As a result of yesterday’s argument it is also possible to show that it is impossible for certain kinds of things to be the cause of anything (and in keeping in line with my earlier use of the word I will call them epiphenomenal).
One type of “epiphenomenal” entity are statements that must be true (necessary truths). I could demonstrate mathematically why this is so, but instead I will provide the epistemological reason here (because I still can’t get formulas formatted properly). Consider then how we come to know that one thing is the cause of another. We observe that whenever event A is present event B always happens later (or is very likely to happen), but that when A is not present B may not occur. Given many occurrences of such a connection, and associated evidence, we come to have good reason to believe that A is the cause of B (there is a high probability of this statement being true). However a statement that is true by necessity can never be missing, it can never “not occur”. Thus we can never have evidence that it is the cause of any event.
(If you really want to prove it though here are the basics: First set up a “description” that represents the necessary truth. This description must always hold, hence its dependence set is empty. Even if this wasn’t a contradiction consider what it would mean; namely that the necessary truth is the cause of every event. Even an event and the non-occurrence of the same event (A and ~A). This is clearly just as contradictory.)
This might seem to be contrary to common sense. For example given the axioms of set theory (or one of many other axiomatic systems) 2+2=4 is a necessary truth. When we put two rocks on top of two more rocks we have four rocks. Doesn’t this prove that 2+2=4 had a causal effect on the physical world (i.e. the processes of putting together two things with two things resulted in four things because 2+2=4 is a necessary truth)? Not at all. What guaranteed that we would have four rocks was the physical properties of the rocks and the structure of nature, neither of which are necessary truths. For example if rocks were liquid we would only have had one rock in the end. It is true that the idea of 2+2=4 may have a causal effect on our reasoning, but clearly the idea is not necessary since not everyone is born knowing that 2+2=4. (you can read a bit more on this idea here)
Something else that can’t have a causal effect is anything that is omnipresent. As with necessary truths, the inability for an omnipresent substance to be “taken away” leads to contradictions. Unlike necessary truths there are few things that we think are omnipresent. Space is one of them, but space by itself doesn’t have a casual effect. The curvature of space has an effect, but a given curvature isn’t omnipresent; there are different curvatures at different locations.* Of course many definitions of god consider he/she/it to be omnipresent. Clearly such an entity is impossible, unless you admit that it is also powerless, which I suspect few people who maintain god’s omnipresence would accept. Of course this doesn’t destroy every idea of god. For example if you thought that god’s omnipresence was guaranteed by “god particles” that shadow every physical particle then it is logically possible for god to be “taken away”, and hence no longer impossible for he/she/it to have causal powers. However, as I mentioned before, it is unlikely that god is necessary, so perhaps this entire argument was unnecessary.
* I know some smart aleck is going to object by supposing a hypothetical universe that had non-zero constant curvature. However this would imply that the universe contains no matter or energy, hence no causal effect.