Basic Logic Unit 4Necessary and Sufficient Conditions The concepts of necessary and sufficient conditions help us understand and explain the different kinds of connections between concepts, and how different states of affairs are related to each other.Necessary conditionsTo say that X is a necessary condition for Y is to say that it is impossible to have Y without X. In other words, the absence of X guarantees the absence of Y. A necessary condition is sometimes also called “an essential condition“.Having four sides, being a square.Being brave, being a good soldier.Not being divisible by 4, being a prime number.Necessary ConditionsTo show that X is not a necessary condition for Y, we simply find a situation where Y is present but X is not. Examples :Being rich, being happy.Being Chinese, being a Hong Kong permanent resident.Sufficient ConditionsTo say that X is a sufficient condition for Y is to say that the presence of X guarantees the presence of Y. In other words, if X is present, then Y must also be present.Being a square, having four sides.Being divisible by 4, being an even number.Four PossibilitiesGiven two conditions X and Y, there are four ways in which they might be related to each other:X is necessary but not sufficient for Y.X is sufficient but not necessary for Y.X is both necessary and sufficient for Y. (or “jointly necessary and sufficient“)X is neither necessary nor sufficient for Y.Four PossibilitiesThis classification is very useful in when we want to clarify how two concepts are related to each other. Here are some examples :Having four sides is necessary but not sufficient for being a square.Having a son is sufficient but not necessary for being a parent.Being an unmarried man is both necessary and sufficient for being a bachelor.Being a tall person is neither necessary nor sufficient for being a successful person.Different Kinds of Possibilities It is impossible to be a tall man without being tall.It is impossible to dissolve gold in pure water.It is impossible to travel from Chengdu to New York in less than ten minutes.It is impossible to visit the army barracks without a permit.Different Types of Necessity and SufficiencyHaving four sides, being a square.Being a father, being a parent.The presence of oxygen, the proper functioning of the brain.Passing current through a resistor, generation of heat.Being an adult of over 18, having the right to vote.The presence of a witness, a valid marriage.EXCLUSIVE AND EXHAUSTIVE POSSIBILITIES A possibility including another: “There being rain tomorrow” includes the possibility of a heavy rainstorm, and the possibility of just a light drizzle. A possibility can exclude another: If Cinta is in Spain right now, that excludes the possibility that she is in Brazil.Two possibilities can be independent: Whether it will rain tomorrow presumably does not depend on what you ate for breakfast this morning.EXCLUSIVE AND EXHAUSTIVE POSSIBILITIESTwo possibilities that are neither exhaustive nor exclusive: x 3, x 4. They are not exhaustive because the possibility that x = 2 is not included. They are not exclusive because both of them can be correct, as when x 5. Exhaustive but not exclusive: x 4, x 4, x = 1 Exclusive and exhaustive: x 0, x = 0, x0