So a conditional statement is basically a formula to set up a hypothesis and a conclusion. The basic formula is
IF (hypothesis), THEN (conclusion).
These statements do not need to be true to be considered a conditional statement. For example, the statement If Casey likes teaching FOM, then we will all get A’s, is a conditional statement, even though it is (unfortunately) not true.
How do we know then if these statements are true or false? Simple.
To make a conditional statement true:
- Any hypothesis that is true should be followed by any conclusion that is true. So, a statement like If I like pancakes, then my name is Kate. They don’t necessarily have to be connected in any way, both parts just must be true. It seems weird, but logical causality is different from straight-forward causality.
- Any hypothesis that is false that is followed by any sort of conclusion. So, a statement like If the Earth is flat, I’ll legally change my name to Beyoncé. It doesn’t matter what the conclusion is, because the condition of this statement will never be proven true, and therefore the conclusion will never be used. This condition is vacuously true.
To prove a conditional statement false:
- The hypothesis must be true, while the conclusion proves false. For example, a statement like If I am in a FOM class, I am a math major. The conclusion is false because I am a math minor, but I am in a FOM class. Therefore the hypothesis is true, but not the conclusion.
With this knowledge, I want to talk about the problem we were given in class today that I’m still trying to figure out. The problem is:
You see four cards and are told that each one of them has a letter on one side and a number on the other. You see the four cards as pictured here:
A 3 T 8
Which of the four cards must you turn over to verify the claim:
“On the opposite side of every vowel is an even number”
First, we’ll make it a conditional statement. If a card has a vowel on one side, then there will be an even number on the other side.
So to prove this right or wrong, we must focus on the conclusion, not the hypothesis. If we were to focus on the hypothesis and prove that wrong, we would be focusing on a whole other statement, not the one at hand. We don’t need to worry about what happens when the card doesn’t have a vowel on one side, we’re focusing on what happens when it does, so we’ll make the condition that the hypothesis is true.
When the hypothesis is true, the statement is true if the conclusion is true, and false if the statement is false. Casey also mentioned in class that it is much easier to prove something false than true. To prove something false, you only need to bring up one instance that proves it false, but to prove something unquestionably true, you would need to flip all the cards.
Or would we? Let’s say we focus on only vowels. T would then be irrelevant to the statement all together because it is not mentioned in the hypothesis. To prove this statement right then, we would have to flip both the 8 card and the A card, to see that there is a vowel on the other side of the 8 card and an even number on the other side of the A card.
To prove this statement wrong, we would need to flip A to see if there is an odd number, and flip 3 to see if there is a vowel on the other side. If either of these cases happened, the statement would be false.
It seems to me that A is the crucial card. Either way, no matter if you are trying to prove it true or prove it false, you need to flip A to see what is on the other side.
Whew, that was confusing. I hope it makes any sort of sense.