Logic of quantified statements: for all vs if then

[jreelawg]

Statement 1: If n is any prime number > 2, then n+1 is even.
Statement 2: For all prime numbers p, if p>2, then p is even.

The above two statements seam equivalent, but have different converses, etc.

converse 1: If n+1 is even, then n is a prime number > 2.
converse 2: For all prime numbers p, if p+1 is even, then p > 2.

If you have a statement that begins with if, is it illegal to write as a for all?

 

[Stephen Tashi]

If you have a statement that begins with if, is it illegal to write as a for all?

This is a question about translating ordinary speech into precise mathematical statements. There aren't any rigid rules for doing this and people's opinions about using "for all" may vary.

In my opinion "for all" is an ambiguous expression. For example if we say "For all real numbers m, there exists a real number k such that k > m" this could be interpreted as the (incorrect) assertion that there is a largest real number, one that is greater than all real numbers. It is clearer to say "For each real number m, there exists a real number k such that k > m", unless you are actually intend to convey a different meaning.

In mathematics it is often true that an if-statement is intended to have some universal truth. For example, if the statement is made that "If a > b then a + c > a + c", there are a lot things that are "understood" about the meaning of that statement. One possible interpretation of that statement is: "For each real number a and for each real number b and for each real number c, if a > b then a + c > a + b."

However, strictly speaking, an if-statement need not be any sort of universal claim. An expression which has variables in it, but no quantifiers for the variables ( i.e. no "for each" or "there exists") is technically not a "statement", it is a "statement function". A "statement" must be true or false. A statement function is only true or false when specific values are substituted for the variables.

So, strictly speaking, "if p > q then r > s" is not a statement. You can't say if it is true or false.

When we quantify the variables in various ways, we get statements. One example is:

"For each p, there exists a q such that for each r, there exists an s such that if p > q then r > s".

I find that statement confusing to interpret! However, it demonstrates how the quantifiers "for each" and "there exists" can be employed.

If we fix the typo "then p is even" to be "then p+1 is even", the statements you gave are both equivalent to "For each integer p, if p is a prime and p > 2 then p+1 is even".

Talking about the "converse" is a tricky matter. Strictly speaking, "converse" is a term that applies to if-statements. I don't know whether logicians apply it to "if-statement functions". Ordinary mathematicians do talk about the converse of theorems. What they would mean in your example is the statement "For each integer p, if p+1 is even then p is a prime and p > 2". So they wouldn't be worried about whether the "p+1 is even" implied that p was an integer.

 

[jreelawg]

This is a question about translating ordinary speech into precise mathematical statements. There aren't any rigid rules for doing this and people's opinions about using "for all" may vary.

In my opinion "for all" is an ambiguous expression. For example if we say "For all real numbers m, there exists a real number k such that k > m" this could be interpreted as the (incorrect) assertion that there is a largest real number, one that is greater than all real numbers. It is clearer to say "For each real number m, there exists a real number k such that k > m", unless you are actually intend to convey a different meaning.

In mathematics it is often true that an if-statement is intended to have some universal truth. For example, if the statement is made that "If a > b then a + c > a + c", there are a lot things that are "understood" about the meaning of that statement. One possible interpretation of that statement is: "For each real number a and for each real number b and for each real number c, if a > b then a + c > a + b."

However, strictly speaking, an if-statement need not be any sort of universal claim. An expression which has variables in it, but no quantifiers for the variables ( i.e. no "for each" or "there exists") is technically not a "statement", it is a "statement function". A "statement" must be true or false. A statement function is only true or false when specific values are substituted for the variables.

So, strictly speaking, "if p > q then r > s" is not a statement. You can't say if it is true or false.

When we quantify the variables in various ways, we get statements. One example is:

"For each p, there exists a q such that for each r, there exists an s such that if p > q then r > s".

I find that statement confusing to interpret! However, it demonstrates how the quantifiers "for each" and "there exists" an be employed.

The statements you gave are both equivalent to "For each integer p, if p is a prime and p > 2 then p is odd".

Talking about the "converse" is a tricky matter. Strictly speaking, "converse" is a term that applies to if-statements. I don't know whether logicians apply it to "if-statement functions". Ordinary mathematicians do talk about the converse of theorems. What they would mean in your example is the statement "For each integer p, if p+1 is odd then p is a prime and p > 2". So they wouldn't be worried about whether the "p +1 is odd" implied that p was an integer.

