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 forthese 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, containsfirst, 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:
(1) All (middle)
|
true teachers are (major)
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sympathetic,
(minor)
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You are a (middle)
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true teacher,
∴ (minor)
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You are (major)
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sympathetic.
(2) No (major)
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shell fish are (middle)
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vertebrates,
All (minor)
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trout are (middle)
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vertebrates,
∴ No (minor)
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trout are (major)
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shell fish.
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.
FIG. 8.
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 writtenfirst, 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 I understand 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.
All who enter this school are examined, Major premise.
You are planning to enter this school, Minor premise.
You must be examined. Conclusion.
(2b) No well bound book goes to pieces, Major premise.
Some of these books are going to pieces, Minor premise.
Some of these books are not well bound. Conclusion.
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 standardof comparison. This becomes clear when recourse is made to the circles.
Illustration.
Syllogism in which the middle term is not distributed:
All men are mortal,
All trees are mortal,
∴ All trees are men.
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:
FIG. 9.
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:
All men are mortal,
No stones are mortal,
∴ No stones are men.
FIG. 10.
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 wouldfollow 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:
All trees grow,
No men are trees,
∴ No men grow.
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:
FIG. 11.
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 representsthe 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:
All true teachers are just,
All true teachers are sympathetic,
∴ All the sympathetic are just.
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:
FIG. 12.
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 tothe 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,
FIG. 13.
“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 ofeach other. The attending syllogism and its “circled” representation will throw light upon this:
No men are immortal,
All Americans are men,
∴ No Americans are immortal.
FIG. 14.
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:
(1) All the possible combinations of the two particular premises I and O are, (1) IO, (2) OI, (3) II, (4) OO.
“IO” considered.
(2) Since O is a negative premise the conclusion would have to be negative according to rule 6. (If one premise is negative, the conclusion must be negative.)
(3) If the conclusion is negative, then its predicate, which is the major term, must be distributed. (All negative propositions distribute their predicates.)
(4) If the major term is distributed in the conclusion, it must be distributed in the major premise, rule 4. (No term must be distributed in the conclusion, which is not also distributed in one of the premises.)
(5) Hence two terms must be distributed in the premises, the major term according to (4) and the middle term according to rule 3.
(6) But I distributes neither term and O distributes its predicate only; I and O together, then, distribute but one term.
(7) To draw a negative conclusion the premises must distribute two terms, the middle and the major, according to the foregoing.
(8) Hence a conclusion from I and O is untenable. The same may be said of “OI.”
“II” considered.
(1) The I proposition distributes neither subject nor predicate, hence the premises “II” would distribute no term.
(2) But the middle term must be distributed at least once according to rule 3.
(3) Therefore no conclusion can be drawn from “II.”
A valid conclusion from “OO” is impossible according to rule 5.
(8) If one premise be particular the conclusion must be particular. Proof: The possible combinations conditioned by rule 8 are AI, AO, EI, EO, IO, II, OO.
“AI” considered.
(1) Proposition A distributes its subject, proposition I neither; hence “AI” together distribute but one term.
(2) According to rule 3 this one term must be the middle term.
(3) The minor term must, therefore, be undistributed in the minor premise, and in consequence undistributed in the conclusion.
(4) But this undistributed minor term is the subject of the conclusion; hence said conclusion must be particular, as only particulars have an undistributed subject.
“AO” and “EI” considered.
Proof:
(1) “AO” distribute two terms; so do “EI.”
(2) Both “AO” and “EI” must have negative conclusions according to rule 6.
(3) A negative conclusion distributes its predicate which is the major term.
(4) The major term and the middle term must be distributed in the premises. Rules 4 and 3.
(5) Thus the third term, which is the minor, cannot be distributed in the minor premise and, consequently, the minor cannot be distributed in the conclusion.
(6) This necessitates a particular conclusion.
Premises EO and OO, being negative, cannot yield a conclusion according to rule 5; similarly, neither can the particulars IO and II because of rule 7.