I've studied a little bit about logical statements, but apparently not enough. Can you recommend a good reference on how to translate logical statements in English?
For instance, when reading the Axiom of Extensionality on Wikipedia, there is a translation into plain English. Like below,
$$\forall A \, \forall B \, ( \forall X \, (X \in A \iff X \in B) \implies A = B)$$
"Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B."
After $$\forall A \, \forall B $$ how does the next group of parenthesis, $$ ( \forall X \, (X \in A \iff X \in B) \implies A = B)$$
relate to first two quantifiers?
I get confused on how to read these statements correctly with all the parenthesis and brackets, why is there a "such that" when there is no symbol for such that, etc...
For instance, in the Axiom of Pairing, there is a "such that," like below,
$$\forall A \, \forall B \, \exists C \, \forall D \, [D \in C \iff (D = A \lor D = B)]$$
"Given any set A and any set B, there is a set C such that, given any set D, D is a member of C if and only if D is equal to A or D is equal to B."
Or, perhaps my question should be, "how do you read these statements?"
Thank you!
With logic, we want to go from the outside in, so addressing
$(∀X(X∈A⟺X∈B)⟹A=B)$
Let's ignore the innermost parentheses for a second - then this statement is essentially saying
$(∀X($Something$)⟹A=B)$
The forward implication symbol $⟹$ denotes an if-then statement, so we can translate this to: "If, for all X, (Something), then A = B.
Now we can unpack the statement in the innermost parenthesis, which is saying that $X \in A$ if and only if $X \in B$. So putting this back into our last statement, we have "If, for all $X$, $X \in A$ if and only if $X \in B$, then $A = B$.
What does this have to do with our $\forall A \forall B$ before? well this is telling us that this above statement is true for ANY sets $A$ and $B$
Actually looking to see if this makes sense helps, essentially we're saying that if we pick any two sets at random, we're confirming that these two sets are equal, and this happens if every element of $A$ is an element of $B$ and vice versa, which is what the $\forall X(X\in A \Leftrightarrow X \in B)$ statement is saying.
When it comes to the $\exists X$ statement, as far as I am aware, linguistically translates to "There exists X such that...", whose conditions follow. Without the "such that", $\exists$ really means nothing at all.