Part Two: Clarifying Arguments

Chapter Six: Structuring

Order and simplification are the first steps toward the mastery of a subject.

—Thomas Mann, The Magic Mountain

TOPICS

  • Variables and Constants
  • Translating Stylistic Variants
  • Matching Wording

An argument is not typically offered as a series of independent sentences, but as a structured unit—not as scattered planks, but as a building in which the planks are nailed together according to a design. As you clarify any argument, you should strive from the start to see its underlying design—that is, its logical form.

Identifying the logical form of an argument helps in two important ways. First, it helps in the clarifying process; once you identify the argument’s form, you are prepared for the process of —that is, organizing your paraphrase of the argument so as to make its logical form as obvious as possible. Second, it ultimately helps in the evaluating process; for logical form is one of the keys to determining whether an argument is logically successful.

Consider this experiment, described in Scientific American, which has to do with the system of flight-enhancing air sacs distributed throughout the bodies of most birds:

A crucial experiment was performed by the French scientist J. M. Soum, who admitted carbon monoxide into the air sacs of birds. If the air sacs had played any major role in their breathing, the birds would have been rapidly poisoned by the carbon monoxide. But they remained completely unaffected. We can therefore conclude that the air sacs of birds play no direct role in their breathing.

This argument depends on the following simple form:

  1. If P then Q.
  2. Not Q.
  3. Not P.

The structured paraphrase (which has also been subjected to the guidelines for streamlining and specifying, where appropriate) looks like this:

  1. If air sacs of birds play a role in their breathing, then birds are poisoned by carbon monoxide introduced into their air sacs.
  2. Birds are not poisoned by carbon monoxide introduced into their air sacs.
  3. Air sacs of birds do not play a role in their breathing.

Seeing the underlying form enables you to organize the paraphrase much more clearly. And, as a bonus, as soon as you recognize the form of this particular argument you can anticipate your evaluation of the argument’s logic—for it turns out that this form is one that is known always to be logically successful.

The vast majority of simple arguments rely on one of only a dozen or so common forms. For all of them, logical success is at least partly a matter of correct form. For many of them (those identified later in the text as deductive arguments) logical success is entirely a matter of correct form. This chapter uses just a few of these argument forms as examples to illustrate the main guidelines for structuring. For that reason, the chapter is short. The point here is to be clear about the notion of logical form because it plays such an important role in clarifying and evaluating.

6.1 Variables and Constants

The form of the air sac argument, as noted, is this:

  1. If P then Q.
  2. Not Q.
  3. Not P.

Not only is this schema far shorter than the argument itself, but also it is uninformative—that is, it is empty; it includes none of the argument’s interesting information about birds, air sacs, and carbon monoxide. That is because the content of the argument has been eliminated.

An argument’s is the part of the argument that can vary without varying the argument’s logical form. Thus, in a description of an argument’s form, the placeholders such as P and Q that indicate where the content can vary are termed . You should be able to intuitively see that the following argument has the same logical form as the air sac argument, but it is about something entirely different—that is, it has different content:

If you are to make an A in this course, you must show that you have a basic understanding of the material. But you have shown no such thing. So you will certainly not make an A in this course.

Something about grades and understanding, rather than about air sacs and carbon monoxide, is filling in for the variables P and Q.

The physical world includes many types of structures. A swimming pool, its form defined by its concrete shell, is one sort of structure. A tent, framed by its poles and pegs, is another. A wide variety of liquids—water, wine, milk, gasoline—can fill out the form of a swimming pool; that is, they can all serve as content for such a structure. (Swimming in some of them might prove to be a memorable experience, but that is a different point.) And a wide variety of fabrics—canvas, nylon, cotton, linen, silk—can serve as content to be structured into the form of a tent. Liquids, however, don’t make for good tents and fabrics do not pour very well into swimming pools.

Similarly, in this text we will encounter two sorts of logical forms, each of which requires a certain sort of linguistic material—and only that sort of material—for its content: those forms that require statements, and those that require names and predicates.

