This lesson is from Bachhuber, Andrew H., S.J. 1957. "Chapter 9: Special Types of Syllogism," An Introduction to Logic. New York: Appleton-Century-Crofts, Inc. Pp. 160-73.
An enthymeme is a syllogism in which one of the premises or the conclusion is omitted. There are three orders of enthymemes. An enthymeme is of the first order if the major premise is omitted, of the second order if the minor premise is omitted, and of the third order if the conclusion is omitted. The enthymeme is not a distinct form of syllogism, but an incomplete statement of any of the forms we have already studied.
Let us examine some of the ways in which the following categorical syllogism can be expressed in enthymemes.
Major: What is spiritual is immortal.
Minor: But the human soul is spiritual.
Concl: Therefore the human soul is immortal.
1. Minor: The human soul is spiritual.
Concl: Therefore the human soul is immortal.
2. Concl: The human soul is immortal
Minor: because it is spiritual.
3. Major: What is spiritual is immortal.
Concl: For this reason the human soul is immortal.
4. Concl: The human soul is immortal,
Major: since whatever is spiritual is immortal.
5. Minor: The human soul is spiritual,
Major: and whatever is spiritual is immortal.
Examples 1 and 2 are enthymemes of the first order; 3 and 4 are of the second order; 5 is of the third order.
We can recognize an enthymeme as categorical as soon as we discover three syllogistic terms. Thus, “A is B; therefore A is C” is obviously categorical, for it has the three syllogistic terms A, B, and C.
An enthymeme is generally hypothetical if neither the subject nor the predicate of the conclusion occurs in the antecedent. For instance, “It is raining; therefore Peter is not working” is obviously hypothetical, the unexpressed member being “If it is raining, Peter is not working.”
The enthymeme is the most natural way of applying a general principle to a particular case and the commonest expression of syllogistic reasoning. Outside of logic books you will find very few completely expressed syllogisms, but you will find enthymemes on almost every page you read.
The weakness of arguments is sometimes concealed by the suppression of false or doubtful premises. Often the only way to test the validity and truth of an enthymeme is to express the omitted member.
Since the enthymeme is not a distinct form of syllogism but merely an abridged statement of the usual forms, it has no special rules.
Notice that many “because” clauses are not intended to be a proof that a thing took place but an explanation of why it took place. The same is true of the antecedents of many “therefore” clauses.
An epichireme is a syllogism in which a proof is joined to one or both of the premises. The proof is often expressed by a causal clause (“for,” “because,” “since,” and so on). The premise to which a proof is annexed is an enthymeme. Sometimes the main syllogism is also an enthymeme.
We must be careful to distinguish the main syllogism from the proofs of a premise. In the following example the proofs of the premises are enclosed in parentheses.
Major: If man has spiritual activities, he has a spiritual soul, (because every activity requires an adequate principle).
Minor: But (since man knows immaterial things), man has spiritual activities.
Concl: Therefore man has a spiritual soul.
In the example given below, the antecedent is a proof of the unexpressed minor premise. The major premise of the main syllogism of this argument is also unexpressed because it is considered too obvious to require statement.
What atrophies those national traits which make America big, virile, and wealthy is bad; hence, we oppose all additions to federal power.
Fully expressed, this argument is as follows (the original argument is written in capital letters):
Major: We oppose whatever is bad.
Minor: But all additions to federal power is bad.
Proof: WHAT ATROPHIES THOSE NATIONAL TRAITS WHICH MAKE AMERICA BIG, VIRILE, AND WEALTHY ARE BAD; but all additions to federal power atrophy those national traits which make America big, virile, and wealthy; therefore all additions to federal power are bad.
Concl: HENCE, WE OPPOSE ALL ADDITIONS TO FEDERAL POWER.
A polysyllogism, as the name suggests (poly is the Greek word for "many"), is a series of syllogisms so arranged that the conclusion of one is the premise of the next. Each individual syllogism must adhere to the rules of the simple syllogism.
