The Art of Logical Thinking Part 7

It will be seen that the above Syllogism, whether expressed in words or symbols, is logically valid, because the conclusion must logically follow the premises. And, in this case, the premises being true, it must follow that the conclusion is true. Whately says: "A Syllogism is said to be valid when the conclusion logically follows from the premises; if the conclusion does not so follow, the Syllogism is invalid and const.i.tutes a Fallacy, if the error deceives the reasoner himself; but if it is advanced with the idea of deceiving others it const.i.tutes a Sophism."

The reason for Rule I is that only three propositions--a Major Premise, a Minor Premise, and a Conclusion--are needed to form a Syllogism. If we have more than _three_ propositions, then we must have more than two premises from which to draw one conclusion. The presence of more than two premises would result in the formation of two or more Syllogisms, or else in the failure to form a Syllogism.

II. _That there should be three and no more than three Terms._ These Terms are (1) The Predicate of the Conclusion; (2) the Subject of the Conclusion; and (3) the Middle Term which must occur in both premises, being the connecting link in bringing the two other Terms together in the Conclusion.

The _Predicate of the Conclusion_ is called the _Major_ Term, because it is the greatest in extension compared with its fellow terms. The _Subject of the Conclusion_ is called the _Minor_ Term because it is the smallest in extension compared with its fellow terms. The Major and Minor Terms are called the _Extremes_. The Middle Term operates between the two Extremes.

The _Major Term_ and the _Middle Term_ must appear in the _Major Premise_.

The _Minor Term_ and the _Middle Term_ must appear in the _Minor Premise_.

The _Minor Term_ and the _Major Term_ must appear in the _Conclusion_.

Thus we see that _The Major Term_ must be the Predicate of the Conclusion; the _Minor Term_ the Subject of the Conclusion; the _Middle Term_ may be the Subject or Predicate _of either of the premises_, but _must always be found once in both premises_.

The following example will show this arrangement more clearly:

In the Syllogism: "Man is mortal; Socrates is a man; therefore Socrates is mortal," we have the following arrangement: "Mortal," the Major Term; "Socrates," the Minor Term; and "Man," the Middle Term; as follows:

_Major Premise_: "Man" (_middle term_) is mortal (_major term_).

_Minor Premise_: "Socrates" (_minor term_) is a man (_major term_).

_Conclusion_: "Socrates" (_minor term_) is mortal (_major term_).

The reason for the rule that there shall be "_only three_" terms is that reasoning consists in comparing _two terms_ with each other through the medium of a _third term_. There _must be_ three terms; if there are _more_ than three terms, we form two syllogisms instead of one.

III. _That one premise, at least, must be affirmative._ This, because "from two negative propositions nothing can be inferred." A negative proposition a.s.serts that two things differ, and if we have two propositions so a.s.serting difference, we can infer nothing from them. If our Syllogism stated that: (1) "Man is _not_ mortal;" and (2) that "Socrates is _not_ a man;" we could form no Conclusion, either that Socrates _was_ or _was not_ mortal. There would be no logical connection between the two premises, and therefore no Conclusion could be deduced therefrom. Therefore, at least one premise must be affirmative.

IV. _If one premise is negative, the conclusion must be negative._ This because "if one term agrees and another disagrees with a third term, they must disagree with each other." Thus if our Syllogism stated that: (1) "Man is _not_ mortal;" and (2) that: "Socrates is a man;" we must announce the Negative Conclusion that: (3) "Socrates is _not_ mortal."

V. _That the Middle Term must be distributed; (that is, taken universally) in at least one premise._ This "because, otherwise, the Major Term may be compared with one part of the Middle Term, and the Minor Term with another part of the latter; and there will be actually no common Middle Term, and consequently no common ground for an inference." The violation of this rule causes what is commonly known as "The Undistributed Middle," a celebrated Fallacy condemned by the logicians. In the Syllogism mentioned as an example in this chapter, the proposition "_Man_ is mortal," really means "_All_ men," that is, Man in his universal sense. Literally the proposition is "All men are mortal,"

from which it is seen that Socrates being "_a_ man" (or _some_ of _all_ men) must partake of the quality of the universal Man. If the Syllogism, instead, read: "_Some_ men are mortal," it would not follow that Socrates _must_ be mortal--he might or might not be so. Another form of this fallacy is shown in the statement that (1) White is a color; (2) Black is a color; hence (3) Black must be White. The two premises _really_ mean "White is _some_ color; Black is _some_ color;" and not that either is "_all_ colors." Another example is: "Men are bipeds; birds are bipeds; hence, men are birds." In this example "bipeds" is not distributed as "_all_ bipeds" but is simply not-distributed as "_some_ bipeds." These syllogisms, therefore, not being according to rule, must fail. They are not true syllogisms, and const.i.tute fallacies.

