1

$$x\in(\cap F)\cap(\cap G)=[\forall A\in F(x\in A)]\land[\forall A\in G(x\in A)]$$

Since the variable $A$ is bounded by universal quantifier, it is regarded as bounded variable, according to the rules, the variable is free to change to other letters while the meaning statement remains unchanged. But,the above statements mention two different families of sets, $F$ and $G$, why is it correct to state the sets of $F$ and $G$ by using the same letter $A$, for the first $A$ in the first part of the conjunction stands for sets in $F$ while the latter stands for sets in $G$? Isn't different letters should be used to refer those sets($A$ for $F$ while $B$ for $G$)? I am extremely confused with the usage of bound variables. Please explain, thanks!

user91500
  • 5,606
Dave Clifford
  • 999

3 Answers3

1

The reason for this is that when you take the conjunction of two statements, its truth depends on the truth of each of the two components, and you don't look any further than that.

For example, the statements

$$\forall x \geq 5 \quad 2x \geq 10$$ and
$$\forall x \leq - 3 \quad x^2 \geq 9$$ are both true, taken separately. Therefore the conjunction $$(\forall x \geq 5 \quad 2x \geq 10) \wedge (\forall x \leq - 3 \quad x^2 \geq 9)$$ must be true, regardless of the fact that $x$ represents different possible numbers in each of the two statements.

Edit: Nonetheless, it is true that $x$ is taken as belonging to some overarching "universe" of all objects. In my example, this is the set of all real numbers, and in your example, $A$ belongs to the universe of all sets, regardless of whether they belong to $F$ or $G$ or not. A statement such as $$ \forall A \in F \quad x \in A$$ really means $$\forall A \quad A \in F \Longrightarrow x \in A,$$ where the part "$\forall A$" means "for all sets $A$." Analyzed this way, you see that really $A$ represents the same type of object, namely a set.

user121173
  • 516
1

In a quantified expression like $\forall x A(x)$ the occurrences of the variable $x$ into the formuala $A(x)$ are bounded by the quantifier (i.e. $\forall x$) becuase they occur into the scope of the quantifier.

So, in the example :

$\forall x (A(x) \land B(x))$

both the occurence of $x$ in $A$ and the occurrence of $x$ in $B$ are bound.

In the expression :

$\forall x A(x) \land B(x)$

I have renoved the parentheses, so "restricting" the scope of the quantifier: now the $x$ in $A(x)$ is bound, but the $x$ in $B(x)$ is free.

When you consder the formula :

$\forall x A(x) \land \forall x B(x)$

two different quantifiers act one on $A(x)$ and one on $B(x)$; so, both occurences of $x$ are bound, but the fact that the two quantifiers use both $x$ is immaterial. The "effect" of the quantifier "vanish" outside its scope.

The last formula is equivalent to :

$\forall x A(x) \land \forall y B(y)$

because the variable in the quantifier preceding $A(x)$ is not "linked" to the variable in the quantifier preceding $B(x)$.

Mauro ALLEGRANZA
  • 94,169
0

The answer to your specific question is that $A$ occurs as a bound variable in two different expressions. When a variable is bound, it has no meaning outside the expression it is in, so you can use it again as a variable in another expression, and that is what is done here. (But if I were writing a book or paper about logic aimed at beginners, I would use a different letter for the two bound variable anyway.)

SixWingedSeraph
  • 695