We often meet with a series of propositions describing the qualities or circumstances of the one same thing, and we may combine them all into one proposition by the process of substitution. This case is, in fact, that which Dr. Thomson has called “Immediate Inference by the sum of several predicates,” and his example will serve my purpose well.‍[63] He describes copper as “A metal—of a red colour—and disagreeable smell—and taste—all the preparations of which are poisonous—which is highly malleable—ductile—and tenacious—with a specific gravity of about 8.83.” If we assign the letter A to copper, and the succeeding letters of the alphabet in succession to the series of predicates, we have nine distinct statements, of the form A = AB (1) A = AC (2) A = AD (3) . . . A = AK (9). We can readily combine these propositions into one by substituting for A in the second side of (1) its expression in (2). We thus get

A = ABC,

and by repeating the process over and over again we obviously get the single proposition

A = ABCD . . . JK.

But Dr. Thomson is mistaken in supposing that we can obtain in this manner a definition of copper. Strictly speaking, the above proposition is only a description of copper, and all the ordinary descriptions of substances in scientific works may be summed up in this form. Thus we may assert of the organic substances called Paraffins that they are all saturated hydrocarbons, incapable of uniting with other substances, produced by heating the alcoholic iodides with zinc, and so on. It may be shown that no amount of ordinary description can be equivalent to a definition of any substance.

Fallacies.

I have hitherto been engaged in showing that all the forms of reasoning of the old syllogistic logic, and an indefinite number of other forms in addition, may be readily and clearly explained on the single principle of substitution. It is now desirable to show that the same principle will prevent us falling into fallacies. So long as we exactly observe the one rule of substitution of equivalents it will be impossible to commit a paralogism, that is to break any one of the elaborate rules of the ancient system. The one new rule is thus proved to be as powerful as the six, eight, or more rules by which the correctness of syllogistic reasoning was guarded.

It was a fundamental rule, for instance, that two negative premises could give no conclusion. If we take the propositions

Granite is not a sedimentary rock,(1)
Basalt is not a sedimentary rock,(2)

we ought not to be able to draw any inference concerning the relation between granite and basalt. Taking our letter-terms thus: