These two so-called rules are not properly speaking rules for the validity of an argument. They simply serve to define the syllogism as a particular form of argument. A reasoning which does not fulfil these conditions may be formally valid, but we do not call it a syllogism.[307] The four rules that follow 288 are really rules in the sense that if, when we have got the reasoning into the form of a syllogism, they are not fulfilled, then the reasoning is invalid.[308]
[307] For example, B is greater than C, A is greater than B, therefore, A is greater than C.
Here is a valid reasoning which consists of three propositions. But it contains more than three terms; for the predicate of the second premiss is “greater than B,” while the subject of the first premiss is “B.” It is, therefore, as it stands, not a syllogism. Whether reasonings of this kind admit of being reduced to syllogistic form is a problem which will be discussed [subsequently].
[308] Apparent exceptions to these rules will be shewn in sections [205] and [206] to result from the attempt to apply them to reasonings which have not first been reduced to syllogistic form.
(3) No one of the three terms of a syllogism may be used ambiguously; and the middle term must be distributed once at least in the premisses.
This rule is frequently given in the form: “The middle term must be distributed once at least, and must not be ambiguous.” But it is obvious that we have to guard against ambiguous major and ambiguous minor as well as against ambiguous middle. The fallacy resulting from the ambiguity of one of the terms of a syllogism is a case of quaternio terminorum, that is, a fallacy of four terms.
The necessity of distributing the middle term may be illustrated by the aid of the Eulerian diagrams. Given, for instance. All P is M and All S is M, we may have any one of the five following cases:—
Here all the five relations that are à priori possible between S and P are still possible. We have, therefore, no conclusion.
If in a syllogism the middle term is distributed in neither premiss, we are said to have a fallacy of undistributed middle.