# A question on quantifiers

1. May 26, 2015

### amilapsn

1. The problem statement, all variables and given/known data

This question may seem as an axiom to some. I also feel the same.
Prove:
$\forall x\forall y\ p(x,y)\Leftrightarrow\forall y\forall x\ p(x,y)$

3. The attempt at a solution

$Assume\ \forall x\forall y\ p(x,y)$
$Let\ x_0\in \mathbb{R}$
$\ \ \therefore \ \forall y \ p(x_0,y)$
$\ \ Let\ y_0\in \mathbb{R}$
$\ \ \ \ \therefore\ p(x_0,y_0)$

2. May 26, 2015

### WWGD

Do you have a rule for existential generalization to conclude first $\forall x p(x,y_0)$ and then do the same for $y_0$?

3. May 26, 2015

### amilapsn

I didn't get you....
Do you mean this.....?

$Assume\ \forall x\forall y\ p(x,y)$
$Let\ y_0\in \mathbb{R}$
$\ \ \therefore \ \forall x \ p(x,y_{0})$
$\ \ Let\ x_0\in \mathbb{R}$
$\ \ \ \ \therefore\ p(x_0,y_0)$
$\ \ \therefore \forall x\ p(x,y_0)$
$\therefore \forall y\ \forall x\ p(x,y)$

BTW: I don't know what existential generalization could do here. Universal specification and universal generalization are the only rules of inference I can think of.
Thank You.

4. May 31, 2015

### amilapsn

Somebody tell me whether I'm right or wrong....

5. Jun 1, 2015

### WWGD

Could you please tell us the details of the rule of universal generalization that you use?

6. Jun 1, 2015

### amilapsn

universal generalization:if P(a) is true for all a in universe of discourse then we can say $$\forall x P(x)$$

7. Jun 1, 2015

### WWGD

Then it seems like from $p(x_0,y_0)$, since each of $x_0,y_0$ is arbitrary, you could generalize to either $\forall x p(x,y_0)$ or $\forall y p(x_0,y)$ and then generalize again. I think that does it.

8. Jun 1, 2015

Me too...