If we say that A causes B, when will this be false? Had it not been A, it would not have been B. Is it only false in the case when it is not A, but is B?
What you've said is correct, in the sense that if we lay out the closest possible situation to what you have described in a truth table (according to the limits of what truth tables are usually used to represent), reading the third column as an expression of agreement or disagreement with the truth-combinations represented in the preceding two columns, of the 4 truth-combinations, the only expression of disagreement (F) stands next to the truth-combination of A not holding, with B holding. However, this truth table does not represent a direct, strong sense of causality, as in your question - rather, it represents the situation in which the truth of A is guarantor for the truth of B, such that whenever B is the case A must also be the case - where A and B are propositions, not objects.
- A B "if B holds then A must also hold"
- T T T
- T F T
- F T F
- F F T
We could also express this by taking the third column, and writing it sideways, so that we represent the relationship between propositions A and B like this: (TTFT)(A,B)
If A causes B then on the usual understanding of causality then it must be true that B cannot cause A.
The only case in which this is not true is if B is exactly A, that is we are considering the situation that A is self-caused. Traditionally speaking, this is only possible for God.
To know when causality is false, we need to know when causality is true. Logical analysis of causality is unfruitful, as material implication can never capture causality. Your counterfactual analysis cannot carry us far either. Presently, the best work on causality is done by Judea Pearl of UCLA CS Dept, in his book entitled Causality. He explains causality by means of statistical, Bayes net method.
In philosophy causality is examined in a course relating to scientific reasoning. So my answer is based on studies from this field. Causality becomes a focal point when a scientist designs an experiment since experiment is to identify a causal relationship between two events (random variables or parameters). The goal in designing an experiment is to show that only X (the independent variable) is the cause of Y (the dependent variable) (e.g., X = the amount of caffeine taken, and Y = the degree of attention ability). When this is successfully shown, the experiment is called internally valid.
When a scientist says that X causes Y, his statement can turn out to be false for many reasons. Consider the following example.
A scientist observes that tall people generally read more proficiently than short people. So he sets up an experiment to study the relationship between height and reading comprehension skill. The result was that the taller a person is, the more proficient his reading is. He concludes that height is the cause of reading ability. His experiment lacks internal validity since the scientist overlooked the fact that tall people are 5th graders and short people are kindergartners. The real cause of reading proficiency is the years of schooling, not height. The variable that the scientist failed to control or eliminate is called a confounding variable, which frequently turns out to be the real cause.
Even when confounding variables are eliminated, a causal statement can be false for the following reasons.
- causal reversal: Y actually is the cause of X.
- common cause: Itchy eyes (X) and a runny nose (Y) are due to a cold(Z).
- bidirectional causation: Bad economy and high unemployment rate.
- indirect causation: "Guns don't kill people: bullets kill people!"
- spurious connection: Since all the people who ate bread had died, eating bread must be the cause of death