
Obvious examples of the former category are DNA evidence in identification of criminals and paternity cases, fingerprint evidence, and ballistic evidence; and I will say a little more about this class of case in the next section.
As for the second category, consider two much-discussed hypothetical examples: the gatecrasher case and the blue bus case. The former concerns a rodeo attended by 1,000 people, when only 400 tickets were sold, and the promoter sues all 1,000 for the price of admission; and the latter concerns a case brought by a person injured at night by a negligently-driven bus which the plaintiff genuinely cannot identify, and the only evidence involving the defendant is that 80 percent of buses operating in that area are operated by the defendant’s Blue Bus Company. The consensus seems to be that in neither case would the defendant(s) have to give evidence in order to escape liability; but I think there is an important difference between them.
In the gatecrasher case, I think it is clear that a suggested 0.6 probability that each defendant did not pay for admission could not justify a decision in favour of the promoter, even in the absence of evidence from the defendants; because it would be unreasonable for the promoter to seek to make out a case in this way, without evidence of the ticketing system (or explaining its absence), of the possible means of access without payment, of challenges to and explanations from particular spectators, and so on.
However, in the blue bus case, it may be that no other evidence is reasonably available to the plaintiff, in which case I think the Blue Bus Company should have to give evidence if it is to escape liability. Suppose that the plaintiff has advertised without success for witnesses; and subpoenas the records of the Blue Bus Company, and also of the Red Bus Company which runs the other 20 percent of buses - and that these records leave the odds unchanged when applied to the place and time range of the accident. Despite the absence of evidence relating to the particular case, and of evidence excluding a maverick outsider bus, it seems to me that if the Blue company does not call its relevant drivers to deny that they were involved (or call some and explain inability to call the rest), then the court could draw an inference against it, on the balance of probabilities. If the Blue company does call evidence, then the case becomes one in which the quantities are just part of the overall material which the court has to assess.
And despite my general sympathy with Dixon J’s approach, I can envisage cases in which a bare mathematical probability would justify a decision, even without assistance from a defendant's failure to give evidence. Suppose that a woman lives with two men and her child, and although it is clear that one of the men is the child’s father, there is nothing in the circumstances (or the birth certificate) to suggest which one. The three adults are killed in an accident, and actual paternity is relevant to succession. If blood tests give respective probabilities of paternity of 0.55 and 0.45 for the two men, and there is just no other evidence reasonably available to the child (or anyone else), I think a court should find paternity on the basis of this bare mathematical probability, rather than make no finding.
(c) The role of quantitative rules
In those cases where mathematical probabilities are relevant, application of the correct quantitative rules can help avoid fallacies of inference, just as logical rules can do.
In this general area, the work of Kahneman and Tversky (see Kahneman, Slovic, and Tversky 1982, and Cohen 1977), suggesting how common-sense reasoning can go astray through illogicality and unconscious biases, is relevant. One of their famous examples illustrates the ‘conjunction fallacy’: the failure to understand that the likelihood of a conjunction of events or states of affairs can never be greater than the likelihood of any element of that conjunction. This is the case of Linda.
Linda is 31 years old, single, outspoken and very bright. She majored in philosophy. As a student she was deeply concerned with issues of discrimination and social justice, and also participated in antinuclear demonstrations.
Experiments were conducted in which subjects were asked to rank, in order of probability, a number of statements about Linda. Perhaps the most telling result concerned 142 subjects, asked to rank in order of probability the statements ‘Linda is a bank teller’ and ‘Linda is a bank teller and is active in the feminist movement’. 85% ranked the latter as more likely (Tversky and Kahneman 1973).
Of course, this is a logical error - and yet even this example suggests a strength of common-sense reasoning. The example is highly artificial: no such question is likely to present itself in actual decision-making. What is likely to arise is a question of the most appropriate categorisation of this young woman, with a view to assessing her character and conduct. On that question (admittedly not the question asked), the common-sense response is not inappropriate.
In legal fact-finding, Bayes’ theorem can alert tribunals to the necessity of taking account of ‘prior probabilities’ when dealing with statistical evidence. For example, if DNA evidence shows that only one person in ten thousand could have the DNA markers of the perpetrator of a crime and the accused has those markers, that does not of itself mean that there is 0.9999 probability that the accused committed the crime. On the other hand, if there are (say) one million people who conceivably could have committed that crime, and thus about 100 with the same DNA markers, it is also wrong to argue that the DNA evidence is irrelevant.
