2. THE SYLLOGISM.

Just as the judgment is expressed by means of the proposition, so mediate inference is best expressed by means of the syllogism.9 The following are syllogisms:

3. THE RULES OF THE SYLLOGISM.

All syllogistic reasoning is conditioned by the following eight rules:

These rules are exceedingly important, as their observance is necessary in all mediate reasoning. The student needs, not only to understand the meaning of these rules, but he needs to commit them to memory so thoroughly that they may be recalled without hesitation or mistake. To aid the memory, the eight rules may be divided into these four groups:

4. RULES OF THE SYLLOGISM EXPLAINED.

(1) A syllogism must have three and only three terms.

It is common to represent the various syllogistic forms by symbols, the same symbols always standing for the same terms. In this treatment we shall let capital G stand for the major term, as “major” means greater; capital S for the minor term, as “minor” means smaller, and capital M for the middle term. G, S and M, the initial letters of greater (major), smaller (minor) and middle, will be the constant symbols for these terms; just as A, E, I and O are used as the constant symbols for the four logical propositions.

Illustration.

Syllogism written in full:

All men are mortal,

Socrates is a man,

(Therefore) Socrates is mortal.

Syllogism symbolized:

All M is G

S is M

? S is G

The major term is always the predicate and the minor term the subject of the conclusion. The conclusion of the foregoing syllogism is, “Socrates is mortal.” Since G stands for the predicate of every conclusion, then it stands for “mortal,” the predicate of the above conclusion. For a similar reason, S stands for the subject, namely, “Socrates”; while M represents the middle term, “man.”

Since every syllogism must have three propositions, and since it takes two terms to form a proposition, then it follows that every syllogism must contain six terms. But, as no syllogism can have more than three different terms, we conclude that each term of the syllogism must be used twice. In the foregoing example, G thus appears, not only in the last proposition, or conclusion, but in the first proposition also. Similarly, both S and M occur twice. Every logical syllogism, then, contains first, a major term, which is always the predicate of the conclusion and appears once in the premises; second, a minor term, which is always the subject of the conclusion and appears once in the premises; and third, a middle term to which the other two terms are referred.

There are two ways of locating the middle term; first, it is the term which is used in both the premises; second, it is the term which never appears in the conclusion. Likewise, there are two ways of locating the major and minor terms; first, the major term is always the predicate and the minor term the subject of the conclusion; second, the major term is usually the broader and the minor term the narrower of the two. If the major and minor terms seem to be of about the same extension or breadth, then the term in the first proposition, which is not the middle term, is the major.

In the attending syllogisms the three terms are designated:

The necessity of having but three different terms in any syllogism may be understood by supposing that there are four different terms; then it would follow that there could be no standard or common link. In the axiom, “Things equal to the same thing are equal to each other,” the same thing is the common standard or link. Two things which equal two different things are not equal to each other.

The impossibility of reasoning from four terms may be shown by circles.

All men are mortal.
All trees grow.

These circles show that no connection can be established between either group. Using four terms in any syllogism is known as the fallacy of four terms.

(2) A syllogism must have three and only three propositions. The proposition containing the major term is called the major premise, while the one containing the minor term is called the minor premise. In a strictly logical syllogism the major premise is written first, the minor premise second and the conclusion third. In common parlance, however, the minor premise or even the conclusion may appear first.

The conclusion of a syllogism is always preceded by therefore, or its equivalent, which may be written or understood. The premises always answer the question, Why is the conclusion true? The premises are often preceded by such words as for and because.

The attending irregular syllogisms are arranged logically and the premises and conclusions indicated:

(1a) Illogical.

“You must take an examination because all who enter the school are examined and you, as Iunderstand it, are planning to enter.”

(2a) “Some of these books are not well bound, for they are going to pieces as no well bound book would do.”

(1b) Logical.

(2b) No well bound book goes to pieces, Major premise.

