Well, all we know something about material implication, at least we may know how it works. But it isn’t obvious, why it returns True for two False values and returns False for True and False. What on the Earth is going on here?
Actually, this concept is not as abstract as it seems. I prefer to consider it as a tool, which helps you to check whether your theorem is True, or not. Just a tool, like a wrench or a hammer. You have a belief? You may exam it with implication.
The best way to understand material implication is to imagine its using in real life. And here are some examples.
I have a belief, that all the oranges are sweet. At least my grandmother told me so. But I’ve decided to do an experiment. I go to a supermarket, take a fruit and taste it. Hmm, that’s an apple. And it’s sweet. This doesn’t affect my theory, because I intended to taste oranges. OoK. I take another fruit. It’s bitter, what’s it? Oh, lime. That doesn’t prove or disprove my theory as well, unfortunately.
Another fruit I get is an orange. Great! I taste it — it’s sweet. I take another orange — it’s sweet too. Another orange is sweet as well. My theorem is still true. But the last orange I taste isn’t sweet — it’s rotten. Now I know that not oranges are sweet — at least one of them was rotten. Some of them may be green probably. My belief is broken. I need to visit hospital.
|I believe, that all the oranges are sweet.|
|Apple||Sweet||This fruit is sweet, but it’s not an orange. I still think all oranges are sweet.||True|
|Lime||Bitter||This doesn’t prove or disprove my theorem. I still think all oranges are sweet.||True|
|Orange||Sweet||That’s an orange and it’s sweet. I still think all oranges are sweet.||True|
|Orange||Rotten||That’s an orange and it is not sweet at all. I have to admit my theorem was wrong.||False|