Hi, LM!
You say the closest fallacies you can find are affirming the consequent and denying the antecedent and that one of these may in fact be the right classification. I will argue that this diagnosis is incorrect and then suggest a simple method for critiquing the view that there is only one cause of an effect. This is lengthy because I address your example-arguments in turn.
Here is your first example-argument. "I will go out of my way to not get [cancer] so I won’t die. I will not get cancer. Therefore, I won’t die."
The first sentence has the word "so" in it, which suggests a conclusion. Read in this way, the first sentence states that I will not get cancer and concludes that I will not die . But that is just what the second and third sentences do. So really, it seems that all you have given us with the first example is this:
1. I will not get cancer.
2. Therefore, I will not die.
This is not obviously affirming the consequent or denying the antecedent. After all, there is no conditional statement here and a conditional statement is required for affirming the consequent or denying the antecedent. Supposing that a conditional statement is tacit in this argument, which is likely the case, I think it is fair to decide that the tacit conditional statement is this: if and only if I get cancer, I will die (which is actually a bi-conditional). I think this is a fair decision because the OP stipulates that the person giving the argument believes that getting cancer causes and is necessary for death (that the given thing, which in this example is cancer, is the only thing that can produce the given outcome, which in this example is death). The tacit bi-conditional that I have suggested reflects this belief by stating that getting cancer is necessary and sufficient for death. Yet if the argument has this tacit statement, then the argument still does not deny the antecedent or affirm the consequent. What's more, this tacit statement, when made explicit, allows for a modus tollens inference, if I did my proof correctly, which is a valid inference.
Take your next example-argument.
"I won’t even drive anywhere. I won’t drive my car, therefore I won’t get into an accident. Therefore, I won’t get killed." This seems to say:
1. I won't drive anywhere.
2. Therefore, I won't get into [a car] accident.
3. Therefore, I won't get killed.
This argument does not apparently affirm the consequent or deny the antecedent. Again, there are apparently no conditional statements. Supposing that the conditional statements are implicit, which they probably are, it is fair to say that they are these: a.) if I won't drive anywhere, I won't get into an accident. And b.) if and only if I get into a car accident, I will get killed. The bi-conditional reflects the person's belief, as stipulated in the OP, that W is the only thing that can produce outcome O (W is necessary and sufficient for O). But this argument, once these conditional statements are drawn out, still does not affirm the consequent or deny the antecedent. What's more, it is a valid argument, as well. 2 follows from 1 and a. 3 follows from 2 and b.
Your first two example-arguments provide us with no grounds for classifying the issue under denying the antecedent or affirming the consequent. And there are reasons to not classify the issue in this way. What needs to be considered is that a person who assumes that there is only one particular thing or way W to bring about a given outcome O, assumes that W is necessary and sufficient for O. By assuming both the necessity and sufficiency of W, this person validly infers that given that W will not happen, O will not happen. If and only if W, O. Not W. Therefore, not O does not deny the antecedent or affirm the consequent.
Take the third example-argument:
1. If I take cyanide, I will die.
2. I did not take cyanide.
3. Therefore, I will not die.
This argument does deny the antecedent and is invalid because of this. The flu argument also has the same problem. Nonetheless, this argument does not represent the views of someone who believes that taking cyanide is the only way to die (that W is the only way for outcome O to occur). Contrary to your assessment, the claims in the argument are not committed to the view that cyanide is the only way to die. What the first premise states, rather, is that taking cyanide is sufficient for death. Whereas a person who believes that cyanide is the only way to die would have stated something like "If and only if I take cyanide, I will die", not "if I take cyanide, then I will die". For the former statement, not the latter, is the one that expresses the person's view that taking cyanide both causes and is necessary for death (that taking cyanide is the only way to bring about her death). So although the argument you present to us is fallacious, it is not an argument that a person who believes cyanide is the only way to die would have made in the first place. The argument made by a person who believes cyanide is the only way to die would be something like the following argument:
1. If and only if I take cyanide, I will die.
2. I will not take cyanide.
3. Therefore, I will not die.
This argument, however, neither affirms the antecedent nor denies the consequent, and the argument is valid! A similar answer can be given to the flu-argument as was given to the cyanide-argument. Next, I will offer a way of finding fault in the view that your OP attacks.
I think that one of the best ways to oppose the causal assumption represented in your OP is by showing that the assumption is false with a counterexample. It is false that cancer is necessary for dying; it is false that getting into an accident is necessary for dying; it is false that taking cyanide is necessary for dying; it is false that getting the flu is necessary for dying. My brief counterexample is this: Abraham Lincoln died but not because of cancer, a car accident, taking cyanide, or the flu. The reason this counterexample shows that the assumption is false, is that it is a real case in which the outcome--death--occurred but each of the four factors (cancer, car accident, cyanide, the flu) were absent. Since death occured without any of these four factors, neither of these four factors is necessary for death. End counterexample. Whatever W may be, LM, you can attack the claim that W is necessary and sufficient for outcome O by giving and explaining a counterexample.
Thank you, LM.
From, Kaiden.