To be "_distributed_," the Middle Term must be the Subject of a Universal Proposition, or the Predicate of a Negative Proposition; to be "_undistributed_" it must be the Subject of a Particular Proposition, or the Predicate of an Affirmative Proposition. (See chapter on Propositions.)

VI. _That an extreme, if undistributed in a Premise, may not be distributed in the Conclusion._ This because it would be illogical and unreasonable to a.s.sert more in the conclusion than we find in the premises. It would be most illogical to argue that: (1) "All horses are animals; (2) no man is a horse; therefore (3) no man is an animal." The conclusion would be invalid, because the term _animal_ is distributed in the conclusion, (being the predicate of a negative proposition) while it is not distributed in the premise (being the predicate of an affirmative proposition).

As we have said before, any Syllogism which violates any of the above six syllogisms is invalid and a fallacy.

There are two additional rules which may be called derivative. Any syllogism which violates either of these two derivative rules, also violates one or more of the first six rules as given above in detail.

The _Two Derivative Rules of the Syllogism_ are as follows:

VII. _That one Premise at least must be Universal._ This because "from two particular premises no conclusion can be drawn."

VIII. _That if one premise is Particular, the Conclusion must be particular also._ This because only a universal conclusion can be drawn from two universal premises.

The principles involved in these two Derivative Rules may be tested by stating Syllogisms violating them. They contain the essence of the other rules, and every syllogism which breaks them will be found to also break one or more of the other rules given.

CHAPTER XVII.

VARIETIES OF SYLLOGISMS

The authorities in Logic hold that with the four kinds of propositions grouped in every possible order of arrangement, it is possible to form nineteen different kinds of valid arguments, which are called the _nineteen moods of the syllogism_. These are cla.s.sified by division into what are called _the four figures_, each of which figures may be known by the position of the middle term in the premises. Logicians have arranged elaborate and curious tables constructed to show what kinds of propositions when joined in a particular order of arrangement will make sound and valid syllogisms. We shall not set forth these tables here, as they are too technical for a popular presentation of the subject before us, and because they are not necessary to the student who will thoroughly familiarize himself with the above stated Laws of the Syllogism and who will therefore be able to determine in every case whether any given argument is a correct syllogism, or otherwise.

In many instances of ordinary thought and expression the _complete_ syllogistic form is omitted, or not stated at full length. It is common usage to omit one premise of a syllogism, in ordinary expression, the missing premise being inferred by the speaker and hearer. A syllogism with one premise unexpressed is sometimes called an _Enthymene_, the term meaning "in the mind." For instance, the following: "We are a free people, therefore we are happy," the major premise "All free people are happy" being omitted or unexpressed. Also in "Poets are imaginative, therefore Byron was imaginative," the minor premise "Byron was a poet"

is omitted or unexpressed. Jevons says regarding this phase of the subject: "Thus in the Sermon on the Mount, the verses known as the Beat.i.tudes consist each of one premise and a conclusion, and the conclusion is put first. 'Blessed are the merciful: for they shall obtain mercy.' The subject and the predicate of the conclusion are here inverted, so that the proposition is really 'The merciful are blessed.'

It is evidently _understood_ that 'All who shall obtain mercy are blessed,' so that the syllogism, when stated at full length, becomes: 'All who shall obtain mercy are blessed; All who are merciful shall obtain mercy; Therefore, all who are merciful are blessed.' This is a perfectly good syllogism."

Whenever we find any of the words: "_because_, _for_, _therefore_, _since_," or similar terms, we may know that there is an argument, and usually a syllogism.

