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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!

Asaf Karagila
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watchy
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1 Answers1

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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.

Michael
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  • That is helpful. Thank you! – watchy Jan 31 '21 at 23:05
  • How does the innermost parenthesis, (Something), relate to the preceding "for all X" before it? Is there another way to rephrase that relationship besides just continuing like "If, for all X, (?) X∈A if and only if X∈B, then A=B." – watchy Feb 01 '21 at 01:17
  • Also, another question! How is it that the sets A and B are equal to each other and not just two different sets that both happen to have the same elements of X? I found another SE post that asks this same question, but I'm still not convinced. https://math.stackexchange.com/questions/2943449/axiom-of-extensionality-word-definition – watchy Feb 01 '21 at 05:44
  • For your first question - the statement inside the parenthesis basically denotes what's true for the thing on the outside. For instance, that first set of parenthesis is true for ALL A and for ALL B. In that same vein, that innermost (something) is saying that if [because of the implication symbol], for all X, this something is true, then A = B – Michael Feb 01 '21 at 10:34
  • To your second question, what makes these sets equal is that for all X (AKA if we choose any X), if it is an element of A, it is also an element of B, and vice versa. Also, the symbol for if and only if means the inverse is true, AKA, if X is not in A, then it is not in B. This tells us that the sets have to be the same - any element of A is in B, any element of B is in A, and any element that isn't in A isn't in B, vice versa. – Michael Feb 01 '21 at 10:38
  • "and any element that isn't in A isn't in B" -- is this due to universal quantifier for all A and for all B? – watchy Feb 01 '21 at 19:32
  • Almost! This is actually a consequence of the two sided arrow (equivalence). A forward arrow says (for generic statements X and Y): “if X, then Y”. In logic, the contra positive “If not Y, then not X” is also true. However, the two sided arrow makes the inverse and converse statements (“if not X, then not Y”, and “if Y, then X”, respectively), true. In this case “any element that isn’t in A isnt in B” is the inverse of the statement “any element in A is in B”, which is true because of the two sided arrow – Michael Feb 02 '21 at 20:36
  • OK, that makes sense! Thank you. I have one more question. I have had trouble answering this question. When we say "for every x in A", are we saying that every possible element is in A -- and there can be no other elements except x? – watchy Feb 02 '21 at 21:01
  • Also, could you briefly explain what universe (or domain) of discourse means? – watchy Feb 02 '21 at 21:03
  • I’m not entirely clear as to what your first question is asking, when we say “for every x in A”, we’re referring to basically picking any element in A, and denoting it x. (This statement is usually followed by a criteria. Ex. If the set A is the positive reals, then we can make the statement: “for every x in A, x>0”). To your second, the domain of discourse is the set that we’re working in when making a statement, as in, a statement that is true over the reals might not be true over the integers. An example is. “There exists x such that x^2 = 2”. True over the Reals, untrue over Intergers – Michael Feb 04 '21 at 15:01