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Truth tables and the short-cut truth-value assignment method enable us to determine the validity or invalidity of a variety of complex arguments. But for one species of argument those methods won’t work. That species of argument is not a rare or exotic one; it includes such workaday, run-of-the-mill arguments as this:

All lions are mammals. All mammals are friendly. Therefore, all lions are friendly.

And also this:

No penguin is fond of walruses. Some penguins are psychics. Therefore, some psychics are not fond of walruses.

And again:

All rock stars are wealthy. No logicians are wealthy. Therefore, no logicians are rock stars.

Of course we don’t want to just ignore such arguments; sometimes it may be quite important to determine whether they are valid or invalid—and their validity or invalidity may not be intuitively obvious. Indeed, it is notoriously true that such arguments may often sound valid when they are invalid. For example:

All dogs have fur. All cats have fur. Therefore, all dogs are cats.

That somehow sounds right, even though we know it can’t be (its premises may be true, but its conclusion is certainly false). And at times, the invalidity of such arguments may be even less obvious:

All projects that are really valuable require sacrifices. Fighting this war certainly requires sacrifices. So fighting this war must be really valuable.

That argument is invalid, of course; but it sounds valid and may well pass for valid—especially in the U.S. Senate. In any case, it is not immediately obvious why it is invalid. So how are we to prove that such arguments are invalid and that valid arguments—such as the following—really are valid?

All lions are mammals. All mammals are friendly. Therefore, all lions are friendly.

Certainly not by means of truth-value assignments. For we would have to diagram it thus:

• L (for “All lions are mammals”)

• M (for “All mammals are friendly”)

• ∴ P (for “All lions are friendly”)

And that argument obviously would be invalid: It is a simple matter to assign P false, L true, and M true, resulting in all true premises and a false conclusion. But the argument is valid, not invalid. So that representation of the argument leaves a lot out. In this chapter, we work on a way to analyze what was omitted.

Arguments like the examples above deal with classes, with categories. “All lions are mammals”: That means that the class of lions is contained in the class of mammals; everything that is a member of the lions class is also a member of the mammals class. We call such statements categorical propositions. We call them that because that is what they are: propositions that make claims about categories and the members of categories.

In order to facilitate our discussion of categorical logic, it will be convenient to have some common terminology. First, we need terms for the parts of a categorical proposition. When we have a statement such as “All lions are mammals,” we want to be able to refer to the lions part of the statement and to distinguish it from the mammals part. In that statement, “lions” is the subject term; “mammals” is the predicate term. And it is important not to mix them up. It’s one thing to say that all lions are mammals; it’s something quite different to claim that all mammals are lions.

Types of Categorical Propositions

It will also be useful to divide categorical propositions into four sorts. The first is the universal affirmative (UA): All lions are mammals, all college students are wealthy, all men are pigs, all the children in Lake Wobegon are above average, all lovers are star-crossed, all bridges in Idaho are structurally sound. These are universal because the subject term deals with all members of its class: all lions, all college students, all the children in Lake Wobegon, all bridges in Idaho. Notice that the class may be rather narrowly defined. The last proposition does not say something about all bridges; rather, it makes a claim about a narrower class: bridges-in-Idaho. The class might be narrower still: railroad-bridges-in-Idaho, or railroad-bridges-in-Idaho-constructed-since-1970. The proposition is still universal if it makes a claim about all members of the class (however small or limited that class might be): “All U.S. Presidents named Barack like basketball” is a universal proposition, though the subject class has only one member. In fact, the subject class of a universal proposition may be empty: “All students over 12 feet tall are psychology majors.” It is universal affirmative, simply because it is an affirmative rather than a negative statement. All college students are wealthy. A universal affirmative proposition states that all members of the subject class are also members of the predicate class: All members of the class of college students are also members of the class of wealthy persons.

The second type of categorical proposition is the universal negative (UN): No college students are wealthy. It is a universal, because it makes a claim about all members of the subject class: In the entire class of college students, not one—not a single member of that whole class—is wealthy. And of course it is negative because it claims that no (rather than all) college students are wealthy.

The third type of categorical statement is the particular affirmative (PA): Some (particular) college students are (affirmative) wealthy. Perhaps all college students are wealthy, or perhaps not; the particular affirmative asserts only that at least some college students are wealthy (actually, at least one). And you have probably already guessed that the fourth type of categorical proposition is the particular negative (PN): Some college students are not wealthy (perhaps none of them are; this particular negative statement asserts only that at least one college student is not wealthy).

