Singletap said:
When you evaluate the "truth" of a statement, you aren't evaluating whether it's
useful. You're evaluating whether the logic is
sound.
If A, then B. In this case, A is "medicine," and B is "feel better." And you have four possible permutations of truth/falsehood here:
The information we are given tells us that medicine will make you feel better. Knowing this, if:
1. A and B are both true, the statement is "Take medicine, feel better." This is true.
2. A is true, B is false, so the statement is "Take medicine, don't feel better." This doesn't line up with the information we are originally given, so it's false.
3. A is false, B is true, so the statement is "Don't take medicine, feel better." Since the information doesn't tell us medicine is the ONLY way to feel better (it only tells us that taking the medicine will definitely do the job), we can't call this statement "false." It's true, if only on a technicality--it's logically sound.
4. Both A and B are false, so the statement is "Don't take medicine, don't feel better." Also lines up with the information we're given, so it is logically sound, and therefore true.
Your teacher is using a useful technique when teaching students about logic--the usefulness of a statement, or whether or not you
agree with the statement, is separate from whether or not the logic behind the statement is internally consistent with the information upon which it is based.
For a better understanding of If/Then, and the logic behind why certain statements are true (sound) or false (unsound), consider this example:
"If the animal is a dog, it is a mammal." - This is the information we are given, which is therefore assumed to be true for the purposes of these operations. Assumptions are useful, just as you assume a chair will support your weight without extensive testing each time you sit. Conveniently, we also know this to be scientifically true (dogs are, in fact, mammals), but that isn't important except to help you understand how if/then statements work.
So, knowing this, let's assume each of the following:
1. A and B are both true: "The animal is a dog, therefore it is a mammal." Makes sense. Dogs are mammals.
2. A is true, B is false: "The animal is a dog, and it is not a mammal." This conflicts with the information we are given, so it is false. There are no such things as non-mammal dogs.
3. A is false, B is true: "The animal is not a dog, and it is a mammal." This doesn't conflict with our information--we aren't told that ONLY dogs can be mammals, just that dogs themselves must be mammals. This statement is consistent with our information, so it is true. It could be a cat, after all.
4. A is false, B is false: "The animal is not a dog, and it is not a mammal." This also does not conflict with the information we are given. It could be a lizard--not a dog, not a mammal. This statement isn't
useful, but it is logically sound.
THE ONLY way for an 'if/then' statement to be false is if we satisfy the "IF," and the "THEN" result does not occur. This would mean that the original statement would be false. Since you cannot falsify the GIVEN, it is instead the CONCLUSION that is false.
). I know that after me not understanding a good 50% of Mechanics 1, when Mechanics 2 referenced M1 I struggled an awful lot. Its easier to be understood now, while its fresh than to have incorrect things for future problems. 
