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shaoen01

Member

Registered: 2008-01-26

Posts: 18

How do we determine if statement is vacuously true?

Hi all,

I was reading through my textbook and did understand the explanation of what it means for a statement to be vacuously true. But what i do not understand is the formal mathematical definition of it. Below is the scenario:

Suppose you have 5 blue and red balls and you place 2 blue balls and 1 red ball into a bowl. Consider the statement "All balls in the bowl is blue", is this true or false? The negation of this statement is "There is at least one ball in the bowl which is not blue". This requires at least one ball to be in the bowl, but there is none. So the negation is false; thus, the statement is true.

If P(x) = "x is in the bowl", Q(x) = "x is blue" then "for all x in D, P(x) --> Q(x)" is vacuously true if P(x) is false for every x in D.

My Question:
So what i don't understand is why must P(x) be false for every x in D for it to be vacuously true? Could we consider P(x) to be false when there is no ball in the bowl? Is this how you interpret it?

william blanchard

Guest

Re: How do we determine if statement is vacuously true?

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mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: How do we determine if statement is vacuously true?

If a statement is vacuously true, then it's obvious.
That's quite ambiguous, but as a guide I like to think that the proof of such a statement would likely be only one or two lines.

In your case, P(x) --> Q(x) can only be false if you can find an x such that P(x) is true and Q(x) is false. You're told that P(x) is always false (and thus no x that fits those conditions exists), and so P(x) --> Q(x) is vacuously true.

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: How do we determine if statement is vacuously true?

If a statement is vacuously true, then it's obvious.

I don't like that, and here's why:

If R is a finite division ring that is not a field, then R contains the element pi.

The way to think of vacuously true is in terms of right and wrong.  If I say absolutely nothing, how could that statement (of nothing) be wrong?  Simply put, it can't.  Similarly, a statement only says something when it's preconditions are met.  If it's preconditions aren't met, then the statement says absolutely nothing.  You can't be wrong when you don't say anything.

Could we consider P(x) to be false when there is no ball in the bowl?

You are.  If P(x) = "x is in the bowl", and there isn't anything in the bowl, then P(x) must be false.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

shaoen01

Member

Registered: 2008-01-26

Posts: 18

Re: How do we determine if statement is vacuously true?

mathsyperson: So basically what you are saying is that since P(x) is always false, there is no way for P(x) --> Q(x) to be false right. Hence, it is vacuously true?

Ricky: I think i kind of get what you mean by the saying absolutely nothing part. In the case of my example, since the negation of the statement ("There is at least one ball in the bowl which is not blue" and the precondition is that there must be at least a ball inside) is false, so i just assume that the statement is true because the precondition could not be met. Am i right?

Thank you both for your replies.