Can you help me wrap my head around conditional reasoning?
I understand how conditional reasoning works symbolically, with arrows and contrapositives and the like. However, I don’t get it. It sort of doesn’t make sense in my mind. Specifically, I don’t get sufficient and necessary conditions, other than the order in which they are supposed to appear. Here’s how one of my study books describes sufficient conditions:
A sufficient condition can be defined as an event or circumstance whose occurrence indicates that a necessary condition must also occur.
And here’s necessary conditions:
A necessary condition can be defined as an event or circumstance whose occurrence is required in order for a sufficient condition to occur.
Huh? They sound almost exactly the same! A sufficient condition makes a necessary condition occur, and a necessary condition is required for a sufficient condition to occur. This almost makes them sound interchangeable when I know they definitely are not. Reading those definitions makes me feel like I’m going in a circle. Like saying “Apples come from apple trees because apple trees make apples.”
Maybe I’m thinking to hard about the meanings of the words “sufficient,” and “necessary,” or maybe I’m tired, but something isn’t clicking for me…
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6 Answers
Sufficient must occur for the necessary to occur.
Apples and apple trees really are biconditional statements. Because in real life you need apples to make apple trees (A -> T), and apple trees to make apples (T -> A). You can shorten biconditionals to just A = T
What my book says about sufficient and necessary conditions:
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Event A is said to be a Sufficient condition for event B whenever the occurrence of A is all that is required for the occurrence of B. On the other hand, event A is said to be a necessary condition for event B whenever B cannot occur without the occurrence of A. For example, having the flu is a sufficient condition for feeling miserable, whereas having air to breathe is a necessary condition for survival. Other things besides having the flu might cause a person to feel miserable, but that by itself is sufficient; other things besides having air to breathe are required for survival, but without air survival is impossible. In other words, air is necessary.
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Oh, I so miss text books!
I’m just going to hazard a guess. Maybe it means like, if a car pulls out in front of you, you’re going to slam on the brakes. Car pulling out in front of you=“sufficient condition.” Slamming on the brakes =necessary condition.
This could be so far from right, because I don’t know what arrows and symbols have to do with it.
I think it would be more like
“If it’s an apple, then it grew on a fruit tree.” with apple as the sufficient condition and the fruit tree as the necessary condition. Apple -> fruit tree.
And if you tried to say that only apples can grow on fruit trees, you’ve got a mistaken reversal. Fruit tree -> apple.
Positng that if it’s not an apple (say it’s a pear) it couldn’t have grown on a fruit tree would be a mistaken negation: fruit tree > -apple
If a condition is necessary for the result, the result CANNOT occur without the condition.
If a condition is sufficient for the result, the result might be able to exist in other ways, but as long as that condition is true, the result will happen as well.
if X is greater than 15, then X is greater than 10. “If X is greater than 15” is a sufficient condition. You know that so long as that condition is true, the result (“x is greater than 10”) is also true. You don’t need to check anything else. The condition sufficiently indicates the result. But X being greater than 15 isn’t required. The result could happen without X being greater than 15. (X could equal 11, 12, 13, or 14). So it’s not a necessary condition.
If |X| = 4, then X = 4.
“If |X| = 4” is a necessary condition. There is no way for the result (X = 4) to be true without the condition also being true. But it’s not a sufficient condition, because there is a way for the condition to be true (|X| = 4), but the result to be false (X could be -4). You would have to add in another condition (“and X > 0”, for example) for the entire condition to be sufficient.
If X is even, then X + 1 is odd. In order for the result to be true, the condition HAS to be true. There’s no way for X + 1 to be odd if X isn’t even. “If X is even” is therefore a necessary condition. But it’s also a sufficient condition. Because if X is even, the result has to be true as well.
To illustrate with truth tables:
If A, then B:
For A to be sufficient:
A is true => B is true
A is false => B is unknown
For A to be necessary:
A is true => B is unknown
A is false => B is false
For A to be sufficient AND necessary:
A is true => B is true
A is false => B is false
Sufficient means: “Just enough to fill the need.”
Necessary means: “Required to fill the need.”
Compare these two tasks:
1) Filling a gas tank
2) Filling a gas tank with Gasoline
“Filling a gas tank” can be sufficiently accomplished by filling a gas tank with Gasoline, Water, Sand, or any other substance. Gasoline is merely a sufficient substance to accomplish the first task since ANY substance could in fact “fill” the tank.
“Filling a gas tank with Gasoline,” on the other hand can only be accomplished by filling it with Gasoline specifically. Filling the tank with sugar or sand would not accomplish the requested task. Gasoline is the “necessary” substance required to accomplish the second task.
“Filling the tank” is a necessary condition as well. In either case, if you left the tank only half full, the task would not be considered accomplished.
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