With the four types of categorical propositions in mind, we can begin to think about the relations among them. If a universal affirmative (UA) statement is true, what does that imply? Suppose that it’s true that all college students are wealthy. Then the corresponding particular negative (PN) statement—some college students are not wealthy—must be false. And going in the other direction, if the PN statement is true (some college students are not wealthy), then the corresponding UA (all college students are wealthy) statement must be false. That is, UA and PN are contradictories: If one is true, the other must be false; if one is false, the other must be true. The same relation holds between universal negative (UN) statements and particular affirmative (PA) statements: They are contradictories. If it is true that no college students are wealthy, then it must be false that some college students are wealthy. If it is true that some college students are wealthy, then it must be false that no college students are wealthy.

What about some other relations among UA, UN, PA, and PN propositions? Well, it would seem that if (UA) all college students are wealthy, then it must also be true that (PA) some college students are wealthy; and that if (UN) no college students are wealthy, then (PN) some college students are not wealthy. Do those implications hold? That is a vexed question. Its answer depends on what sort of existential presupposition we ascribe to universal statements. By “existential presupposition,” we do not mean that universal statements are experiencing bad faith or are worried about their essences or their place in the cosmos. Rather, the question concerns whether a universal statement implies actual existence of members of the subject class. That is, if we say “All college students are wealthy,” does that imply that any college students really exist (i.e., does it imply that there exist some members of the subject class)? And if we say “No college students are wealthy,” does that imply that there are some college students?

That question gets a trifle thorny. Universal statements usually—in ordinary conversation—do seem to carry existential presuppositions. Suppose, for example, I played baseball yesterday, and you are inquiring about my performance: “Did you have a good day at bat?” I reply, with zest: “All my hits were home runs!” Very impressive, right? But when you inquire further about how many hits I had, you discover that in fact I had no hits at all. You might suggest that I had been a bit deceitful: Saying that all my hits were home runs would seem to suggest the presence of at least one hit. On the other hand, some of our universal statements do not seem to carry existential presuppositions. At the beginning of a baseball game, the announcer warns: All persons throwing objects onto the playing field will be immediately ejected. But that does not imply that there are such persons; in fact, it is hoped that there are not. And (to borrow an excellent example from I. M. Copi) when a physicist asserts Newton’s law that “all bodies not acted upon by external forces persevere in their state of rest or of uniform motion in a straight line,” certainly the physicist is not suggesting that there actually exist any bodies-not-acted-upon-by-external-forces; indeed, they would claim that there are no such bodies. (Incidentally, that statement is a universal affirmative: The subject is bodies-not-acted-upon-by-external-forces. The subject class is defined by a negative characteristic; but the statement makes an affirmative statement about all members of the class so defined.) So contemporary logicians have usually found it easier to say that universal propositions do not carry an existential presupposition. Particular propositions do imply existence of members of the subject class; universal propositions do not. As noted above, that is not always exactly consistent with ordinary usage; but on the whole, for logical analysis, it is better not to assume that universal propositions have existential presuppositions. (After all, if we want to assert an existence claim along with a universal proposition, we can always add the appropriate particular proposition: “All ducks have feathers, and some ducks have feathers.”)

Now let’s get to the fun part: Venn diagrams, developed by John Venn, an English mathematician. By using them, we can not only visualize the implications of categorical propositions, but also determine the validity or invalidity of many arguments using categorical propositions, and we can have a lot of fun doing so.

Let’s start with a universal affirmative proposition: All seals are pessimists. What does that really mean? It means that the class of seals is fully contained within the class of pessimists. That’s a mouthful. But using Venn diagrams, we can illustrate it clearly and elegantly: Venn’s pictures may not be worth a thousand words, but they are worth at least a few dozen.

