Is for every x in domain D, Q(x) false if Q(x) is false for all x in D?

[Ling Min Hao]

"For every x in domain D , Q(x) "is false if Q(x) is false for some x in D .

"For every x in domain D , Q(x) "is false if Q(x) is false for all x in D .

Which of the following is correct ? Or both are correct ?

 

[BvU]

Hi,

Which of the following is correct ?

You mean: "Which of the preceding is correct ?"
The first is not correct
THe second is correct: "every" and "all" mean one and the same thing, so the statement reads more or less like: "if A then A"

 

[Ling Min Hao]

Hi,
You mean: "Which of the preceding is correct ?"
The first is not correct
THe second is correct: "every" and "all" mean one and the same thing, so the statement reads more or less like: "if A then A"

No , the first one basically mean that if there is some value of x in D causing Q(x) to be false , then the whole statement "for all x in D , Q(x)" is certainly false . So I think the first one is correct . Yet , it is quite confused whether the second one is correct or not since it is not necessary that we need all x in D to be false so that the statement is false . One counterexample is enough.

 

[fresh_42]

No , the first one basically mean that if there is some value of x in D causing Q(x) to be false , then the whole statement "for all x in D , Q(x)" is certainly false . So I think the first one is correct . Yet , it is quite confused whether the second one is correct or not since it is not necessary that we need all x in D to be false so that the statement is false . One counterexample is enough.

Yes. So why do you ask?
If $Q(x)$ is false for a single $x$ then it cannot be generally true as $Q(x)$ which might(!) mean for all $x$ unless specified otherwise.
If $Q(x)$ is false for all $x$, then "$(not \; Q)\, (x)$" is true, which isn't the case in the first statement.

However, as you can see, logic and language don't always fit well. Therefore it is important to be as precise as possible, e.g. by stating "$Q(x)$ is true for all $x$". Otherwise there will be room for misunderstandings. One reason, mathematicians often use formulas instead of words, e.g. $\forall x \; Q(x)$ instead of simply $Q(x)$.

Other prominent examples are the usage of or: "Will you come over tomorrow or not?" of which "yes" is the only possible answer without lying and is rarely expected as such, or the double negation: "Didn't you say not tomorrow?" which is nothing but confusing.

Thus it is better to be precise in the first place.

 

[Ling Min Hao]

Yes. So why do you ask?
If $Q(x)$ is false for a single $x$ then it cannot be generally true as $Q(x)$ which might(!) mean for all $x$ unless specified otherwise.
If $Q(x)$ is false for all $x$, then "$(not \; Q)\, (x)$" is true, which isn't the case in the first statement.

However, as you can see, logic and language don't always fit well. Therefore it is important to be as precise as possible, e.g. by stating "$Q(x)$ is true for all $x$". Otherwise there will be room for misunderstandings. One reason, mathematicians often use formulas instead of words, e.g. $\forall x \; Q(x)$ instead of simply $Q(x)$.

Other prominent examples are the usage of or: "Will you come over tomorrow or not?" of which "yes" is the only possible answer without lying and is rarely expected as such, or the double negation: "Didn't you say not tomorrow?" which is nothing but confusing.

Thus it is better to be precise in the first place.

So by what you mean the second one is correct and more precise ?

 

[fresh_42]

So by what you mean the second one is correct and more precise ?

As I have read it: the first one. However I have read your

"For every x in domain D , Q(x) "is false if Q(x) is false for some x in D .

as

"For every x in domain D , Q(x)" is false if "Q(x) is false for some x in D" .

I assume @BvU has read

"For every x in domain D , Q(x) is false" if "Q(x) is false for some x in D" .

as the first statement, which would be false.

Look out for the subtle differences.

It is not precise as you did not clearly mention what $x$ is / are meant to be for $Q(x)$ being false. Simply saying $Q(x)$ leaves it open, whether $x$ has to be "all $x$ from a set" or a single instance. As you may have seen in post #2, it is ambiguous. The hidden parentheses are important here to determine what the all-quantifier $\forall x\in D$ is applied to.

What I meant is - to say it clear - that either isn't a good wording and is typically used by sloppy authors or people who want to hide the weakness of their argument. At best it can be used when it's obvious by context what is meant.

 

[Ling Min Hao]

As I have read it: the first one. However I have read your

as

I assume @BvU has read

as the first statement, which would be false.

Look out for the subtle differences.

It is not precise as you did not clearly mention what $x$ is / are meant to be for $Q(x)$ being false. Simply saying $Q(x)$ leaves it open, whether $x$ has to be "all $x$ from a set" or a single instance. As you may have seen in post #2, it is ambiguous. The hidden parentheses are important here to determine what the all-quantifier $\forall x\in D$ is applied to.

What I meant is - to say it clear - that either isn't a good wording and is typically used by sloppy authors or people who want to hide the weakness of their argument. At best it can be used when it's obvious by context what is meant.