A sorites is a polysyllogism consisting of a series of simple syllogisms whose conclusions, except for the last, are omitted. It is either categorical or conditional.
a. Categorical Sorites
A categorical sorites consists of a series of simple categorical syllogisms of the first figure whose conclusions, except for the last, are omitted. It links or separates the subject and predicate of the conclusion through the intermediacy of many middle terms.
There are two kinds of categorical sorites, the Aristotelian (or progressive) and the Goclenian (or regressive). In the Aristotelian sorites the predicate of each premise is the subject of the following premise, and the subject of the first premise is the subject of the conclusion. In the Goclenian sorites the same premises occur, but their order is reversed. Hence, the two types differ from one another only accidentally. The following diagrams reveal the differences in their construction and manner of procedure. The first diagram dis¬ plays the arrangement of the premises and indicates which of them may be particular and which negative.
There are TWO SPECIAL RULES FOR THE SORITES. For the Aristotelian sorites they are:
All but the last premise must be affirmative. If a premise is negative, the conclusion must be negative.
All but the first premise must be universal. If the first premise is particular, the conclusion must be particular.
Obviously, for the Goclenian sorites the rules are the reverse of these, and only the first premise may be negative and only the last particular. If the first rule is violated, there is an illicit process of the major term. As the predicate of an affirmative proposition the term “E” is particular in the premise; but as the predicate of a negative proposition it is universal in the conclusion.
If the second rule is violated, there is an undistributed middle.
These two rules are corollaries of the rules of the first figure of the categorical syllogism.
b. Conditional Sorites
A conditional sorites is one whose premises contain a series of conditional propositions, each of which (except the first) has as its antecedent the consequent of the preceding premise. Sometimes all the premises, including the last, are conditional propositions, and then the conclusion must be a conditional proposition. Sometimes the last premise is a categorical proposition, and then the conclusion must be a categorical proposition.
Keep in mind that to posit an antecedent is to posit not only its proximate consequents but also its remote consequents, and to sublate a consequent is to sublate not only its proximate antecedents but also its remote antecedents. Thus in the series If A, then B; if B, then C; if C, then D; if D, then E” to posit A is to posit B, C, D, and E; and to sublate E is to sublate D, C, B, and A. Hence, we can argue in any of the following ways.
The dilemma is a syllogism that is both conditional and disjunctive. The major premise is a compound conditional proposition consisting of two or more simple conditional propositions connected by ‘‘and” or its equivalent. The minor premise is a disjunctive proposition that alternatively posits the antecedents (constructive dilemma), or sublates the consequents (destructive dilemma), of each of these simple conditional propositions.
In the constructive dilemma the disjunctive proposition is commonly placed first; in the destructive dilemma, however, the conditional propositions are commonly placed first. The conclusion is either a categorical or a disjunctive proposition.
If the disjunctive premise has three members, the syllogism is a trilemma; if it has many members, the syllogism is a polylemma. But the name “dilemma” is also applied to these.
a. Forms of the Dilemma
The dilemma has four forms. It is either constructive or destructive, and each of these is either simple or complex. The schema on Page 170 displays the structure of these four forms.
1) In the SIMPLE CONSTRUCTIVE DILEMMA the conditional premise infers the same consequent from all the antecedents presented in the disjunctive proposition. Hence, if any antecedent is true, the consequent must be true. This form is illustrated by the reflections of a man trapped in an upper story of a burning building.
I must either jump or stay--there is no other alternative.
But if I jump, I shall die immediately (from the fall);
But if I stay, I shall die immediately (from the fire);
Therefore I shall die immediately.
2) In the COMPLEX CONSTRUCTIVE DILEMMA the conditional premise infers a different consequent from each of the antecedents presented in the disjunctive proposition. If any antecedent is true, its consequent is likewise true. But since the antecedents are posited disjunctively and since a different consequent flows from each of them, the consequents must likewise be posited disjunctively. The men who brought to Jesus the woman caught in adultery had this form of dilemma in mind.
Jesus will either urge that she be stoned to death or that she be released without stoning.
But if he urges the first, he will make himself unpopular with the people because of his severity;
But if he urges the second, he will get into trouble with the Jewish authorities for disregarding the law of Moses.
Therefore he will either become unpopular with the people or get into trouble with the Jewish authorities.
You will recall how Jesus slipped between the horns of this dilemma by writing on the sand and saying, "Let him who is without sin cast the first stone" (John 8: 1-11).
3) In the SIMPLE DESTRUCTIVE DILEMMA the conditional premise infers more than one consequent from the same antecedent. If any of the consequents is false, the antecedent is false. Hence, since the disjunctive sublates the consequents alternatively, at least one of them must be false, and consequently the antecedent must also be false. This type is not distinct from a conditional syllogism in which the consequent is sublated in the minor premise and the antecedent is sublated in the conclusion. Still, on account of the disjunctive premise, it is generally called a dilemma. The following example illustrates this form.
If I am to pass the examination, I must do two things--I must study all night and I must also be mentally alert as I write.
But either I will not study all night,
or I will not be mentally alert as I write.
Therefore I will not pass the examination.
4) In the COMPLEX DESTRUCTIVE DILEMMA the conditional premise infers a different consequent from each antecedent. The disjunctive premise sublates these consequents alternatively, and the conclusion sublates their antecedents alternatively. For instance:
If John were wise, he would not speak irreverently of holy things in jest; and if he were good, he would not do so in earnest.
But he does it either in jest
or in earnest.
Therefore John is either not wise or not good.
b. Rules of the Dilemma, Answering a Dilemma
The dilemma is subject, first of all, to the general rules of the conditional syllogism. The minor premise, as in the conditional syllogism, must either posit the antecedents or sublate the consequents of the conditional propositions. If the minor premise has posited the antecedents, the conclusion must posit the consequents, either absolutely or disjunctively—depending on the type. If the minor premise has sublated the consequents, the conclusion must sublate the antecedents, and so on, as explained above.
The dilemma also has the following special rules:
The disjunction must state all pertinent alternatives.
The consequents in the conditional proposition must flow validly from the antecedents.
The dilemma must not be subject to rebuttal.
The names traditionally given to the ways of ANSWERING A DILEMMA will not puzzle us if we keep in mind that the alternatives presented in a dilemma are called “horns” and that a dilemma is sometimes called a syllogismus cornutus or “horned argument.”
If you show that the first rule is violated, you escape between the horns, as in the following example:
I must either devote myself to the interests of my soul, or to secular pursuits. If I devote myself to the interests of my soul, my business will fail; if I devote myself to secular pursuits, I shall lose my soul. Therefore either my business will fail, or else I shall lose my soul.
There is a third alternative, to devote myself both to the interests of my soul and to secular pursuits with the proper subordination of the latter to the former.
If you show that the second rule has been violated, you take the dilemma by the horns. For instance, in the following example you can show that at least one of the consequents does not flow from its antecedent. This dilemma is attributed to the Caliph Omar, and is quoted in many logic books.
The books in the library of Alexandria are either in conformity with the Koran or not in conformity with it. If they are in conformity with it, they are superfluous and should be burned; if they are not in conformity with it, they are pernicious and likewise should be burned. Therefore the books in the library of Alexandria should be burned.
A book might be in conformity with the Koran and still be useful in that it can explain the Koran and treat of subjects not mentioned in the Koran.
If you show that consequents other than the unfavorable ones given in the dilemma flow from the antecedents, you butt back or or make a rebuttal. The argument of the Athenian mother who tried to dissuade her son from entering public life is a classic example. The mother argued:
If you say what is just, men will hate you; if you say what is unjust, the gods will hate you. But you must either say what is just or what is unjust. Therefore you will be hated.
The son replied:
If I say what is just, the gods will love me; if I say what is unjust, men will love me. But I must say either the one or the other. Therefore I will be loved.