What has to be taken into account is the effect of the other evidence in the case. For example, if there is no other evidence which picks the accused out from (say) 5,000 other persons who could have committed the crime, then Bayes’ theorem shows that the DNA statistic will only give an overall probability of about O.67, far too low for proof beyond reasonable doubt. If other evidence indicates that the accused is one of only 500 who could have committed the crime, then, even in the absence of any further evidence isolating the accused, Bayes’ theorem shows that the DNA evidence will increase the probability that he committed the crime to about 0.95, and it may be that not much more is needed for proof beyond reasonable doubt.
This is just an example: there are many ways in which the quantitative rules of probability can help ensure that evidence of probabilities is not misapplied.
(d) Combining probabilities
A very significant difference between the non-quantitative approach of Dixon J, and the quantitative approach recommended by RV, concerns the combination of probabilities.
RV make the sweeping assertions (pp472-3) that ‘the task of the court is to determine the odds that the defendant is liable’; and that if this liability can arise through any of two or three different events, then it will be sufficient to prove the requisite degree of probability that one or more of these events occurred, without proving which one. Similarly, they assert (p464n) ‘that in civil cases where liability must be proved on the balance of probabilities the essential elements of liability, e.g., duty of care, breach of duty and loss or damage must each be proved to a higher standard’.
So, they would say, if a plaintiff was entitled to succeed on any of three entirely independent grounds, and each is shown to have an independent probability of 0.25, then the probability of the defendant being liable is about 0.25 + (0.25 x 0.75) + (0.25 x 0.5625), or about 0.58; and the plaintiff wins. But if the plaintiff needs to show duty, breach, and damage, and shows a probability of each (independent of the others) of 0.65, then the probability of the defendant being liable is only 0.65 x 0.65 x 0.65, or about 0.275, and the plaintiff loses.
Looking first at the latter case, in relation to damages it may be necessary to distinguish between those events which allegedly did in fact occur, and those events advantageous to the plaintiff which allegedly would have occurred but for the breach of duty. In so far as probability concerning damages relates to the latter class of events, any damages awarded to the plaintiff will be reduced proportionately to the extent to which this probability is less than 1: where the probability is small, this is called the loss of a chance. There could thus be no justification for taking this probability into account against the plaintiff winning the case. But quite apart from this, contrary to RV’s contention, the law seems to be that normally the tribunal finds each element, and indeed each fact, on the balance of probabilities, and then treats it as certain (169 CLR at 642-3); so that there is generally no question of a reduced overall probability being given by multiplying component probabilities.
I contend that this is generally justified, because on the Dixon approach the tribunal is not assessing numerical probabilities at all; and strict application of RV’s view would require the tribunal to bring into account in the plaintiff’s favour all possible permutations and combinations of all more or less probable facts which could establish liability, making the decision process impossibly complex. Furthermore, I think RV’s view would have questionable results in certain cases, such as cases of alternative defendants. If a plaintiff proved a 0.65 probability that someone was liable, plus a 0.65 probability that it was one defendant and a 0.35 probability that it was the other, then arguably she should succeed against the former; but RV would say that she should fail completely, because her best case was only a 0.42 probability against the first defendant. Consider also the case of alternative plaintiffs: if in two cases heard together, the court finds (1) a 0.65 probability that one plaintiff owned certain property, (2) a 0.35 probability that the other (competing) plaintiff owned it, and (3) a 0.65 probability that the defendant negligently destroyed it, RV’s view would mean that the defendant would win both cases.
However, I accept that courts should be well aware that if it is less than certain that each of two independent events occurred, then it is even less certain that both occurred; and courts should also be aware of the underlying mathematical rules. And I accept that it will sometimes be appropriate to bracket aspects of a plaintiff's case, and look for satisfaction, on the balance of probabilities, about the combination of these aspects. But I think it will rarely be appropriate to express this quantitatively, in terms (say) of a 0.65 probability for each of two independent elements giving a 0.42 probability for their combination; and I think that whether or not this approach should be taken will depend, not so much on rules of probability, as on considerations of reasonableness or rules of law.
As for the other case, that of adding probabilities of alternative grounds of liability, again I think RV’s general proposition is incorrect as a matter of law. For example, suppose that a purchaser of some property seeks to set aside the purchase on three independent grounds: undue influence presumed from a solicitor-client relationship, misrepresentation that the property has feature A (which it certainly doesn't have), and fundamental breach in that feature B (which was certainly promised) is totally defective. The court decides that the probability of each alternative (i.e. the existence of the solicitor-client relationship and consequent presumed undue influence, the making of the misrepresentation and reliance on it, and the fundamental defectiveness of property B) is 0.25. 1 think it is clear that each ground would be decided against the plaintiff, and the defendant would win the case even if the probabilities are independent.
