Combining causal and identity relations

Question

Suppose I have the causal relation C causes E, and the symmetric identity relation E is (E1 ^ E2 ^ E3), which have the following probability functions: E = bC and E = E1 * E2 * E3 where E, C, E1, E2 and E3 are variables and b is a constant. Combining these equations yields bC = E1 * E2 * E3.

Assuming that b, E2 and E3 are 1, could one conclude that C causes E1? If not, why? What are the limits for how causal and identity relations can be combined?

Thank you in advance!

Answer 1

TL;DR

What you can say is that, if E2 and E3 are certain, and C causes E, then C causes E1.

To dissect what is going on, let’s review a little of probability theory, then address the subtleties of your question.

Probablity theory

You want to know if E is true or not. You know that if C has happened, then there is s chance that E has happened as well. We normally write this as P(E | C) = b for some number b between 0 and 1. You read this equation as “the probability of E given C is b”. So now if you want to know the probability of E happening you just need the probability of $C$:

P(E) = P(E | C) P(C) + P(E | not C) ( 1 - P(C))

Note this is Bayes’s theorem and it’s a statement of correlation not of causation. Distinguishing between correlation and causation using probabilities is quite a subtle matter. I will use the two terms loosely, generally implying correlation.

Now there is another way of getting at the probability of $E$. You know that $E$ is actually the combination of 2 events (you can use three but like this is clearer):

E = E_1 \wedge E_2

We have to distinguish two cases: the cases where the 2 events are dependent and independent. An example of the first case is “I am a woman named Alex” which is the conjunction of the two statements “I am a woman” and “my name is Alex”. Not that these propositions are not independent: if I know your name is Alex, I might assume you are a man, since most people named Alex are men. The probability of your name being a man is not the same as the probability of you being a man, given your name is Alex. We have the following.
An example of independent events is getting heads at my first coin flip and heads at my second one. The probability of getting heads at the second flip is 50% regardless of the ouctome of the first one.
In both cases we have

P(E) = P( E_2 | E_1 ) P(E_1) = P(E_1 | E_2) P(E_2)

But only in the case of independent outcomes we can write

P(E) = P(E_1) P(E_2)

A couple of issues with your question

You seem to be confusing statements with their probabilities. Statements are sentences while probabilities are numbers. So writing E = bC is meaningless, because even if you can define what multiplying a statement by a number means, it probably won’t meant what you want it to mean, which is instead the perfectly sensible equation between numbers P(E) = b P(C).

Similarly, when you wrote E = E_1 * E_2 you meant P(E) = P(E_1)P(E_2). The second issue here is that this equation only holds if E_1 and E_2 are independent, or if one of the two equals 1. Actually, let's quickly prove that this last clause is true. From earlier we have

P(E_2 | E_1) P(E_1) = P(E_1 | E_2) P(E_2)

But if E_2 is certain, we have P(E_2) = 1. We also have P(E_2 | E_1) = 1 since E_2 happens no matter what! The equation above then reduces to:

P(E_1) = P(E_1|E_2)

and thus:

P(E) = P(E_1|E_2)P(E_2) = P(E_1)P(E_2)

Finally, an answer

Now you have two equations between numbers, so you can apply algebra to it and obtain:

bP(C) = P(E_1|E_2)P(E_2)

Then as we said, if E_2 is certain we write

bP(C) = P(E_1)

And thus we have the same relationship between C and E_1 that we had at the beginning with C and E_2.