What is a vacuous truth?

Question

anonymous · Answer

It is not a lie

thomas5267 · Answer

The cake? @kevinkeegan

anonymous · Answer

if 2 + 2 = 5, then the moon is made of green cheese

anonymous · Answer

statement that looks like 
"if P then Q" where P is always false

anonymous · Answer

@satellite73 i like ur previous answer better

unklerhaukus · Answer

$$A \implies B:\quad \& ~~~A \quad \text{     is false}$$

anonymous · Answer

could be used to say something about an empty set.

thomas5267 · Answer

Wait, the following statement is true?  I have read it somewhere on the internet.
If there exist a n that is both even and odd, n=17.

That does not make any sense to me...

blockcolder · Answer

Yes, that's true. Weird, I know. =))

unklerhaukus · Answer

it is vacuously true

parthkohli · Answer

All cats who are expert chess-players are also fluent in ancient Sanskrit. This is vacuously true as there are no such cats. :P

thomas5267 · Answer

Why we have to define vacuous truth as so?  What are the consequence of defining such statements as false?

thomas5267 · Answer

Still anxiously waiting for SmoothMath's reply..., as well as pinging my self for no particular reasons...  @thomas5267

anonymous · Answer

The easiest way to think of it, in my opinion, is trying to disprove such a statement. If I make a statement like, "All doors are wooden," which could be restated as "If an object is a door, then that object is wooden," consider how you would go about disproving my statement. The method would be pretty simple. You would show me a non-wooden door. That is, you would show me an object, verify that the object is a door, and then show that it is not made of wood. This is called counter example.

Here's where your proof could go wrong though. If you showed me an object that was not wooden, but it was also not a door, then you haven't proved me wrong. To disprove me, you need only one counterexample, but you have to satisfy my if statement and show that my then statement is false. This is because the opposite of "If P, then Q" would be "P and not Q.'

So in general, if I make a statement "If P, then Q" and there are NO cases where P is true, then you could not disprove this. And if I were to sit down to prove it, I would need to prove it for all possible cases of P, which happens to be no cases at all, so I am done.

anonymous · Answer

Political truth

thomas5267 · Answer

Wait, you can not prove or disprove it, then we say it is true?  Then god exists...

anonymous · Answer

No, that's not what I'm saying. I'm saying that if P is certainly never true, then the statement "If P, then Q" is immediately proven.

anonymous · Answer

"All time machines are wooden"

thomas5267 · Answer

Certainly never true...How can you know that time machines does not exists...

thomas5267 · Answer

I found a rather controversial statement on the web.

All red elephants live in loaves of bread.

It is vacuously true, yet we can construct a similar statement like this.

All red elephants live in loaves of bread and cannot live in loaves of bread.

Still true, isn't it?  So does red elephants live in loaves of bread or not in loaves of bread?

unklerhaukus · Answer

none of the red elephants i know live on loaves of bread

unklerhaukus · Answer

but so do all the red elephants i know live on loaves of bread,

anonymous · Answer

Still true. The second statement, the elephant both living in a loaf of bread and NOT living in a loaf of bread is never true, always false. However, the first statement, an elephant being red is also always false.

If P is always false, then the truth values of the second statement are irrelevant.

thomas5267 · Answer

What now?  You can have red elephants everywhere then?