Everything on the square so far used the traditional assumption that the subject class has members. On the modern reading, universal statements have no existential import, and most of the square comes apart.
Key idea
On the modern (Boolean) square, only the contradictories hold: A–O and E–I still have opposite values. Contrariety, subcontrariety and subalternation can all fail when the subject class is empty.
Why it works
Suppose there are no dragons. "All dragons breathe fire" and "No dragons breathe fire" say only that two regions are empty, and with no dragons both are true: contrariety fails. "Some dragons breathe fire" and "Some dragons do not" both say a dragon exists, so both are false: subcontrariety fails. And the true A does not send truth down to the false I: subalternation fails. The diagonals survive because each pair still talks about the same region, empty or not.
Worked examples
Worked example: Subalternation on the modern square
On the Boolean reading, "All campers are dancers" is true. What is "Some campers are dancers"?
- A and I are not contradictories, so the modern square links them by nothing.
- If there are no campers, I is false; if there are, I is true.
Answer: undetermined
Worked example: A contradictory still works
On the Boolean reading, "Some actors are poets" is false. What is "No actors are poets"?
- E and I are contradictories, which hold on every reading.
- I is false, so E is true.
Answer: true
Why keep both squares? The traditional square is right whenever we already know the subject class has members, which is true of most everyday talk. The modern square is safer in general, because it makes no assumption, and it is the one built into modern symbolic logic.
Common mistake
Keeping subalternation on the Boolean reading: from "All dragons breathe fire" it does not follow that some dragon breathes fire. Check: could the subject class be empty? If so, the universal is true and the particular false.