The fact that all syllogisms must have three and only three premises follows from rule “1.” One premise must compare the middle term with the “major”; another premise must compare the middle term with the “minor”; while the conclusion links together the “major” and the “minor.”

(3) The middle term must be distributed at least once. The rule is usually given in this way, “The middle term must be distributed once at least, and must not be ambiguous.” In this treatment the last part of the rule has been omitted because it must be apparent to the student that a middle term used in two senses is virtually equivalent to two different terms; such an “ambiguous middle” would, in consequence, give a syllogism of four terms.

Rules 3 and 4 are of greater importance than the others because they are more frequently violated. If the middle term is not distributed at least once, the fallacy is referred to as “undistributed middle.” If the distributed major term of the conclusion is not distributed in the major premise, then the fallacy is called, “illicit process of the major term”; and finally, if the distributed minor term of the conclusion is not distributed in the minor premise the fallacy is denominated an “illicit process of the minor term.” These two illicit processes may be abbreviated to illicit major and illicit minor.

Recall that any term is distributed when it is referred to as a definite whole. Unless the whole of the middle term is considered it fails to become a common standard of comparison. This becomes clear when recourse is made to the circles.

Illustration.

Syllogism in which the middle term is not distributed:

All the propositions are A’s and consequently the predicates of each are undistributed, as A distributes the subject only. Therefore the middle term, “mortal,” is not distributed in either of the premises and thus the fallacy.

Fallacy shown by circles:

These circles indicate the correct meaning of the two premises. By them it is seen that all of the “men” circle belongs to the “mortal” circle and all of the “tree” circle belongs to the “mortal” circle, but in this case there is no connection between the “men” and “tree” circles. Thus, to say that “All trees are men,” is fallacious. We have no right to either affirm or deny the connection between men and trees. If “mortal” were distributed we would have this right as the following will make clear:

Here the middle term mortal is distributed in the second premise as in it the subject “stones” is excluded from the entire mortal territory. This conclusion is verified by the formal statement that “E” distributes both subject and predicate. Since all of the “men” circle belongs to the “mortal” circle and none of the “stones” circle belongs to the “mortal” circle then none of the “stones” circle can belong to the “men” circle.

(4) No term must be distributed in the conclusion which is not also distributed in its premise.

It has been affirmed that a term is distributed when it is referred to as a definite whole. To put it in another way, a term is distributed when it is employed in its fullest sense. It is obvious that we should not employ a term in its fullest sense in the conclusion when it has been used only in a partial sense in its premise. What is said of the part cannot necessarily be said of the whole. For example: Because some men are honest it does not follow that all men are honest. Of course the converse of this is true, namely, if it could be proved that all men are honest then surely it would follow that some of the men are honest. To put it briefly: What is true of all is true of some but what is true of some is not necessarily true of all.

To distribute a term in the conclusion when it is not distributed in the premise where it occurs is equivalent to saying, “what is true of some is true of all.” This error which violates rule “4” leads to the two fallacies of illicit process of the major and minor terms. The following illustrate the two fallacies.

Syllogism illustrating illicit major:

The first premise is an A and consequently its subject is distributed. The second premise and conclusion being E’s have both subject and predicate distributed. Thus grow, as used in the conclusion, is distributed, but, as used in the major premise, it is not distributed. Fallacy shown by circles:

Here all of the “tree” circle belongs to the “grow” circle and none of the “men” circle belongs to the “tree” circle, hence the diagram correctly represents the meaning of the two premises and shows the fallacy of concluding that no men grow. The “men” circle, being entirely within the “grow” circle, indicates that all men grow. Syllogism illustrating illicit minor:

Each proposition being an A distributes its subject. But the subject of the conclusion which is “the sympathetic” is not distributed in the minor premise, as an A proposition distributes its subject only. Hence the fallacy of illicit minor.