We have seen that there are three special kinds of Propositions, namely, (1) Categorical Propositions, or propositions in which the affirmation or denial is made without reservation or qualification; (2) Hypothetical Propositions, in which the affirmation or denial is made to depend upon certain conditions, circ.u.mstances, or suppositions; and (3) Disjunctive Propositions, in which is implied or a.s.serted an _alternative_.

The forms of reasoning based upon these three several cla.s.ses of propositions bear the same names as the latter. And, accordingly the respective syllogisms expressing these forms of reasoning also bear the cla.s.s name or term. Thus, a Categorical Syllogism is one containing only categorical propositions; a Hypothetical Syllogism is one containing one or more hypothetical propositions; a Disjunctive Syllogism is one containing a disjunctive proposition in the major premise.

_Categorical Syllogisms_, which are far more common than the other two kinds, have been considered in the previous chapter, and the majority of the examples of syllogisms given in this book are of this kind. In a Categorical Syllogism the statement or denial is made positively, and without reservation or qualification, and the reasoning thereupon partakes of the same positive character. In propositions or syllogisms of this kind it is a.s.serted or a.s.sumed that the premise is true and correct, and, if the reasoning be logically correct it must follow that the conclusion is correct, and the new proposition springing therefrom must likewise be Categorical in its nature.

_Hypothetical Syllogisms_, on the contrary, have as one or more of their premises a hypothetical proposition which affirms or a.s.serts something provided, or "if," something else be true. Hyslop says of this: "Often we wish first to bring out, if only conditionally, the truth upon which a proposition rests, so as to see if the connection between this conclusion and the major premise be admitted. The whole question will then depend upon the matter of treating the minor premise. This has the advantage of getting the major premise admitted without the formal procedure of proof, and the minor premise is usually more easily proved than the major. Consequently, one is made to see more clearly the force of the argument or reasoning by removing the question of the material truth of the major premise and concentrating attention upon the relation between the conclusion and its conditions, so that we know clearly what we have first to deny if we do not wish to accept it."

By joining a hypothetical proposition with an ordinary proposition we create a Hypothetical Proposition. For instance: "_If_ York contains a cathedral it is a city; York _does_ contain a cathedral; therefore, York is a city." Or: "If _dogs_ have four feet, they are quadrupeds; dogs _do_ have four feet; therefore dogs _are_ quadrupeds." The Hypothetical Syllogism may be either affirmative or negative; that is, its hypothetical proposition may either hypothetically _affirm_ or hypothetically _deny_. The part of the premise of a Hypothetical Syllogism which conditions or questions (and which usually contains the little word "if") is called the Antecedent. The major premise is the one usually thus conditioned. The other part of the conditioned proposition, and which part states what will happen or is true under the conditional circ.u.mstances, is called the Consequent. Thus, in one of the above examples: "If dogs have four feet" is the Antecedent; and the remainder of the proposition: "they are quadrupeds" is the Consequent. The Antecedent is indicated by the presence of some conditional term as: _if_, _supposing_, _granted that_, _provided that_, _although_, _had_, _were_, etc., the general sense and meaning of such terms being that of the little word "_if_." The Consequent has no special indicating term.

Jevons gives the following clear and simple _Rules regarding the Hypothetical Syllogism_:

I. "If the Antecedent be affirmed, the consequent may be affirmed. If the Consequent be denied, the Antecedent may be denied."

II. "Avoid the fallacy of affirming the consequent, or denying the antecedent. This is a fallacy because of the fact that the conditional statement made in the major premise _may not be the only one_ determining the consequent." The following is an example of "Affirming the Consequent:" "_If_ it is raining, the sky is overclouded; the sky _is_ overclouded; therefore, it _is raining_." In truth, the sky may be overclouded, and still it may _not_ be raining. The fallacy is still more apparent when expressed in symbols, as follows: "_If_ A is B, C is D; C _is_ D; therefore, A is B." The fallacy of denying the Antecedent is shown by the following example: "_If_ Radium were cheap it would be useful; Radium is _not_ cheap; therefore Radium _is not_ useful." Or, expressed in symbols: "_If_ A is B, C is D; A is _not_ B; therefore C _is not_ D." In truth Radium may be useful although not cheap. Jevons gives the following examples of these fallacies: "If a man is a good teacher, he thoroughly understands his subject; but John Jones thoroughly understands his subject; therefore, he is a good teacher."

Also, "If snow is mixed with salt it melts; the snow on the ground is _not_ mixed with salt; therefore it does _not_ melt."

Jevons says: "To affirm the consequent and then to infer that we can affirm the antecedent, is as bad as breaking the third rule of the syllogism, and allowing an undistributed middle term.... To deny the antecedent is really to break the fourth rule of the syllogism, and to take a term as distributed in the conclusion which was not so in the premises."

Hypothetical Syllogisms may usually be easily reduced to or converted into Categorical Syllogisms. As Jevons says: "In reality, hypothetical propositions and syllogisms are not different from those which we have more fully considered. _It is all a matter of the convenience of stating the propositions._" For instance, instead of saying: "If Radium were cheap, it would be useful," we may say "Cheap Radium would be useful;"

or instead of saying: "If gla.s.s is thin, it breaks easily," we may say "Thin gla.s.s breaks easily." Hyslop gives the following _Rule for Conversion_ in such cases: "Regard the antecedent of the hypothetical proposition as the subject of the categorical, and the consequent of the hypothetical proposition as the predicate of the categorical. In some cases this change is a very simple one; in others it can be effected only by a circ.u.mlocution."

The third cla.s.s of syllogisms, known as _The Disjunctive Syllogism_, is the exception to the law which holds that all good syllogisms must fit in and come under the Rules of the Syllogism, as stated in the preceding chapter. Not only does it refuse to obey these Rules, but it fails to resemble the ordinary syllogism in many ways. As Jevons says: "It would be a great mistake to suppose that all good logical arguments must obey the rules of the syllogism, which we have been considering. Only those arguments which connect two terms together by means of a middle term, and are therefore syllogisms, need obey these rules. A great many of the arguments which we daily use are of this nature; but there are a great many other kinds of arguments, some of which have never been understood by logicians until recent years. One important kind of argument is known as the Disjunctive Syllogism, though it does not obey the rules of the syllogism, or in any way resemble syllogisms."

The Disjunctive Syllogism is one having a disjunctive proposition in its major premise. The disjunctive proposition also appears in the conclusion when the disjunction in the major premise happens to contain more than two terms. A disjunctive proposition, we have seen, is one which possesses alternative predicates for the subject in which the conjunction "_or_" (sometimes accompanied by "_either_") appears. As for instance: "Lightning is sheet _or_ forked;" or, "Arches are _either_ round or pointed;" or, "Angles are either obtuse, or right angled, or acute." The different things joined together by "or" are called Alternatives, the term indicating that we may choose between the things, and that if one will not answer our purpose we may take the other, or one of the others if there be more than one _other_.

The _Rule regarding the Use of Disjunctive Syllogisms_ is that: "If one or more alternatives be denied, the rest may still be affirmed." Thus if we say that "A is B or C," or that "A is either B or C," we may _deny_ the B but still affirm the C. Some authorities also hold that "If we affirm one alternative, we must deny the remainder," but this view is vigorously disputed by other authorities. It would seem to be a valid rule in cases where the term "either" appears as: "A is _either_ B _or_ C," because there seems to be an implication that one or the other alone can be true. But in cases like: "A is B _or_ C," there may be a possibility of _both being true_. Jevons takes this latter view, giving as an example the proposition: "A Magistrate is a Justice-of-the-Peace, a Mayor, or a Stipendiary Magistrate," but it does not follow that one who is a Justice-of-the-Peace may not be at the same time a Mayor. He states: "After affirming one alternative we can only deny the others _if there be such a difference between them that they could not be true at the same time_." It would seem that both contentions are at the same time true, the example given by Jevons ill.u.s.trating his contention, and the proposition "The prisoner is either guilty or innocent" ill.u.s.trating the contentions of the other side.

A _Dilemma_ is a conditional syllogism whose Major Premise presents some sort of alternative. Whately defines it as: "A conditional syllogism with two or more antecedents in the major, and a disjunctive minor." There being two mutually exclusive propositions in the Major Premise, the reasoner is compelled to admit one or the other, and is then caught between "the two horns of the dilemma."