Vacuously true statements and why false implies truth

In summary: This does not mean that the statement I start with is true. But, it means that I can claim that it is true if it follows from a set of statements that I know are true.In summary, vacuous implication states that if p is false, then any statement of the form "if p then q" is considered true. This is because in a hypothetical world where p is false, there is no way to verify the truth or falsity of q. This concept is important in logic and is also known as the paradox of material implication.
  • #1
Rishabh Narula
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TL;DR Summary
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.
 
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  • #2
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.
 
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  • #3
The definition of "a implies b" is [itex] \bar{a}\vee b[/itex] ((not a) or b)
 
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@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.
 
  • #5
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##.
 
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  • #7
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.
 
  • #8
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.
 
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Hornbein said:
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.
 
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  • #10
I can't help noting the subtle reference. "If Berlin is bombed my name is Meyer." -- Hermann Goering.
 
  • #11
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.
 

FAQ: Vacuously true statements and why false implies truth

What is a vacuously true statement?

A vacuously true statement is a logical assertion that is deemed true because the condition under which it would be false never occurs. This typically involves statements with a false premise. For example, the statement "All unicorns have five legs" is vacuously true because there are no unicorns to contradict the statement.

Why is a statement considered true if it is vacuously true?

A statement is considered true if it is vacuously true because of the rules of logic that govern implications. In logic, an implication (if P then Q) is considered true if the premise P is false, regardless of the truth value of Q. This is because there is no instance where P is true and Q is false, which is the only condition that would make the implication false.

What does it mean when we say 'false implies truth'?

The phrase 'false implies truth' refers to a principle in classical logic where an implication statement (if P then Q) is automatically true if the premise P is false. This is irrespective of whether the conclusion Q is true or false. The rationale is that there can be no counterexample where P is true and Q is false, thus fulfilling the condition of the implication.

Can you give an example of a vacuously true statement?

An example of a vacuously true statement is: "If the moon is made of green cheese, then I can fly." Since the premise "the moon is made of green cheese" is false, the entire implication is considered true regardless of the truth about the ability to fly.

How does understanding vacuous truths help in logical reasoning or mathematics?

Understanding vacuous truths is crucial in logical reasoning and mathematics as it helps in correctly interpreting statements and proofs. It ensures that conclusions drawn from hypotheses are logically sound, even in cases where the hypotheses are not applicable. This understanding prevents incorrect conclusions in theoretical scenarios and aids in the construction of rigorous mathematical proofs.

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