Bivalent Probabilities

The aim is to understand evidence and proof, and thereby ensure more accurate factfinding. A possible route to that end is probability theory. The traditional probability with which we are all familiar is actually built on assumptions that are much less familiar. Its logical basis assumes bivalence, so that everything is either true or false and so that the probabilities of truth and falsity add to one. This assumption leads inevitably to all the logical problems we encounter when using probability to explain factfinding, problems such as the law’s strange willingness to accept weak evidence on both sides and the seemingly asymmetrical burden put on plaintiffs to prove a whole series of elements. After critiquing all of probability’s problems, this chapter, whose length merits a separate Analysis of its sections, concludes that probability theory is not the way to understand evidence and proof. Section 3.1: Traditional probability systems are built on bivalent logic, with an excluded middle, so that any proposition must be either true or false. The systems are additive in the sense that the chances of truth and falsity add to one. The most useful of those systems for factfinding is subjective probability, which relies on the image of the factfinder’s willingness to bet on the disputed fact. Section 3.2: However, the use of any system of traditional probability for factfinding runs into six major problems, which all result from its failure to account for epistemic uncertainty: resorting to an unrealistic way to express how humans measure proof; fumbling in its advice when there is weak evidence on both sides; allowing the argument to be made that infinite possibilities prevent proving anything to be more likely than not; lacking accurate tools for combining uncertain evidential items or found facts; creating the appearance of asymmetries in the treatment of plaintiffs and defendants; and failing conceptually to capture what we intuitively think of as adequacy of legal proof. Section 3.3: Law professors also anguish over, or revel in, the use of naked statistical evidence and its resulting paradoxes, such as finding a bus company liable for an injury solely because it owns 80% of the buses running that route. Although there are hazards of improper use of mathematics in evidence, statistical evidence is not unique in nature. And in the law’s treatment of statistical evidence, there is no real logical problem. Section 3.4: Over the decades, the problems of traditional probability have become obvious to theoreticians, leading some of them to alter probability in ways that account for epistemic uncertainty. They have produced a variety of dazzling but clumsy ways for probability to do what multivalent logic does easily and clearly. These ways include so-called logical, imprecise, and inductive probability theories.