6.1.1 Statements as Content

In the air sac and course grade arguments, only statements can replace the variables P and Q—one for each occurrence of P, another for each occurrence of Q. For convenience, in describing an argument’s form we will indicate statements by capital letters from P to Z. In the air sac argument, the variables are filled in by the following statements:

P: Air sacs of birds play a role in their breathing.
Q: Birds are poisoned by carbon monoxide introduced into their air sacs.

But in the course grade argument, these statements would replace the variables:

P: You make an A in the course.
Q: You show that you have a basic understanding of the material.

The key point is that in each of these cases, the form of the argument depends solely on relationships among statements.

These sorts of forms are usually described as belonging to , since they have to do with logical relationships among entire sentences (and a statement, as we have defined it, is a kind of sentence). This is also sometimes called propositional logic (since statements are sometimes also called propositions). Here is an argument with a different logical form that, nevertheless, relies on sentential logic:

I can either support the union or support management. But I can’t bring myself to trust any manager. So that leaves me on the side of the union.

Its form is identical to the following argument.

Either the CIA or Oswald killed Kennedy. The CIA didn’t do it. Thus, Oswald killed Kennedy.

Again, the content of these two arguments is completely different. But the statements in each of them are arranged in exactly the same way. They share the following logical form:

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

For the animals argument, the content can be represented in this way:

P: I can support the union.
Q: I can support management.

For the CIA argument, these are the relevant statements:

P: The CIA killed Kennedy.
Q: Oswald killed Kennedy.

Exercises Chapter 6, set (a)

Given the proposed form, identify which statement is represented by each variable.

Sample exercise. It did not rain on our parade. (Not P.)

Sample answer. P: It rained on our parade.

  1. She could not remember the phone number. (Not P.)
  2. If you had told me about the exam, then I would have passed the class. (If P then Q.)
  3. She will either buy the paperback or download the digital version. (P or Q.)
  4. If you pay me enough, then I will take the job. (If P then Q.)

6.1.2 Names and Predicates as Content

In many cases, it is not the relationships between statements but the relationships within statements that matter most from a logical point of view. Take the following familiar argument:

All men are mortal. Socrates is a man. So, Socrates is mortal.

You can intuitively see that it has the same logical form as this one:

All highways are paved. Route 66 is a highway. So, Route 66 is paved.

If we attempted to deal with these two as though they were arguments of sentential logic, then we could only represent their form as three separate statements, as follows:

  1. P.
  2. Q.
  3. R.

This does nothing to display the form that the arguments are depending on for logical success. That form, rather, is this:

  1. All F are G.
  2. A is F.
  3. A is G.

Here, the variable A can be filled in by a —that is, an expression that identifies something that has properties attributed to it. (These do not have to be proper names; for example, the expression my teacher in My teacher is no slave to fashion can serve as a name for our purposes.) We’ll use the letters A through E as variables for names when describing an argument’s form. In the mortality argument, the name fills in for the variable thus:

A: Socrates.

And in the highway argument, it is as follows:

A: Route 66.

The variables F and G, on the other hand, can each be filled in by a —that is, an expression that identifies a property or attribute that can be ascribed to the thing named. We’ll use the letters F through O as variables for predicates when describing an argument’s form. In the mortality argument, the predicates are these:

F: a man.
G: mortal.

And in the highway argument, these are the predicates:

F: a highway.
G: paved.

The branch of logic that deals with relationships among names and predicates is called or, alternatively quantifier logic (since these arguments include quantity terms like all).

Exercises Chapter 6, set (b)

Given the proposed form, identify which statement is represented by each variable.

Sample exercise. Montana is a large state. (A is F.)

Sample answer. A: Montana. F: a large state.

    1. All large states have mountains in them. (All F are G.)
    2. My dog is a beagle. (A is F.)
    3. The band will sign with Geffen Records. (A is F.)
    4. Dolphins are mammals. (All F are G.)

6.1.3 Constants

Regardless of how you vary the fabric of a tent, the pegs and poles determine its form; and no matter which liquid you pour into a swimming pool, the concrete shell fixes its form. The pegs and poles, or the concrete shells, of arguments are called (or connectives). These are the expressions that cannot vary without varying the form of the argument.

