On the square of opposition, the diagonals join A with O and E with I. These pairs are contradictories.
Key idea
Contradictories cannot both be true and cannot both be false: exactly one of them is true. The contradictory of "All S are P" is "Some S are not P"; the contradictory of "No S are P" is "Some S are P".
Why it works
Each pair talks about the same region of a Venn diagram and says opposite things about it. "All S are P" says region 1 (S outside P) is empty; "Some S are not P" says region 1 has something in it. A region is either empty or not, so exactly one of the two is right. The same goes for E and I with region 2. Because this reason uses no assumption about existence, the contradictories hold on every reading.
Worked example
Worked example: Denying a universal
What is the contradictory of "All the cookies are burned"? (Standard form: All cookies on the tray are burned cookies.)
- It is an A statement, so its contradictory is the O with the same terms.
- To show that not all the cookies are burned, you only need one cookie that is not.
Answer: Some cookies on the tray are not burned cookies.
Worked example: Denying a negative
What is the contradictory of "No hikers are poets"?
- It is an E statement, so its contradictory is the I with the same terms.
- To show that "none" is wrong, one hiker who is a poet is enough.
Answer: Some hikers are poets.
Common mistake
Giving "No S are P" as the contradictory of "All S are P". That is the contrary, and both can be false at once (some cookies burned, some not). To deny "all", you need only "some are not". Check: could your two statements both be false? If yes, they are not contradictories.