I have a problem I encountered in a logic textbook that I cannot figure out after multiple tries.
Say we assume that "All S is P" is true.
Does this allow us to conclude the truth value of "No Non-S is Non-P", where non-X is the complementary class of X.
The textbook answer is that the truth value cannot be determined. However, I seem to be able to prove the statement is false. This is how I do it:
- If "All S is P" is true, it is also true that S refers to a collection of objects that is smaller than or equal to the collection of objects referred to by P.
- If S<=P, then No Non-P can be S.
- All Non-P must therefore be Non-S.
- By subalternation, since All Non-P is Non-S, there must be some Non-P that is Non-S.
- If there is some Non-P that is Non-S, then there is some Non-S that is Non-P.
- If there is some Non-S is Non-P, then the statement "No Non-S is Non-P" must necessarily be false, because it is contradictory with the former statement, which has been arrived at via valid inferences from true premises and must therefore be true.
Yet, when I draw it on a Venn diagram for "All S is P", there is a case where P refers to the collection of ALL objects, which means that Non-P does not exist, hence all Non-S must be P. This admits a rare case where the statement holds, hence the statement's truth value is undetermined.
Both lines of reasoning seem correct, yet contradictory. What went wrong?