I have already given one account of causation (see here), which I suggest you read before this piece. In this post I will update my previous account, present a second formalization of causation, and finally describe the differences between the two.
I have modified my previous formalization of causation, yielding the following formula:
This version, besides being somewhat shorter, leaves the causes of an event slightly more separated then they were before. To clarify the differences between the new definition and the old definitions I present the causes of the four examples from my previous article under the new definition:
Example 1: 
Example 2: 
Example 3: 
Example 4: 
Upon reflection I realized that this definition did not attempt to characterize sufficient causation, that is the complete states of affairs that will bring about a certain event. To fix this omission here is a second formalization that captures this idea.
As before n is the time between the event we are considering the causes of it, and w1 is the set representing the world at the initial time. Additionally:
Now let us define a relation called IC as follows:
With this relation we can now describe the set of sufficient causes as follows:
To illustrate how this description of causation differs from my first account I have worked out the examples again, this time under the new definition.
Example 1: 
Example 2: 
Example 3: 
Example 4: 
Of course both these definitions may need to be modified as described here if you want to apply them to more complicated situations.
Why should we prefer one of these definitions to the other? Although it seems clear to me that one of these definitions may not necessarily fit every need I generally find the first definition given here more useful. It has the benefit of breaking the casual contributors to an event up into different sets, which I find handy. Secondly it captures some information about causation that the second definition does not (see example 3). Of course this doesn’t mean that the second definition is invalid, the choice of when to use one over the other should be based on how the argument relies on causation.
On Philosophy 