To start with, let’s represent the class of seals, as shown in Figure 19-1. That’s easy enough, right? Everything in the circle is a seal, and all seals are in the circle; and anything that is not in the circle is not in the class of seals: it is non-seal. Another way of describing the class of non-seals is to say that it is the complement of the class of seals. Every class has a complement; the complement of class A is simply the class of all things that are not members of class A. (Incidentally, the complement of class A—the class of non-A—also has a complement: the class of everything that is not non-A, in other words, A. That is, the complement of the complement of class A is simply class A.) We’ll use a special symbol to represent class complements: a bar over the symbol used for the class. Thus the complement of A is “A bar” (). (Walruses, volcanoes, skateboards, and U.N. ambassadors all fall into the realm of non-seal; they are members of the complement of S; they are members of .) Within the circle, there may still be a lot of variety: pessimistic seals and optimistic seals, friendly seals and haughty seals, clever seals and dull seals. But what all members of the class have in common is that they are seals.

So we’ve got the subject term represented, now for the predicate. The predicate deals with the class of things that are pessimistic: pessimistic seals, pessimistic football coaches, pessimistic taxpayers; pessimistic pine trees, if there are such. And we represent the predicate class just as we did the subject (see Figure 19-2). Everything within the circle is a pessimist; everything outside the circle is a non-pessimist. (Notice that everything outside the circle need not be an optimist; many people and seals—and almost all rocks, turnips, volcanoes, and radioactive isotopes—are neither pessimistic nor optimistic; but they are still non-pessimists.)

So now we have a circle representing the class of seals and another representing the class of pessimists. How do we put them together?

There are several possibilities. We could just draw two circles as shown in Figure 19-3. That may be esthetically appealing, but it has some problems: the two classes are completely separate, and that makes it impossible for any seals to be pessimists. We need some way for the classes to overlap. So we could try another arrangement (Figure 19-4). Now we have the opposite problem: With these concentric circles, the class of seals is wholly contained within the class of pessimists, and we may want to represent the claim that some seals are not pessimists. The solution is a Venn diagram to overlap the circles

First, it is essential to understand what each part of this picture represents. With all the labels applied, the diagram looks like Figure 19-6. What does all that mean? We already know that everything in the circle on the left is a seal, and everything in the circle on the right is a pessimist. But those circles overlap; and so now we must discuss the intersection of the two circles. That intersection, or product, represents everything that is a member of both classes; that is, the intersection represents pessimistic seals, or the class of pessimistic seals: things that are both seals and pessimistic, and it is written as simply SP. The part of the S circle that does not intersect P represents all members of the class that are not members of P: that is, all seals that are not pessimistic. That part of the circle represents the product (the intersection) of S and the complement of P: the product of the classes seal and non-pessimist. And it is written as . The pattern is repeated in the other circle. The part of the P circle that does not intersect the S circle represents all pessimists that are not seals: all members of the class P that are not members of S. That is, it represents the product of P and non-S: P. Finally, there is that part of the diagram that falls outside both circles; and indeed, I imagine that’s where most of us fall, since we are neither seals nor pessimists. That represents everything that is non-seal and non-pessimist: the product or intersection of the complements of S and of P, which is .

With that in mind, we can represent just about anything we might wish to say about the relationships between the class of seals and the class of pessimists. We can, for example, represent the claim that: All seals are pessimists. Exactly what does that claim assert? Simply this: that there is nothing belonging to the class of seals that does not also belong to the class of pessimists; or more positively, all members of the seal class are also members of the pessimist class. Or one more way of stating the same thing: There are no members of the intersection of S; nothing exists in that section of circle S.

So how do we represent that universal affirmative proposition? Easy; we just shade out that part of the diagram, to show that nothing is there (Figure 19-7). Notice exactly what that means. There are no seals that are non-pessimists; all seals are members of the class of pessimists. But it does not imply that there really exist any pessimistic seals (any members of SP), or that there exist any pessimists who are not seals (P), or that anything exists that is a non-pessimistic non-seal (). It says only that any seals that do exist (if there should be any) are all pessimists. (As you recall, universal propositions do not carry existential presuppositions. Remember the signs your high school coach put up on the walls of the locker room? Things like “All quitters are losers.” But coach certainly was not suggesting that any quitters really exist—at least not in his locker room.) So this diagram implies only that there is nothing in S; it doesn’t make any claims about what might or might not exist anywhere else.

What about a universal negative proposition, such as: No seals are pessimists. “No seals are pessimists” simply means that there is nothing that is both a seal and a pessimist; no members of the class of seals are also members of the class of pessimists; nothing exists in the intersection of those two classes; SP is empty. And so we just shade out that section of the diagram (Figure 19-8). Again, just as with the universal affirmative proposition, this says nothing about what actually exists or does not exist anywhere else in the diagram; it says only that there is nothing in SP.

