Inference with a Simple and a Partial Identity.
A form of reasoning somewhat different from that last considered consists in inference-between a simple and a partial identity. If we have two propositions of the forms
A = B,
B = BC,
we may then substitute for B in either proposition its equivalent in the other, getting in both cases A = BC; in this we may if we like make a second substitution for B, getting
A = AC.
Thus, since “The Mont Blanc is the highest mountain in Europe, and the Mont Blanc is deeply covered with snow,” we infer by an obvious substitution that “The highest mountain in Europe is deeply covered with snow.” These propositions when rigorously stated fall into the forms above exhibited.
This mode of inference is constantly employed when for a term we substitute its definition, or vice versâ. The very purpose of a definition is to allow a single noun to be employed in place of a long descriptive phrase. Thus, when we say “A circle is a curve of the second degree,” we may substitute a definition of the circle, getting “A curve, all points of which are at equal distances from one point, is a curve of the second degree.” The real forms of the propositions here given are exactly those shown in the symbolic statement, but in this and many other cases it will be sufficient to state them in ordinary elliptical language for sake of brevity. In scientific treatises a term and its definition are often both given in the same sentence, as in “The weight of a body in any given locality, or the force with which the earth attracts it, is proportional to its mass.” The conjunction or in this statement gives the force of equivalence to the parenthetic phrase, so that the propositions really are
Weight of a body = force with which the earth attracts it.
Weight of a body = weight, &c. proportional to its mass.
A slightly different case of inference consists in substituting in a proposition of the form A = AB, a definition of the term B. Thus from A = AB and B = C we get A = AC. For instance, we may say that “Metals are elements” and “Elements are incapable of decomposition.”