Vacuously true statements and why false implies truth

[Rishabh Narula]

TL;DR

I was looking up the meaning of vacuously true statement and why false implies truth.
Have I understood it correctly?

We say that an implication p --> q is vaccuously true if p is false.
Since now it's impossible to have p true and q false.
That is we can't check anymore whether the contrary, p being true and q being false,can be.Since p being true is non-existent.
So we take the implication as true.

For eg. If 3 squared = 27,then 2+2=5.
Can we check if it is indeed true that 3 squared equals 27 then 2+2 is not 5.
No.
Because 3 squared equals 27 is non-existent. Or false.
So we can't check if the statement is false.
Hence it must be true.

 

[pbuk]

I think that is a good enough understanding, but note the correct spelling: vacuous. Note also that if p is false, both $p \rightarrow q$ and $p \rightarrow \neg q$ are true.

 

[Svein]

The definition of "a implies b" is \bar{a}\vee b ((not a) or b)

 

[Rishabh Narula]

@pbuk thanks for verifying.
@Svein currently its hard for me to digest that,though i had noticed that also while looking this up.will someday try to understand that as well.

 

[FactChecker]

I imagine that there are many automated theorem provers that would not work if this were not true. Suppose that an automated system looked at two cases, $p$ and $\lnot p$, where it later determined that $p$ was false. It would be an error to conclude that the logic statement '$\lnot p \land$ (if $p$ then $g$)' implied $\lnot q$.

 

[WWGD]

You may be interested in reading about The Paradox of material Implication:

https://duckduckgo.com/?q=paradox+of+material+implication&t=hs&va=l&ia=web

 

[Hornbein]

Vacuous implication is weird. Basically it is saying "in this hypothetical world where this false thing happens, what else would happen?" If the moon is made of green cheese, is my name still Meyer? Well, who knows. There really isn't any answer. Giving it the value of "true" was better than the complication of giving it a value of "undefined," I suppose. Sort of a lesser evil thing. True things remain true regardless of some weird hypothetical. That seems reasonable enough.

 

[nuuskur]

From falsehood one may conclude anything they wish. However, $P\Rightarrow Q$ being true has no bearing on the truth value of $Q$.

In fact, True statements can Only be derived from True statements. Also referred to as modus ponens.

 

[FactChecker]

Vacuous implication is weird. Basically it is saying "in this hypothetical world where this false thing happens, what else would happen?" If the moon is made of green cheese, is my name still Meyer? Well, who knows. There really isn't any answer. Giving it the value of "true" was better than the complication of giving it a value of "undefined," I suppose. Sort of a lesser evil thing. True things remain true regardless of some weird hypothetical. That seems reasonable enough.

Just to be clear, if p is false, the "if p then q" statement is true. That is not the same as saying that q is true.

 

[Hornbein]

I can't help noting the subtle reference. "If Berlin is bombed my name is Meyer." -- Hermann Goering.

 

[WWGD]

Notice too, we don't allow True then False, since reasoning preserves true statements. This says that if I start with a true statement and reason correctly( using logic rules), I will not end up with a false statement.