Vacuous Truths

I was wondering how much we can rely on such statements. According to Wikipedia: (http://en.wikipedia.org/wiki/Vacuous_truth#Arguments_that_only_some_vacuously_true_statements_are_true)

I think the statement P->Q needs more resrictions on being true.

For instance, let P be R & S.  Also, suppose S!->Q, and R and Q are independent of each other.  That should imply that P!->Q.  But as you pointed out, just because all rhinos are herbivores and being pink vs. being a herbivore are independent, it doesn't stop us from asserting that all pink unicorns are carnivores.  Logic is weird.

All invisible pink unicorns are hornless.

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Look at this:

Let P be : If this statement is true, pigs fly.

Let Q be: Pigs Fly.

Then P=P->Q.  So ~P=P^~Q, giving a contradiction, meaning P is true, making Q true, so pigs fly.

pawn_slayer66 could you please clarify what you mean by  ~P=P^~Q and P=P->Q for that matter!? .. And isn't this a case of principii petitio? Q is contained in P which is the root of your apparent paradox ..

Pawnslayer I think it should be

If this statement is true pigs can fly, which can then be divided into

P: this statement is true

Q: pigs can fly

thus we have

P=>Q

non Q => non P ...

An example could be:

The day I win a million dollars is the day pigs can fly ..

No pigs are flying around, so I didn't win a million bucks ... This may be a stupid example, but I am tired..

Just get rid of bivalence and throw all the troublesome vacous truths and such in the niether true nor false bag, then get yourself a nice shiny new logic, problem solved!  I kid, I kid.  Annoyingly this was topic that I was told about at uni but didn't get covered in any depth due to lack of time.  Seems to me you either accept the wierd conclusions or develop another an alternative logic, which some have done as you mention in the opening post.

"If that's a Rolex, then I'm a monkey's uncle" is an example showing that the notion of a false statement implying any other false statement is a concept that the average man in the street uses without worries.

Thank you all for your posts and I will be studying further.

@Elroch: We have a famous example of a false statment implying a false conclusion in Danish Literature. It goes:

A stone can't fly

Mother can't fly

ergo Mother is a stone!

That opens up to a whole new universe of yo momma jokes.. =)

It is true that "a stone can't fly => mother can't fly",which is an example of "vacuous truth". However, I would classify that example as being one of the classical fallacies (errors of inference) given by Aristotle, called "affirming the consequent" http://en.wikipedia.org/wiki/Affirming_the_consequent

It can be expressed very precisely in logical symbols. We have two facts:

A->B

B

and from these two facts we falsely infer A is true. In symbols it is clear that the mistake is related to confusion about the direction of the arrow in the first statement.

In the example given by Summun_Malum, A is "being a stone", B is "being unable to fly" and Mother is an example of an object (no offense) with property B, which is falsely inferred to have propery A.

..

"there are a number of justifications for saying that vacuously true statements are indeed true. there's still something odd about the choice. there seems to be no direct reason to pick true; it's just that things blow up in our face if we don't. we say S is vacuously true; it is true, but in a way that doesn't seem entirely free from arbitrariness. the fact that S is true doesn't really provide us with any information, nor can we make useful deductions from it; it is only a choice we made about how our logical system works"

knowledgerush.com

So I take it there is nothing one can do to disagree with the validity of a vacuously true statement, correct?

Well, if falsity is equivalent to the existence of a counterexamples, and the set of objects from which you can choose a counterexample is empty, you are not going to be able to find one!

I'm just thinking this....1+1=2 We all accept this as the truth...but why? My understanding is that there can only be "one thing"  "One thing" represent its own situation within time and space...but suddenly out of the blue, someone make a statement and say...."One thing"(standing for itself)plus"One thing"(where does this one come from???)=TWO!!!!....I mean...how is it possible, can the majority be wrong?  It is more likely that 0.999999999999999999999999+1.000000000000000=???????

infinity+infinity=two

how can be there two infinities?

Were do we get the other infinity from.....perhaps we can only add mirror images e.g.

-1 plus 1=0

Can it be that numbers spiral into infinity, and that the only thing that separate, the one from the other, is a point in space...like a fingerprint...no one number can stand at another space...it can only stand for itself...if something is equal to something else, then it have to be in the opposite dimension, at  exactly the opposite point in time....but we just make it 1+1=two, and that is it! No questions asked!  

To make something equal to something else, you need to measure it against something that can never change...zero is such a point in time...Anything that is not zero, would be relative to zero...and therefore towards each other...to say that something is equal to something else is almost impossible....unless there is a mirror image of it, to balance it...therefore one can say, that for everything that is present, there must be something that is equal, but not present...else we would not be...

the pig that would fly in our dimension, would not be equal to another flying pig , unless it is the direct opposite within another dimension.  It can be considered that  a herbivore in one dimension, can be a mirror carnivore in another dimension, so the one can be equal to the other.....from the following statement, plus=minus, the two dimensions can overlap and create nothing, or zero...-1=+1=0  (anything is relevant to zero, the difference between things is the distance between a mutual point of understanding.  If you can imagine such a point, then one can actually imagine anything, and anything would be time travel! would that make time relevant? I wonder why such a thought would present itself if it was to be ignored.  Perhaps it is the glue that would make sense to another opinion- in a boiling pot of understanding...why it present itself...i do not know