Fourth R

The Square of Opposition

The four forms of categorical statements would appear, even at a casual glance, to have some implicit interrelationships in play among them. The relationships are defined by two opposed sets of qualities: universality versus particularity, and affirmation versus negation.

Form

Relationship

Structure (Plural)

Structure (Singular)

A

universal

affirmative

All S are P

or

Every S is P

E

universal

negative

No S are P

or

No S is P

I

particular

affirmative

Some S are P

or

At least one S is P

O

particular

negative

Some S are not P

or

At least one S is not P

Aristotle (384-322 bce), the Greek philosopher credited with organizing principles of rational thinking into a clear and methodical system of rules, devised a conceptual diagram to illustrate the interactions among categorical relationships. As an exercise in learning by doing, let's attempt to duplicate his process of developing the classic square of opposition.

THE BASIC SQUARE

If we arrange the four relationships as the four corners of a square, we should be able to illustrate those opposing forces, and analyze the implications between each pair of relationships. We'll begin by laying out the square with universal forms at the top and particular forms at the bottom, and with affirmations to the left and negations to the right. Then we'll indicate all possible interconnections with a line drawn between each pairing of forms.

Next, we'll identify the interactions between each pair of relationships, and use symbolic notation to describe them. In assessing the compatibility of each pair of relationships, we need to answer the following four questions:

1. Is it possible for both relationships to be simultaneously true?
2. Is it possible for both relationships to be simultaneously false?
3. Is it possible for either relationship to be true when the other is false?
4. Is is possible for either relationship to be false when the other is true?

Once we've determined the answers to these questions, we can identify the logical pattern that it follows, and use standard symbolic notation to express it.

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THE AFFIRMATIONS: A AND I

Our first comparison is between the A statement, All S are P, and the I statement, Some S are P. Now, a condition that applies to some S might or might not apply to all S. But any condition that applies to all S must apply to some S. In other words, if some things in category S are also in category P, this tells us nothing about whether all S are P (e.g., some signs are no-parking signs does not imply all signs are no-parking signs); but if all things in category S are also in category P (e.g., all squids are predators), then we can be certain that some S are P (some squids are predators) So we get the following pattern of answers:

1. It's possible for both A and I to be true, because I must be true whenever A is true.
2. It's possible for both A and I to be false, because A must be false whenever I is false.
3. It's possible for A to be false when I is true, because some does not imply all.
4. It's impossible for A to be true when I is false, because all implies some.

The pattern is such that whenever A is true, it's implied that I must also be true. This pattern corresponds exactly to the logic of IF A THEN I, or in symbolic notation, A → I. Furthermore, a false value for I implies that A must also be false. And indeed, this corresponds to the contraposition of A → I, which (as we learned in the Symbolic Logic section) is IF NOT I THEN NOT A, or ~I → ~A, which neatly confirms our assessment. Pairs of categorical statements, the second of which must be true if the first is true, and the first of which must thus be false if the second is false, are called implications. An implication is a condition that works in one direction only; that is, "all S are P" implies "some S are P," but "some S are P" does not imply "all S are P."

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THE NEGATIONS: E AND O

Although we're here dealing with negations rather than affirmations, the same principle holds: A condition that applies to some S might or might not apply to all S, but any condition that applies to all S must also apply to some S. Thus, we get a similar pattern of answers to our four questions:

1. It's possible for both E and O to be true, because O must be true whenever E is true.
2. It's possible for both E and O to be false, because E must be false whenever O is false.
3. It's possible for E to be false when O is true, because some does not imply all.
4. It's impossible for E to be true when O is false, because all implies some.

And again, this pattern corresponds to the conditional relation, E → O, and is confirmed by its contraposition, ~O → ~E. Again, these relationships are implications, and are thus one-directional: "No S are P" implies "some S are not P," but "some S are not P" does not imply "no S are P."

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THE UNIVERSALS: A AND E

The universal forms are represented by statement A, All S are P, and statement E, No S are P. When all of category S is contained in category P, then there can be nothing in category S that is outside category P. Likewise, if all of S is outside category P, then there can be none of S inside category P. In this situation, the only impossible configuration is that both statements A and E are true; all situations in which one or both statements are false are possible.

1. It's impossible for both A and E to be true, because the two statements are mutually exclusive.
2. It's possible for both A and E to be false.
3. A must be false whenever E is true.
4. E must be false whenever A is true.

This pattern corresponds to the logical negation of the conjunction of statements A and E, expressed as NOT (A AND E), or symbolically as ~(A * E). Pairs of categorical statements, at least one of which must be false, are called contraries.

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THE PARTICULARS: I AND O

Here we are dealing with statement form I, "Some S are P," and statement form O, "Some S are not P." Obviously, it's possible for some of S to be in category P while some other S is outside category P. It's also possible for some (all) of S to be within category P while none of S is outside P, or for some of S to be outside P when there is no S inside P. The only impossibility is for all of S to be both entirely inside and entirely outside category P at the same time.

