philosophy question about necessary and sufficient conditions?

Wilfred Santacruce: A mathematician encounters this on a daily basis. Suppose you have a hypothesis H and you get a conclusion C. Restating this we have H implies C which means that whenever you have H, then C necessarily follows. Then C is called a necessary condition for H. This is clear enough but mathematicians always want more. They ask: Is the conclusion C actually a sufficiently strong image of H so that C in some sense contains the complete H and not just part of H? In other words, is C sufficient to get a conclusion H? If so, then C is a sufficient condition for H. To recap: If H implies C then C is necessary for H and H is sufficient for C. It's just an english language description of what's going on. Another way is " H is strong enough " to get C. I'll give you an example after this idea: It's quite possible and advantageous for C to be both a necessary and sufficient condition for H. This makes C and H equal strength or in some sense logically equivalent. Thus! if C is both necessary and sufficient for H then H is both necessary and sufficient for C, they "contain" each other. Example: If a number is divisible by 4(call this H), then it is divisible by 2 (call this C). H implies C but C is not strong enough to imply H for example 6 is divisible by 2 but not by 4 Therefore C is necessary for H but not sufficient for H. Example: In Euclidean geometry, a triangle has equal sides (call this H) if and only if the triangle has equal angles (call this C). This means H if C: H is necessary for C since C implies H H only if C: C is necessary for H since H implies C H if and only if C: H and C imply each other and are both necessary and sufficient conditions for each other...Show more

Ardath Templer: I think it is not justified to assume that all life must require the conditions that support OUR life here on earth. We just don't know enough about it, or have looked in enough places. Like, virtually none other.Further we don't know ! whether the big bang was necessary. Perhaps a Lesser Bang wou! ld have done the trick - or a Big Squeeze - or something else altogether. This is all so above our pay grade, we look silly even talking about it, imho.But I always appreciate when someone contrasts necessary with sufficient. Those are two great concepts and the ability to differentiate bteween them is sharp....Show more

Irma Poiter: There is a difference between necessary and sufficient conditions.A necessary condition is one that which has to be true in order to have the proper consequence. To use the example you provided, if the planet is at the proper distance to a host star, the planet can develop life; if the planet is too close or too far to the host star, the planet cannot develop life. Thus "proper distance to a host star" is a necessary condition. That said, a planet at the proper distance may not develop life if the planet has no water, or other conditions. Sufficient conditions is the set of conditions that produce the proper consequence, wher! e subtracting one of those conditions would not produce the proper consequence, and adding conditions would have no effect on the proper consequence. To use the example you provided, if the planet has water, a stable rotation, and so on, the planet would develop life. The set of such conditions is sufficient to produce life. As you see, the term "necessary condition" applies to one statement, while the term "sufficient condition" applies more to groups of statements, or a set of statements. It is possible to have one condition that is both necessary and sufficient to produce a proper consequence, but in most cases, more than one condition is required....Show more