Defeasible Reasoning and Probability

Probability calculus—especially its versions based on the idea of subjective probability—provides an attractive alternative to defeasible reasoning as a method for dealing with limited and provisional information. It has a rich history of successful applications in many domains of science and practice, including legal practice (though its legal applications are still controversial: see Fenton, Neil, and Berger 2016) and has recently found many applications in artificial intelligence.
Consider, for instance, a case where Tom was run over by a car carrying Mary and John, and in which it is not clear who was driving at the time of the accident.
On the probabilistic approach, conflicting evidence does not lead us to incompatible belief —like the belief that John was driving the car when the car ran over Tom, and the belief Mary was driving the car on the same occasion— between which a choice is needed. We rather come to the consistent view that incompatible hypotheses have different probabilities. For instance, on the basis of the available evidence, we may consistently conclude that there is a 40 percent probability that John was driving, and a 60 percent probability that Mary was doing it. Probabilistic inference uses probability calculus to determine the probability of an event on the basis of the probability of other events. For instance, if there is an 80 percent probability that Tom will have problems walking because he has been run over, there is a 32 percent probability (40 percent * 80 percent) that Tom will have such problems having been run over by John, and a 48 percent chance (60 percent * 80 percent) that he will have such problems having been run over by Mary. Here I cannot enter probability calculus or discuss the many difficult issues related to it, especially when ideas of probability and causation are combined, or when Bayesian reasoning is used to determine the probability of a hypothesis in light of the evidence. I will merely highlight three issues that make probability calculus inadequate as a general approach for dealing with uncertainty in legal reasoning.
The first issue is that of practicability: we often do not have enough information to assign numerical probabilities in a sensible way. For instance, how do I know that there is a 40 percent probability that John was driving and a 60 percent probability that Mary was driving? In such circumstances, it seems that we must attribute probabilities arbitrarily or, no less arbitrarily, we must assume that all alternative ways in which things may have turned out have the same probability.
The second issue is conceptual: although it makes sense to ascribe probabilities to factual propositions, it makes little sense to assign probabilities to legal rules and principles, unless we are making predictions. A legal decision-maker does not usually decide to use a normative premise by assessing the probability that the premise holds.
The third issue relates to psychology: humans tend to face situations of uncertainty by choosing to endorse hypothetically one of the available epistemic or practical alternatives (while keeping open the chance that other options may turn out to be preferable), and by applying their reasoning to this hypothesis (while possibly, at the same time, exploring what would be the case if things turn out to be different). We do not usually assign probabilities and then compute what further probabilities follow from such an assignment. When we have definite beliefs or hypotheses, we are usually good at developing inference chains, storing them in our minds (keeping them dormant until needed), and then retracting any such chains when one of its links is defeated. Conversely, we are bad at assigning numerical probabilities, and even worse at deriving further probabilities and revising probability assignments in light of further information.
Our inability to work with numerical probabilities certainly figures among the many failures of human cognition (like our inability to quickly execute large arithmetical calculations). In fact, computer systems exist which can handle efficiently complex probability networks (otherwise termed belief networks, Bayesian networks ). They perform very well in certain domains by manipulating numerical probabilities much faster and more accurately than a normal person (see Russell and Norvig 2010, chap. 13). However, our bias toward exploring alternative scenarios, and defeasibly endorsing one of them, does have some advantages: it focuses cognition on the implications of the most likely situations, it supports making long reasoning chains, it facilitates building scenarios (or stories) which may then be evaluated according to their coherence, it enables us to link epistemic cognition with binary decision-making (it may be established that we have to adopt decision Q if P is the case, and NON-Q if P is not the case). There is indeed psychological evidence that humans develop theories even under situations of extreme uncertainty, when no reasonable probability assignment can be made.
The limited applicability of probability calculus in many domains does not exclude that there may be various practical and legal issues where statistics and probability provide decisive clues, as when scientific evidence is at issue.
Recently, approaches have been developed that try to combine defeasible reasoning and probability by working out the likelihood that different premises and combinations of them will be used in making arguments and that these will interact with other arguments. Such approaches would lead to probabilistic refinements of the IN and OUT labelling previously considered: rather than just saying that an argument is IN or OUT, we could establish that it has a certain probability of being IN or OUT relative to an argumentation basis whose premises or combinations of them are assigned certain probabilities (Riveret, Rotolo, and Sartor 2012; Hunter 2013).