Lotfi, I believe that there are a number of nonsequiturs in your note below.
> Dear All: > > There are few, if any, concepts that are as fundamental as > the concept of causality. Perceptions of causal dependencies govern our > behavior and underlie our decisions. Identification of causal > dependencies plays a pivotal role in the realms of law, medicine, > economics, data mining, AI and many other fields. Indeed, a case could > be made for requiring most students in law, medicine and economics to > take a course in which causality is an object of substantive attention. > > Our discussions of causality, with insightful contributions > by prominent members of the legal profession, have clarified many > issues. But a question that remains unsettled is: Does there exist a > formalized, bivalent-logic-based, theory of causality which is capable > of dealing with realistic problems exemplified by the raincoats > example? I believe the answer is yes. > In earlier messages, I suggested that existing theories do > not have this capability. The principal source of difficulty is that > when we activate an event A0, call it the nominal event, and observe a > consequent event B, it is almost always the case that B is a consequence > of a multiplicity of other events, call them a network of collateral > events, N(A1, A2, �). What this implies is that we cannot answer the > question: Was B caused by A0? with a categorical yes or no. You seem to imply that because, given some information, you cannot answer the question "was B caused by A0" with a categorical yes or no, then there can't be a bivalent-logic-based theory of causality. That simply doesn't follow. It is perfectly consistent to say that, in any given model of a situation, either A0 is a cause of B or it's not, while at the same time saying that your information does not determine which is the right causal model (and so you do not know whether A0 is a cause of B). > In other > words, we have to assume that causality and related concepts such as > responsibility and propensity are a matter of degree�with the degree of > causal dependence representing our perception of materiality of the role > of A0 in causing B. I agree that it is of interest to have a notion of degree of resonsibility although, as I said, I don't believe it follows from your earlier statement that causality must be a matter of degree. > In turn, this implies that the assumption that > causality is a matter of degree should be the point of departure in any > formalized theory of causality which aspires to provide a body of > operational concepts and techniques for dealing with causal dependencies > in realistic settings. Here again I disagree. Our notion of degree of responsibility (which is a number between 0 and 1) is simply a refinement of the 0-1 notion of causality. (A has a positive degree of responsibility for B iff A is a cause of B.) > If we accept this postulate as a basic premise, the next > question is: How can the degree of materiality be assessed? > Unfortunately, there is no simple or obvious answer to this question. > Furthermore, it should be noted that existence of the network of > collateral events raises serious questions regarding the validity of use > of counterfactual conditionals in causal reasoning. Again, I believe you're confounding the difficulty of determining what is the correct model of a given situation (which is often admittedly quite difficult) with the difficulty of giving a useful definition. > In the case of the raincoats example, the assumption is that > we are dealing with a single experiment and have a single datapoint: > increase in advertising: 20%; increase in sales: 10%) Furthermore, I do > not have a model of the network of collateral events. The only other > information that I have is (a) world knowledge, e.g., rainy weather > increases demand for raincoats; and (b) case-based knowledge, that is, > knowledge about other experiments which in some sense are similar to my > experiment. Both (a) and (b) are imprecise, uncertain and not totally > reliable. A further complicating factor is that an event may have a > positive or negative polarity. To illustrate, in the raincoats example, > rainy weather has positive polarity, while dry weather has negative > polarity. The issue of polarities makes it much more difficult to > aggregate contributions of collateral events to the consequent event, > and expressing the aggregate as a weighted combination in the manner > suggested by Marianne Belis. The raincoat example is indeed one in which it may indeed be difficult to determine which causal model correctly describes the situation. But I don't see what the example shows beyond that. We also often have difficulty in determining the true situation even when causality is not an issue. > A conclusion which emerges is that in realistic settings it > would be unrealistic to aim at expressing the degree of causality or > materiality as a sharply defined number. That may well be. > What is necessary is a recourse > to granulation of variables and their probabilities, resulting in what > may be called a bimodal distribution, with �bimodal� signifying that > granulation is applied to both variables and their probability > distributions. More specifically, if granulation is coarse, the granular > values of degree may be zero, low, medium and high, and likewise for the > values of probabilities. As an example, a bimodal distribution of degree > may be of the form ((low, low), (high, medium), (low, high)), meaning > that the granular probabilities of low, medium and high are low, high > and low, respectively. I don't see why the need for granulation of variables (which I don't understand) follows from the difficulty in modeling causal situations. There may be many other useful ways of modeling the uncertainty. > What this suggests is that, in realistic > settings, causal dependence is certainly not a matter of yes or no; > rather, it is a matter of degree. Again, I don't see why it suggests this at all. Although I do agree that it's useful to speak of degree of responsibility, that's perfectly compatible with also have 0-1 notion of causality. > However, in most cases, the degree > cannot be represented as a number, an interval or a probability > distribution. It may be representable as a fuzzy interval or, more > generally, as a bimodal distribution. While I agree that one can certainly debate what the most useful representaion of degree of resonsibility should be, I don't see why it follows at all that it "cannot be represented as a number, an interval, or a probability distribution". > Obviously, more can be said when the experiment can be > repeated�as in the realm of medical experimentation, and/or when at > least a partial model of the collateral network is known. But what could > be said would not be in the spirit of theories based on bivalent logic > and bivalent-logic-based probability theory. I hope to have an > opportunity to say more about this important issue at a later time. > > Regards to all > > Lotfi -- Joe
