(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_.
Illustrations of Reduction; and Comparison of Hypothetical and Categorical Fallacies:
(1) If it has rained, the ground is damp; It has rained, ∴ The ground is damp.
M G A The falling rain makes the ground damp, ―――――――――――――――― ―――――――――――
S M A In this case rain has fallen, ―――――――――――――――――
S G A ∴ In this case the ground is damp ground. ――――――――――――
It is seen that the argument in the hypothetical form is valid as the minor premise affirms the antecedent. Reducing to the categorical gives { A to the argument the mode { A in the first figure which we know to be { A valid.
(2) If one were wise, he would study; But you will not study, ∴ You are not wise.
G M A A wise person would study, ―――――――――――
In the hypothetical form the argument is valid since the minor premise { A denies the consequent. Reducing to the categorical gives mood { E in { E the second figure. This is valid.
(3) If the wind blows from the south, it will rain; But the wind is not blowing from the south, Hence it is not going to rain.
S M E This wind is not a south wind, ――――――――― ――――――――――
S G E ∴ This wind will not bring rain. ――――――――― ――――――――――
Hypothetically considered, the minor premise denies the antecedent and consequently the argument is invalid. Reducing to the categorical form, it is found that the major term is distributed in the conclusion, but is not distributed in the major premise; hence the fallacy of _illicit major_ is committed.