1. It's possible for both I and O to be true.
2. It's impossible for both I and O to be false, because there's no S that both is and isn't P.
3. It's possible for E to be false when O is true.
4. It's possible for O to be false when E is true.

Pairs of categorical statements, at least one of which must be true, are called subcontraries.

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THE OPPOSING CORNERS: A AND O, E AND I

The statements, "All S are P" and "Some S are not P," directly contradict each other. Likewise, the two statements, "No S are P" and "Some S are P" are mutually contradictory. It is logically impossible for two mutually contradictory statements to have the same truth value; one must be true, and the other most be false.

1. It's impossible for both A and O to be true, and impossible for both E and I to be true.
2. It's impossible for both A and O to be false, and impossible for both E and I to be false.
3. A must be true when O is false, and vice versa.
4. E must be true when I is false, and vice versa.

This pattern illustrates an exclusive disjunction, "either one option or the other, but not both, and not neither." As we saw in the Symbolic Logic section, this pattern is succinctly expressed as the negation of a bi-condition, in these cases ~(A ↔ O) and ~(E ↔ I). Pairs of categorical statements, the truth values of which must always be opposite, are mutually contradictory. Contradictories illustrate the exclusive OR (XOR) relationship, which we briefly discussed in symbolic logic, and which can be symbolically expressed as the negation of a bi-condition. The truth values of contradictory statements must be opposite to each other; if one is true, then the other must be false, and vice versa.

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THE CLASSIC SQUARE OF OPPOSITION

We've now explored all possible pairings of the four categorical statement types. If these relationships did not seem commonsense before, they should after having considered the reasoning behind them. If they still aren't burned into your psyche, you might want to make a copy of our pretty square and review it until it becomes second nature, because the relationships it illustrates are a very fundamental part of reasoning. Once you fully understand it, you're not likely to forget it—or if you do, it shouldn't require much effort to figure out again.

Aristotle's principles of reasoning held firm, essentially unchanged and unchallenged, for nearly two and a half millennia. By the 18th century, this impressive record of durability had long been considered convincing evidence that his system of logic was perfect and could not be improved upon. Relatively recently, though, a few adjustments have been made as a result of an explosion of intellectual innovation in the modern era.

THE MODERN SQUARE OF OPPOSITION

One of these latter-day tweaks derives from the mathematical concept of zero, which didn't exist in Aristotle's time and culture. The ancient view was that, if a category contained nothing at all, then the category itself didn't exist. Modern thinking, though, considers categories as concepts, rather than as collections of actual things. To speak metaphorically, even if there isn't a molecule of tea in it, the teapot itself still exists. This shift in thinking has the following effect on categorical reasoning. Consider: If the current number of humans on our planet is seven billion, then "all humans now alive" equates to "seven billion humans." And if the number of planets inhabited by humans is one, then "all planets inhabited by humans" equates to "one human-inhabited planet." To follow the same pattern of thought, if the number of humans on the surface of Venus is zero, then "all humans on Venus" equates to "no humans on Venus." So, in an empty category, "zero" is equivalent to "all."

Under this condition, some of the relationships in the classic square of opposition break down. When any subject category S could conceivably be empty, then the classic contraries "All S are P" and "No S are P" are no longer mutually exclusive, for both become simultaneously true if category S is empty. "All S are P" no longer implies "Some S are P," and "No S are P" no longer implies "Some S are not P," if there's a possibility that there is no S. Even the subcontrary relationship, that either "Some S are P" or "Some S are not P" must be true, ceases to hold up if no S exists. The only relationships left standing when category S is empty are the contradictories: "All S are P" and "Some S are not P" continue to have opposite truth values, as do "No S are P" and "Some S are P." So, the modern square of opposition boils down to this:

Now, it would be a mistake to propose that the modern square of opposition renders the classic model worthless. From a practical standpoint, in most real-world cases concerning non-conjectural things, it's safe to assume that most subject categories contain at least one actual example. And as long as the subject category S isn't empty, then all of Aristotle's logical relationships remain intact; we can rely on them. But we should be aware of what happens in the special case when there's a significant chance that the actual number of S could be zero. This is especially important when S is something uncertain or unverifiable, like phlogiston,* God-given rights, or hard-working philosophers. We'll see the implications of this in the next lesson.

*Phlogiston is a now obsolete concept of a massless fluid once believed to be a component in all flammable substances and released as heat and flame during combustion.

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TERMS

Click a term to review its definition or explanation in the text.

implications

contraries

subcontraries

contradictories

classic square of opposition

modern square of opposition

Aristotle's thinking and his square of opposition dominated the principles of logic until the 1800s. Since then, however, innovation has devised new ways to express, visualize, and evaluate logical concepts. In the next lesson, we'll consider John Venn's eye-catching graphic approach to categorical reasoning, and we'll experience visually how the possibility of an empty category limits what we may legitimately infer from a line of reasoning.

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► Next: Venn Diagrams