Relations Amongst Propositions: The Square of Oppositions

Having understood the basics of predication in terms of the A, E, I, and O forms, there is one other very fundamental sort of predication to comment on before we move on to understanding the relationships that obtain between the various standard forms.

When we talked about predication as saying something about something, we forgot to pay very much attention to the fact that very many times, the something we are saying something about is not a group of at least one or a group of all; sometimes it’s just a single individual thing. Like

Pluto is not a planet or

Verizon is better than Cingular or

Stephen Colbert didn’t invent “truthiness.”

In these statements, the subject is what we call singular, so these are known as singular statements or predications. Later in this course, when we have some suitable symbolic conventions to play with, we’ll represent this kind of predication as a unique one. But at the moment, working without anything outside of our ordinary language conventions, it makes best sense to see these as a sub-species, if you will, of Universal predication.

No thing identical to Pluto is a planet

has a pretty strange ring to it at first. It raises the immediate question: what do you mean by “thing identical to…?” The answer is just the obvious one: each and every thing that there is (or is not –anything that can be named) is identical to itself and distinct from every other thing. Talk about your tivial insight, huh? Pluto is identical to itself. Stephen Colbert is identical to himself too. Verizon is identical to itself as well. If you can see that, then you can see that “all members of the class of things identical to Stephen Colbert” is another way to say “Stephen Colbert.” What’s nice about it, from the point of view of Logic, is that this convention allows us to represent our predications about Stephen Colbert, and to understand that whatever is true about any A statement as a mere function of its being universal and affirmative is going to also be true about every statement made about Stephen Colbert (and Pluto, and Verizon).

With this somewhat inelegant strategy for handling singular statements, it is possible for us to represent any simple predication as a categorical proposition, and to fit it into the rigid format:

Quantifer—Subject term—Copula—Predicate term.

Every simple statement or predication you can think of is one that it is possible to render in one of the four forms of Universal Affirmative, Universal Negative, Particular Affirmative or Particular Negative: A, E, I, or O.

It is likely that Leonardo’s play drive disappeared in his maturer years and was absorbed into his research activity, which represented the last, supreme unfolding of his personality. Freud, Leonardo daVinci and a Memory of his Childhood from The Uncanny (Penguin Books)

From the point of view of categorical logic, this becomes an A statement, the subject of which is “things identical to Leonardo’s play drive.” The predicate can be represented as the complex one: “personality traits that disappeared in his maturer years, absorbed into his research activity, which represented the last, supreme unfolding of his personality.”

Now let’s consider the relations between these four predication patterns. Since each is characterized, itself, by two descriptive terms, and since our four descriptive terms are actually two pairs of terms of opposite meaning, it is easy to spot the relationship between two sets of pairs.

Refresh your grasp of the descriptive terms:

A Universal Affirmative

E Universal Negative

I Particular Affirmative

O Particular Negative

A and O have the same relation to each other that E and I have. One of each pair is universal, the other particular; and one of each pair is affirmative, the other negative. The A and O are opposites in both respects, the same holds for the E and I. Intuitively, one might well expect that this will have implications for their truth values: if they are completely opposites, then surely if one of them is true, the other must be false. This is indeed the case, and the relationship between any two statements of these forms (about the same subject and predicate terms) is called “Contradiction.”

No rationalists are empiricists T

Some rationalists are empiricists F

All fundamentalists are atheists F

Some fundamentalists are not atheists T

All husbands have wives T

Some husbands do not have wives F

No lawyers are philosophers F

Some lawyers are philosophers T

It is traditional to lay out the following square, with A and O at opposite corners, and E and I at the other opposite corners:

(Source: Stanford Encyclopedia of Philosophy)

This powerpoint with narration walks you through the same material as the text has just presented and that then follows. I hope it is helpful.

This is the last two minutes:

Contradiction is represented as the diagonal relationships. But as you can see, there is room for commenting on the relationships between A and E, between A and I, between E and O, and between I and O. I think of this as the Square of Oppositions (with an “s”), though most Logic books call it the Square of Opposition (without the “s”). I prefer this just because there are multiple relationships, not a single one, even though they are not all a matter of incompatibility (which “opposition” connotes). We can even call it the Square of Propositions. The name is less important than the relationships the square calls to our attention:

Between any A and E statements about the same subject and predicate terms, you see that both are universal, but one is negative, the other affirmative. Not completely different from each other. Consider the consequences for their truth values, with a few examples. As you read these statements, note whether each is true or false, and see what you can determine as a general statement about the relationship between A and E.

The A- E relation:

1 No talking eggs are rude

2 All talking eggs are rude

3 No Popes are Catholics

4 All Popes are Catholics

5 No Senators are Communists

6 All Senators are Communists

7 No Senators are Democrats

8 All Senators are Democrats

Before I comment on the A-E relation here, let me provide you with examples of the other three. See what you can come up with as generalizations from considering the patterns of truth-value compatibility that you identify:

The A- I relation:

9 All cognitive scientists are neurobiologists

10 Some cognitive scientists are neurobiologists

11 All atheists are agnostics

12 Some atheists are agnostics

13 All clarinets are trumpets

14 Some clarinets are trumpets

The E- O relation:

15 No comedians are singers

16 Some comedians are not singers

17 No grandfather is someone’s sister

18 Some grandfather is not someone’s sister

19 No chickens are birds

20 Some chickens are not birds

The I- O relation:

21 Some punk rockers are shoegazers

22 Some punk rockers are not shoegazers

23 Some husbands are married

24 Some husbands are not married

25 Some husbands have wives

26 Some husbands do not have wives

Think about these sets of statements, and what you know about their truth-values. Can you determine what is always the case, what can always be inferred when one of an A-E pair is True? False?

When one of a I-O pair is True?  False?

When one of a Universal and a Particular of the same quality (A-I,  E-O) is True? False?

. To follow up, you can go to the next section of this chapter on Categorical Logic, called “Finishing the Square and Immediate Inferences.”

Immediate inferences and Square