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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?

Yes the two propositions are contradictory in Aristotelian ogic. Your reasoning is not even close to WHY the two propositions are contradictory. You are accidentally correct. You must understand there are different types of logic with different rules. So in Mathematical logic this would not be a question at all. It would never be asked. Rules of inference in Aristotelian logic would show that the 2 propositions you state are indeed contradictory. That is, both propositions can't be true at the same time both can't be false. If 1 is true the other must be false & vice versa. – Logikal – 2020-09-06T07:31:17.113

In Aristotelian logic you can use inference rules such as obversion, conversion, etc to prove that "No non-s is non-p" is IDENTICAL (not a logical equivalence) to the O type of proposition: Some s are not p. The square of Opposition shows that A type propositions are contradictory to O type propositions. Your reasoning should have been close to the subject material of deductive reasoning--not your own invention. Perhaps you are confusing logic as all logics are the same thing. Perhaps you thought logic is discrete mathematics or something. There is more subject matter to logic than math. – Logikal – 2020-09-06T07:39:30.403