What “necessary assumption” means
Yesterday, I discussed one specific type of Assumption question--the "Sufficient Assumption." Today, I'll switch gears and talk about Necessary Assumptions. Warning: I'm going to use math again. And once again, if you passed third grade you're going to do just fine.
Premise: Anything times zero equals zero. Conclusion: Therefore A times B is not zero.
Question: "Which one of the following is an assumption on which the argument depends?" Or, stated another way, "Which one of the following is necessary support for the argument's conclusion?" Or, stated still another way, "Which one of the following is an assumption required by the argument?"
These three questions are all asking you to do the same thing. Your task is to identify an answer that must be true, in order for the argument to even conceivably be true. In other words, it's asking you to identify an answer that, if untrue, would cause the argument to fail. In other words, it's asking for a necessary condition.
There are two answers here. "A does not equal zero" is one of them, and "B does not equal zero" is the other. If either of these statements are untrue, then the conclusion of the argument would fail (because anything times zero equals zero). It is necessary that A not be zero, and it is necessary that B not be zero, because if either A or B are zero, the argument is complete nonsense. Since "A does not equal zero" and "B does not equal zero" were unstated, they are assumptions. So "A does not equal zero" and "B does not equal zero" are both necessary assumptions of the argument.
Note! Both of these statements are necessary, but neither are sufficient. If A does not equal zero, that doesn't prove that A times B is not zero (because B might equal zero). Likewise, if B does not equal zero, that doesn't prove that A times B is not zero (because A might equal zero). So neither of these would be correct answers to the question "Which one of the following, if assumed, would allow the conclusion to be properly drawn," which is a sufficient assumption question.
But the questions listed above were all asking for Necessary Assumptions, and "A does not equal zero" and "B does not equal zero" must be true or else the argument will fail--thus we know they are "necessary."
The definition of a necessary assumption is "something that must be true, or else the argument will fail." (Alternatively: "something that, if untrue, will prove the argument invalid.") Go ahead and memorize that. Thanks for bearing with my math examples... I'll put away my abacus for a while now.