Unfortunately this calculus is exceedingly complicated and I haven't even seen a definition of "a causes b" in terms of this calculus. One problem is that Pearl and others make use of the notion of "d-separation". This allows for elegant proofs but is hard to understand. I once found a paper which replaced d-separation with equivalent but more intuitive assumptions about common causes, but I since forgot the source.

By the way, there is also an alternative to causal graphs, namely "finite factored sets" by Scott Garrabrant. Probably more alternatives exist. Though I don't know more about (dis)advantages.