The Hypothetical Syllogism

By an examination of examples we shall draw up the basic laws governing the relationship of an antecedent and its consequents in both valid and invalid inference. These laws are basic principles of all inference—of immediate and mediate inference, and of formal and material inference as well. We are taking them now because they are the immediate foundation of the rules governing the conditional syllogism.

Note that an antecedent is false when only one premise is false, as well as when both premises are false

Note, too, that where the sequence is invalid, there is, strictly speaking, no sequence, antecedent, or consequent at all; there is, however, an apparent, or pseudo, sequence, antecedent, and consequent. We give them these names just as we give the name “money” to counterfeit money.

Never forget that when the sequence is invalid, the apparent premises and conclusions are not related to one another at all. Consequently, in an argument whose sequence is invalid, anything can come after anything—truth after truth, truth after falsity, falsity after truth, or falsity after falsity. If the conclusion is true (or false), it is true (or false) independently of the premises.

First, we shall consider the relationship of an antecedent to its consequent and then of a consequent to its antecedent.

a. Antecedent to Consequent

1. If the antecedent is true and the sequence valid, the consequent is true.

This law is a particularized statement of the principle of contradiction and is the basic principle of all inference without exception. A denial of this law is an implicit denial of the principle of contradiction. If a true antecedent could have a false consequent, the antecedent would have to be both true and false at the same time— for if it were not false in any respect at all, it would be impossible to derive falsity from it.

This law has been illustrated by every example we have had of syllogisms whose premises are true and whose sequence is valid; hence, there is no need of additional examples now.

2. If the antecedent is true and the sequence invalid, the consequent is doubtful.

A consideration of the following two examples will make this law clear.8 The premises of both examples are obviously true, and the sequence is obviously invalid. Yet one conclusion is true, and the other conclusion is false. If both truth and falsity can come after true premises when the sequence is invalid, the consequent must be doubtful (unless it can be known—as here—from some other source).

1. Every dog is an animal; but no cat is a dog; therefore no cat is an animal.

In Example 1 the premises are true; the sequence is invalid because of an illicit process of the major term; the conclusion is obviously false. Example 2 has the same defect as Example 1, but the conclusion is true by accident.

2. Every dog is an animal; but no stone is a dog; therefore no stone is an animal.

Because the sequence of both examples is invalid, the pretended conclusion is unrelated to the premises. It has no more connection with the premises than has the make-believe conclusion of the following pseudo syllogism.

3. Cows give milk; but horses pull wagons; therefore it will rain tomorrow.

3. If the antecedent is false and the sequence valid, the consequent is doubtful.

In the following examples the antecedent is false, since at least one premise is false, and the sequence is formally valid. The conclusion of the Example 4 is true by accident; the conclusion of Example 5 is false. This shows that either truth or falsity can flow from falsity and that therefore the consequent is doubtful unless known from some other source.

4. Every dog is an animal; but every cat is a dog; therefore every cat is an animal.

5. Every dog is a rhinoceros; but every cat is a dog; therefore every cat is a rhinoceros.

4. If the antecedent is false and the sequence invalid, the consequent is doubtful.

We shall now examine two syllogisms whose premises are false and whose sequence is invalid. In each example there is an illicit process of the major term. The conclusion of Example 6 is true; but the conclusion of Example 7 is false.

6. Every cat is a monkey; but no cat is a dog; therefore no dog is a monkey.

7. Every cat is a dog; but no cat is a terrier; therefore no terrier is a dog.

If both truth and falsity can come after false premises in a syllogism whose sequence is invalid, the conclusion—it is only a pseudo conclusion—is doubtful (unless it is known from some other source).

b. Consequent to Antecedent

1. If the consequent is false and the sequence valid, the antecedent is false.

This is a corollary of the first law on the relationship of an antecedent to its consequent. If only truth can flow from truth, every antecedent from which a false consequent can flow must itself be false. Falsity can come only from falsity (supposing, of course, that the sequence is valid).

