# A Logic Problem

1. Feb 23, 2009

### soopo

1. The problem statement, all variables and given/known data
There exists y > 0 such that [$$y^{2} = x$$ if and only if $$x > 0$$].

This means that "there is some positive number whose square equals all positive
numbers." - St. John College, Oxford

3. The attempt at a solution
all positive numbers", and particularly about the word "all".

I would read the statement as
If $$\exists y > 0$$, then $$\exists [ y^{2} = x$$ if and only if $$x > 0]$$

It seems that the statement should be read as
If $$\exists y > 0$$, then $$\forall [ y^{2} = x$$ if and only if $$x > 0]$$

Is there always "for all" after "such that"?

2. Feb 23, 2009

### HallsofIvy

Staff Emeritus
You don't say "there exists" a statement. "There exists" and "for all" only apply to variables.

Not necessarily. There exist x> 0 such that x2= 4. That has no "for all". Try thinking about what "for all" means rather than looking for general rules.

3. Feb 23, 2009

### soopo

It seems that we need to make statements true for a given context.

For example, the above example with "for all" is false, whereas right with the "exists". It is nonsense to say that there exists one positive real number whose square equals all positive numbers.

The quantifiers apply to the variables. I agree with you about that.