Suppose we now want to represent a particular affirmative proposition: Some seals are pessimists. That statement asserts that there really does exist something that is both a seal and a pessimist: There exists something that is a member of the class of seals and also a member of the class of pessimists; something exists in the intersection of the class of seals and the class of pessimists. How do we represent that? We can simply place an X in that SP section of the diagram, as shown in Figure 19-9. That means there is at least one seal that is also pessimistic; at least one member of the class of seals is also a member of the class of pessimists. Maybe there are lots of pessimistic seals, maybe just one, but there is at least one really existing individual occupying the intersection of the seal and pessimist classes. Are there some seals that are non-pessimists? Are there some pessimists that are non-seals? Maybe, maybe not; no claim is made about any other part of the diagram.

That leaves only negative particular statements, such as: Some seals are not pessimistic. That means that something really exists that is a seal and is non-pessimistic; there is at least one member of the intersection of the class of seals and the complement of the class of pessimists; something is S. And that is represented by placing an X in the appropriate section of the diagram (Figure 19-10). Again, this statement makes no claims

There’s another useful way of representing categorical propositions: in terms of their existence claims. For example, we might say, There are no passenger pigeons. Unfortunately, passenger pigeons are now extinct: That class has no members; the class is empty. That fact could be represented (using P to stand for passenger pigeons) thusly: P = 0. If we should be so fortunate as to discover that in some remote area there remains a small population of passenger pigeons, we would then say, There are some passenger pigeons; that would mean that the class of passenger pigeons is not empty, which would be represented as: P ≠ 0. Taking this process a step further, we can represent standard-form categorical propositions—all seals are pessimists, no seals are pessimists, some seals are pessimists, some seals are not pessimists—using this technique. The universal affirmative proposition—all seals are pessimists—implies (as shown in Figure 19-7) that the intersection of S is empty; that is, S = 0. The universal negative proposition—no seals are pessimists—implies that the intersection of S and P is empty: SP = 0. The particular propositions are just as simple. The particular affirmative proposition—some seals are pessimists—means that there does exist something that is a seal and is also a pessimist: The intersection of the class of seals and the class of pessimists is not empty. Thus the particular affirmative is represented as: SP ≠ 0. And finally, a particular negative proposition (some seals are not pessimists) means that there does exist something that is a member of the class of seals and is not a member of the class of pessimists: The intersection, or product, of the class of seals and the complement of the class of pessimists is not empty. That is represented as S ≠ 0.

about whether there are or are not any pessimistic seals or any non-seal pessimists; it says only that there exists at least one seal non-pessimist: The intersection of the class of seals and the class of non-pessimists is occupied.

So we now have ways of representing categorical propositions, but a single categorical proposition does not an argument make. We need a way to represent arguments using categorical propositions and a way of determining whether those arguments are valid or invalid. Fortunately, a simple extension of the diagrams used to represent categorical propositions will speed us on our way to diagramming and testing such arguments. It may not be handy for diagramming and testing very long and complex categorical arguments, but we can use it on the most popular sort of categorical argument: the famous categorical syllogism. Categorical syllogisms consist of two propositions as premises, a third as conclusion, and three terms or categories or classes (such as seals or pessimists), each of which occurs twice in the argument.

Consider an example of a categorical syllogism: one that we shall now be able to diagram and analyze with amazing alacrity.

• All seals are pessimists.

• All pessimists cheat at cribbage.

• Therefore, all seals cheat at cribbage.

Here we have three categorical propositions, each of which we can represent with a two-circle diagram, thusly:

All seals are pessimists (Figure 19-11).

All pessimists cheat at cribbage (i.e., All pessimists are cribbage-cheaters) (Figure 19-12).

All seals are cribbage-cheaters (Figure 19-13).

Figure 19-11

All seals are pessimists.

Figure 19-12

All pessimists are cribbage-cheaters.

Figure 19-13

All seals are cribbage-cheaters.

Figure 19-14

Seals, pessimists, and cribbage-cheaters.

