Chapter Twelve: Either–Or Arguments and More

1. P or Q.
2. Not P.
3. ∴Q

1. P or Q.
2. Not Q.
3. ∴P

1. P or Q or R.
2. Not P.
3. ∴Q or R.

1. P or Q or R.
2. Not P.
3. Not Q.
3. ∴R

1. The master took the first path or the master took the second path or the master took the third path.
2. The master did not take the first path.
3. The master did not take the second path.
3. ∴The master took the third path.

1. P
3. ∴P or Q.

1. Not (P or Q).
3. ∴Not P and not Q.

1. Not (P and Q)
3. ∴Not P or not Q.

1. P or Q.
2. P
3. ∴Not Q.

1. The student can come to class more often or the student can do the exercises.
2. The student can come to class more often.
3. ∴The student cannot do the exercises.

1. P or Q and only one.
2. P
3. ∴Not Q.

1. P or Q.
2. If P then R.
3. If Q then R.
3. ∴R

1. P or Q.
2. If P then R.
3. If Q then S.
3. ∴R or S.

1. P or Q.
2. If R then not P.
3. If R then not Q.
3. ∴Not R.

1. P or Q.
2. If R then not P.
3. If S then not Q.
3. ∴Not R or not S.

1. 1. [Nixon knew about Watergate or Nixon didn’t know about Watergate.]
   2. If Nixon knew about Watergate, then Nixon did not deserve to be president.
   3. If Nixon did not know about Watergate, then Nixon did not deserve to
      be president.
   3. ∴[Nixon did not deserve to be president.]

EVALUATION

(The anti-Nixon view, for illustration’s sake.)

TRUTH

Premise 1. Certainly true, since the form P or not P is the law of the excluded middle.

Premise 2. Probably true; if he knew about it then he lied repeatedly about it, and we should hold our presidents to higher moral standards than this. An objector might protest that we have come to expect presidents to lie to us whenever it is politically expedient, even that they need to do so due to the peculiar demands of the office. But political leadership is strongest when it incorporates moral leadership, and only the strongest political leaders deserve to be president.

Premise 3. Probably true; a president should have enough influence over his inferiors either to insure that they are in line with his policies or to be quickly informed when they are seriously out of line. An objector might argue that a president surely cannot be expected to be fully informed about every unauthorized activity of his associates. This is true, but the Watergate break-in involved people at a high enough level that this does not excuse him.

LOGIC

The argument is valid, since it has the correct form for an either–or dilemma.

SOUNDNESS

Probably deductively sound.

Note that dilemmas don’t require us to learn much that is new. Evaluating the logic is simple, and evaluating the truth of the premises—given that they combine if–then and either–or premises—is already a familiar process.

12.3.2 If–Then Dilemmas

If–then dilemmas are like either–or dilemmas, but for one exception. Whether you take one of the alternatives presented in the either–or premise depends on something else; whether you are stuck with the outcome in the conclusion hangs on the same condition. For any either–or dilemma, prefix If T, then . . . to the either–or premise and to the conclusion (and leave alone the two if–then premises); that makes it an if–then dilemma. Below is the scheme for the first, most common, form of if–then dilemma:

For example: “If I decide on psychobiology as my major, then I’m definitely going to take either Psychology 101 or Biology 101 next term. If I take Psychology 101, I’ll satisfy a requirement for my major. If I take Biology 101, I’ll satisfy a requirement for my major. So, if I decide on psychobiology as my major, then I’ll satisfy a requirement for my major next term.”

The other forms of if–then dilemmas follow the same pattern.

12.4 Categorical Arguments

12.4.1 Categorical Syllogisms

Until the last century or so, the study of logic was the study of the categorical syllogism. We will not, however, need to cover this style of argument in any detail. An important goal of this book is to equip you with the tools for handling arguments that you are likely to encounter in your own experience, but you will probably never meet a categorical syllogism anywhere outside of a textbook. This section is included primarily for the historical context it provides and for the opportunity to introduce a common argument form that is related to the categorical syllogism.

The categorical syllogism, as codified by Aristotle in the fourth century bc, is made up of two premises and a conclusion, each of which takes one of these four forms:

A:   All F are G.
I:   Some F are G.
E:   No F are G.
O:   Some F are not G.

The statements identified as A and I, which are affirmative, are named for the first two vowels of the Latin affirmo; E and O, the negative statements, are named for the first two vowels of the Latin nego.

Since these statements have to do entirely with whether things that are in one category also belong to some other category (whether things that are F are also G), they are categorical statements. And an argument, or syllogism, made up entirely of categorical statements is a categorical syllogism. (Recall from Chapter 6 that the variables here are predicates, not sentences; so categorical syllogisms fall under the heading of predicate logic rather than sentential logic.)

