Necessary and Sufficient Conditions

The following are equivalent:

· If A, then B

· All A’s are Bs

· A is sufficient for B and B is necessary for A

· A Venn diagram in which A is in the inner circle and B is in the outer circle

Sufficient

When A is sufficient for B, A’s existence guarantees B's existence. If A exists and B does not exist, then A is not sufficient for B.
To figure out whether A is sufficient for B ask yourself the following question: Can I, just by knowing A, infer that B is the case?

Necessary

When B is necessary for A, B's absence guarantees A's absence. If A exists and B does not exist, then B is not necessary for A.
To figure out whether B is necessary for A ask yourself the following question: Can I, just by knowing that B is not the case, infer that A is not the case?

Examples of Necessary and Sufficient Conditions:

1. Paying your tuition bill is necessary but not sufficient for graduation from MSU.

Necessary: You can't graduate without paying your tuition bill.
Not sufficient: Paying your tuition bill is not enough to ensure that you graduate.

2. Being able to run a 4-minute mile is sufficient but not necessary for being physically fit.

Sufficient: Being able to run a 4-minute mile guarantees that you are physically fit.
Not Necessary: Being able to run a 4-minute mile is not required for being physically fit: You can be physically fit without being able to run a 4-minute mile.

3. Satisfying all the academic requirements and paying all your bills is both necessary and sufficient for being eligible to graduate.

Necessary: You can't graduate unless you satisfy all the academic requirements and pay all your bills.
Sufficient: If you do satisfy all the academic requirements and pay all your bills, then you are eligible to graduate.

Necessary and Sufficient Conditions for Being a Mammal

Necessary Conditions for being a Mammal - Being warm blooded

If we know that some creature is not warm blooded we can infer that this creature is not a mammal, as being warm blooded is necessary for being a mammal. In other words every mammal is warm blooded.

Another way of making this point is to say that we accept the following statements: If x is a mammal, then x is warm blooded

Being warm blooded is not sufficient for being a mammal, however, because there are creatures who are warm blooded who are not mammals (birds).

Sufficient Conditions for being a Mammal - Being an elephant

If we know that there is an elephant in the next room we can infer that there is a mammal in the next room, as we believe that in order for a creature to be a mammal it is sufficient that that the creature be some specific kind of mammal. In other words, some mammals are elephants.

Another way of making this point is to say that we accept the following statements: If x is an elephant, then x is a mammal.

Being an elephant is not necessary for being a mammal because there are mammals who are not elephants (e.g., cats, dogs and humans).

* Note that some conditions may be both necessary and sufficient. For example, nursing one’s young (as a species) is both necessary and sufficient for being a mammal.