| A = A b | (1) |
|---|---|
| C = C b . | (2) |
The reader will find it impossible by the rule of substitution to discover a relation between A and C. Three terms occur in the above premises, namely A, b, and C; but they are so combined that no term occurring in one has its exact equivalent stated in the other. No substitution can therefore be made, and the principle of the fifth rule of the syllogism holds true. Fallacy is impossible.
It would be a mistake, however, to suppose that the mere occurrence of negative terms in both premises of a syllogism renders them incapable of yielding a conclusion. The old rule informed us that from two negative premises no conclusion could be drawn, but it is a fact that the rule in this bare form does not hold universally true; and I am not aware that any precise explanation has been given of the conditions under which it is or is not imperative. Consider the following example:
| Whatever is not metallic is not capable of powerful magnetic influence, | (1) |
|---|---|
| Carbon is not metallic, | (2) |
| Therefore, carbon is not capable of powerful magnetic influence. | (3) |
Here we have two distinctly negative premises (1) and (2), and yet they yield a perfectly valid negative conclusion (3). The syllogistic rule is actually falsified in its bare and general statement. In this and many other cases we can convert the propositions into affirmative ones which will yield a conclusion by substitution without any difficulty.
To show this let
The premises readily take the forms
| b = bc , | (1) |
|---|---|
| A = A b , | (2) |
and substitution for b in (2) by means of (1) gives the conclusion
A = A bc . (3)
Our principle of inference then includes the rule of negative premises whenever it is true, and discriminates correctly between the cases where it does and does not hold true.
The paralogism, anciently called the Fallacy of Undistributed Middle, is also easily exhibited and infallibly avoided by our system. Let the premises be
| Hydrogen is an element, | (1) |
|---|---|
| All metals are elements. | (2) |
According to the syllogistic rules the middle term “element” is here undistributed, and no conclusion can be obtained; we cannot tell then whether hydrogen is or is not a metal. Represent the terms as follows
The premises then become
| A = AB, | (1) |
|---|---|
| C = CB. | (2) |