For example, if A denotes the event "The person is a smoker," B denotes the event "The person now has or will have cancer at some time in the future" and C denotes the event "The person now has or will have emphysema some time in the future," then the following three relationships hold: P{B|A} ≥ P{B}, P{C|A} ≥ P{C} and P{B|C} ≥ P{B}. The last relationship states that knowing that the person has emphysema increases the likelihood that he will have cancer. The reason for this is that having the information that the person has emphysema increases the likelihood that the person is a smoker, thus indirectly increasing the likelihood that the person will have cancer. However, we would not want to conclude that having emphysema causes cancer. Thus, we need additional conditions such as temporal relationship of A to B and a rational explanation as to the mechanism of action. It is hard to quantify this last requirement and thus different authors prefer somewhat different definitions. Causal calculus
When experimental interventions are infeasible or illegal, the derivation of a cause-and-effect relationship from observational studies must rest on some qualitative theoretical assumptions, for example, that symptoms do not cause diseases, usually expressed in the form of missing arrows in causal graphs such as Bayesian networks or path diagrams.
According to the counterfactual view, X causes Y if and only if, without X, Y would not exist. Hume interpreted the latter as an ontological view, i.e., as a description of the nature of causality but, given the limitations of the human mind, advised using the former (stating, roughly, that X causes Y if and only if the two events are spatiotemporally conjoined, and X precedes Y) as an epistemic definition of causality. Having an epistemic concept of causality is needed to distinguish between causal and noncausal relations. The contemporary philosophical literature on causality can be divided into five big approaches to causality. These include the (mentioned above) regularity, probabilistic, counterfactual, mechanistic, and manipulationist views. The five approaches can be shown to be reductive, i.e., define causality in terms of relations of other types. According to this reading, they define causality in terms of, respectively, empirical regularities (constant conjunctions of events), changes in conditional probabilities, counterfactual conditions, mechanisms underlying causal relations, and invariance under intervention. Geometrical significance
Causality has the properties of antecedence and contiguity. These are topological, and are ingredients for space-time geometry. As developed by Alfred Robb, these properties allow the derivation of the notions of time and space. Max Jammer writes "the Einstein postulate ... opens the way to a straightforward construction of the causal topology ... of Minkowski space." Causal efficacy propagates no faster than light.Thus, the notion of causality is metaphysically prior to the notions of time and space.