From previous experience, the ideas of a necessary condition, vs. a sufficient condition, and a necessary and sufficient condition are often not easily grasped by students. The handout in IVLE Files (“Everything you always wanted to know about arguments”) covers the relevant topics and your tutors will also be reinforcing the ideas. Below is a refresher.

For our purposes, take note of the following equivalences (everything in the same column are logically equivalent to each other):

 P is a necessary condition for Q P is a sufficient condition for Q P is a necessary and sufficient condition for Q P is necessary for Q P is sufficient for Q P is necessary and sufficient for Q Q only if P Q if P P if and only if Q Only if P, then Q If P then Q Q if and only if P If Q then P P only if Q If not P then not Q If not Q then not P

(Additional: Do note that the above has (almost) nothing to do with the idea of a necessary truth. Best to keep them as separate as possible. Watch out for “necessary condition” and “necessary for“.)

Ok, on with the refresher:

P is necessary for Q = If you don’t have P, then you definitely won’t have Q; but even if you do have P, you still might not have Q.

• Example: Water is necessary for plant growth–your plant just won’t grow if you don’t give it water; but even if you give it water, it still might not grow (it needs other stuff).

P is sufficient for Q = if you have P, then you definitely have Q; but when you don’t have P, you might still have Q.

• Example: Rain is sufficient to get the road wet–if it rains the road is going to get wet; but even if it doesn’t rain, the road can still get wet (there are other ways for it to get wet),

P is both necessary and sufficient for Q = combine the above, so, if you have P, then you definitely have Q, and if you have Q, then you definitely have P; they go together perfectly.

• Example: Being an unmarried male person is necessary and sufficient for you to be a bachelor; the two things go together perfectly.

P is neither necessary nor sufficient for Q = the opposite of the previous–you might get one without the other, both ways.

• Example: Being an unmarried male person is neither necessary nor sufficient for being a violinist–you might be one without being the other, and vice versa.

And to the above, we can also add:

P unless Q = If not Q, then P

• Example: This plant will die unless there is water, i.e., if there’s no water, this plants will die.

Now, what happens when you deny one of these conditions? The first thing to note is that when you deny that P is a necessary condition for Q, you aren’t implying that P is sufficient condition for Q, and vice versa. When we say that is a necessary condition for Q, what it means is that if not P then not Q. For instance, rice is a necessary condition for making chicken rice–if you don’t have any rice, you ain’t making any chicken rice. But what happens when we deny that something is a necessary condition for something else? Here’s a way to think about it–having access to an electric rice cooker is not a necessary condition for making chicken rice. What this means is that even if you don’t have access to an electric rice cooker, it’s not going to follow that you ain’t making any chicken rice. You can always just use a pot over a stove–like in the old days.

 Deny: P is a necessary condition for Q Deny: P is a sufficient condition for Q Deny: P is a necessary and sufficient condition for Q P is not a necessary condition for Q P is not a sufficient condition for Q P is neither necessary nor sufficient for Q Even when not P, it could still be Q Even when P, it might not be Q Even when not P, it could still be Q, and even when P, it might not be Q Not P and Q is a possible combination P and not Q is a possible combination P and not Q, and Q and not P, are both possible combination

Bonus material. For those who read Classical Chinese, the below is how the ancient Mohist logicians would have defined the various concepts (we have fragments of some of them in surviving texts):

Necessary condition = 有之不必得, 無之必不得 (having it, you need not get the other; not having it, you definitely will not get the other)

Sufficient condition = 有之必得, 無之不必不得 (having it, you definitely will get the other; not having it, you need not not get the other)

Necessary and Sufficient Condition = 有之必得, 無之必不得 (having it, you definitely will get the other; not having it, you definitely will not get the other)

Neither Necessary nor Sufficient Condition = 有之不必得, 無之不必不得 (having it, you need not get the other; not having it, you need not not get the other)