Many persons may think that the choice of a symbol is a matter of slight importance or of mere convenience; but I hold that the common use of this sign = in so many different meanings is really founded upon a generalisation of the widest character and of the greatest importance--one indeed which it is a principal purpose of this work to explain. The employment of the same sign in different cases would be unphilosophical unless there were some real analogy between its diverse meanings. If such analogy exists, it is not only allowable, but highly desirable and even imperative, to use the symbol of equivalence with a generality of meaning corresponding to the generality of the principles involved. Accordingly De Morgan’s refusal to use the symbol in logical propositions indicated his opinion that there was a want of analogy between logical propositions and mathematical equations. I use the sign because I hold the contrary opinion. I conceive that the sign = as commonly employed, always denotes some form or degree of sameness, and the particular form is usually indicated by the nature of the terms joined by it. Thus “6,720 pounds = 3 tons” is evidently an equation of quantities. The formula - × - = + expresses the equivalence of operations. “Exogens = Dicotyledons” is a logical identity expressing a profound truth concerning the character and origin of a most important group of plants. We have great need in logic of a distinct sign for the copula, because the little verb *is* (or *are*), hitherto used both in logic and ordinary discourse, is thoroughly ambiguous. It sometimes denotes identity, as in “St. Paul’s is the *chef-d’œuvre* of Sir Christopher Wren;” but it more commonly indicates inclusion of class within class, or partial identity, as in “Bishops are members of the House of Lords.” This latter relation involves identity, but requires careful discrimination from simple identity, as will be shown further on. When with this sign of equality we join two nouns or logical terms, as in Hydrogen = The least dense element, we signify that the object or group of objects denoted by one term is identical with that denoted by the other, in everything except the names. The general formula A = B must be taken to mean that A and B are symbols for the same object or group of objects. This identity may sometimes arise from the mere imposition of names, but it may also arise from the deepest laws of the constitution of nature; as when we say Gravitating matter = Matter possessing inertia, Exogenous plants = Dicotyledonous plants, Plagihedral quartz crystals = Quartz crystals causing the plane of polarisation of light to rotate. We shall need carefully to distinguish between relations of terms which can be modified at our own will and those which are fixed as expressing the laws of nature; but at present we are considering only the mode of expression which may be the same in either case.