Let's break down the problem step by step.

The given statement is: "Some shoes on display are new."

You are asked to determine whether each statement below is the negation of this statement.

The negation of the statement "Some shoes on display are new" would imply that no shoes on display are new. Let's evaluate each option accordingly.

1. Some shoes on display are not new.
   This statement is not the negation of "Some shoes on display are new." It just says that some shoes are not new, but it still allows for the possibility that some shoes are new.
   Answer: No

2. No shoe on display is new.
   This statement is the negation of "Some shoes on display are new." If no shoe is new, it directly contradicts the original statement that says some are new.
   Answer: Yes

3. Every shoe on display is new.
   This statement is not the negation of "Some shoes on display are new." In fact, it is a stronger statement in the same direction, stating that all shoes are new rather than just some.
   Answer: No

4. Not all shoes on display are new.
   This is not the negation either. It suggests that some shoes may not be new, but it doesn't deny the possibility that some shoes are new.
   Answer: No

Would you like more details or have any other questions? Here are 5 related questions you can think about:

1. What would be the negation of "Every shoe on display is new"?
2. How would you negate the statement "No shoe on display is new"?
3. Can two statements both be true but not be negations of each other? Why or why not?
4. How can logical quantifiers (like "some" and "all") change the meaning of statements?
5. What are some other examples of negations in everyday language?

Tip: When dealing with negations, pay close attention to universal terms like "all," "none," or "some," and how changing them affects the meaning of the statement.