Having established the boundaries of our domain of logically well-behaved natural language, we turn now to an investigation of the properties of its inhabitants. The four types of categoricals are related to one another in systematic ways; we will look at those relationships.
The relationships are inferential: we can often infer, for example, from the truth of one of the four categoricals, whether the other three are true or false. These inferential relationships among the four categorical propositions are summarized graphically in a diagram: The Square of Opposition. The diagram looks like this:
The four types of categorical propositions are arranged at the four corners of the square, and along the sides and diagonals are marked the relationships that obtain between pairs of them. We take these relationships up in turn.
Contradictory pairs of categorical propositions are at opposite corners from one another on the Square of Opposition. A and O propositions are contradictory; E and I propositions are contradictory. What it means for a pair of propositions to be contradictory is this: they have opposite truth-values; when one is true, the other must be false, and vice versa.
This is pretty intuitive. Consider an A proposition—all sailors are pirates. Suppose I make that claim. How do you contradict me? How do you prove I’m wrong? “My brother’s in the Navy,” you might protest. “He’s a sailor, but he’s not a pirate.” That would do the trick. The way you contradict a universal affirmative claim—a claim that all S are P—is by showing that there’s at least one S (a sailor in this case, your brother) who’s not a P (not a pirate, as your brother is not). At least one S that’s not a P—that’s just the particular negative, O proposition, that some S are not P. (Remember: ‘some’ means ‘there is at least one’.) A and O propositions make opposite, contradictory claims. If it’s false that all sailors are pirates, then it must be true that some of them aren’t; that’s just how you show it’s false. Likewise, if it’s true that all dogs are animals (it is), then it must be false that some of them are not (you’re not going to find even one dog that’s not an animal). A and O propositions have opposite truth-values.
Likewise for E and I propositions. If I claim that no saints are priests, and you want to contradict me, what you need to do is come up with a saint who was a priest. It’s not hard: Saint Thomas Aquinas (who was the most prominent medieval interpreter of Aristotle, by the way, and a terrific philosopher in his own right) was a priest. So, to contradict universal negative claim—that no S are P—you need to show that there’s at least one S (a saint in this case, Thomas Aquinas) who is in fact a P (a priest, as Aquinas was). At least one S that is a P—that’s just the particular affirmative, I proposition, that some S are P. (Again, ‘some’ means ‘there is at least one’.) E and I propositions make opposite, contradictory claims. If it’s false that no saints are priests, it must be true that some of them are; that’s just how you show it’s false. Likewise, it’s true that no cats are dogs (it is), then it must be false that some of them are (you’re not going to find even one cat that’s a dog). E and I propositions have opposite truth-values.
The two universal propositions—A and E, along the top of the Square—are a contrary pair. This is a slightly weaker form of opposition than being contradictory. Being contrary means that they can’t both be true, but they could both be false—though they needn’t both be false; one could be true and the other false.
Again, this is intuitive. Suppose I claim the universal affirmative, “All dogs go to heaven,” and you claim the corresponding universal negative, “No dogs go to heaven.” (Those sentences aren’t in standard form, but the translation is easy.) Obvious observation: we can’t both be right; that is, both claims can’t be true. On the other hand, we could both be wrong. Suppose getting into heaven, for dogs, is the way they say it is for people: if you’re good and stuff, then you get in; but if you’re bad, oh boy—it’s the Other Place for you. In that case, both of our claims are false: some dogs (the good ones) go to heaven, but some dogs (the bad ones, the ones who bite kids, maybe) don’t. But that picture might be wrong, too. I could be right and you could be wrong: God loves all dogs equally and they get a free pass into heaven. Or, I could be wrong and you could be right: God hates dogs and doesn’t let any of them in; or maybe there is no heaven at all, and so nobody goes there, dogs included.
Along the bottom of the Square we have the two particular propositions—I and O—and they are said to be subcontraries. This means they can’t both be false, but they could both be true—though they needn’t be; one could be true and the other false.
It’s easy to see how both I and O could be true. As a matter of fact, some sailors are pirates. That’s true. Also, as a matter of fact, some of them are not. It’s also easy to see how one of the particular propositions could be true and the other false, provided we keep in mind that ‘some’ just means ‘there is at least one’. It’s true that some dogs are mammals—that is, there is at least one dog that’s a mammal—so that I proposition is true. In fact, all of them are—the A proposition is true as well. Which means, since A and O are contradictories, that the corresponding O proposition—that some dogs are not mammals—must be false. Likewise, it’s true that some women are not (Catholic) priests (at least one women isn’t a priest); and it’s false that some women are priests (the Church doesn’t allow it). So O can be true while I is false.
It’s a bit harder to see why both particular propositions can’t be false. Why can’t ‘Some surfers are priests’ and ‘Some surfers are not priests’ both be false? It’s not immediately obvious. But think it through: if the I (some surfers are priests) is false, that means the E (no surfers are priests) must be true, since I and E are contradictory; and if the O (some surfers are not priests) is false, that means the A (all surfers are priests) must be true, since O and A are contradictory. That is to say, if I and O were both false, then the corresponding A and E propositions would both have to be true. But, as we’ve seen already, this is (obviously) impossible: if I claim that all surfers are priests and you claim that none of them are, we can’t both be right.
The particular propositions at the bottom of the table—I and O—are subalterns of the universal propositions directly above them—A and E, respectively. (And the universal propositions are called superalterns) This means that the pairs have the following relationship: if the universal proposition is true, then the particular proposition (its subaltern) must also be true. That is, if an A propositions is true, it’s corresponding I proposition must also be true; if an E proposition is true, its corresponding O proposition must also be true.
