I need to know if these two statements are independent:
All winners are losers.
All losers are winners.
I am thinking that they are because I don't see that the truth or falsity of one tells anything about the truth or falsity of the other.
Also, if "Some Americans are children" is false, is the statement "Some nonchildren are nonAmericans" true, false, or undetermined.
I know that the statements are not logically equivalent, but I don't know what that says about the contrapositive in terms of truth value.
I agree with your first statement.
Your second statement about nonchildren would be undetermined.
To use another example:
Some bears are purple. (false)
Some nonbears are not purple. (undetermined)
sub+ob
sub+ob=
To determine if the two statements "All winners are losers" and "All losers are winners" are independent, we need to examine their relationship and dependencies.
The statement "All winners are losers" implies that there are no winners who are not losers. If we assume this statement is true, we can conclude that there is no winner who is excluded from being a loser. However, this statement says nothing about whether all losers are winners.
Similarly, the statement "All losers are winners" implies that there are no losers who are not winners. Assuming this statement is true, we can conclude that there is no loser who is not also a winner. But again, this statement does not tell us anything about whether all winners are losers.
From this analysis, we can say that the two statements are indeed independent. The truth or falsity of one statement does not determine the truth or falsity of the other.
Now, regarding the second question, let's analyze the statement "Some Americans are children" and its implications.
If this statement is false, it means that there are no Americans who are children. Therefore, the statement "Some nonchildren are nonAmericans" refers to the possibility that there may be non-children who are also non-Americans.
In this case, we cannot determine the truth or falsity of "Some nonchildren are nonAmericans" based solely on the falsity of "Some Americans are children." The statement is undetermined because there might be non-children who are not Americans, or it might not be the case.
The contrapositive of a statement involves switching the positions as well as the negation of both the subject and the predicate. However, in terms of truth value, the contrapositive does not necessarily inform us. It only guarantees that if the original statement is true, the contrapositive will also be true, and if the original statement is false, the contrapositive will also be false.
To determine the truth value of the contrapositive statement, you would need additional information or context.