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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.

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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 classified 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 Beatitudes 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, circumstances, or suppositions; and (3) Disjunctive Propositions, in which is implied or asserted an alternative .

The forms of reasoning based upon these three several classes of propositions bear the same names as the latter. And, accordingly the respective syllogisms expressing these forms of reasoning also bear the class 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 asserted or assumed 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 asserts 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 circumstances, 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."