Logic: sufficient condition vs. necessary condition?

"A" is said to be a sufficient condition for "B" whenever the occurrence of "A" is all that is needed for the occurrence of "B".

Example:

Being a wolf is a sufficient condition for being an animal.

--The above idea can be expressed in this conditional statement:

If something is a wolf, then it is an animal.

On the other hand, "B" is said to be a necessary condition for "A" whenever "A" cannot occur without the occurrence of "B".

Example:

Being an animal is a necessary condition for being a wolf.

--This example can be expressed in the same conditional statement.

Thus, every conditional statement expresses both a necessary condition and a sufficient condition.

Which is why our current obsession with hand sanitizers and surface sterilizing products makes no sense. We are actually increasing our likelihood of getting sicker over time because our immune systems weaken. Then when we do encounter a pathogen we can't fight it off. The little illnesses we experience help keep us from getting big ones.

Hi

P is a sufficient condition for Q just in case P implies Q. P is a necessary condition for Q just in case Q implies P. Let's write 'P > Q' for 'P implies Q'. Then:

(1) P is a sufficient condition for Q just in case P > Q;

and

(2) P is a necessary condition for Q just in case Q > P.

Now, to say that P implies Q is to say that if P, then Q. For example, to say that an argument's being sound implies that it's valid is to say that if an argument is sound, then it's valid. 'If P, then Q' says that P suffices for Q; it says that P is a sufficient condition for Q. So, for example, Jessica's being at least 30 years old is a sufficient condition for her being at least 25 years old: if Jessica is at least 30, then she's at least 25.

However, Jessica's being at least 30 years old is not a necessary condition for her being at least 25 years old: it is possible for Jessica to be at least 25 years old but not at least 30 years old (imagine Jessica is 27). So:

(a) If Jessica is at least 30, then she's at least 25;

(b) Jessica is at least 25 only if she's at least 30.

In (a), Jessica's being at least 30 is treated as a sufficient condition for her being at least 25. (a) is correct: being at least 30 does suffice for being at least 25. In (b), Jessica's being at least 30 is treated as a necessary condition for her being at least 25. (b) is incorrect: it is possible for someone to be at least 25 yet not be at least 30, and so being at least 30 is not necessary for being at least 25.

What you wrote down is correct: If X is necessary for Y, then without X, there is no Y, but X does not guarantee Y. Let's consider another example. All Texans are Americans. In other words, everyone who is from Texas is from America. So, being Texan is a sufficient condition for being American: 'If you're from Texas, then you're from America' is true. However, being Texan is not a necessary condition for being American: there are millions of people who are from America but not from Texas (the chances are that you're one such person :-)), and so 'You're from America only if you're from Texas' is false.

If P is a sufficient condition for Q, then whenever you have P, you have Q. If P is a necessary condition for Q, then whenever you don't have P, you don't have Q.

Consider the following four arguments:

(i) All Texans are Americans. George Bush is a Texan. Therefore, George Bush is an American.

(ii) All Texans are Americans. Barack Obama is an American. Therefore, Barack Obama is a Texan.

(iii) All Texans are Americans. Tony Blair is not an American. Therefore, Tony Blair is not a Texan.

(iv) All Texans are Americans. Jennifer Aniston is not a Texan. Therefore, Jennifer Aniston is not an American.

Arguments (i) and (iii) are fine, but there is clearly something wrong with (ii) and (iv). In (ii), being American is held to be a necessary condition for being Texan. That's correct: if you're not American, you're not Texan. However, being American is not sufficient for being Texan: you can be American without being Texan. So, the fact that someone is American does not guarantee that he's Texan; you can see this in the case of Obama, who's American but not Texan.

In (iv), being Texan is held to be sufficient for being American. That's correct: someone's being Texan guarantees that she's American. But, as in the case of Jennifer Aniston, the fact that someone is not Texan does not guarantee that she's not American: being Texan is not a necessary condition for being American.

So, the following argument forms are valid (as before, 'P > Q' means 'If P, then Q'):

(i) P > Q. P. Therefore, Q.

(iii) P > Q. ~Q ('~Q' means 'not Q'). Therefore, ~P.

But the following are invalid:

(ii) P > Q. Q. Therefore, P.

(iv) P > Q. ~P. Therefore, ~Q.

In other words, it is possible for an argument of the form of (ii) or (iv) to have true premises and a false conclusion (an argument is valid if and only if it is logically impossible that its premises should be true yet its conclusion false).

So, in summary:

- 'If P, then Q' says that P is a sufficient condition for Q.

- 'If Q, then P' says that P is a necessary condition for Q.

- P is a sufficient condition for Q if and only if Q is a necessary condition for P. So, whenever P is a sufficient condition for Q, Q is a necessary condition for P; and whenever Q is a necessary condition for P, P is a sufficient condition for Q.

- Whenever P is a sufficient condition for Q and it is the case that P, it is the case that Q.

- Whenever P is a necessary condition for Q and it is not the case that Q, it is not the case that P.

The following expressions are all equivalent:

- If P, then Q

- P only if Q

- Only if Q, P

- If not Q, then not P

- P is a sufficient condition for Q

- Q is a necessary condition for P

- P implies Q

- Not P unless Q.

I really hope that this helps a bit. Good luck, and don't worry - as you do more logic, things will b

Seems like Albert Einsteins first rough draft of E=MC 2....

Answer my Q!

http://answers.yahoo.com/question/index?qid=200909...

seems fine to me