Q&A: From a Contradiction You Can Prove Anything
From a Contradiction You Can Prove Anything
Question
Hello. I saw that you wrote in several places that, as is well known to those in the know, if you assume something and its opposite, you can prove anything.
How does that work? Why, if my cat is black and also red, is the conclusion that I am a 13th-century Indian elephant residing on Mars??
Answer
You want an instant crash course? 🙂 Logical implication can be defined in several ways, because this operator is different from all the other logical operators. An operator like "and," "or," negation, NAND, XOR, and the like are all defined formally. If I know the truth value of A and of B, I know unambiguously the truth value of A AND B, and also of A OR B, and so on. But with implication, even if you know the truth value of A and B, you still cannot determine whether there is a relation of entailment between them, because that depends on the content of the statements and not only on their truth values. Therefore it is customary to define implication materially, meaning in a way that depends only on the truth values of A and B and not on their contents. The definition is as I presented in the previous message: always true unless A is true and B is false.
I understand. Thanks. A.H., see also here: https://mikyab.net/%D7%A9%D7%95%D7%AA/%d7%a2%d7%9c-%d7%aa%d7%a4%d7%99%d7%A1%D7%AA-%D7%94%D7%90%D7%9C%D7%95%D7%A7%D7%95%D7%AA-2/
No. A proof is a material implication. You are right on the level of content, but mathematics does not deal with contents; it deals with formal relations.
Thanks.
So the argument is basically this? Nice.
This is actually an open question in philosophical logic: does implication implicitly assume existence? It seems to me there are several paradoxes that arise from this view. What is the problem with "the set of all people who are currently ruling France is contained in the set of bald people"? That set could also be the empty set (the wording you used, "the current king," is more problematic). Because if the first statement is true only on the assumption that there is no king (that is, that the set is empty), then it cannot be applied to a situation in which the set is not empty. Got it, thanks.
Discussion on Answer
But from this you still can't conclude that A.H. is a 13th-century Indian elephant residing on Mars, only that it's possible; that is, the opposite has not been proven (it makes no difference, in your terms)?
1. My cat is black. –> so it is definitely not red.
Therefore we can say that in any situation where the cat is red, (which never happens according to assumption 1) I am an Indian elephant (because it never happens).
(That's like saying that all the dragons in this room are pink — always true.)
2. The cat is red. –> we showed earlier that in any such situation, I am an Indian elephant. Therefore I am an Indian elephant.
One example I remember (maybe from Kripke?) is the statement: The current king of France is bald. If you check the set of bald people you will not find him there, but not in the set of hairy people either. The reason is of course that France has no current king. In such a case, is it still correct to say that if France now has a king, he is bald? And at the same time, that if there is a king, he is hairy? In other words: is this a hypothetical statement (conditional), or a factual claim?
And then if I claim that there is a king, I am claiming that he is bald.
There is a difference between saying concretely that all the fairies in our world have three wings, which is said only because there are no fairies, and the hypothetical claim that if there were a situation in which fairies existed in the world, they would necessarily have three wings. The second statement, even if said in a world without fairies, commits you to the conclusion in a world where fairies do exist. But the first statement, which merely summarizes the existing situation, does not.
What does "materially" mean?