In developing the results of the Principle of Inference we require to use an appropriate language of signs. It would indeed be quite possible to explain the processes of reasoning by the use of words found in the dictionary. Special examples of reasoning, too, may seem to be more readily apprehended than general symbolic forms. But it has been shown in the mathematical sciences that the attainment of truth depends greatly upon the invention of a clear, brief, and appropriate system of symbols. Not only is such a language convenient, but it is almost essential to the expression of those general truths which are the very soul of science. To apprehend the truth of special cases of inference does not constitute logic; we must apprehend them as cases of more general truths. The object of all science is the separation of what is common and general from what is accidental and different. In a system of logic, if anywhere, we should esteem this generality, and strive to exhibit clearly what is similar in very diverse cases. Hence the great value of *general symbols* by which we can represent the form of a reasoning process, disentangled from any consideration of the special subject to which it is applied. The signs required in logic are of a very simple kind. As sameness or difference must exist between two things or notions, we need signs to indicate the things or notions compared, and other signs to denote the relations between them. We need, then, (1) symbols for terms, (2) a symbol for sameness, (3) a symbol for difference, and (4) one or two symbols to take the place of conjunctions. Ordinary nouns substantive, such as *Iron*, *Metal*, *Electricity*, *Undulation*, might serve as terms, but, for the reasons explained above, it is better to adopt blank letters, devoid of special signification, such as A, B, C, &c. Each letter must be understood to represent a noun, and, so far as the conditions of the argument allow, *any noun*. Just as in Algebra, *x*, *y*, *z*, *p*, *q*, &c. are used for *any quantities*, undetermined or unknown, except when the special conditions of the problem are taken into account, so will our letters stand for undetermined or unknown things. These letter-terms will be used indifferently for nouns substantive and adjective. Between these two kinds of nouns there may perhaps be differences in a metaphysical or grammatical point of view. But grammatical usage sanctions the conversion of adjectives into substantives, and *vice versâ*; we may avail ourselves of this latitude without in any way prejudging the metaphysical difficulties which may be involved. Here, as throughout this work, I shall devote my attention to truths which I can exhibit in a clear and formal manner, believing that in the present condition of logical science, this course will lead to greater advantage than discussion upon the metaphysical questions which may underlie any part of the subject.