Dennis:

If c1 necessitates e, then in every possible world where c1 occurs it brings about e. If it's not the case that in every possible world where c1 occurs it brings about e, then c1 doesn't necessitate e.

In the gunshot case, perhaps the laws of nature are contingent and God suspends them so that you live. Then even though you were struck by a bullet in a specific place, the laws of nature that determine your death from it fail and it's not true that you died.

Perhaps, in some possible world, you're a freak with a second heart. Then even though it's true that your heart1 was struck by a bullet, it's not true that you died. Hence, your being struck in a specific part of the heart1 by a bullet doesn't necessitate your death.[1]

In the match case, perhaps the struck match fails to light in the wind. Then even though it's true that you struck the match, it's not true that the match lit. Hence, your striking the match doesn't necessitate the match lighting.

Mumford's contention is that, for every cause, something extra could happen that prevents e from being brought about by c1 even though c1 occurs.

. . . which I don't really find determinate enough an example.

Suppose we include every purely positive truth in your antecedent. There are almost certainly still contingently negative truths that could have been positive and thereby stopped e from occurring even though c1 occurred[2]. So, you can make your antecedent detailed enough to necessitate your consequent only if you add negative truths to it.

But negative truths aren't causes and can't be converted to causes. Hence, they can't be included in the kind of antecedent you need to stop Mumford and Anjum's argument against causal necessitarianism. Hence, in replying to the Mumford-Anjum argument, you can't make your antecedent detailed enough to necessitate your consequent. Thus, it doesn't matter how detailed the cause in your example is.


[1]In the second example, the added factor is your heart2 pumping enough blood to keep the rest of your organs running even though heart1 stops.
[2]Even if it's something as simple as "God didn't prevent e from occurring even though c1 occurred", which seems like it would be a required truth in most cases.

So, would you say that I'm making an error if I reply with, "The necessary conditions would simply be wrong, it would mean the removal of 2-hearts, or rather, all hearts."

The claim was that a bullet striking heart1—in most cases your only heart—in a specific place necessarily causes your death. All that's needed to break that claim is the mere possibility that it doesn't. (No, changing the claim after the fact won't help you.)

Would I necessarily have to rely on (resort to) something like the laws of nature being contingent in order to say that causes do not produce their effects out of necessity?

No. God can simply withdraw His consent from the effect occurring. The same goes for seemingly any effect.