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An Affirmative Proposition is one in which the Predicate is affirmed to agree with the Subject. A Negative Proposition is one in which the agreement of the Predicate and Subject is denied . Examples of both of these classes have been given in this chapter.

Another classification of Propositions divides them in three classes, as follows (1) Categorical; (2) Hypothetical; (3) Disjunctive.

A Categorical Proposition is one in which the affirmation or denial is made without reservation or qualification, as for instance: "Man is an animal;" "the rose is a flower," etc. The fact asserted may not be true , but the statement is made positively as a statement of reality.

A Hypothetical Proposition is one in which the affirmation or denial is made to depend upon certain conditions, circumstances or suppositions, as for instance: "If the water is boiling-hot, it will scald;" or "if the powder be damp, it will not explode," etc. Jevons says: "Hypothetical Propositions may generally be recognized by containing the little word 'if;' but it is doubtful whether they really differ much from the ordinary propositions.... We may easily say that 'boiling water will scald,' and 'damp gunpowder will not explode,' thus avoiding the use of the word 'if.'"

A Disjunctive Proposition is one "implying or asserting an alternative," and usually containing the conjunction "or," sometimes together with "either," as for instance: "Lightning is sheet or forked;" "Arches are either round or pointed;" "Angles are either obtuse, right angled or acute."

Another classification of Propositions divides them in two classes as follows: (1) Universal; (2) Particular.

A Universal Proposition is one in which the whole quantity of the Subject is involved in the assertion or denial of the Predicate. For instance: "All men are liars," by which is affirmed that all of the entire race of men are in the category of liars, not some men but all the men that are in existence. In the same way the Proposition: "No men are immortal" is Universal, for it is a universal denial .

A Particular Proposition is one in which the affirmation or denial of the Predicate involves only a part or portion of the whole of the Subject, as for instance: " Some men are atheists," or " Some women are not vain," in which cases the affirmation or denial does not involve all or the whole of the Subject. Other examples are: "A few men," etc.; " many people," etc.; " certain books," etc.; " most people," etc.

Hyslop says: "The signs of the Universal Proposition, when formally expressed, are all , every , each , any , and whole or words with equivalent import." The signs of Particular Propositions are also certain adjectives of quantity, such as some , certain , a few , many , most or such others as denote at least a part of a class.

The subject of the Distribution of Terms in Propositions is considered very important by Logicians, and as Hyslop says: "has much importance in determining the legitimacy, or at least the intelligibility, of our reasoning and the assurance that it will be accepted by others." Some authorities favor the term, "Qualification of the Terms of Propositions," but the established usage favors the term "Distribution."

The definition of the Logical term, "Distribution," is: "The distinguishing of a universal whole into its several kinds of species; the employment of a term to its fullest extent; the application of a term to its fullest extent, so as to include all significations or applications." A Term of a Proposition is distributed when it is employed in its fullest sense; that is to say, when it is employed so as to apply to each and every object, person or thing included under it . Thus in the proposition, "All horses are animals," the term horses is distributed; and in the proposition, "Some horses are thoroughbreds," the term horses is not distributed. Both of these examples relate to the distribution of the subject of the proposition. But the predicate of a proposition also may or may not be distributed. For instance, in the proposition, "All horses are animals," the predicate, animals , is not distributed, that is, not used in its fullest sense , for all animals are not horses —there are some animals which are not horses and, therefore, the predicate, animals , not being used in its fullest sense is said to be " not distributed ." The proposition really means: "All horses are some animals."

There is however another point to be remembered in the consideration of Distribution of Terms of Propositions, which Brooks expresses as follows: "Distribution generally shows itself in the form of the expression, but sometimes it may be determined by the thought. Thus if we say, 'Men are mortal,' we mean all men , and the term men is distributed. But if we say 'Books are necessary to a library,' we mean, not 'all books' but 'some books.' The test of distribution is whether the term applies to ' each and every .' Thus when we say 'men are mortal,' it is true of each and every man that he is mortal."

The Rules of Distribution of the Terms of Proposition are as follows:

1. All universals distribute the subject .

2. All particulars do not distribute the subject .

3. All negatives distribute the predicate .

4. All affirmatives do not distribute the predicate .

The above rules are based upon logical reasoning. The reason for the first two rules is quite obvious, for when the subject is universal , it follows that the whole subject is involved; when the subject is particular it follows that only a part of the subject is involved. In the case of the third rule, it will be seen that in every negative proposition the whole of the predicate must be denied the subject, as for instance, when we say: "Some animals are not horses ," the whole class of horses is cut off from the subject, and is thus distributed . In the case of the fourth rule, we may readily see that in the affirmative proposition the whole of the predicate is not denied the subject, as for instance, when we say that: "Horses are animals," we do not mean that horses are all the animals , but that they are merely a part or portion of the class animal—therefore, the predicate, animals , is not distributed.

In addition to the forms of Propositions given there is another class of Propositions known as Definitive or Substitutive Propositions , in which the Subject and the Predicate are exactly alike in extent and rank. For instance, in the proposition, "A triangle is a polygon of three sides " the two terms are interchangeable; that is, may be substituted for each other. Hence the term "substitutive." The term "definitive" arises from the fact that the respective terms of this kind of a proposition necessarily define each other. All logical definitions are expressed in this last mentioned form of proposition, for in such cases the subject and the predicate are precisely equal to each other.

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In the process of Judgment we must compare two concepts and ascertain their agreement of disagreement. In the process of Reasoning we follow a similar method and compare two judgments, the result of such comparison being the deduction of a third judgment.