12.4.2 Early Methods of Evaluating Logic

Logic students in the Middle Ages were asked to evaluate categorical syllogisms like this one:

They would have determined that the argument was valid based on the following verse—made up of personal names of the era—which identifies all valid categorical syllogisms:

Barbara, Celerant, Darii, Ferioque, prioris;
Cesare, Camestres, Festino, Baroco, secundai;
Tertia, Darapti, Disami, Datisi, Felapton,
Bocardo, Ferison habet. Quarta insuper addit
Bramantip, Camenes, Dimaris, Gesapo, Fresison.

The first three vowels of each name pick out which three categorical statements make up the syllogism. Our argument is AOO, so it matches the name Baroco. Does the verse deem it valid? Yes. There are four different ways in which the Fs and Gs can be arranged in any categorical syllogism. The terms prioris, secundai, tertia, and quarta in the doggerel indicate whether the Fs and Gs are in the first, second, third, or fourth arrangement. Our argument is in the second arrangement, and Bacoro is indeed grouped with the names labeled secundai. The student thus ascertains that the argument is valid by solving this puzzle.

This system, unfortunately, did little to inspire a deep understanding of logic within those who memorized it. A better approach was to have students memorize a series of rules followed by any valid categorical syllogism. Our sample argument, for example, conforms to rules such as these:

There must be exactly three terms. (Our example has prophet, mortal,
and men.)
At least one premise must be of the all or none variety. (Premise 1 is.)
It cannot have two negative premises. (Ours has only one, premise 2.)
If the conclusion is particular (I or O), it cannot have two universal premises (A or E). (Ours, appropriately, does not.)
If one premise is negative, then the conclusion must be negative.
(Ours is.)

Because it follows these rules, it can be evaluated as valid.

12.4.3 The Venn Diagram Method

The best system—because it is simple and because it promotes understanding of the underlying logic of any categorical syllogism—was introduced by mathematician John Venn in the 19th century. Alas, he made this contribution when Aristotelian logic was about to be eclipsed by the vastly superior system of logic founded by Gottlob Frege.

Venn’s system, as it is normally implemented, requires the drawing of three overlapping circles, one circle representing each of the three categories. (See the accompanying figure.) For our sample argument, we can represent the first premise by shading the entire area of the prophet circle that does not overlap with the man circle (showing that there is no such thing as a prophet that is not a man, that is, All prophets are men). We then represent the second premise by putting an x in the part of the prophet circle that does not overlap the mortal circle (showing that there do exist some prophets that are outside the category of mortality, that is, Some prophets are not mortal); we avoid the shaded area, of course, since premise 1 has precluded our putting anything there. We can then see that the argument is valid, since it can be read off the diagram; we find an x in the man circle that is outside the mortal circle, that is, Some men are not mortal.

12.4.4 Singular Categorical Arguments

There is, however, one near relative of the categorical syllogism that does occur with some frequency and thus does merit a more careful look. Recall this familiar example:

Strictly speaking, this does not count as a categorical syllogism because neither the second premise nor the conclusion are in the form of A, E, I, or O.^[6] The insertion of Socrates into the second premise and conclusion moves the argument from the universal to the particular. Following the pattern of the preceding chapter, we might term this form a singular categorical argument. Not all singular categorical arguments are valid. The preceding argument is valid and can be defended on the grounds that it has correct form for a singular categorical argument. The following argument, however, would not be valid:

The evaluation of the logic of this argument should say that it has incorrect form for a singular categorical syllogism and should provide a validity counterexample.

For the sake of simplicity, however, I suggest that such arguments ordinarily be paraphrased so that they become examples of singular if–then arguments. As noted in Chapter 11, All F are G can be a stylistic variant for If anything (or anyone or anytime) is an F, then it is a G. The example above, then, is the exact logical equivalent of this:

And it is valid by singular affirming the antecedent.

If the first premise had been No men are mortal, it would have validly followed that Socrates was immortal; the clarification would have looked like this:

There are a few easy-to-spot variants of categorical statements that also can be translated into if–then statements. Only G is F, for example, is logically equivalent to All F are G and can typically be paraphrased as If anything is F, then it is G. Here is an example from ordinary English:

Now they know that she was a real princess, since she had felt the pea that was lying on the bedstead through twenty mattresses and twenty eiderdown quilts. Only a real princess could be so sensitive. —Hans Christian Andersen, “The Princess and the Pea”

The clarification would look like this:

It is valid by singular affirming the antecedent.

