If we put A for Neptune, B for planet, and C for “having retrograde motion,” then by the corresponding negative term c, we denote “not having retrograde motion.” The premises now fall into the forms
| A = AB | (1) |
|---|---|
| B = B c , | (2) |
and by substitution for B, exactly as before, we obtain
What is called in the old logic a particular conclusion may be deduced without any real variation in the symbols. Particular quantity is indicated, as before mentioned (p. 41), by joining to the term an indefinite adjective of quantity, such as some, a part of, certain, &c., meaning that an unknown part of the term enters into the proposition as subject. Considerable doubt and ambiguity arise out of the question whether the part may not in some cases be the whole, and in the syllogism at least it must be understood in this sense.7 Now, if we take a letter to represent this indefinite part, we need make no change in our formulæ to express the syllogisms Darii and Ferio. Consider the example—
| Some metals are of less density than water, | (1) |
|---|---|
| All bodies of less density than water will float upon the surface of water; hence | (2) |
| Some metals will float upon the surface of water. | (3) |
| Let | A = some metals, |
|---|---|
| B = body of less density than water, | |
| C = floating on the surface of water |
then the propositions are evidently as before,
| A = AB, | (1) | |
|---|---|---|
| B = BC; | (2) | |
| hence | A = ABC, | (3) |
Thus the syllogism Darii does not really differ from Barbara. If the reader prefer it, we can readily employ a distinct symbol for the indefinite sign of quantity.
| Let | P = some, |
|---|---|
| Q = metal, |
B and C having the same meanings as before. Then the premises become
| PQ = PQB, | (1) |
|---|---|
| B = BC; | (2) |
hence, by substitution, as before,
Except that the formulæ look a little more complicated there is no difference whatever.