Without any knowledge of the rules of the hypothetical syllogism let us strive to determine how many of the foregoing are valid. Relative to the first, it would be impossible for any rain to fall without making the ground somewhat damp; a few drops would be sufficient. In short, if the antecedent happens, the consequent _must_ follow. It seems, therefore, that the first argument is _valid_. Considering the second: rain is not the only cause for the dampness of the ground, as it might result from the falling of dew, or from a dense fog; _no rain_ does not necessarily mean _no dampness_. It is clear that if the antecedent does not happen, the consequent may or may not follow. Thus it appears that the second argument is _invalid_. Attention to the third makes evident a condition similar to the second: the ground may be made damp by agencies other than rain, such as fog and dew. Thus the third argument is likewise _invalid_. But in the fourth argument it is obvious that if the ground is not damp, then there could have been neither rain, nor fog, nor dew. No dampness shuts out _all_ of the conditions, including the rain. Therefore the fourth argument is _valid_. This investigation suggests a rule for hypothetical arguments. Since only the first and fourth arguments are valid, this is the rule which must obtain: _The minor premise should either affirm the antecedent or deny the consequent._ Any violation of this rule would result in the fallacies of _denying the antecedent_ or _affirming the consequent_. There is one exception to this rule which must not be overlooked; viz.: If the antecedent and consequent of the hypothetical proposition are _co-extensive_ then both may be either affirmed or denied. ILLUSTRATIONS: (1) If the rectangle is equilateral, then it is a square; The rectangle is equilateral, ∴ It is a square. (2) If the rectangle is equilateral, then it is a square; The rectangle is not equilateral, ∴ The rectangle is not a square. (3) If the rectangle is equilateral, then it is a square; It is a square, ∴ The rectangle is equilateral. (4) If the rectangle is equilateral, then it is a square; It is not a square, ∴ The rectangle is not equilateral. =6. HYPOTHETICAL ARGUMENTS REDUCED TO THE CATEGORICAL FORM.= The hypothetical syllogism so closely resembles the categorical that it may be changed to it by a slight alteration in the wording. After testing the hypothetical by its own rule, it may be expedient to reduce the argument to the categorical form, and subject it to a second test in which the categorical rules are applied. This reduction usually necessitates two steps; first, _change the propositions which represent the antecedent and consequent to a subject term and a predicate term respectively and then unite them to form the major premise_; second, _supply a new minor term, if necessary_.