Introduction to Necessary and Sufficient Conditions

Sandra LaFave

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A is necessary for B if and only if B can’t occur without A (or “unless” A). In logic the word “unless” means “or”, so one way to express that A is necessary for B is to say “not-B unless A”, (~ B v A).  If A is necessary for B, furthermore, then all Bs are As; and only As are Bs. Do some examples to convince yourself of these equivalences.

 

A is sufficient for B if and only if A guarantees B. Whenever you have A, you have B. Anything A is B. All As are Bs.  You never have A without B.

 

A	B	Analysis
Being female	Being pregnant	Being female is necessary for being pregnant.
Getting a grade of A	Passing the class	Getting an A is sufficient for passing.
Gas in car	(Gas powered) Car runs	Gas in the car is necessary for the car to run.
Decapitation	Death	Decapitation is sufficient for death.

Table 1: Ordinary Examples

 

Make sure you understand the “Analysis” column above.

 

Now, if you switch A and B in Table 1, you can see the interesting result that

1. If A is sufficient for B, B is necessary for A.

   AND

2. If A is necessary for B, B is sufficient for A.

 

A	B	Analysis
Being pregnant	Being female	Being pregnant is sufficient for being female, i.e., being pregnant guarantees being female.
Passing the class	Getting grade of A	Passing is necessary for getting an A, i.e., you can’t get an A without passing.
Car runs	Gas in car	Car running is sufficient for gas in car, i.e., the car running guarantees there is gas in the car.
Death	Decapitation	Death is necessary for decapitation, i.e., you can’t be decapitated without dying.

Table 2: Reversed Examples

 

Review the definitions of necessary and sufficient conditions and satisfy yourself that the analyses in Table 2 are correct.