2. If the consequent is false and the sequence is invalid, the antecedent is doubtful.

When the sequence is invalid, anything can come after anything, since the consequent and antecedent are not related to one another at all. Consider:

1. Every cat is a dog; but no cat is a terrier; therefore no terrier is a dog.

2. No cat is a dog; but no terrier is a cat; therefore no terrier is a dog.

The consequent "no terrier is a dog " is false. The sequence of each example is invalid, since Example 1 contains an illicit process of the major term and Example 2 has two negative premises. Yet in one instance the pretended conclusion is preceded by true premises and in the other by false premises.

3. If the consequent is true and the sequence is valid, the antecedent is doubtful.

This law is a corollary of the third law on the relationship of an antecedent to its consequent. Since a true consequent can flow from a false antecedent as well as from a true antecedent, you cannot infer that an antecedent is true because its consequent is true. The conclusion might be true only by accident; that is, for reasons other than those given in the premises. This is illustrated in the following example.

3. Squares have three sides; but triangles are squares; therefore triangles have three sides.

Triangles do have three sides, but not for the reason given here.

4. If the consequent is true and the sequence is invalid, the antecedent is doubtful.

Obviously, if the antecedent of a true consequent is doubtful even when the sequence is valid, it is also doubtful when the sequence is invalid.

Synopsis

The results of our observations on the relationship of the antecedent and consequent as to truth and falsity in both valid and invalid inference may be tabulated as follows.

Only the first law in each group can serve as a basis of valid inference. Hence, the two basic laws with which we are especially concerned are:

1. If the antecedent is true and the sequence valid, the consequent is true.

2. If the consequent is false and the sequence valid, the antecedent is false.

A conditional syllogism is one whose major premise is a conditional proposition. There are two general types: the MIXED CONDITIONAL SYLLOGISM, whose minor premise is a categorical proposition—this is the commonest and most important type—and the PURELY CONDITIONAL SYLLOGISM, both of whose premises are conditional propositions.

First we shall explain the nature of the conditional proposition; then we shall give a thorough treatment of the mixed conditional syllogism; and, finally, we shall give a brief treatment of the purely conditional syllogism. The following outline will be of great help both in remembering the rules and in seeing their relationship to one another:

a. Conditional Proposition

A conditional proposition is a compound proposition [Note: Compound propositions express the relationship of clauses toward one another by the use of adverbs and conjunctions, such as "therefore," "because," "if ..., then ...," and so on.], of which one member (the “then” clause) asserts something as true on condition that the other member (the “if” clause) is true; for instance, “If it is raining, then the roof is wet.” The “if” clause or its equivalent is called the antecedent; the “then” clause or its equivalent is called the consequent. The assent in a conditional proposition does not bear on either the antecedent or the consequent taken by itself but on the connection between them—that is, on the sequence. Thus, if the truth of the consequent really follows upon the fulfillment of the condition stated in the antecedent, the proposition is true even if, taken singly, both the antecedent and the consequent are false. And if the truth of the consequent does not follow upon the fulfillment of the condition stated in the antecedent, the proposition is false even if, taken singly, both the antecedent and the consequent are true.

The proposition “If God exists, the world exists” is false, although each member, taken singly, is true—God does exist and the world does exist. But the existence of the world is not a necessary consequent of the existence of God, since God could exist without having created the world. On the other hand, the proposition “If God did not exist, the world would not exist” is true, although the members, taken singly, are false. The proposition is true because the non¬ existence of the world would really follow upon the non-existence of God.

A conditional proposition, then, is an assertion of a sequence (and nothing else), and is true if this sequence is valid. It makes no difference whether the validity is formal or merely material, as long as the truth of the antecedent necessitates the truth of the consequent.

Not every “if” proposition is a conditional proposition. Sometimes “if” is synonymous with “when,” “although,” “granted that,” and so on. For instance, the proposition “If John is a scoundrel, his brother is a virtuous man” is equivalent to “Although John is a scoundrel, his brother is a virtuous man.” Neither does every conditional proposition have an “if” or “unless.” The following, for instance, are conditional propositions: “Destroy this temple, and in three days I will rebuild it,” “Had I been there, it would not have happened,” and “Eat too many green apples, and you’ll get sick.”

“Unless” is equivalent to “if .. . not.” For instance, the proposition “Unless you do penance, you shall all likewise perish” is equivalent to “If you do not do penance, you shall all likewise perish.”

b. The Rules of the Mixed Conditional Syllogism

The rules governing the conditional syllogism are direct applications of the laws governing the relationship of an antecedent and its consequents. As we have seen, these laws are:

1. If the antecedent is true and the sequence is valid, the consequent is true.

2. If the consequent is false and the sequence valid, the antecedent is false.

Hence, to posit an antecedent is to posit its consequent, and to sublate a consequent is to sublate its antecedent. Accordingly, sup¬ posing that the major premise is a genuine conditional proposition (that is, a conditional proposition whose consequent flows from its antecedent with valid sequence), we may proceed in either of two ways:

1. We may posit the antecedent in the minor premise and posit the consequent in the conclusion,

2. or we may sublate the consequent in the minor premise and sublate the antecedent in the conclusion.

All other procedures are invalid.

To posit a member is to assert it as true. To sublate a member is to deny it by asserting either its contradictory or a proposition implying its contradictory. For instance, the proposition “Every man is seated” is sublated not only by its contradictory (“Some man is not seated”) but also by its contrary (“No man is seated”). We use the words “posit” and “sublate” rather than “affirm” and “deny” because it is inconvenient, not to say confusing, to speak of affirm¬ ing a negative member by means of a negative proposition or of denying a negative member by means of an affirmative proposition.

The schema and examples on Page 144 will help us understand and remember both the valid and invalid forms of the conditional syllogism. Procedures 1 and 2 (in heavy type) are the valid forms. Procedures 3 and 4 (in light type) are invalid forms, as is clear from the examples. You can be very sick from countless other causes besides acute appendicitis—for instance, from yellow fever, ptomaine poisoning, or diphtheria. Hence, if you are very sick, it does not follow that you have acute appendicitis; and if you do not have acute appendicitis, it does not follow that you are not very sick.

The syllogism “If he is not a thief, you will get your purse back; but he is not a thief; therefore you will get your purse back” illustrates the first form. Note that the minor premise posits the antecedent even though it is a negative proposition. We will make this clear by indicating the antecedent with an ellipse and the consequent by a rectangle.

Notice that conditional syllogisms do not have minor, middle, and major terms. Hence, we should not call the subject of the conclusion the minor term or the predicate the major term. This terminology is restricted to the categorical syllogism.

Notice, too, that (unless you have a disguised categorical syllogism ) when you posit an antecedent in the minor premise, you must posit it in its entirety. Example 1 (below) is invalid because in the minor premise the antecedent is posited only partially.

1. If every A is a B, every X is a Y; but some A is a B; therefore some X is a Y.

“Every A is a B” is an A proposition. “Some A is a B” is an I proposition. If I (the minor premise) is true, A (the antecedent) is doubtful. If the antecedent is doubtful, its consequent is doubtful too. Hence, no conclusion can be drawn. The children of a certain grade school recognized the invalidity of this form. They got a quarter holiday when all the children were present and on time for a period of two weeks. No child ever tried to get an extra holiday by arguing; as in Example 2.

2. If all were present and on time for two weeks, all will get a quarter holiday; but some were present and on time for two weeks; therefore some will get a quarter holiday.

To sublate a member, when we are proceeding from the minor premise to the consequent of the major premise, means to posit either its contradictory or some proposition implying its contradictory. In Example 3 the minor premise sublates the consequent by positing its contradictory:

3. If every A is a B; every X is a Y; but some X is not a Y; therefore some A is not a B.

In Example 4 the minor premise sublates the consequent by posit¬ ing its contrary, which implies its contradictory. Notice that both Example 3 and 4 have the same conclusion.

4. If every A is a B, every X is a Y; but no X is a Y; therefore some A is not a B.

To sublate a member, when we are proceeding from the sublation of the consequent to the sublation of the antecedent, means only to posit the contradictory of the antecedent. For instance. Example 4 cannot conclude validly in the universal proposition “No A is a B,” but only in the particular proposition “Some A is not a B.” Let us go through Example 4 step by step. The minor premise is an E proposition. If E is true, then the consequent “every X is a Y” (A) is false. If the consequent is false, the antecedent ( every A is a B ) is likewise false. If “every A is a B” is false, its contradictory, but not necessarily its contrary, must be true.

Syllogisms like the one given below in Example 5 need a word of explanation, since they resemble conditional syllogisms. In the major premise a statement is made about any man; in the minor premise a statement is made only about some men, namely about us. What is true of any (or every) man is obviously also true about us. This syllogism does not incur the defect alluded to above where we stated that the minor premise must posit the antecedent in its entirety because this is a disguised categorical syllogism—and in a categorical syllogism it is permissible to proceed from any (or every) to some. The syllogism given below applies a general principle to a particular instance, just as does a categorical syllogism.

5. If a man is convinced that virtue is rewarded and vice punished in the next world, he is less likely to follow every impulse; but we have that conviction; therefore we are less likely to follow every impulse.

“If a man is ..., he is ...” is equivalent to “Whoever is ... is ...” In spite of the fact that these are disguised categorical syllogisms, they are subject, with the qualification made above, to the general rules of the conditional syllogism.

c. The Purely Conditional Syllogism

The purely conditional syllogism, which has conditional propositions for both its premises, has exactly the same forms and the same rules as the mixed conditional syllogism except that the condition expressed in the minor premise must be retained in the conclusion. For instance,

If A is a B, then C is a D; but if X is a Y, then A is a B; therefore, if X is a Y, then C is a D.

A disjunctive syllogism is one whose major premise is a disjunctive proposition, whose minor premise sublates (or posits) one or more members of the major premise, and whose conclusion posits (or sublates) the other member or members. A study of the logical forms of the categorical syllogism will serve several purposes. At present its chief fruit will be to deepen our understanding of the general rules of the syllogism and to give us practice in applying them. Later on it will serve as a background for the consideration of the principles underlying the syllogism and for the reduction of syllogisms of the second, third, and fourth figures to syllogisms of the first figure.

a. Disjunctive Proposition

A disjunctive proposition is one that presents various alternatives and asserts that an indeterminate one of them is true. It consists of two or more members joined by the conjunctions “either ... or.” It is sometimes called an alternative proposition.

1) STRICT DISJUNCTIVE. In a disjunctive proposition in the strict or proper sense, only one member is true and the others are false. If all the members except one are false, the remaining member must be true; and if one member is true, the remaining members must be false. For instance, “Every proposition is either true or false,” “Every number is either one hundred or more than one hundred or less than one hundred,” and “It is either raining or not raining.” A proposition and its contradictory may always be asserted in a disjunctive proposition in the strict sense.

2) BROAD DISJUNCTIVE. In a disjunctive proposition in the broad or improper sense, at least one member is true but more than one may be true. For instance, the proposition “Either my brother or I will go” can mean that at least one of us will go, but possibly both of us will go. Often we must decide whether a proposition is a disjunctive in the strict or in the broad sense by a consideration of the matter and by the context.

Disjunctive propositions are reducible to a series of conditional propositions. To avoid repetition we shall postpone our treatment of the reduction of disjunctive to conditional propositions until we take the reduction of disjunctive syllogisms to conditional syllogisms.

b. Kinds and Rules

There are two kinds of disjunctive syllogisms, corresponding to the two kinds of disjunctive propositions. Each has its own rules.

1) DISJUNCTIVE SYLLOGISM IN THE STRICT SENSE. In a disjunctive syllogism in the strict sense, the major premise must be a disjunctive proposition in the strict or proper sense. The minor premise either posits or sublates one (or more—but not all) of the members of the major premise. In the conclusion there are two possible procedures:

a. If the minor premise posits one or more members of the major premise, the conclusion must sublate each of the other members. For instance,

The number is either one hundred or more than one hundred or less than one hundred; but the number is one hundred; therefore the number is neither more nor less than one hundred.

b. If the minor premise sublates one or more of the members of the major premise, the conclusion posits the remaining members, one of which must be true. If more than one member remains, the conclusion must be a disjunctive proposition in the strict sense. For instance,

The number is either one hundred or more than one hundred or less than one hundred; but the number is not one hundred; therefore the number is either more than one hundred or less than one hundred.

Every procedure besides those indicated under a and b is invalid. You may not posit one member and then posit another or sublate one member and then sublate all the others.

A diagram of the following “brain teasers” will make clear what we mean by positing or sublating a member. Note that in each example the minor posits one member, and the conclusion sublates the other.

2. DISJUNCTIVE IN THE BROAD SENSE. In a disjunctive syllogism in the broad sense, the major premise is a disjunctive proposition in the broad or improper sense. There is only one valid procedure: to sublate one (or more—but not all) of the members in the minor and posit the remaining member (or members) in the conclusion. If more than one member remains, the conclusion itself must be a disjunctive proposition in the broad sense. For instance:

It is either A or B, or C, or D--at least one of them; but it is neither A nor B; therefore it is either C or D--at least one of them.

A conjunctive syllogism is one whose major premise is a conjunctive proposition, whose minor premise posits one member of the major, and whose conclusion sublates the other member of the major.

a. Conjunctive Proposition

A conjunctive proposition is one that denies the simultaneous possibility of two alternatives; for instance, “You cannot eat your cake and have it,” “No man can serve both God and Mammon,” “A thing cannot both be and not be in the same respect.” A conjunctive proposition can be expressed in the formula “Not both A and B,” as well as in the formula “Either not A or not B—and maybe neither.”

b. Rule for Conjunctive Syllogism

There is only one valid procedure: to posit one member in the minor premise and sublate the other in the conclusion. Two examples will make this rule clear.

1. He cannot be in Chicago and St. Louis at the same time; but he is now in Chicago; therefore he cannot now be in St. Louis.

2. He cannot be in Chicago and St. Louis at the same time; but he is not in Chicago; therefore he is in St. Louis.

Example 1 is valid. Example 2 is invalid; obviously there are millions of places in which he might be besides Chicago and St. Louis. Hence, his not being in the one does not prove that he is in the other.

Disjunctive and conjunctive propositions are reducible to a series of conditional propositions. Notice that a disjunctive in the strict sense says all that both a disjunctive in the broad sense and conjunctive say; the negative implication (“but not both”) is expressed by the conjunctive. The following schema will manifest the relationship of disjunctives and conjunctives to conditional propositions as well as to one another.

Disjunctive and conjunctive syllogisms are reducible to compound conditional syllogisms whose major premises consist of the entire series of conditional propositions to which the major premises of the disjunctives and conjunctives are equivalent respectively. Thus, the syllogism “A is either C or D-but not both; but A is C; therefore A is not D” is reducible to the following:

Major:----------(1) If A is not C, A is D.

The minor (“A is C”) posits the antecedent of Number 3; hence, in the conclusion we validly posit its consequent. The same minor also sublates the consequent of Number 4; hence, in the conclusion we validly sublate its antecedent. These are the only valid procedures, and each of them gives us the same conclusion “A is not D.”

The syllogism “A is either C or D—maybe both; but A is not C; therefore A is D” is reducible to the following:

Major

The minor (“A is not C”) posits the antecedent of Number 1; hence, in the conclusion we validly posit its consequent. The same minor also sublates the consequent of Number 2; hence, in the conclusion we validly sublate the antecedent. Both processes give us the same conclusion.