Fallacy shown by circles:

The diagram correctly represents the two premises since all of the “true teacher” circle belongs to both the “just” and “sympathetic” circles. But all of the “sympathetic” circle does not belong to the “just” circle. Hence the fallacy.

(5) No conclusion can be drawn from two negative premises.

When two terms are both denied of a third term, it is quite impossible to draw any conclusion relative to the two terms, as the absolute exclusion of the third term eliminates any possibility of a common link or standard.

The circles will make this apparent:

No men are immortal,
No trees are immortal,

“No trees are men” is the conclusion represented by Fig.13.

Other possible conclusions are, “All trees are men,” “All men are trees” and “Some men are trees.”

It is thus seen that no definite conclusion can be drawn. It may now be said that when the major and minor terms are used in two negative premises the connection between them is indeterminate. This violation of rule “5” may be termed the fallacy of two negatives.

(6) If one premise be negative the conclusion must be negative; and conversely, to prove a negative conclusion one of the premises must be negative.

Referring to the first part of this rule, it may be said of two terms that if one is affirmed and the other denied of a third term, then the two terms must be denied of each other. The attending syllogism and its “circled” representation will throw light upon this:

Since none of the “men” circle belongs to the “immortal” circle and all of the “American” circle is inside the “men” circle, it is evident that none of the “American” circle can belong to any part of the “immortal” circle. Thus it is manifest that an affirmative conclusion like, “All Americans are immortal,” is invalid.

The converse of rule 6, “To prove a negative conclusion, one of the premises must be negative,” may be explained by the general principle in logic that when two terms are known to disagree, one must agree with a third term while the other must disagree. If both agreed with a third, then the conclusion would of necessity be affirmative. If both disagreed no conclusion could be drawn. A violation of rule 6 may be called the fallacy of negative conclusion.

(7) No conclusion can be drawn from two particular premises. Proof:

(8) If one premise be particular the conclusion must be particular. Proof: The possible combinations conditioned by rule8 are AI, AO, EI, EO, IO, II, OO.

Premises EO and OO, being negative, cannot yield a conclusion according to rule5; similarly, neither can the particulars IO and II because of rule7.

5. THE DICTUM OF ARISTOTLE.

Aristotle gives an axiom on which all syllogistic inference is based. Indeed from this fundamental principle the significant rules of the syllogism could be derived. The dictum is stated in this wise: “Whatever is predicated, whether affirmatively or negatively, of a term distributed may be predicated in the manner of everything contained under it.” The following statements represent various ways of explaining this dictum:

6. CANONS OF THE SYLLOGISM.

The dictum of Aristotle is ostensibly a self-evident truth, and some logicians have put this truth in the form of three axiomatic statements which are known as the canons of the syllogism. These are as follows:

Making use of the symbols as explained on a previous page of this chapter, it will be seen that the first canon conforms to this syllogistic type:

The two terms are S and G, while M is the third term.

The attending symbolizations illustrate, respectively, the second and third canons:

7. THREE MATHEMATICAL AXIOMS.

Analogous to the three canons treated in “6,” there are certain mathematical axioms which are here stated:

Illustrations of the three axioms:

8. OUTLINE.

MEDIATE INFERENCE.

9. SUMMARY.

(1) Inference is a term used to denote a process as well as a product. As a process reasoning and inference are in reality synonomous terms.

Inference is a thought process of deriving a judgment from one or two antecedent judgments.

Mediate inference is inference by means of a middle term. Mediate inference makes use of three terms, two of which are compared with a third term as a standard. This third term is called the middle term.

(2) The syllogism is the common mode of expression for mediate inference.

(3) Valid syllogistic reasoning is conditioned by eight rules. The first and second relate to the composition of the syllogism; the third and fourth to the distribution of terms; the fifth and sixth to negative premises; the seventh and eighth to particular premises.

(4) All syllogisms must have three terms: the major, the minor, and the middle. The middle term occurs twice in the premises but never appears in the conclusion. The minor term is always the subject, and the major term the predicate of the conclusion. The major term is usually broader than the minor.

