Notions of Causality?

I brought up the idea in class that there are various notions of causality. What concepts of causality can you find? Find a particular notion of causality, define it and describe it. The concept of cause and effect is fundamentally intuitive, yet defining causality has been an active area of debate for thousands of years in philosophy, science, and mathematics. You should be able to find any number of definitions of causal relationships, but I am asking you to find just one and describe some details about it.

For example, I mentioned Granger causality in class. This idea was proposed in the 1960's as a tool of probabilistic causality for analyzing economic data and it has been widely used. The definition of Granger causality relies on two assumptions. The first assumption is that earlier events cannot be caused by later events. The second assumption is that all possible causal factors are accounted for. Let's pretend that our question is whether April showers bring May flowers. We'll call April showers X and May flowers Y, and ask whether X causes Y. We need to quantify all other possible causes of Y and lump them together in a variable called Z (sunshine, temperature, pollination, etc). We could try to predict future values of Y based on past values of Y (perhaps past flowers beget new flowers) and past values of Z. However, if knowing past values of X gives us an even better prediction of Y, then X "Granger causes" Y. The intuition is that if a change in X is observed (more showers) followed by a change in Y (more flowers), and all other causes have been factored out; then there are only two reasons this could have happened: 1) X causes Y; or 2) the coincidental change in X and Y is completely random. The 2nd explanation, that it is a coincidence, is a null hypothesis that can be tested then accepted or rejected to support the causality hypothesis.

It is interesting to note that the first assumption of Granger causality is reasonable in many situations (events of the future cannot have effects on the past), however the second assumption is rarely true (everything that could possibly cause Y has been controlled or quantified into the variable Z). For the most part, this operational definition of causality is only useful in closed systems where all potential causal factors are contained within the system.