What is the difference between "necessary" and "sufficient"?

Question

What is the logical difference between something being necessary in order for something else to be true; as opposed to something being sufficient to make something else true. i.e. Fuel is sufficient to make an internal combustion engine run. vs Fuel is necessary to make an internal combustion engine run. and what are some subtle examples of how the difference between these two things can greatly impact the meaning of a sentence, discussion, or conclusion.

Answer 1

Re: Hien's explanation: If you're trying to think about the difference between necessary and sufficient conditions in context to a course on classical truth-functional logic, I'm afraid that Hien's explanation might be confusing to some. This is because, given the truth-functional definition of conditional statements, many statements of the form

If P then Q

will wind up true even though most of us would be more than hesitant to think that P is sufficient for Q or that Q is necessary for P. For example, consider the sentences

A Socrates is a philosopher.

and

B Some animals are not in Texas.

Because B is true, the truth-functional conditional

If A then B

is true. Yet, hardly anyone would think A, though true, is sufficient for B, or that B is necessary for A.

All this to say that, if you're trying to learn truth-functional logic, thinking that, for any conditional of the form, if P then Q, is true if and only if P is sufficient for Q or Q is necessary for P, will be a good reason to think you're not on the right track in your logic course.

Answer 2

The logical difference between p being necessary for q and p being sufficient for q is the following: the former means that p is true in all cases that q is true (but not vice versa) while the latter means that q is true in all cases that p is true (but not vice versa). It might be easier to understand this in the following way:

1. 'p is necessary for q' is equivalent to saying 'in order for it to be even possible that q to be true, p must be true'
2. 'p is sufficient for q' is equivalent to saying 'in order to guarantee that q is true, p must be true'

This demonstrates that when p is necessary for q, p isn't enough to guarantee that q is true, since it only makes it possible that q be true. However, it does entail that without p being true, q cannot be true. On the other hand, when p is sufficient for q, p guarantees that q is true, but q could still be true without p being true.

Here are some examples:

1. It raining outside is sufficient, but not necessary, for the sidewalk being wet. (since the sidewalk could be wet as a result of water coming from sprinklers)
2. Listening in class is necessary, but not sufficient, for doing well on the exam. (since one also needs to retain what one has learned in class to do well on the exam)
3. Fuel is necessary, but not sufficient, for driving a car. (since the car might have fuel but the engine doesn't work or nobody is in the car to drive it)

In these examples, you can see how replacing every instance of 'sufficient' with 'necessary' and every instance of 'necessary' with 'sufficient' would make all these statements go from true to false.

It's worth pointing out that p being sufficient for q will always entail that q is necessary for p, and vice versa. On a more formal note, for any conditional in the form of "If p, then q", p here is a sufficient condition for q while q is a necessary condition for p. This is also the case for "p only if q" or "q if p", since both of these are equivalent to "If p, then q".

It's also important to note that it's possible for p to be both sufficient AND necessary for q, where this is represented by the biconditional, "p if and only if q". Since p being sufficient for q entails that q is necessary for p, and vice versa, then the p being both sufficient and necessary for q also entails that q is both sufficient and necessary for p.