Subject: Falsity implies anything.
Date: Sat 11/28/98 4:30 AM
Hi. I've been discussing this issue (or something along those lines) with someone from Tulane university, and he happened to point me to cut-the- knot, which incidentally happens to be one of my favorite sites. On the page Falsity implies everything is the following text:
>> This is a definition and the only criteria
>> to establishment of the falsity or veracity
>> of a particular implication however
>> paradoxical it may sound. For example,
>> If you are not reading this sentence then I have not
>> written it.
>> The premise A in this sentence ("you are
>> not reading this sentence") is
>> obviously false or have you managed to
>> skip it? For this reason only the
>> implication is true even though its conclusion B
>> ("I have not written it") is
My question is: If logic leads to such paradoxes and absurd statements, then why do we have these definitions. This comes from a problem on a local math contest which perhaps I should include here to make my question clearer. The problem reads:
Three men are accused of a crime, Abel, Baker, and Cain, of which exactly one is guilty. They each make a statement, of which exactly one is true.
- Abel - If I'm guilty, then so is Baker
- Baker - Cain is guilty
- Cain - Baker lied when he said I was guilty
Well, who's guilty?
Now, the debate is when Abel's statement is false. Logic says that it is false only if Abel is guilty (and therefore Baker obviously is not, and so it's a T=>F). But take the statement in general. Isn't it a lie as a whole, and therefore false?
To further my question, take the statement "If I have a computer, then you have a computer." Suppose that I do not have a computer. Was the original statement true? Logic tells us that it is, but isn't that absurd? I don't know if it was true, because I don't have a computer to test it out. Certainly my not having a computer does not prove the statement false, but it does not prove it true either. So why do we say that the statement is true? In my opinion, it's foolishness. So far, no one has been able to give me a satisfactory answer, but I'd appreciate any suggestions or comment you could provide. Thanks in advance.