My book has a problem in it:

For each of the true statements, show that their converses are false.

...

"If n is any prime number greater than 2, then p+1 is even."

My reaction was that the converse would be- "for all prime numbers p, if p+1 is even, then p is greater than 2."

The problem is that this converse is also true. But in the back of the book, the answer is that the converse is - "If n+1 is even, then n is a prime number greater than 2." which is false.

This made me kind of uncomfortable, so I asked my professor this morning, and it kind of threw him off too.

 

[SW VandeCarr]

Statement 1: If n is any prime number > 2, then n+1 is even.
Statement 2: For all prime numbers p, if p>2, then p is even.

The above two statements seam equivalent, but have different converses, etc.

converse 1: If n+1 is even, then n is a prime number > 2.
converse 2: For all prime numbers p, if p+1 is even, then p > 2.

If you have a statement that begins with if, is it illegal to write as a for all?

Well, for "converse" 1, it's certainly not the case that if n+1 is even, then n is a prime number > 2. The correct converse of your original statement 1 is: If n+1 is even, then n is any prime number > 2 (which is not true either).

In general Q implies P does not follow from P implies Q.

 

[jreelawg]

Well, for "converse" 1, it's certainly not the case that if n+1 is even, then n is a prime number > 2. The correct converse of your original statement 1 is: If n+1 is even, then n is any prime number > 2 which is not true either.

In general Q implies P does not follow from P implies Q.

Yes the exercise was to show that the converse of statement 1 is false.

I thought about wether the converse should be n is any prime number, but decided it didn't make sense for something to be any number. I would say maybe is an element of the set of all prime numbers. I guess to say it is any element of the set of prime numbers just sounds strange. The word is implies that it is a specific number.

And the books answer uses the form "is a prime number".

My problem is that which converse is correct seams inconsistent depending on how you interpret or format a statement, and who you ask.

Also the book considers that the negation of a for all (universal statement) is the existential, some are, or at least one is. But it seams like the two options for negating a for all, are not equivalent. The difference seams to be >= 1 vs >1.

 

[Stephen Tashi]

"If n is any prime number greater than 2, then p+1 is even."

Let's make it "If p is any prime number..."

My reaction was that the converse would be- "for all prime numbers p, if p+1 is even, then p is greater than 2."

If we define the statement functions:

P(x) to mean "x is a prime number"
E(x) to mean "x is even"
G(x) to mean "x is greater than 2"

then we can consider the following statements:

Statement 1: For each x, if ( P(x) and E(x)) then G(x).
Statement 2: For each x, if P(x) then (if E(x) then G(x)).

I think the statements are logically equivalent. However if you look at statement 2, you can see why "converse" may not be a well-defined concept for expressions with statement -functions. Do we reverse the sides of both the if-then expressions? Do we reverse the sides of one of them and not the other?

"Converse" is unambiguous when applied to expressions of the form

if (Statement A) then (Statement B)

However, when applied to expressions like

For each x, if ( statement function A) then (statement function B)

things can become unclear since the statement functions themselves may have other if-statement functions inside them.


I think what you did was convert statement 1 into statement 2 and then you applied one possible definition of "converse" to statement 2. Given that you did that, some confusion is appropriate.

 

[jreelawg]

Let's make it "If p is any prime number..."
If we define the statement functions:

P(x) to mean "x is a prime number"
E(x) to mean "x is even"
G(x) to mean "x is greater than 2"

then we can consider the following statements:

Statement 1: For each x, if ( P(x) and E(x)) then G(x).
Statement 2: For each x, if P(x) then (if E(x) then G(x)).

I think the statements are logically equivalent. However if you look at statement 2, you can see why "converse" may not be a well-defined concept for expressions with statement -functions. Do we reverse the sides of both the if-then expressions? Do we reverse the sides of one of them and not the other?

"Converse" is unambiguous when applied to expressions of the form

if (Statement A) then (Statement B)

However, when applied to expressions like

For each x, if ( statement function A) then (statement function B)

things can become unclear since the statement functions themselves may have other if-statement functions inside them.I think what you did was convert statement 1 into statement 2 and then you applied one possible definition of "converse" to statement 2. Given that you did that, some confusion is appropriate.

That makes sense, although I'm still uncomfortable with the level of ambiguity. Part of why this was so confusing, is that some of the exercises in the section prior to this one, involved converting verbal statements into formal statements using only symbols, and it seamed as though we were expected to put everything in the form of either (for all), or a (there exists) form.