Introduction to Necessary and Sufficient Conditions

Sandra LaFave

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.



B Analysis

Being female

Being pregnant Being female is necessary for being pregnant.

Getting a grade of A

Passing the classGetting an A is sufficient for passing.

Gas in car

(Gas powered) Car runs Gas in the car is necessary for the car to run.


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.


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


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. 





Sandy's X10 Host Home Page | Sandy's Google Sites Home Page
Questions or comments?