There are some rare cases in which it is better to leave a categorical statement as it is rather than paraphrase it as an if–then statement. Standard format requires, in certain cases, that you represent a conclusion in the form All F are G. In Chapter 14, for example, we will look at arguments like this: “All computer programs I ever used have had bugs in them. I conclude, then, that all computer programs have bugs.” As we will see, clarifying the argument in standard format gives us this:

What makes these arguments distinctive is that they move from a sample to the general population; this move is made clear when the conclusion looks exactly like the premise but for the omission of the word sampled. This clarity of structure would be lost if we paraphrased the conclusion’s categorical statement as an if–then statement, as you can see:

Keep the categorical statement in such cases. This means that you will still have the occasional singular categorical argument to evaluate. Suppose, for example, I further argued, “So, the word processing program I’m using right now has a bug in it.” The clarified complex argument would look like this:

The inference from 1 to 2 will be dealt with in Chapter 14; the inference from 2 and 3 to C is valid on the grounds that it has correct form for a singular categorical argument.

12.5 Summary of Chapter Twelve

Two valid forms of either–or arguments are the process of elimination and disjunction. An invalid form is the fallacy of affirming an alternative; if, however, an argument with this form has an exclusive either–or premise (which can be paraphrased as P or Q and only one) it is a valid example of affirming an exclusive alternative.

Several other valid forms use either–or premises in combination with both–and premises (DeMorgan’s laws) or if–then premises (dilemmas).

All either–or statements express the core idea that at least one of the alternatives is true. Thus, if all alternatives are false then the either–or statement is false; otherwise, it is true. When thinking in terms of probabilities, a helpful guideline is to add the probabilities of both alternatives (but then to subtract the probability that both are true) to arrive at the probability for the entire either–or statement.

Some arguments require an either–or statement that, in addition to the core idea, expresses the idea that only one of the alternatives is true—that is, that the alternatives are exclusive. In such cases, paraphrase as P or Q and only one.

Historically, logic has mainly been concerned with categorical syllogisms. These arguments occur rarely in ordinary language, and logicians in the last century have abandoned the study of the categorical syllogism for much more sophisticated approaches that are taught in formal logic courses. But the singular categorical argument is related to the categorical syllogism and does occur in ordinary language from time to time. It is usually best, however, to treat categorical statements like All F are G and No F is G as stylistic variants of if–then statements and to paraphrase—and evaluate—them accordingly.

12.6 Guidelines for Chapter Twelve

• Translate stylistic variants for the either–or premise into the standard constant. Also, be alert for implicit statements, including the either–or premise.
• When the context and the logic of the argument call for it, paraphrase exclusive alternatives by including and only one as part of the standard constant.
• When the alternatives of an either–or statement are merely probable, tentatively assign them a probability (even if the result is misleadingly precise) so that you can apply the rules of probability. Convert the numbers back into everyday language for your final evaluation.
• Convert categorical statements, in most cases, into if–then statements. Retain the categorical statement, however, when standard format for the argument calls for it, as with the conclusion in some inductive generalizations.

12.7 Glossary for Chapter Twelve

Affirming an exclusive alternative—valid argument form, as follows:

Historically known as modus ponendo tollens, Latin for “the method of
affirming in order to deny.”

Alternative—a statement connected to another by or. Also known as disjunct.

Categorical statement—statement of the form All F are G, No F are G, Some F are G, or Some F are not G.

Categorical syllogism—one of a family of deductive arguments, some valid and others invalid, each with three categorical statements—two as premises, one as conclusion.

DeMorgan’s laws—valid deductive forms, as follows:

Dilemma—a valid argument form that points out the consequences, whether good or bad, of two inevitable alternatives. The word comes from the Greek words di, for two, and lemma, for proposition. Dilemmas come in two varieties: either–or dilemmas and if–then dilemmas.

Disjunction—valid deductive form, as follows:

The term is sometimes also used for an either–or statement.

Either–or argument—one of a loosely defined group of arguments that has an either–or premise. Also called disjunctive syllogism.

Either–or dilemma—a valid dilemma that begins with an either–or premise. The most common forms are these:

Either–or statement—statement of the form P or Q. Also sometimes called a disjunction.

Exclusivity premise—either–or premise that includes the notion that only one of the alternatives is true; it has the form P or Q and only one.

Fallacy of affirming an alternative—an invalid argument form, as follows:

If–then dilemmas—valid dilemmas that are constructed like any either–or dilemmas, except for an if–clause prefixed both to the either–or premise and to the conclusion. The most common form is as follows:

Process of elimination—a valid form of either–or argument in which premises eliminate alternatives, while the conclusion includes all the alternatives that have not been eliminated by a premise. Examples include:

Also called modus tollendo ponens, Latin for “the method of denying in order to affirm.”

Singular categorical argument—a deductive argument, either valid or invalid, with a universal categorical statement as a premise, but with a conclusion about a single instance that is included in the universal category. For example, a valid form is: