You may think of all the preceding chapters as preliminaries to the detailed and precise task of evaluating arguments. Now it is time to put arguments to the test. The book's remaining chapters will take up deductive arguments, inductive arguments, and then the difficult kinds of arguments concerning moral, legal, and aesthetic matters.
We begin with deductive arguments, which we have characterized as being either valid or invalid. Chapters 8-9 present two methods for testing for validity. In the present chapter, we use the logic of categorical statements, which dates back to Aristotle, and is a powerful tool for handling one large group of arguments. Those arguments are distinguished by the form of claims that make up their premises and conclusions—roughly, claims beginning with the words "all" and "some."
Chapter 8 first shows how to work with such categorical statements by themselves: translating them into equivalent forms, separating them from superficially identical claims, and so on. Then we look at the standard arrangement of such statements into arguments, called categorical syllogisms. Two methods test syllogisms for validity; either one will let the reader evaluate every standard categorical syllogism.
Corresponding A- and E-claims are contrary claims: They are never both true.
If "All rooms are vacant" is true, then "No rooms are vacant" is false.
Both claims might be false. Neither "All cars are Toyotas" nor "No cars are Toyotas" is true.
So if an A- or E-claim is true, you know that its contrary is false. If it's false, you can't draw any conclusions.
Corresponding I- and O-claims are subcontrary claims: They are never both false.
If "Some rooms are vacant" is false, then "Some rooms are not vacant" must be true.
Both might be true: "Some cars are Toyotas" and "Some cars are not Toyotas."
So if an I- or O-claim is false, you know its subcontrary is true. If it's true, you can't draw any conclusions about its subcontrary.
Exceptions to both these rules occur when the subject class is empty. We will assume, however, that it is not.
Corresponding A- and O-claims are contradictory claims, as are corresponding E- and I-claims: They have opposite truth values.
If "All boxers are left-handed people" is false, "Some boxers are not left-handed people" must be true.
If "Some left-handed people are boxers" is true, "No left-handed people are boxers" must be false.
When you have a true A- or E-claim (a claim at the top of the square), or a false I- or O-claim (at the bottom), you can infer the truth values of all corresponding claims.
Say that "All windows are glass objects" is true. Then:
"No windows are glass objects" (contrary) is false;
"Some windows are not glass objects" (contradictory) is false;
"Some windows are glass objects" (contradictory of the contrary) is true.
Say that "Some cars are boats" is false. Then:
"Some cars are not boats" (subcontrary) is true;
"No cars are boats" (contradictory) is true;
"All cars are boats" (contradictory of the subcontrary) is false.
However, when you have a false A- or E-claim, or a true I- or O-claim, you can only infer the truth value of its contradictory.
From the false claim "All politicians are men," all that follows is the truth of its contradictory, "Some politicians are not men."
From the true claim "Some politicians are men," all that follows is the falsity of "No politicians are men."
5. Together with the square of opposition, three operations help us draw simple inferences from categorical claims.
To produce the converse of a categorical claim (a process called conversion), simply switch the subject and predicate terms.
E- and I-claims are equivalent to their converses, but A-and O-claims are not.
When you say, "No cats are dogs," you equally say, "No dogs are cats."
When you say, "Some doctors are men," you equally say, "Some men are doctors."
To produce the obverse of a standard-form claim (a process called obversion), change it from affirmative to negative, or vice versa, and replace the predicate term with its complementary term.
A term's complementary term names every member of the universe of discourse that is not in the original class.
Usually, the complementary term can be formed with the prefix "non" in front of the original term: "democracy" and "nondemocracy."
Sometimes you need to take care to restrict the universe of discourse. The term complementary to "drivers" is "people who are not drivers."
Non" is safer than common opposites. The complement to "people who are happy" is not "people who are sad" but "people who are not happy."
Changing a claim from affirmative to negative, or vice versa, is the same as going across the square of opposition.
All claims are equivalent to their obverses.
"All blessings are mixed things" becomes, through obversion, "No blessings are unmixed things."
Similarly, saying, "Some athletes are pros" amounts to saying, "Some athletes are not nonpros."
To produce the contrapositive of a claim (a process known as contraposition), switch the subject and predicate terms and replace both by their complementary terms.
A- and O-claims are equivalent to their contrapositives, but E-and I-claims are not.
That is, claims that are not equivalent to their converses are equivalent to their contrapositives.
"All poodles are dogs": "All nondogs are nonpoodles."
"Some employees are not guards"? Then "Some nonguards are not nonemployees."
6. We begin the study of categorical syllogisms with vocabulary.
In a categorical syllogism all claims are categorical claims and three terms appear, each one twice.
A syllogism is a deductive argument with two premises.
Each term in a categorical syllogism occurs in two of the claims (whether premises or conclusion), once in each.
"All poodles are dogs. All dogs are mammals. Therefore, all poodles are mammals." Note the distribution of terms.
In the interest of clear and uniform labeling, we refer to these three terms by standard names:
The major term is the conclusion's predicate term; we abbreviate it P.
The minor term is the conclusion's subject term; we abbreviate it S.
The middle term occurs in both premises, but not in the conclusion; we abbreviate it M.
Although it is easy to confuse "major" and "minor" here, what matters most is to see the function of the middle term in a syllogism. The middle term links the major term to the minor.
We now possess two ways of telling categorical syllogisms from impostors.
Make certain that all three claims have been presented in standard form.
Make certain that exactly three terms appear in the syllogism, each one twice. Watch out for complementary terms sneaking in.
7. Venn diagrams provide one method for testing categorical syllogisms for validity.
Keep certain basic ideas in mind when using this method.
Recall the meaning of validity. If the premises are all true, the conclusion is true, too.
The method extends the Venn diagrams for single claims to illustrate how premises and conclusion work together.
The point of this method is simple: Once you diagram the premises, the diagram should reveal the conclusion. (This is, after all, what validity amounts to, that stating the premises of an argument means stating its conclusion.)
In certain cases of arguments with I- and O-claims for conclusions, the conclusion will not simply appear in the finished diagram. See below for this exception.
With only a few additional guidelines, the work of diagramming a syllogism will look exactly like the work of diagramming two separate claims; that is, shading and Xs will work much as they do for separate claims.
Every diagram begins with three overlapping circles:
Add four rules to the diagramming rules you have already learned for individual categorical claims.
To keep all diagrams uniform, put the minor term on the left, the major term on the right, and the middle term a little below them, in the middle.
When one premise is an A- or E-claim and the other is an I- or O-claim, diagram the A- or E-claim first.
In other words, shade before putting in Xs. Do not put an X in a shaded area.
This usually helps solve the problem of where to put an X, when more than one area is available.
If it is still not clear where to put the X—if it could go in either of two spaces—put the X on the line between them.
After you have finished diagramming the premises, see if any circle has only one unshaded area; if so, put an X in that area.
Remember our assumption that no class is empty.
This final step permits you to go from two premises that are A- or E-claims to a conclusion that is an I- or O-claim; that step would otherwise be impossible.
Once the premises have been diagrammed, see if the diagram for the conclusion appears in the circles.
If the argument is valid, the finished Venn diagram will "state" the conclusion.
When the conclusion is an I- or O-claim, you need the relevant X to appear completely within its space. An X only partly in the right area does not give you your conclusion.
8. Many ordinary-language syllogisms can be brought within this formal structure and evaluated with Venn diagrams (or the rules method).
In some cases, you need to supply unstated premises.
Many categorical syllogisms we encounter in ordinary reading and conversation are missing premises, usually because a premise is considered too obvious to assert.
Find a reasonable claim you can add to the syllogism's argument to make it valid. (See Chapter 7.)
The subject and predicate terms of the missing premise will be the two terms that each occur once in the stated premise and conclusion.
That is, if you are given a premise of the form "No A are B" and a conclusion of the form "Some C are not A," you know that the missing premise will contain the terms B and C.
It takes a little more work to see how to arrange those terms to form a missing premise (in this instance, more than one form works) to produce a valid argument. Syllogisms in ordinary language usually suggest their missing premises.
In all cases, begin your evaluation of the completed syllogism by abbreviating its terms.
Write down a clear abbreviation key, for example, "B = baseball players."
Rewrite the argument using your abbreviations. Both methods of evaluation go faster when the form is more perspicuous.
9. A simple set of three rules provides the second test of a categorical syllogism's validity.
In most respects the rules method is better than Venn diagrams.
It is easier to understand the principles behind Venn diagrams, where the three rules do not wear their rationale on their sleeve.
However, applying Venn diagrams can be cumbersome and slow, and often provides opportunities for clerical errors.
To use the rules, you first need to feel comfortable with the distinction between affirmative and negative claims.
A- and I-claims are affirmative; E- and O-claims are negative.
Negative claims either begin with "no" or contain "not."
Second, you need to become familiar with distributed terms.
A term is a distributed term if the claim it appears in says something about all members of the class in question.
"All dogs are mammals" distributes the term "dog," because it speaks of all dogs.
The same claim does not distribute "mammal," because it tells you nothing about all mammals.
Memorize the distributed terms of each form of claim (distributive terms are in boldface):
A: All S are P.
E: No S are P.
I: Some S are P.
O: Some S are not P.
With this terminology in place, we can state the rules. A syllogism is valid if and only if all the following are true:
The number of negative claims in the premises is the same as the number of negative claims in the conclusion.
At least one premise distributes the middle term.
All terms distributed in the conclusion are distributed in the premises.
When applying these rules, first go through the syllogism circling all distributed terms. Then you will find rules 2 and easy, even pleasant, to apply.