I think we can make it simple like
"For all x in domain D , Q(x) " this statement is false if we can produce one counterexample of x which does not fix in Q(x) right ? But what if all x in domain D is false(meaning that all x does not fix in Q(x) )? Then the statement"For all x in domain D , Q(x) " still false ? Or just unrelated ?

Sorry I don't quite understand your comment , need to clarify more .After I see my question , I realize that this is actually unrelated to english , maybe I put the wrong feed topic , sorry again

 

[fresh_42]

"For every x in domain D , Q(x) "is false if Q(x) is false for some x in D .

If $D \neq \emptyset$ then:

$\lnot \; [\; \forall \, x \in D : Q(x) \;]\; \Leftrightarrow \; [ \; \exists \, x \in D : \lnot \; Q(x) \; ] \;$

$[\; \forall \, x \in D : \lnot \; Q(x) \;]\; \Rightarrow \; [ \; \exists \, x \in D : \lnot \; Q(x) \; ] \; \nRightarrow \; [\; \forall \, x \in D : \lnot \; Q(x) \;]\; \vee \; [ \; \lnot \; [\; \forall \, x \in D : \; \lnot \; Q(x) \;] \;] \;$

$[\; \forall \, x \in D : \lnot \; Q(x) \;]\; \vee \; [ \; \lnot \; [\; \forall \, x \in D : \; \lnot \; Q(x) \;] \;] \; \Leftrightarrow \; [\; \forall \, x \in D : \lnot \; Q(x) \;]\; \vee \; [\; \exists \, x \in D : \; Q(x) \;]$

"For every x in domain D , Q(x) "is false if Q(x) is false for all x in D .

$[\; \forall \, x \in D : \lnot \; Q(x) \;]\; \Leftrightarrow \; [\; \lnot \; Q(x) \; \forall \, x \in D \;] \;$

If $D \neq \emptyset$ then:

$\; \forall \, x \in D : \lnot \; Q(x) \; \wedge \; Q(x)$

If you want to write this in words, you will have to make sure, where the brackets are, where the "nots" and "falses" apply to and where the quantifiers apply to.

 

[Stephen Tashi]

"For every x in domain D , Q(x) "is false if Q(x) is false for some x in D .
"For every x in domain D , Q(x) "is false if Q(x) is false for all x in D .

From your use of quotation marks, I interpret your question as follows.

Let A be the the statement "For each x in domain D, Q(x)".
Assertion 1) Statement A is false if there exists a y in domain D such that Q(y) is false.
Assertion 2) Statement A is false if for each y in domain D, Q(y) is false.

We must further clarify whether you are using "if" in the literal sense or using it as "if" as it is sometimes used in making definitions. Sometimes, in mathematical writing, "if" is used in definitions with the understanding that it actually means "if and only if". For example, someone might define a "positive number" by saying "A number x is positive if x is greather than zero". The intended meaning is "A number x is positive if and only if x is greater than zero".

If you intend to use "if" literally and D is not the null set then both assertions are true. If we know Q(y) is false for each y in domain D then we know there exists at least one y in domain D such that Q(y) is false.

If you intend to use "if" in the sense of a definition, to mean "if and only if" then assertion 2) is not true for an arbitrary domain D. (There could be special domains where assertion 2) is true. For example, a domain consisting of just a single element.)

 

[Ling Min Hao]

From your use of quotation marks, I interpret your question as follows.

Let A be the the statement "For each x in domain D, Q(x)".
Assertion 1) Statement A is false if there exists a y in domain D such that Q(y) is false.
Assertion 2) Statement A is false if for each y in domain D, Q(y) is false.

We must further clarify whether you are using "if" in the literal sense or using it as "if" as it is sometimes used in making definitions. Sometimes, in mathematical writing, "if" is used in definitions with the understanding that it actually means "if and only if". For example, someone might define a "positive number" by saying "A number x is positive if x is greather than zero". The intended meaning is "A number x is positive if and only if x is greater than zero".

If you intend to use "if" literally and D is not the null set then both assertions are true. If we know Q(y) is false for each y in domain D then we know there exists at least one y in domain D such that Q(y) is false.

If you intend to use "if" in the sense of a definition, to mean "if and only if" then assertion 2) is not true for an arbitrary domain D. (There could be special domains where assertion 2) is true. For example, a domain consisting of just a single element.)

My lecturer gave me a question but he didn't clarify whether the "if" mean "if" or "if and only if ", so, generally , which one is a better answer ?

 

[Stephen Tashi]

My lecturer gave me a question but he didn't clarify whether the "if" mean "if" or "if and only if ", so, generally , which one is a better answer ?

You must be the mind-reader who figures out what the lecturer means. I don't know him.

If it is supposed to be a simple question about the definition (meaning) of "for each x .." then my guess is that he means "if and only if".