Today we realize that there are clear and unavoidable errors in the Traditional Square of opposition. For this reason, only presuppositionalists rely on it (a point I will clarify later). So why are we learning the Traditional Square of opposition? Because learning why it commits errors teaches us a good deal about classical logic.
The Square - A Study in EquivalencesEdit
The Traditional Square of Opposition is a diagram specifying logical relations among the four types of Categorical Propositions we just learned about in the preceding section. I think nothing makes it more clear that deductive categorical arguments deal with equivalences than the Traditional Square.
As noted in the Categorical Propositions section, these propositions can be referred to as: A and I (based on the word Affirmo, we refer to the affirmative Universal and particular propositions as A and I and based on the term Nego, we refer to the the Negative universal and particular propositions as E and O:
(A) All S are P (E) No S are P (I) Some S are P (O) Some S are not P
Standard form categorical propositions having the same subject and same predicate terms may differ from each other in quantity or quality, or both. For example:
All men are poets (A) Some men are not poets (O)
...differ in both quantity and quality. This kind of differing was given the technical name of opposition by classic logicians and certain important truth relations were correlated with various knds of opposition. The square of opposition is a diagram that of the four types of opposition, and it was held by classical logicians that these four types of oppositions allowed the truth value of one categorical proposition to be determined on the basis of the truth value of another of the categorical propositions. We call this type of argument an immediate argument , because we can refer immediately or directly from one type of categorical proposition to another to determine truth value! (When we come to the section on categorical syllogistic arguments, we will see that we need two premises to reach a conclusion, making these sort of arguments mediate arguments.)
There are four types of immediate arguments, or oppositions: Contradictories, Contraries, Subcontraries and Subalterns.
Two propositions are contradictory if one is the denial or negation of the other: that is, they cannot both be true and cannot both be false. A and O propositions are contradictory, as are I and E... one of the pair MUST be true and the other MUST be false. If the statement "All S are P",(A) is true, then the statement "some S are not P", (O) must be false. Example: If "All dogs are animals" is true, then "Some dogs are not animals" must be false. Or vice versa.
Two propositions are contrary if they cannot both be true but they might both be false. A and E are contrary. It can't be that "all dogs are animals" and "no dogs are animals" at the same time, but it may be that only some dogs are animals (and others are stuffed toys) making both Universal statements false. Cool, huh? (Note: recall from above that the traditional square contains errors discovered by modern logicians. The case of contraries will be revealed to include an unjustified presupposition in the section concerning the modern square of opposition. Take heed!)
Two propositions are subcontrary if they cannot both be false but they might both be true. The particular statements: I and O are subcontrary: "Some S are P" and "Some S are not P" can be true , but both cannot be false. (Note: recall from above that the traditional square contains errors discovered by modern logicians. The case of subcontraries will be revealed to include an unjustified presupposition in the section concerning the modern square of opposition. Take heed!)
A and I propositions are related by subalteration. Subalterns are a different sort of 'opposition', because a subalternation does not imply a contradiction at all. The truth of I may be inferred by the truth of A. If "All S are P" is true, then we can be certain that "Some S are P" must be true. The reverse, from I to A, is invalid. The same goes for the negative propositions E and O . One can infer the truth of O from the validity of E, but not vice versa. (Note: recall from above that the traditional square contains errors discovered by modern logicans. The case of subalterns will be revealed to include an unjustified presupposition in the section concerning the modern square of opposition. Take heed!)
So, Remembering that, (A) All S are P, (E) No S are P, (I) Some S are P, and (O) Some S are not P,Aristotle's Square of opposition states authoritatively:
If A is true, E is false - I is true, and O is false If E is true, then A is false, I is false and O is true If I is true, then E is false, but O and A are undetermined If O is true, then A is false, but E and I are undetermined
If A is false, then O is true, but E and I are undetermined If E is false, then I is true, but A and O are undetermined If I is false, then A is false, E is true and O is true If O is false, then A then is true, E is false, and I is true
Now that we have at least a shakey grasp on this, lets muddle things up a bit by introducing the concepts of Conversion, Obversion and Contraposition.
The operations of conversion, obversion, and contraposition are applied to categorical propositions to yield new categorical propositions - these can become immediate arguments.
Again.. (A) All S are P (E) No S are P (I) Some S are P (O) Some S are not P
We create a conversion by switching the Subject with the Predicate. - the converse. One standard form categorical proposition is said to be the converse of another when it is formed by simply interchanging the subject and predicate terms. Thus: "No pigs are dogs" becomes "No dogs are pigs." The converses of E and I propositions are automatically true and logically equivalent. The converse of A propositions usually are not, unless the Subject and predicate are synonyms. There is however, another way: An A proposition can be made converse through limitation. Recall from the square of opposition that we can create subalterns. The subaltern of an A propositon is an I propositon, and we can always create a converse of an I proposition. So we can create a converse of an A proposition through limitation.
The converse of O propositions are, in general, not valid.
|A All S is P||I Some P is S|
|E No S is P||E No P is S|
|I Some S is P||I Some P is S|
|O Some S is not P||conversion is not valid!|
An argument that offers a conclusion that is the converse of an E or I proposition is valid. We can make a conversion of an A statement through limitation, BUT, such a statement is no longer equivalent, in quantity, to the original A statement. An argument that offers a conclusion that is the converse of an A (without limitation) or O argument commits the formal fallacy of Illicit Conversion.
The next type of immediate inference is called obversion. To best understand an obversion, I will again make the point that categorical propositions deal with categories, or classes of entities. And each class, in theory, has a complimentary class of entities that do not belong to that class. For example, the class of all books has a complimentary class: the class of all nonbooks. So, an obversion is a proposition makes a reference to this complimentary class, in a negative fashion. We create an obversion by changing the quality of the proposition (from affirmative to negative, or vice versa) and then negating the predicate term. To negate the predicate, one attaches the prefix "NON" to it. Oberversion is an easy rule to remember, because the obverse of ANY categorical proposition is equivalent to its original form. "Some fish are not bass" becomes "Some fish are non-bass"
|A All S is P||I No S is Non P|
|E No S is P||E All S is Non P|
|I Some S is P||I Some S is Non P|
|O Some S is not P||Some S is Non P|
We create a contrapositive by switching the subject term with the predicate term while negating both terms. The contraposition of "All dogs are mammals" becomes "all non-mammals are non-dogs". The contrapositives of A and O are logically equivalent to the originals, while E and I are usually not. We can make a contrapositive of an E proposition through limitation - by using the sub altern of an E proposition: an O propositon, but again recall that the O propostion, as a particular proposition, is not equivalent to the orignial universal categorical statment.
An argument that has a conclusion that is a contrapositive of an unlimited E or I premise commits the fallacy of Illicit Contraposition which otherwise sounds like a lot of fun.
You should now have a very good grasp of categorical propositions and how they relate to each other. I also strongly advise those taking the Course in Logic 101 read the section entitled The Modern Square of Opposition - for as I noted at the beginning of this section, there is a serious problem for the traditional square of opposition that modern logicians have identified and corrected!
- Copi, I. M, Cohen, C., (2001), "Introduction to Logic", 11th Edition.