knowledge by deductive reasoning, part 1
The second way we know things is through deductive reasoning. The second thing Descartes was unable to doubt was the fact that he existed. He inferred his existence from the fact that he was thinking. He reasoned that in order to think, he must also exist, because if he didn’t exist, he couldn’t think. He was basically using a syllogism known as a modus tollens, which goes like this:
1. If A, then B.
2. Not B.
3. Therefore, not A.
Descartes’ argument went like this:
1. If I do not exist, then I do not think.
2. I think.
3. Therefore, I exist.
(2) and (3) together are just the contrapositive of the first premise, so the modus tollens is just a different way of saying the modus ponens, which goes like this:
1. If B, then A.
3. Therefore, A.
How do we know that given the first two premises, that the conclusion follows? Well you can kind of see that just by thinking about it. Again, we go back to intuition. It is through our intuitive knowledge that we know the syllogism is valid. You only have to think about it to see it. There are two other syllogisms worth looking at. The next one is the disjunctive, which goes like this:
1. Either A or B.
2. Not A.
3. Therefore, B.
This one is also obviously true. Basically, the disjunctive syllogism is the same thing as the process of elimination. Given a set of live options, if you know first of all that one of the options is true, then all you have to do is eliminate all but one of the options. Then it would follow that the one remaining option is the true one. So you can actually have more than two options. For example, suppose you’re playing that game where somebody hides a pea under one of three shells. He mixes them all up so that you get confused, but at the end, you know the pea is under one of the shells. Suppose you look under two of the shells and the pea isn’t there. Where is the pea? Without looking under the third shell, and assuming nobody has magically removed the pea, you can deduce that the pea is under the third shell before you even lift the shell to see.
The fourth syllogism is the transitive property, which goes like this:
1. A = B.
2. B = C.
3. Therefore, A = C.
The transitive properties takes many different forms. Rather than using equal signs, you could use greater than or less than signs. It works just as well with propositions as it does with math because mathematical statements actually are propositions stated in a different language. Now let’s suppose there’s three people, Jim, Dan, and Bob. Assume you know that Jim is taller than Dan, and Dan is taller than Bob. What can you conclude about the height difference between Jim and Bob? Just think about it for a second. Picture it in your mind. Jim is taller than Dan. Dan is taller than Bob. Who is taller—Jim or Bob? Well, I think you can see that Jim is necessarily taller than Bob. You can see that merely by reflecting on it in your mind. You don’t have to know Jim, Dan, or Bob or even how tall they are. You just have to understand concepts like “taller than” and “shorter than.” You can tell that the transitive property holds across the board, too. It’s not like the scientific method where you have to test it over and over again to see that it’s true. You can already see it in your mind. With one hypothetical example used for illustration, you can know that it will always be true.