It might seem like a silly question-- what could be easier than checking to see if our premise matches the world?-- but it's only really obvious for the simplest of premises.
For intance, if our premise is:
The apple is red.
we simply look to see if the apple actually is red. If it is, the premise is true. If it's not, the premise is false. Easy-peasy.
Things get more complicated if we string multiple propositions together into a single premise. For instance, if our premise is:
The apple is red and the elephant is pink.
both parts must be true for the premise to be true. In the above, the premise must be false, even if the apple is red, because the elephant is definitely not pink.
On the other hand, if our premise is:
The apple is red or the elephant is pink.
the premise might be true (as long as the apple actually is red). When two propositions are connected by "or", only one part needs to be true to make the entire premise true.
The situation is most complicated when the two parts are connected in a conditional sentence. For instance, look at this premise:
If the match is struck, then the match will catch fire.
To decide if the premise is true, we again have to examine the world. And again, we have to examine the world in two parts. First, we look to see whether the match actually was struck. And we also look to see if the match actually caught fire.
If both things happen, then we say the conditional premise was true. This makes sense-- remember, the "if" part of the conditional is supposed to give the sufficient condition for the "then" part. In this case, the match actually was struck, and it did indeed catch fire. Everything the premise claimed would happen did in fact happen. In short, the premise was true.
What if neither thing happens? (That is, what if we don't strike the match, and it doesn't catch fire? Or put another way, what if both parts of the conditional are false?) In this case, we still say that the overall conditional was true. Again, think about what the conditional is saying (this time, from the other direction): The "then" part gives a necessary condition for the "if" part. In our example, the match didn't catch fire, meaning that the necessary condition for it being struck was not fulfilled. And, as it turns out, the match wasn't struck. Our conditional held up again. We say it was true.
What about other combinations? Let's say we don't strike the match, but it catches fire anyway. (Maybe we lit it with a magnifying glass aimed at the sun.) In this case, the sufficient condition for the match catching fire wasn't fulfilled-- but that's OK, because there might be other ways to light the match (and in fact there are). The necessary condition for striking the match was fulfilled-- but that's OK, because that alone doesn't guarantee that the match was struck. In fact, our evidence in this case doesn't tell us anything at all about whether the conditional actually held up. In this case, we give the conditional the benefit of the doubt, and say it was true anyway.
Finally, what if we strike the match, but it doesn't catch fire? In this case, the conditional predicts that striking the match must make it catch fire, but it didn't happen. The conditional lied! When the "if" part turns out true, but the "then" part is false, this is the only time we can say that the overall conditional was false.
All of the above is typically organized into a diagram called a truth table. The truth table takes the truth values for the individual parts, and tells us whether the premise that connects them is true or false. See below:
| p | q | p and q | p or q | if p, then q |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | F | T | T |
| F | F | F | F | T |
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