Therefore it has not rained.
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 isvalid. 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;