Now how do we represent and analyze the argument? Simply by putting the circles together. We are discussing the classes of seals, pessimists, and cribbage-cheaters, so we require three overlapping circles rather than two. But the basic principles remain the same. Let’s start with three overlapping circles, S, P, and C, representing the classes of seals, pessimists, and cribbage-cheaters (Figure 19-14). What does each section of those circles actually represent? Well, the area outside of all the circles represents the intersection of the complements of S, P, and C. (Anything outside all three circles is non-S, non-P, and non-C: It is a non-seal, non-pessimist, non-cribbage-cheater. And I trust that most of us—with perhaps a few unsavory exceptions—fall into that category.) That is, it represents . In contrast, there is the area where all three circles overlap, and that area represents the intersection or product of the three classes. That little area where all three circles overlap is home to (and only to) all pessimistic seals who cheat at cribbage. And that intersection is labeled SPC. So let’s add SPC and to the picture, as shown in Figure 19-15. Where is the area of residence for the seals who are pessimists but who do not cheat at cribbage (SP, for short)? Obviously it must fall within the seal class, and, also, it must fall within the pessimist class. And finally, it must fall outside the class of cribbage-cheaters. There is only one spot that fits those criteria (Figure 19-16).

Now what about the cribbage-cheaters who are non-seals and non-pessimists (indeed, there may be some in our midst; they are known as C to their friends). They must fall With all of that done, mapping out the categorical syllogism is easy. Lest you have forgotten, the argument we were examining was this:

• All seals are pessimists.

• All pessimists cheat at cribbage.

• Therefore, all seals cheat at cribbage.

Consider the first premise: All seals are pessimists. How shall we diagram that in the three-circle diagram? The proposition asserts that all members of the class of seals are contained in the class of pessimists; that is, no seals fall outside the pessimist class. So we shall have to shade out all parts of the seal circle that fall outside the pessimist circle, as shown in Figure 19-19.

That was easy, right? Just be sure that you get all the parts shaded; you’ve got to cover both S and SC (and try to stay within the lines with your shading; those skills you honed in kindergarten should serve you well).

Now we add the second premise to the diagram: All pessimists cheat at cribbage. That means that all members of the pessimist class are also members of the cribbage-cheaters class: There are no pessimists outside the cribbage-cheaters circle. So next we shade out all parts of the pessimist circle that fall outside the cribbage-cheaters circle (notice that again there are two sections to shade: SP and P; that shouldn’t give you any trouble; just focus on shading out all of P that is outside C), as shown in Figure 19-20.

Now we’ve diagrammed all (both) the premises, and this may be a good moment to pause from our labors and reflect on exactly what we are doing, and why. What we want to know is whether the argument is valid; that is, we want to know if the truth of the premises would guarantee the truth of the conclusion: If the premises are true, will the conclusion have to be true? If it is in any way possible for all the premises to be true and the conclusion false, then the argument is invalid; if that is not possible, then the argument is valid. We are trying to determine whether this argument is valid; if it is valid, then, if its premises are true, its conclusion must be true. We have the premises diagrammed into the Venn diagram; now all we have to do is scrutinize that diagram to see if according to that diagram the conclusion must be true.

The conclusion is, All seals are cribbage-cheaters. That is, the conclusion asserts that all members of the class of seals are also members of the class of cribbage-cheaters. Of course if any section of our diagram is shaded out, that means that there is nothing in that section. The question, then, is this: Are all nonshaded parts of the seal circle completely contained within the cribbage-cheaters circle? If they are, then that means that all possible seals must be cribbage-cheaters; that would make the argument valid. But if there is any unshaded section of the seals circle that is not contained in the cribbage-cheaters circle, then there might possibly be a seal lurking there who does not cheat at cribbage; that possibility would be enough to make the argument invalid.

Look back at the diagram in Figure 19-20: Is there any unshaded part of circle S that does not fall within circle C? No! The only unshaded part of circle S is SPC; and SPC falls within circle C. So there is no place for a seal that does not cheat at cribbage; if there are any seals (and remember, universal propositions do not carry an existence presupposition), then they must be cribbage-cheaters. Therefore, the argument is valid: The truth of the premises makes it impossible for the conclusion to be false.

Now you try one. Diagram the premises of this argument onto a Venn diagram, and then examine your diagram to determine the validity or invalidity of the argument:

• All seals are pessimists.

• All cribbage-cheaters are pessimists.

• Therefore, all seals cheat at cribbage.

I’m going to try to work this one out also; but do it yourself before you peek further down the page.

What’s your answer? Invalid, right? First you diagrammed the first premise, “All seals are pessimists”; the resulting diagram looked like Figure 19-21.

