(3) The hypothetical proposition is composed of antecedent and consequent; the former being the limiting condition; while the latter is the direct assertion. As the words indicate the antecedent usually precedes the consequent. The signs of the antecedent are “if,” “though,” “unless,” “suppose,” “granted that,” “when,” etc. (4) The two kinds of hypothetical syllogisms are the constructive and destructive; the former is involved when the minor premise affirms the antecedent; the latter when the minor premise denies the consequent. These two kinds are sometimes referred to as “_modus ponens_” and “_modus tollens_” respectively. (5) Out of the four possible hypothetical syllogisms only _two_ are valid as investigation proves this rule: _The minor premise must affirm the antecedent or deny the consequent._ In the case of the hypothetical proposition being _co-extensive_, the rule does not apply. (6) Hypothetical arguments may be reduced to the categorical by contracting the antecedent of the hypothetical proposition to form the subject-term, and by contracting the consequent of the hypothetical proposition to form the predicate-term of the major premise of the categorical syllogism. If it is necessary, supply a new minor term. Denying the antecedent is a matter of illicit major; whereas affirming the consequent is equivalent to undistributed middle. (7) Hypothetical arguments may be tested by following this outline: (1) Arrange logically. (2) Determine antecedent and consequent. (3) Apply hypothetical rule. (4) Reduce to categorical form. (5) Apply categorical rules. (8) A disjunctive syllogism is one in which the major premise is a disjunctive proposition. (9) The two kinds of disjunctives are those which “_by affirming deny_” and those which “_by denying affirm_.” (10) In testing disjunctive arguments there are _two_ rules involved: _First_, “The major premise must assert a _logical disjunction_.” This necessitates the two requisites “_the alternatives must be mutually exclusive_” and the “_enumeration must be complete_.” The two opinions relative to the nature of an alternative assertion are, first, if one is false, the other must be true and vice versa; and second, if one is false, the other must be true, but _both_ may be true. The first is adopted in this discussion. _Second._ The second rule involved is “When the minor premise affirms or denies _one_ of the alternatives of a logical disjunctive the conclusion must deny or affirm all of the others.” (11) Subjecting the disjunctive arguments to the categorical test gives evidence to the close relation existing between the two forms. A logical disjunctive proves to be logical when reduced to the categorical. The reduction entails the two steps, first, reduce to the hypothetical; second, reduce to the categorical.