Glad you asked this question here. Mumford and Anjum are my favorite contemporary Philosophers, bar none. Their work on causation, I take to be seminal work. First and foremost one must understand that the necessity operator is an operator. Mumford and Anjum argue that the notion of necessity (the operator) is much more obscure than causation itself. Anjum and Mumford put forward what they call Antecedent strengthening. This is something they've gained from Anscombe or Geach's work, I'm not too sure, but I do know that Geach has a paper on it somewhere. I've not read their short introduction, but I am familiar with their work in Getting Causes from Powers and elsewhere.
The key focus of the Anjum-Mumford thesis is that we use the necessity operator recklessly and this obfuscates everything. Quotation from p. 47, Getting Causes from
*Even if you strike the match exactly right, there may be some external factor, outside your control, that prevents the match from
lighting. You still have no doubt that when the match does light, the
striking was the cause, or at least a cause. Does that mean that
striking necessitates the lighting in just some cases, namely, the
successful ones, but fails to do so on other occasions? That seems
like a misuse of the concept of necessity. If someone were to say that
in some cases, being water necessitated being H2O, but in other cases
it did not, prima facie it would seem as if they did not understand
the meaning of necessity. In the causal case, it seems
more plausible in such circumstances to say that particular a caused
or produced particular b without necessitating it. That a caused b
suggests that a made b happen (Woodward 2003), but why should that
automatically mean that a necessitated b? Could there not be an
account of causal production without necessitation? Is necessitation
rather more than is needed for causation? Isn't causation between a
and b already a strong enough connection? The philosophers who try to
account for causation in terms of necessity are, we argue,
over-bidding in their theories.
Anjum and Mumford are targeting what it means for the notions of sufficient conditions as applied to the case of causation. It is a necessary condition for my peanut plant to receive sunlight in order to grow, but it is not a sufficient condition. A necessary condition is some event or condition without which we would not have the effect, and sunlight is a necessary condition. However, this necessary condition doesn't guarantee the effect. It seems that a sufficient condition would guarantee the effect. They then move on to the argument that there is never a sufficient conditions for an effect, and thus against a cause ever necessitating its effect.
Two quick things. Propositions do not cause anything. Propositions entail certain things. Entailment is different from the relation of necessitation, they are simply said, two different categories. A rock (by the traditional definition) means by definition, that it's something which is a non-thinking substance. Being a rock doesn't necessitate that it is a non-thinking substance, it entails it. When you unpack the meaning of words and propositions, certain things entail the premises, but this is not to be confused with necessitation.
Causal Necessitation is the thesis that if certain conditions are met, then the effect follows out of necessity, that there is no chance of it being prevented or interfered with.
Anjum and Mumford take an issue with this because of the following;
(1) Antecedent Strengthening. The challenge is that, If A necessitates B, then, if A + ∅, for any ∅, then B.
To put it in other words, if a cause necessitates any effect, it necessitates it simpliciter. There are no added conditions to it, the moment you do so, you're leaving Causal Necessitarianism (CN). So when they say that it could have been interfered with, and thus stopped the causal production, they take this to go against the thesis of CN above defined. People usually read this and say, "Well, that's an easy challenge to meet, let's just give it enough conditions and remove the kind of conditions that could interfere with an effect, and thus CN will be redeemed."
Suppose an effect e has four causes, c1, c2, c3, c4. An instance of c1-c4 produces e. The question is not whether c1-c4 caused it, but whether it caused it by means of necessitating it. So if there is a possibility of it being interfered with, it cannot be said to have necessitated the effect. Because there could always been a causal interferer (CI). As such, most of all causation seems to fail the test of necessity. It should be added that they are saying something like, "Give us any cause or event, and let us see whether or not that passes our test." If it doesn't pass their test the cause doesn't necessitate its effect, if the effect is ever produced. Suppose that you can show some (for the sake of argument) causes in nature that do pass their test, that would only mean that those specific causes are necessary causes while the rest of all causation is without necessity. CN should also not be confused with determinism which concerns the fixity of the future, and can be achieved or argued for even without CN (though I think that is wrong on different grounds).
(2) Negative state of affairs cannot be converted to causes. Absences are not causes per se. There is no such thing as causation by absence. Suppose I'm suffocating, the absence of oxygen causes my suffocation, or so we say. A lack, or nothingness is not something else in disguise. The problem is that apparent nothingness is involved in causation. Causation by absence claims are false.
(2a) Phil Dowe (2001) invokes counterfactuals in place of absences as causes. Statements like "Absence of A caused *B," what we have in mind is that should A not have occurred, B would not have been the case. So in me being suffocated, it's not that the absence of oxygen that causes me to suffocate, but rather, had I received oxygen, I wouldn't have suffocated. Since Anjum and Mumford subscribe to the Powers theory of causation, it is in virtue of the powers present in my body that cause me to suffocate. Me breathing finely was a state of equilibrium by intaking oxygen among other things to breathe. Certain facts about me always disposed towards my suffocation. So in the cases of the absence of oxygen, the remaining powers of my substance-hood did all the causal lifting to cause me to suffocate (though the absence of oxygen plays an explanatory role, it doesn't play a causal role). You would need truthmakers for the kind of counterfactuals I've mentioned above, but I'll refrain from putting them here.
(3) Analysandum? Suppose someone touches my hand when I'm numb, someone has touched my hand. Isn't it always the case that when someone mentions a cause that a cause and effect are being related? No. Anjum and Mumford deny this and spend their whole book on finding out what a cause is. This example is a prime case of a cause happening, that is, someone holding my hands, but due to my defective senses, fails to have the effect of me feeling their warmth. A cause should not be looked at as always producing an effect, but should be seen as something which makes certain effects possible by increasing their possibility. They spend their whole book on 'What does it mean to be a cause?' after decisively taking down Causal Necessitarianism, there is simply way too much to go into in order to show how they succeed in doing this, so my best bet is to ask you to read the book in due time :). I've only mentioned the things I've suspected you're having problems with, the bulk of it, I wouldn't dare go into over here. It would be too much to do, feel free to ask me questions though.