No conclusion can be drawn from four terms. To attempt this gives rise to the fallacy of four terms.

All syllogisms must have three propositions, the major and the minor premises, and the conclusion. The major premise first and the minor second is the more logical arrangement, although the common conversational form is to use the minor premise first.

Ambiguous middle amounts to the fallacy of four terms.

Unless the middle term is distributed at least once in the syllogism, it fails to become a common standard.

Distributing a term in the conclusion, without its being distributed in its premise, is equivalent to asserting that, “What is true of a part is true of the whole.” This error results in the fallacies of illicit major and minor.

A conclusion from two negatives is impossible, because of the total exclusion of the middle term.

Of two terms, if one is affirmed and the other denied of a third term, then they must be denied of each other; and, conversely, if two terms are to be denied of each other, one must be affirmed and the other denied of a given third term. This fundamental principle necessitates deriving a negative conclusion from two premises when one is negative. It, likewise, compels the converse of this.

A valid conclusion from two particulars is untenable because of the two negative fallacies, or some fallacy relative to the distribution of terms.

One particular premise forces a particular conclusion because of the fallacies of two negatives, two particulars, and illicit minor.

(5) Aristotle’s dictum simplified means, “What is true of the whole is true of the part.”

(6) The canons of the syllogism, three in number, are:

(7) The foregoing canons may be stated as mathematical axioms.

10. ILLUSTRATIVE EXERCISES.

(1a) Make use of the proper symbols and indicate the three terms of each of the attending syllogisms:

I recall that the three terms are the middle, the major and the minor, and that the “middle” does not occur in the conclusion, whereas the “major” is always the predicate and the “minor” the subject of the conclusion. The symbols M, G and S being the initial letters of middle, greater and smaller, Imake use of these in designating the three terms, as the following will illustrate:

(1) All M
fixed stars G
twinkle,

S
Vega is a M
fixed star,

? S
Vega G
twinkles.

“Twinkles” being the predicate of the conclusion is designated as being the major term by putting the letter G above it. Then “G” is placed above the term “twinkle” in the first premise.

“S” is placed above the subject of the conclusion to indicate that it is the minor term. “S” is also placed above “Vega,” the minor term, as found in the second premise.

The remaining term, “fixed stars,” must be the middle term, therefore Iplace “M” above it. The fact that “fixed star” does not occur in the conclusion verifies this.

Using only the symbols, the syllogism takes this form:

All M is G

S is M

? S is G

Using the symbols to represent the other syllogisms, we have

(2) All G is M

No S is M

? No S is G

(3) All M is G

All M is S

? Some S is G

(1b) Indicate by symbols the three terms of the following syllogisms:

(2a) Illustrate by syllogism the fallacy of undistributed middle. An easy way is to use the middle term as the predicate of two A premises. This yields the fallacy because an A proposition does not distribute the predicate.

The illustration: distributed terms underscored.

All true teachers are students,

All scholars are students,

———————————————

? All scholars are true teachers.

(2b) Give two illustrations of undistributed middle.

(3a) Give syllogistic illustrations of the fallacies of illicit major and minor.

Illicit Major.

Use the middle term as the subject of an A proposition, and then as the predicate of an E proposition. This would necessitate a negative conclusion in which the major term is distributed. But the major term is not distributed in the major premise, hence the fallacy.

Illustration in which the distributed terms are underscored:

All men are mortal,

No trees are men,

——————————

? No trees are mortal.

Illicit Minor.

To illustrate this fallacy one may use the middle term as the subject of two A premises. This would give an A conclusion in which the subject is distributed. But this same term is not distributed in its premise because here it is used as the predicate of anA. Illustration:

All earnest students study,

All earnest students desire to succeed,

———————————————————

? All who desire to succeed study.

11. REVIEW QUESTIONS.