This is intuitive if we keep in mind, as always, that ‘some’ means ‘there is at least one’. Suppose the A proposition that all whales are mammals is true (it is); then the corresponding I proposition, that some whales are mammals, must also be true. Again, ‘some whales are mammals’ just means ‘at least one whale is a mammal’; if all of them are, then at least one of them is! Similarly, on the negative side of the square, if it’s true the no priests are women (universal negative, E), then it’s got to be true that some priests are not women (particular negative, O)—that at least one priest is not a woman. If none of them are, then at least one isn’t!
Notice that these relationships are depicted in a slightly different way from the others on the Square of Opposition: there’s an arrow pointing toward the bottom. This is because the relationship is not symmetrical. If the proposition on top is true, then the one on the bottom must also be true; but the reverse is not the case. If an I proposition is true—some sailors are pirates—it doesn’t follow that the corresponding A proposition—that all sailors are pirates—is true. Likewise, the truth of an O proposition—some surfers are not priests—does not guarantee the truth of the corresponding E proposition—that no surfers are priests. (I doubt it’s true; there’s gotta be at least one surfing priest, no? Then again.... Point is, the O doesn’t tell us whether it’s true or not)
Truth, as it were, travels down the side of the Square. Falsehood does not: if the universal proposition is false, that doesn’t tell us anything about the truth or falsehood of the corresponding particular. You could have a false A proposition—all men are priests—with a true corresponding I—some men are priests. But you could also have a false A proposition—all cats are dogs—whose corresponding I—some cats are dogs—is also false. Likewise, you could have a false E proposition—no men are priests—with a true corresponding O—some men are not priests. But you could also have a false E proposition—no whales are mammals—whose corresponding O— some whales are not mammals—is also false.
Falsehood doesn’t travel down the side of the Square, but it does travel up. That is, if a particular proposition—I or O—is false, then its corresponding universal proposition—A or E, respectively—must also be false. Think about it in the abstract: if it’s false that some S are P, that means that there’s not even on S that’s also a P; well in that case, there’s no way all the Ss are Ps! False I, false A. Likewise on the negative side: if it’s false that some S are not P, that means you won’t find even one S that’s not a P, which is to say all the Ss are Ps; in that case, it’s false that no S are P (A and E are contraries). False O, false E.
Given information about the truth or falsity of a categorical proposition, we can use the relationships summed up in the Square of Opposition to make inferences about the truth-values of the other three types of categorical proposition.
Here’s what I mean. Suppose a universal affirmative proposition—an A proposition—is true. What are the truth-values of the corresponding E, I, and O propositions? (By “corresponding”, I mean propositions with the same subject and predicate classes.) The Square can help us answer these questions. First of all, A is in the opposite corner from O—they’re contradictory. That means A and O have to have opposite truth-values. Well, if A is true, as we’re supposing, then the corresponding O proposition has to be false. Also, A and E are contraries. That means that they can’t both be true. Well, we’re supposing that the A is true, so then the corresponding E must be false. What about the I proposition? Three ways to attack this one, and they all agree that the I must be true: (1) I is the subaltern of A, so if A is true, then I must be true as well; (2) I is the contradictory of E, and we’ve already determined that E must be false, so I must be true; (3) I and O are subcontraries, meaning they can’t both be false, and since we’ve already determined that O is false, it follows that I must be true.
Summing up: if an A proposition is true, the corresponding E is false, I is true, and O is false.
Let’s try another one: suppose a universal negative, E proposition is true. What about the corresponding A, I, and O propositions? Well, again, A and E are contraries—can’t both be true— so A must be false. I is the contradictory of E, so it must be false—the opposite of I’s truth-value. And since O is subaltern to E, it must be true because E is.
If an E proposition is true, the corresponding A is false, I is false, and O is true.
Another. Suppose a particular affirmative, I proposition is true. What about the other three? Well, E is its contradictory, so it must be false. And if some S are P, that means some of them aren’t— so the O is also true. And since A is the contradictory of O... WAIT JUST A MINUTE! Go back and read that again. Do you see what happened? “And if some S are P, that means some of them aren’t....” No it doesn’t! Remember, ‘some’ means ‘there is at least one’. If some S are P, that just means at least one S is a P—and for all we know, all of them might be; and then again, maybe not. I and O are subcontraries: they can’t both be false, they could both be true, and one could be true and the other false. Knowing that I is true tells us nothing about the truth-value of the corresponding O, or the corresponding A. That some are, meaning at least one is, leaves open the possibility that all of them are; but then again, maybe not. The fact is, based on the supposition that an I is true, we can only know the truth-value of the corresponding E for sure. If an I proposition is true, then the corresponding E is false, and A and O are of unknown truth- value.
1. Suppose an O proposition is true. What are the truth-values of the corresponding A, E, and I propositions, according to the Square of Opposition?
2. Suppose an A proposition is false. What are the truth-values of the corresponding E, I, and O propositions, according to the Square of Opposition?
3. Suppose an E proposition is false. What are the truth-values of the corresponding A, I, and O propositions, according to the Square of Opposition?
4. Suppose an I proposition is false. What are the truth-values of the corresponding A, E, and O propositions, according to the Square of Opposition?
5. Suppose an O proposition is false. What are the truth-values of the corresponding A, E, and I propositions, according to the Square of Opposition?