Then you added the second premise: All cribbage-cheaters are pessimists. That premise required you to shade all parts of circle C that are not contained in circle P; so you shaded out section C. You would have shaded section SC, but it was already shaded by the first premise, so you didn’t have to shade it again. (Of course, some of you are so compulsive that you could not resist reshading section SC, so now that part of your diagram is double-shaded; that’s OK: compulsive people make great logicians.) Your final diagram looked like Figure 19-22. Then you looked at that diagram to see if there was any way that the conclusion could still be false (after the premises are diagrammed in). The conclusion is, All seals are cribbage-cheaters. Does that follow? Or is it instead possible that the conclusion is false: that there might possibly be some seals who are not cribbage-cheaters? When we examine the diagram, we see that there is such a possibility: Section SP is not shaded, so there might be some seals there; since that section of the seal circle falls outside the cribbage-cheater circle, it is possible for there to be some seals that are not cribbage-cheaters. So when the premises of the argument are true, it is still possible for the conclusion to be false; therefore, this argument is invalid.

Now we need to do some arguments with particular propositions. Consider this argument:

• Some seals are pessimists.

• All seals are cheaters-at-cribbage.

• Therefore, some pessimists are cheaters-at-cribbage.

We do this in basically the same fashion as we did the other categorical syllogisms. Let’s start with the universal affirmative: the second premise. It states that all members of the seal class fall within the cribbage-cheaters class; to represent that proposition in our Venn diagram, we shade out all parts of the seal circle that fall outside the cribbage-cheaters circle, as shown in Figure 19-23. Next we add the first premise: It is a particular proposition, and so it asserts that something actually exists. Specifically, it asserts that there exist some (at least one) pessimistic seals. That is, there is something in the intersection of the classes of seals and pessimists. To represent that premise in our Venn diagram, we place an X at the appropriate place in the diagram. But exactly where? There are two sections of the Venn diagram where seal-pessimists might dwell: SP and SPC. But we already know—from the second premise—that there is nothing in the SP section (that is why that section is shaded out); so the seal-pessimists—whose existence is affirmed by the first premise—must be lodged in the only section that remains open to seal-pessimists: SPC. And it is there that we place an X, as shown in Figure 19-24. Now we simply look at the completed

Venn diagram to see if the argument is valid: That is, when all the premises are drawn in, must the conclusion be true? The conclusion is that some pessimists are cribbage-cheaters; that is certainly guaranteed by our Venn diagram, since there is an X squarely within the intersection of P and C, signifying that there is something that is both a pessimist and a cribbage-cheater. When all the premises are true, it is impossible for the conclusion to be false. So this argument is valid.

In diagramming that argument, we started with the universal rather than the particular premise, and there’s a good reason for doing so. Look what would have happened had we started with the particular premise: Some seals are pessimists. To represent that in our Venn diagram, we must place an X within the intersection of the class of seals and the class of pessimists. But that intersection is made up of two sections: SP and SPC. In which of those sections do our pessimistic seals belong? The premise doesn’t tell us, and we must be very careful not to put any information in the diagram that is not contained in the premises. If we placed the X in SP, that would assert that there are some seals that are pessimists and are not cribbage-cheaters. But the premise doesn’t say that: It says only that some seals are pessimists; it says nothing about whether those seal-pessimists do or do not cheat at cribbage. So obviously we can’t just place our X in the SPC section, either: It is one thing to say (as the premise does) that some seals are pessimists; it is quite another thing to say—and that premise does not say—that those seal-pessimists are cribbage-cheaters. So where do we place the X? Look at Figure 19-25.

The X is on the fence, in the middle, right on the line between sections SP and SPC. And that fence-sitting X indicates that there is something in at least one of those sections (possibly both), but we do not know which. That is, there is at least one pessimistic seal; but we do not know whether it cheats at cribbage.