Consider the arguments of sentential logic, in which statements provide the content. The expressions that remain the same—the constants—are terms like if–then, or, and, and not. As for the arguments of predicate logic, where names and predicates provide the content, the constants are terms like is (which means, roughly, has the property of . . . ) and all. Other constants that might occur in these sorts of arguments are is not and other quantities besides all, such as none, some, and most.

Be very clear: you will never be asked to display the mere form, simply with constants and variables, in your clarified argument. Thinking about the mere form helps you to organize your paraphrase and helps you in evaluating the logic of the argument. But it would normally not help, for example, in evaluating the truth of premises. Is Not Q true, or is it false? How can you tell? As it stands, it is empty. You can’t think about whether it is true or false until you know what statement is being substituted for Q!

Guideline.  Identify the logical form of the argument; this will help you in structuring the argument—that is, in organizing your paraphrase of the argument—and eventually in evaluating its logical success.

Exercises Chapter 6, set (c)

Given the proposed content, write out the form of the sentence, using only constants and variables.

Sample exercise. Most people who go fishing don’t catch a fish.

(F: people who go fishing. G: catch a fish.)

Sample answer. Most F are not G.

  1. Jeff did not catch a fish. (P: Jeff caught a fish.)
  2. Jeff did not catch a fish. (A: Jeff; F: caught a fish.)
  3. Jeff either caught a trout or a bass. (P: Jeff caught a trout; Q: Jeff caught a bass.)
  4. If Jeff caught a fish, then it was the first time. (P: Jeff caught a fish; Q: it was the first time.)

Sentential Logic versus Predicate Logic

Sentential Logic Predicate Logic
Logical Constants If-then, and, or, not Is, is not, all, some, none, most
Variables P through Z A through E; F through O
Content of Variables Statements Names; predicates

6.2 Translating Stylistic Variants

The point of structuring is to paraphrase the argument in ordinary language so that its logical form is close enough to the surface to be clearly visible. This usually requires both doing something with the constants—namely, translating the stylistic variants—and also doing something with the content—namely, matching the wording.

There are many ways that the same logical constants might be phrased in ordinary English. Consider, for example, the sentence If you are to make an A in this course, then you must show that you have a basic understanding of the material. We would ordinarily say that if–then is the constant, and that its form is If P then Q. But there are many ways of saying the same thing without making any change in the content of P and Q. Here are only a few examples:

If you are to make an A in this course, you must show that you have a basic understanding of the material.

You must show that you have a basic understanding of the material if you are to make an A in this course.

Supposing you are to make an A in this course, then you must show that you have a basic understanding of the material.

You are to make an A in this course only if you show that you have a basic understanding of the material.

If P then Q is the —the expression that is conventionally thought to be most effective in bringing to the surface an argument’s logical form, and thus the one that we will adopt for use in the structuring of the argument. The others are on the standard constant—that is, they are alternative styles of saying the same thing. Structuring requires that you translate stylistic variants into the standard constant. Each of the preceding four variants must be paraphrased as If you are to make an A in this course, then you must show that you have a basic understanding of the material.

Here are some additional examples of stylistic variants and the standard constant into which they should be translated.

Stylistic Variants and their Standard Constant

Standard Constant Stylistic Variants
If P then Q. If P, Q.
Q if P.
Supposing that P, then Q.
P only if Q.
So long as P, then Q.
P or Q. P unless Q.
Q or P.
Not P. P is false.
P is not true.
All F are G. 100% of F are G.
Any F is G.
Everything that is F is G.
A is F. One thing that has F is A.
A has the property of being F.

For every form of argument that is covered in future chapters, there will be a more extensive discussion of the stylistic variants for its standard constants.

Guideline.  Translate stylistic variants into the standard constants for the relevant logical form.

Exercises Chapter 6, set (d)

For each sentence, translate the stylistic variants into the standard constant for the given logical form. If you get stuck, check the brief list of sample stylistic variants in the preceding section.

Sample exercise. Russia will not perish so long as we shall drink.

—slogan of pro-sobriety campaign in Poland (If P, then Q.)

Sample answer. If we shall drink, then Russia will not perish.

