Fuzzy Evidential Logic: A Model of Causality for Commonsense Reasoning

This paper proposes a fuzzy evidential model for commonsense causal reasoning. After an analysis of the advantages and limitations of existing accounts of causality, a generalized rule-based model FEL (Fuzzy Evidential Logic) is proposed that takes into account the inexactness and the cumulative evidentiality of commonsense reasoning. It corresponds naturally to a neural (connectionist) network. Detailed analyses are performed regarding how the model handles commonsense causal reasoning. Shoham's Causal Theory The issue of causality has recently received a lot of attentions from various perspectives (cf. Shoham 1987, Iwasaki & Simon 1986, de Kleer & Brown 1986, Pearl 1988, etc.). The issue has wide ranging impact on areas such as learning, control, and recognition. However, most of these logic based models are aimed for modeling truth functional aspects of causal knowledge, and they tend to ignore some important characteristics of commonsense causal reasoning, for example, gradeness of concepts, inexact causal connections, evidentiality of causal rules, etc., while probabilistically motivated models are mainly concerned with the probabilistic aspect of causal events, and they are more computationally complex and oftentime have only marginal cognitive plausibility in terms of mechanisms involved. Connectionism provides a new and di erent kind of models that might be of help in accounting for causality in commonsense reasoning; these models entertain a number of interesting properties that other models lack (for example, massive parallelism, generalization, fault/noise tolerence, and adaptability; see Waltz & Feldman 1986, Sun & Waltz 1991) and present a new perspective of reasoning as a complex process in a dynamic system; it will be worthwhile to look into the question of how such models can deal with the issue of causality. Let us look into Shoham's account of causality (Shoham 1987), which is undoubtedly one of the most notable accounts of causality with rule-based formalisms. His temporal modal logic formalism has a close resemblance to Horn clause logic, and therefore is very suitable for use in rule-based systems. According to Shoham's Causal Theory (CT), causes are primary conditions which, together with other conditions, will bring about the e ect. These \other" conditions are somewhat secondary. In reasoning, as long as we know that the primary conditions (causes or necessary conditions) are true and that there is no information that the secondary conditions (enabling conditions or possible conditions) are false, then we can deduce that e ects will follow. The theory is described in terms of modal logic, with one basic modal operator (2 or necessity) for specifying necessary conditions. and one auxiliary modal operator (3 or possibility) for specifying possible conditions. The formal de nition is as follows: De nition 1 A Causal Theory is a set of formulas of the following form ^ i 2n i a i (t i1 ; t i2 ) ^ j 3n j b j (t j1 ; t j2 ) ! 2c(t 1 ; t 2 ) where n i 's are either : or nothing, t 2 > t i2 for all i's, t 2 > t j2 for all j's, n i a i 's are necessary conditions (causes), and n j b i 's are possible conditions (enabling conditions). C is concluded i all n i a i 's are true and none of n j b j 's are known to be false. 1 From the standpoint of modeling commonsense knowledge, this model has some advantages, such as that it provides a simple and elegant formalism with e cient inference algorithms, that it is easily representable (and implementable), and that it has 1 This process is formally described by a minimization principle in Shoham (1987). compatibility with philosophical accounts of causality (Shoham 1987). On the other hand, the model ignores or discounts many aspects of commonsense causal reasoning; for example, 1. All propositions in this theory are binary: either true or false, and there is no sense of gradedness. Commonsense knowledge is certainly not limited to true/false only (Sun 1991, Hink & Woods 1987). 2. Beside the inexactness of individual concepts, reasoning processes in reality are also inexact and evidential. Speci cally, the evidential combination process is cumulative (as observed in protocol data; Sun 1991, 1991a); that is, it \adds up" various pieces of evidence to reach a conclusion, with a con dence that is determined from the \sum" of the con dences of the di erent pieces of evidence. Moreover, di erent pieces of evidence are weighted, that is, each of them may havemore or less impact, depending on its importance or saliency, on the reasoning process and the conclusion reached. We have to nd a way of combining evidence from di erent sources cumulatively and with weights, without incurring too much computational overhead (such as in probabilistic reasoning or Dempster-Shafer calculus; cf. Pearl 1988). 3. Because of the lack of gradedness, the model will make projections too far along a chain of reasoning (or too far into the future; Sun 1991). An example from Shoham (1987):