Suppose, then, that we had started our Venn diagram by diagramming the first premise: the particular proposition “Some seals are pessimists.” Our Venn diagram would now look like Figure 19-25, with its fence-sitting X. Next we turn to the second premise—all seals are cribbage-cheaters—and insert it into the diagram; to do so, we shade out all parts of S that are not contained in C, and the resulting diagram looks like Figure 19-26. But when we do that, our X gets jolted off the fence. Section SP is closed. Since we have an X that must be lodged in either SP or SPC, we now know that that X belongs to SPC (we don’t have to change it on the diagram; we just have to think about it and realize where it must fall). And so the final Venn diagram looks just like the one we got when we started by diagramming the universal premise; of course, that’s precisely as it should be. The moral of this story is this: If you start with the particular premise (rather than the universal premise) you can still get the right answer; it’s just a bit easier if you start with the universal premise. You may do as you wish, but I prefer the easiest way of doing Venn diagrams.

Now you do one. Represent this argument by means of a Venn diagram, and tell whether it is valid or invalid.

• Some pessimists are seals.

• All cribbage-cheaters are seals.

• Therefore, some cribbage-cheaters are pessimists.

Now that you’ve finished the problem, let’s see if we got the same answer. You started, I hope, with the universal proposition (if some of you started instead with the particular premise, then you’ll have to trudge along on your own: I’m taking the easy way). To diagram “All cribbage-cheaters are seals,” we simply shade out all the parts of the cribbage-cheaters circle that fall outside the seal circle, as shown in Figure 19-27. Next we toss the first premise—some pessimists are seals—into the diagram. To do that, we have to place an X somewhere in the pessimist–seal intersection, made up of sections SP and SPC; but exactly where, and in which section? Are the pessimistic seals cribbage-cheaters or non-cribbage-cheaters? We don’t know, and we can’t say. So we must put the X on the line dividing section SP from section SPC to indicate that there is something in at least one of those sections, but we do not know which one. (Notice, in Figure 19-28, that the X falls completely within the intersection of S and P. It is completely withins both circle S and circle P.

That is as it should be, since the premise asserts that there are pessimistic seals; what we don’t know is whether those pessimistic seals are cribbage-cheaters.) Now we examine our diagram of the premises to see if the diagram makes it impossible for the conclusion to be false (if so, the argument is valid; if it is possible for the conclusion to be false, then the argument is invalid). The conclusion is: Some cribbage-cheaters are pessimists. Does our diagram of the premises show that there must be some cribbage-cheaters that are pessimists? It shows that it is possible that some cribbage-cheaters are pessimists, but it does not show that to be necessary, for—according to our Venn diagram of the premises—there certainly are some pessimists (there is an X squarely within the P circle); but we can not tell whether any cribbage-cheaters are pessimists (because the X in the P circle is on the line of the cribbage-cheaters circle, and not inside the circle; so we don’t know which way it falls). So this argument is invalid.

Let’s do one more, together, and this one is about as hard as these get, which, as you will quickly gather, is not very hard at all.

• Some seals are not pessimists.

• Some pessimists are not cribbage-cheaters.

• Therefore, some seals are not cribbage-cheaters.

In diagramming this argument, we don’t have a universal proposition to do first; so let’s just start with the first premise: a particular negative proposition, “some seals are not pessimists,” like all the propositions in this argument. That means that there is something in the seal class that is not a member of the pessimist class; so we have to place our X completely inside the seal circle and completely outside the pessimist circle. But we don’t know whether our non-pessimistic seals are cribbage-cheaters; so the X must fall on the line of the cribbage-cheaters circle (and inside S, and outside P), as in Figure 19-29. Now we turn to the second premise: Some pessimists are not cribbage-cheaters. To diagram this premise, we must put the X completely inside the pessimist circle and completely outside the cribbage-cheaters circle; since we do not know whether our pessimistic cribbage-cheaters are seals or not, the X must straddle the line dividing seals from non-seals, as in Figure 19-30.

Now we examine the diagram, to see if those premises make it impossible for the conclusion—some seals are not cribbage-cheaters—to be false. Must it be the case that some seals are not cribbage-cheaters? Well, no. It’s possible that some seals are not cribbage-cheaters; but it is also possible that there are no seals that are not cribbage-cheaters. To see that, look closely at both Xs. One X shows that there are seals, but we don’t know if those seals are cribbage-cheaters or not (for that falls on the line of the cribbage-cheater circle). The seals represented by that X might not be cribbage-cheaters, but then again, they might be. The other X definitely represents non-cribbage-cheaters, since it falls completely outside the C circle. The problem is, we don’t know whether those non-cribbage-cheaters are seals or not, since the X straddles the line of the seals circle. Thus if all the premises are true, the conclusion may still be false; so this is an invalid argument.