However, where a liability-creating event (say, a cause of loss under an insurance policy: RV 1993, pp472-3) can occur in more than one way, a court may be justified in bracketing these ways together, if the evidence gives appropriate satisfaction that the event occurred in one or other of these ways. But again, I think that whether or not this bracketing is justified will depend less on rules of probability than on considerations of reasonableness or rules of law.
4 Wider questions
(a) The philosophical debate
It should be recognised that the debate concerning quantitative and non-quantitative probability in legal fact-finding is related to a wider philosophical debate as to whether human rationality can be formalized - and indeed as to the nature of the human mind, in that if human rationality could be formalized, it would seem reasonable to believe in the adequacy of the computational view of the human brain and mind.
One important thread in the wider debate can be traced through Hume’s argument about the circularity of induction (Hume 1739), Popper’s elaboration of this argument (Popper 1959), Hempel’s paradox of confirmation (Hempel 1965), Goodman’s strictures on analogical argument and his new riddle of induction (Goodman 1965, 1970), and Putnam’s critiques of Bayesian analysis and the Carnap theory of confirmation based on it (Putnam 1979, 1981, 1983).
I find convincing Putnam’s arguments to the conclusion that human rationality cannot be formalized, at least without formalizing complete human psychology (and possibly not even then); and that the various puzzles, paradoxes, and riddles of induction are related to the need for prior probabilities for the application of Bayes’ theorem (TMM pp114-26).
Bayes’ theorem can never itself give us the probabilities that it needs to get started, in particular the prior probability of the hypothesis being considered, and the prior probability of each piece of evidence. Since common-sense reasoning is generally required to produce these ‘priors’, there seems little justification for attempting to exclude it entirely, in favour of purely quantitative rules, in later stages of the reasoning process.
(b) An evolutionary slant
Our brains are capable of performing marvellous algorithmic procedures - for instance, in the pre-conscious processes associated with seeing. These capabilities, which have yet to be approached by today’s computers, make possible our access to a vast amount of visual information, continually ‘updated’ in ‘real time’. As examples of the computational complexity required for these processes, one may consider the computations that must be involved in achieving continuous stereoscopic vision, by analysis of similarities and differences between the signals from two eyes; and those involved in achieving the apparent stability of a viewed scene despite movements of one’s eyes, head, and body.
The computational virtuosity and reliability of these processes contrasts starkly with our lack of virtuosity and reliability in consciously performing even simple mental arithmetic. Some gifted (or partially gifted) persons have a facility for tapping into some of the brain's algorithmic capacities - think, for example, of the twins described by Oliver Sacks (1986, pp185-203), who enjoyed exchanging prime numbers of six digits. But most of us perform mental arithmetic in the same clumsy and fallible way as we conduct other conscious common-sense reasoning, at a mathematical level incomparably lower than that of the computations our brains are constantly carrying out in order to enable us to see, hear, balance, catch balls, etc.
Natural selection in evolution has equipped us with this prodigious computing capacity. If satisfactory decisions on matters important for our survival and reproduction could be made by algorithms, one might have expected that evolution would have ensured that they be made by using this capacity, with no interference from our very fallible conscious processes. And yet, we are in fact so constituted that, whenever in life we are faced by a novel situation in which an important decision or action is required, our conscious minds are brought to bear - and we have no alternative but to use our sloppy common-sense reasoning, with all its fallacies and biases.
To me, this suggests that our conscious common-sense reasoning must have some advantages over even the most marvellous of algorithmic procedures, so that efforts to replace common-sense plausible reasoning completely by algorithms of the type discussed by Bayes, or Carnap, or RV, are misguided. A better course is to recognize a more modest role for algorithms, a role of assistance and correction, and to pay careful attention to the actuality of common-sense reasoning as it works in real life.
(c) ‘The mind projection fallacy’
Finally, I note that RV refer (1993, p460) to ‘the fallacy of regarding probability as a property of objects and processes in the real world rather than a measure of our own uncertainty’. If all they mean to say is that probability, as a measure of our uncertainty, does not imply uncertainty in the real world, this would be unexceptionable. But they also seem to be saying that it is a fallacy to think there is in fact any uncertainty or indeterminism at all in the world - and that is a very dubious position. In a footnote, they add: ‘Readers may have encountered references to quantum theory as a non-deterministic foundation for physics ... They should be aware that this theory is by no means uncontroversial’; and they refer to a forthcoming article.
