For the purposes of the following discussion, the Aristotelian view that the affirmative propositions have existential import is used. This position allows that the negative propositions might have empty terms and the relations above then hold. The more modern "Boolean" approach may be more mathematically useful, however for the beginning student who speaks English as a second language, the position that the affirmative propositions carry existential import makes more sense inherently.
The proposition "Every Sepe is Kosraen" or the more accurate equivalent "Every person named Sepe is a person from the island of Kosrae" is a universal affirmative. Whether the proposition is actually true is not at issue. There may be people named Sepe who are not Kosraen. If the proposition is true, then "Some persons named Sepe are not Kosraen" cannot also be true. Put in the terms of the Peripatetics, "Every Sepe is Kosraen" is contradicted by "Not every Sepe is Kosraen." A and O propositions are said to be "contradictory" or "contradictories."
The E and I propositions are also contradictory. "No sakau is alcoholic" is contradicted by "Some sakau is alcoholic."
If one proposition in a contradictory pair is true, the other must be false. The contradictory propositions are the propositions on the diagonal of the square.
Contraries cannot both be true. The A and E propositions are contrary to each other. "All Sepes are Kosraen" and "No Sepes are Kosraen" cannot both be true. Contraries can split true-false, "All dogs are animals" (true) "No dogs are animals" (false).
There are situations in which the A and E contraries can both be false. "All airplanes are Continental Micronesia planes" and "No airplanes are Continental Micronesia planes" are both false propositions. While not all airplanes are Continental Micronesia airplanes, there are some airplanes that are Continental Micronesia planes. Note that this true I statement, "Some airplanes are Continental Micronesia planes" is a contradictory to E, and thus if I is true, E must be factually false.
Another way to consider contraries is that if one is true, then the other cannot also be true. But knowing that one is false does not provide any information on the true/false status of the other.
The I and O propositions are subcontraries. The use of the "sub-" prefix denotes that I and O are a different type of contrary than A and E. I and O propositions cannot both be false. I and O can both be true. Consider the two factually true propositions, "Some airplanes are Continental Micronesia planes" and "Some airplanes are not Continental Micronesia planes." The O statement could also be written, "Not every airplane is a Continental Micronesia plane." Both are true statements. Subcontraries can both be true, they cannot both be false.
"Some cats are dogs" is a false I statement. "Some cats are not dogs" is a true proposition. In fact, the stronger E statement holds for this S and P: "No cats are dogs." If I is false, then there must be S which are not P and therefore O cannot also be false. If O is false, if "Some S are not P" is not true, then there are no S outside of P and thus some S must be included in P.
A and I are subalterns. E and O are subalterns. In the following discussion keep in mind the idea that "truth flows down from heaven above" while "falsity rises up from the earth below." This mnemonic should help remember the subaltern relations.
If A is true, then I must be true. If all S are P, then some S are P is almost trivially true. "All corals are an animal" is true. The I statement, "Some corals are an animal" is also true. Some, in fact all, are animals.
Note that if I is true, then A need not be true. "Some dogs are black" is true. "All dogs are black" is not true. Truth does not flow up the diagram. When I is true, there is no way to determine whether A is true or false. Remember that I and O are subalterns: while they can both cannot be false, they can both be true or they can split with one true and one false. So there is no way to argue the truth status of A by moving from a true I to O.
When E is true, then O is also necessarily true. "No night is a night that lasts forever" guarantees that "Some nights are not nights that last forever."
I being true does not help predict the truth status of A. When I is false, however, then A is most assuredly false. Consider a false I, "Some ferns produce coconuts." Clearly false. A cannot be true in this situation: "All ferns produce coconuts." When I is false, then A must be false. The false status of I flows up the diagram. Note that in this example the copula is formed from the verb "to produce." This can be wordsmithed into a "to be" copula if necessary, "Some ferns are plants that produce coconuts."
O being true does not help to determine the truth of E. Yet when O is false, then E is also false. "Some tuna are not fish" is clearly not true. Note that the Aristotelian version, "Not every tuna is a fish" is also not true. With this being false, the statement "No tuna is a fish" is also false. Falsity rises up the diagram.
Suppose A is known to be true. Then O, by contradiction, must be false. The subalterns I and O cannot both be false, so I must be true. Note that truth flows down, thus the truth of A could have been used to determine that I was true from A. E is a contradictory to I. With I true, then E must be false.
An example of a true A would be "All lightning is electrical." "Some lightning is not electrical" (false). "Some lightning is electrical" (true, in fact not just some but all). "No lightning is electrical" (false).
Running the diagram means to use the relations to move between the four types of propositions. Sometimes knowing the truth or falsity of one proposition allows the other three types to be worked out. Sometimes not all three can be worked out. If A is known to be false, then O must be true. In this case I and O remain undetermined. I and O cannot both be false, but O is known to be true. I and O can both be true or they can split true/false as noted above. O is true, but truth does not flow up the diagram, thus E is not known. And both A and E can be false (they cannot both be true, but they can both be false).
"All dogs are cats" is false. "Some dogs are not cats" is true - there do exist dogs which are not cats. "Some dogs are cats" is actually undetermined from the above information, although in this case the proposition is false. "No dogs are cats" happens to be true, but this too is not demanded by "All dogs are cats" being false. Think of this example as AEIO=FTFT.
"All cars are Toyotas" is a false A proposition, as in the example immediately above. O is clearly true, "Some cars are not Toyotas." In this case I is true, "Some cars are Toyotas," and E is false, "No cars are Toyotas." Think of this example as AEIO=FFTF.
Using the provided true propositions, generate the other forms specified. Note whether the other forms are true or false.
i A small fish found in the rivers of Micronesia. Some flash a bright blue color in the sunlight.