Two propositions are contraries if they cannot both be true, but could both be false. If one is true, then the other one is necessarily false. The truth of the A or E claim forces the contrary to be false.
Two propositions are sub contraries if they cannot both be false but could both be true. This is true of the two particular claims. If one is false then the other must be true.
If it is false that “Some dogs are fish.”
“ Some dogs are not fish” is true.
If it is false that “Some mice are not rodents”
“ Some mice are rodents” is true.
Remember : This only applies when the particular claim is false. It could very well be the case that they are both true. For instance, “Some dogs are not chihuahuas” and “Some dogs are chihuahuas” are both true.
Two propositions are contradictories if they cannot both be true and they cannot both be false. All categorical propositions have contradictories. “All P is Q” is opposite in truth-value to “Some P is not Q.” “No P is Q” has an opposite truth-value to “Some P is Q.”
When a universal claim is true and the subject class is not empty of members, we can conclude that the corresponding particular claim is also true. This is, called the subaltern. The process of going from the universal claim to its corresponding particular claim is called subalternation.
If it is true that “All Persians are cats” and we know there exist Persians,
“ Some Persians are cats” must be true.
If it is true that “All flying saucers are UFOs,” but we don’t know if flying saucers exist then:
“ Some flying saucers are UFOs” cannot be inferred as true.
There are three other key moves you can make in terms of drawing inferences. These are the obverse, the converse, and the contrapositive. For these we need to know one more thing —the complement of a class.
The complement of a class A is the class of those things not in A.
So, for instance, the complement of the set of voters is the set of nonvoters. The complement of the set of non-citizens is the set of citizens. So, given any set A, the complement is the set non-A.
Similarly, given any set non-B, the complement is the set B. (Think of a non-non-B as a double negative, that takes us back to set B) Examples of complements: farmworkers/non-farmworkers; ballerina/non-ballerina; non-workers/workers.
The Obverse of a proposition involves two steps: First, change the quality (from positive to negative or vice versa); then change the predicate to its complement. The result is the Obverse.
It has the same truth-value as the original claim. So, for example, “All A is B” has as its obverse, “No A is non-B.” If the original proposition is true, so is the obverse. If it is false, then the obverse is false. The obverse can be taken on any proposition.
The Converse of a proposition is obtained by switching the subject and the predicate, when possible. We can take a converse of an E or I claim.
However, the converse of an A claim is known as converse by limitation , for we must step down to an I claim. So, “All A is B” has as its converse “Some B is A.” We can’t take the converse of an O claim.
No scuba divers are nonswimmers. No nonswimmers are scuba divers.
Some ice-skaters are hockey players. Some hockey players are ice-skaters.
All pilots are daredevils. Some daredevils are pilots.
Some hikers are not fond of heights. Does not exist (no converse of an O claim!)
To take the contrapositive of a proposition, follow these two steps: First, replace the subject with the complement of the predicate. Second, replace the predicate with the complement of the subject.
The contrapositive cannot be taken on the I claim. It can only be taken on an A, E and O. The E claim is contrapositive by limitation : step down to an O claim.
So the contrapositive of “No A is B” would be “Some non-B is not non-A.” Don't be surprised with a strange-looking result. Once you verify the original sentence is A, O, or E, then just flip the subject and predicate, changing each one to the complement when you do the switch and, in the case of the E claim, move it down to an O claim.
All trout are fish. All nonfish are nontrout.
No non-dancers are non-athletes Some athletes are not dancers.
Some citizens are not nonvoters. Some voters are not non-citizens.
No FBI agent is a spy. Some non-spies are not non-FBI agents.