  1. What goes up must come down. (All F are G.)
  2. More voters would come out to the polls if the candidates ran more positive campaigns. (If P, then Q.)
  3. One city with a rich history is Boston. (A is F.)
  4. Unless I’m mistaken, we’ve met before. (P or Q.)
  5. There is no way that you will catch your plane. (Not P.)

6.3 Matching Wording

Structuring the argument requires you to standardize not only the argument’s logical constants, but also its content. As with logical constants, the same statement, name, or predicate can be expressed in different ways. Socrates, for example, may be referred to as the teacher of Plato; instead of men (for mankind) you might say humans; and for mortal you might say eventually die. The mortality argument, then, might well be expressed this way in ordinary language:

All humans are mortal. The teacher of Plato is a man. So, Socrates must eventually die.

When it seems clear that the arguer really means the same thing by the alternative expressions, you should revise them so that they match. As a result, it becomes much clearer that they are the same, and the
logical form of the argument comes closer to the surface. When
you match wording in this argument, then, your paraphrase might look like this:

  1. All men are mortal.
  2. Socrates is a man.
  3. Socrates is mortal.

Of course, the structured paraphrase might just as well look like this:

  1. All humans must eventually die.
  2. The teacher of Plato is human.
  3. The teacher of Plato must eventually die.

When faced with such a choice, follow the principles of clarifying as already laid out—pick the wording that is clearest and most loyal to the arguer’s intentions or make up your own clearer and more loyal paraphrase. If two choices seem to be on equal footing (such as mortal and must eventually die), then just pick one, be sure to match the wording throughout the argument, and move ahead.

You might suspect that there is really an implicit inference involved in cases where we match wording. For example, you might think that it’s a mistake just to substitute Socrates is a man for The teacher of Plato is a man. Isn’t there actually an implicit premise—namely, The teacher of Plato is Socrates—that allows us to connect one to the other? Perhaps there is. But this is a situation where practical common sense should take precedence over excessive logical scrupulosity. Treating every case of matching wording as an implicit subargument would convert many simple arguments into complex arguments with many subconclusions—each subargument with premises and logic of its own that must be evaluated. Usually, the alternative expressions are reasonable; we’re not normally giving anything important away, for example, by simply granting that Socrates and the teacher of Plato can be used interchangeably. If the alternative expressions are reasonable—if allowing them is not likely to weaken the argument in any way—then don’t clutter the argument with excessive subconclusions; just match the wording and move ahead.

Guideline.  When the same statement, name, or predicate is expressed in more than one way, match the wording in your paraphrase.

Structuring

  1. For logical constants—translate stylistic variants into standard constants.
  2. For content—match wording when the same statements, names, or predicates are expressed in different ways.

Exercises Chapter 6, set (e)

Structure each argument according to the logical form described in parentheses, paying special attention to translating stylistic variants and matching wording.

Sample exercise. General Halftrack, looking at the overflowing “Complaints” box: “Look at those complaints! It’s disgraceful! Either the box is too small or we’re not running this camp right.”

The General, in reply: “Have a bigger box built.” —Beetle Bailey (P or Q. [Not Q.]P.)

Sample answer.