I am astounded by this. Quantum theory is the best-established physical theory we have. And while its interpretation is very controversial, there is nevertheless a substantial consensus among physicists that, at the atomic level, there is irreducible indeterminacy and indeterminism in the world. This is apparent from superior popular expositions such as those by Polkinghorne (1984) and Davies (Davies and Brown 1986), text books such as those by Feynman (Feynman et al 1963) and Dirac (1958), collections of learned articles such as that of Wheeler and Zurek (1983), and deep philosophical considerations such as that by d’Espagnat (1989). What RV have done is to refer to one of the few dissenters from this consensus, so as to suggest a controversy which barely exists - and then completely ignore the consensus itself, by asserting, contrary to this consensus, that it is a fallacy to attribute probability to processes in the world!
What has this to do with legal fact-finding? Not much directly; but such distortions should not go unchallenged - and acceptance of a deterministic view of the world (which I say is unjustified) may lead one more readily to slip into thinking, without question, that reasoning must be algorithmic. But to pursue this would require another article, or a book.
First, I consider the hypothesis ‘William has lied on oath when he believed it would help his case’. I take the prior probability of this to be 0.025.
The first piece of evidence on this is that William told the building society that he had no house. The probability of this, given the hypothesis, I put at 0.5 (remembering that William could have tried other financiers for whom owning another property was no disqualification, and also that, if William has lied, there is some chance that he did not in fact beneficially own any other property); and its probability, if the hypothesis is false, I put at 0.05. So the first step is:
P(H|E) = (0.025 x 0.5) / [(0.025 x 0.5) + (0.975 x 0.05)]
The second piece of evidence on this is that William said on oath that he had no discussion with Jane about the house and or the mortgage. The probability of him saying this, given the hypothesis, I put at 0.3; and its probability, if the hypothesis is false, I put at 0.15. So the second step is:
P(H|E) = (0.2 x 0.3) / [(0.2 x 0.3) + (0.8 x 0. 15)]
The third piece of evidence on this is that William said on oath that he purchased the house as an investment. The probability of him saying this, given the hypothesis, I put at 0.3; and its probability, if the hypothesis is false, I put at 0.15. So the third step is:
P(H|E) = (0.33 x 0.3) / [(0.33 x 0.3)+(0.67 x 0.15)]
The fourth piece of evidence on this is that William said on oath that Jane made no contribution towards his purchase. The probability of him saying this, given the hypothesis, I put at 0.7; and its probability, if the hypothesis is false, I put at 0.3. So the fourth step is:
P(H|E) = (0.5 x 0.7) / [(0.5 x 0.7) + (0 5 x 0.3)]
The fifth piece of evidence on this is that William has said three things on oath with respective probabilities of 0.15, 0.15, and 0.3 (if the hypothesis is false), and 0.3, 0.3, and 0.7 (if the hypothesis is true). I think there is an additional (im)probability in this, if the hypothesis is false, of 0.4; as against a probability of 0.6 if the hypothesis is true. So the fifth step is:
P(H|E) = (0.7 x 0.6) / [(0.7 x 0.6) + (0.3 x 0.4)]
Next, I consider the hypothesis ‘There was an agreement whereby William gave up his interest in the Carlton house for an appropriate consideration’. I take the prior probability of this, in the circumstances of a cordial separation in which William went to a house of his own, to be 0.3.
The first piece of evidence on this is that William told the building society that he had no house. Given that there is a 0.78 probability that William has lied on oath before me, I put the probability of this, given the hypothesis, at 0.6; and its probability, if the hypothesis is false, I put at 0.4. So the first step is:
P(H|E) = (0.3 x 0.6) / [(0.3 x 0.6) +(0.7 x 0.4)]
The second piece of evidence on this is that William told the building society that Jane would provide $8,000. The probability of this, given the hypothesis, I put at 0.4; and its probability, if the hypothesis is false, I also put at 0.4 (since this evidence seems neutral as between the hypothesis, on the one hand, and an intention to lend and or make some other agreement, on the other). So the second step leaves the probability of the hypothesis unchanged.
The third piece of evidence on this is that William cannot account for $4,000 of the price. The probability of this, given the hypothesis (and given that there is a 0.78 probability that William lied on oath), I put at 0.6; and its probability, if the hypothesis is false, I put at 0.3. So the third step is:
P(H|E) = (0.39 x 0.6) / [(0.39 x 0.6) + (0.61 x 0.3)]
The fourth piece of evidence on this is that the evidence as to consideration concerns no more than $12,000, whereas a half interest was worth about $15,000 or a little more. Having regard to the non-availability of witnesses for Jane's estate, I put the probability of this, given the hypothesis, at 0.5; and its probability, if the hypothesis is false (and trying to avoid double counting of the two previous pieces of evidence), I also put at 0.5. So the fourth step leaves the probability of the hypothesis unchanged.
The fifth piece of evidence on this is that no lawyers were consulted and no documentation of the agreement has been produced. The probability of this, given the hypothesis, I put at 0.3 (remembering there could have been some documentation not now available to Jane’s estate); and its probability, if the hypothesis is false, I put at 0.8 (not more, since the falsity of the hypothesis includes the making of a lesser agreement). So the fifth step is:
P(H|E) = (0.56 x 0.3) / [(0.56 x 0.3) + (0.44 x 0.8)]
The sixth piece of evidence on this is that William and Jane, for nine years have lived in separate houses, paying off separate mortgages, and William has had no association with the Carlton house. The probability of this, given the hypothesis, I put at 0.9; and its probability, if the hypothesis is false, I put at 0.3. So the sixth step is:
P(H|E) = (0.32 x 0.9) / [(0.32 x 0.9) + (0.68 x 0.3)]
The final piece of evidence on this is that William has with probability 0.78 lied on oath. The probability of this, given the hypothesis (and trying to avoid double counting), I put at 0.05; and its probability, if the hypothesis is false, I put at 0.025. So the final step is:
P(H|E) = (0.59 x 0.05) / [(0.59 x 0.05) + (0.41 x 0.025)]
Cohen, L. J. (1977), The Probable and the Provable (Oxford: Oxford University Press).
Davies, P. and Brown, J. R. (1986), The Ghost in the Atom (Cambridge: Cambridge University Press).
d’Espagnat, B. (1989), Reality and the Physicist (Cambridge: Cambridge University Press).
Dirac, P. A. M. (1958), The Principles of Quantum Mechanics, 4th ed. (Oxford: Oxford University Press).
Feynman, R., Leighton, R., and Sands, M. (1963), The Feynman Lectures in Physics (Reading MA: Addison-Wesley).
Goodman, N. (1965), Fact, Fiction and Forecast (New York: Bobbs-Merrill).
Goodman, N. (1970), ‘Seven strictures on similarity’, in Foster and Swanson (1970).
Hempel, C. G. (1965), Aspects of Scientific Explanation (London: Macmillan).
Hodgson, D. (1991), The Mind Matters (Oxford: Oxford University Press).
Hume, D. (1739), A Treatise of Human Nature.
Kahneman, D., Slovic, P., and Tversky, A. (eds) (1982), Judgement under Uncertainty: Heuristics and Biases (Cambridge: Cambridge University Press).
Polkinghorne, J. (1984), The Quantum World (London: Longman).
Popper, K. R. (1959), The Logic of Scientific Discovery (London: Hutchinson).
Putnam, H. (1979), Mathematics, Matter and Method (Cambridge: Cambridge University Press).
Putnam, H. (1981), Reason, Truth and History (Cambridge: Cambridge University Press).
Putnam, H. (1983), Realism and Reason (Cambridge: Cambridge University Press).
Robertson, B. and Vignaux, G. A. (1993), ‘Probability: the logic of the law’, Oxford Journal of Legal Studies 13, 457-78.
Sacks, O. (1986), The Man who Mistook his Wife for a Hat (London: Picador).
Tversky, A. and Kahneman, D. (1983), ‘Extensional versus intuitive reasoning: the conjunction fallacy in probability judgment’, Psychological Review 90.
Wheeler, J. A. and Zurek, W. H. (eds) (1983), Quantum Theory and Measurement (Princeton: Princeton University Press).
 I think the fallacy of Murphy J’s view in TNT Management v Brooks (1979) 23 ALR 345 is that the probabilities he uses are based on material inadequate to support an inference in a particular case.
 It follows that I disagree with the reasoning of the majority in SGIC v Laube (1984) 37 SASR 31; cf. Eggleston (1987) and see Rose v Abbey Orchard Property Investments  Aust Torts Reports 80-121.
 P(H|E) = 0.002 x 1 / [(0.002 x 1) + (0.998 x 0.0001)] = 0.9524716.
 The fallacy which, on one view, was suggested by the decision in Chamberlain v The Queen (1984) 153 CLR 521 may be regarded as being a misapplication of quantitative rules of probability: cf. Shepherd v The Queen (1990) 170 CLR 573.
 Malec v J. C. Hutton Pty Ltd (1990) 169 CLR 638. However, where damages are an element of the cause of action, the plaintiff does have to prove, on the balance of probabilities, the loss of a chance or commercial opportunity which itself has some value: Sellars v Adelaide Petroleum NL (1992-4) 179 CLR 332.