  1. The complaints box is too small or the general is
    not running the camp right.
  2. It is not the case that the general is not running
    the camp right.
  3. The complaints box is too small.
  1. Every great university has an excellent library. So Oxford, being one of the best, can be expected to have an outstanding library. (All F are G. A is F.A is G.)
  2. Epidemiological evidence has consistently indicated that severe alcoholism runs in families, and these findings, the researchers say, lend further support to the idea of a hereditary component to the disorder. If alcoholism were exclusively a social or psychological phenomenon, they point out, all of the alcoholic subjects would be expected to have acquired the same biological abnormality; 20% did not. Further studies of alcoholic and nonalcoholic twins will be necessary to verify the genetic contribution, Rutstein says. —Science News (If P then Q. Not Q. ∴ Not P.)
  3. Certainly most who would go into the death hogan would not be Navajo. White men, yes. So the horse thief was a white.—The Ghostway, by Tony Hillerman (Most F are G. [A is F.] ∴ A is G.)
  4. “Mankind, he said, . . . have never, as I think, at all understood the power of Love. For if they had understood him they would surely have built noble temples and altars, and offered solemn sacrifices in his honor; but this is not done. . . .” —Plato, Symposium (If P, then Q. Not Q. ∴ Not P.)
  5. It is a common notion that morality simply means conformity to the customs of one’s group. But this cannot be the case. If it were, we could never criticize and improve the morals of our group, at least we would have no moral basis for doing so. However superstitious, or stupid, or cruel the customs of our community are, they would be, by definition, morally right—for us. The unthinking conformist would be the moral man, the moral reformer the immoral man. There would be no moral progress. But no one really believes this. We all constantly criticize the morals of our group. —Durant Drake, Invitation to Philosophy (If P then Q. Not Q. ∴ Not P.)
  6. “The sentence ‘I know that I am in pain’ makes sense only if ‘I doubt that I am in pain’ does also. The latter sentence does not make sense. Therefore, the former doesn’t either.” —Ludwig Wittgenstein, Philosophical Investigations (If P then Q. Not Q. ∴ Not P.)

6.4 Summary of Chapter Six

Structuring is an important part of clarifying an argument, since it organizes the paraphrase in such a way that the argument’s logical form is close enough to the surface to be clearly visible. Logical form is always relevant to the logical success of an argument; thus, structuring contributes to the overall goal of all argument clarification—it makes it easier to evaluate the argument.

An argument’s form is determined by certain logical constants (like all and or) and by certain content (like statements, names, and predicates) that is linked together by the constants. Structuring requires that when you determine an argument’s logical form, you do something to both the constants and the content. With respect to the constants, translate stylistic variants (like every) into the standard constant (like, all). With respect to the content, match the wording of statements, names, or predicates when they occur more than once and the meaning is clearly the same (such as paraphrasing must eventually die to is mortal when both expressions are used).

6.5 Guidelines for Chapter Six

  • Identify the logical form of the argument; this will help you in structuring the argument—that is, in organizing your paraphrase of the argument—and eventually in evaluating its logical success.
  • Translate stylistic variants into the standard constants for the relevant logical form.
  • When the same statement, name, or predicate is expressed in more than one way, match the wording in your paraphrase.

6.6 Glossary for Chapter Six

Content—the part of the argument that can vary without varying the argument’s logical form. It can be made up of statements, names, or predicates. In a description of an argument’s form, the placeholders—such as P and Q—that indicate where the content can vary are termed variables.

Logical constants—the expressions that provide an argument with its logical form. These cannot vary without varying the form of the argument. Also called connectives.

Name—an expression that identifies something to which, typically, properties are attributed. Names do not have to be proper names; for example, the expression my teacher in My teacher is no slave to fashion can serve as a name for our purposes. We are using the letters A through E as variables for names when describing logical form.

Predicate—an expression that identifies a property or attribute that can be ascribed to the thing named. We are using the letters F through O as variables for predicates when describing an argument’s form.

Predicate logic—the branch of logic that deals with logical relationships among names and predicates. This is also sometimes called quantifier logic, because their arguments include quantity terms like all.

Sentential logic—the branch of logic that deals with logical relationships among entire statements (which are a kind of sentence). This is also sometimes called propositional logic because statements are sometimes called propositions. We are using the letters P through Z as variables for statements when describing the form of an argument of sentential logic.

Standard constants—when there are various ways of expressing the same constant, the one expression that we are adopting for use in the structuring of the argument. The alternative expressions are termed stylistic variants and are to be translated into the standard constants. For example, if–then is the standard constant, assuming–then is a stylistic variant of it.

Structuring—organizing your paraphrase of the argument so as to make its logical form as obvious as possible. This typically requires two procedures: translating stylistic variants into their standard constants, and matching the wording of statements, names, or predicates when they are expressed more than once in different language.

Stylistic variants—when there are various ways of expressing the same constant, the expressions that are to be translated into the standard constants. For example, if–then is the standard constant, assuming–then is a stylistic variant of it.

Variables—placeholders such as P and Q that indicate where the content can vary. These appear only in a description of an argument’s form, not in its clarification.

Share This Book