Sometimes, but much less frequently, we require a symbol to indicate difference or the absence of complete sameness. For this purpose we may generalise in like manner the symbol ~, which was introduced by Wallis to signify difference between quantities. The general formula B ~ C denotes that B and C are the names of two objects or groups which are not identical with each other. Thus we may say Acrogens ~ Flowering plants. Snowdon ~ The highest mountain in Great Britain. I shall also occasionally use the sign ᔕ to signify in the most general manner the existence of any relation between the two terms connected by it. Thus ᔕ might mean not only the relations of equality or inequality, sameness or difference, but any special relation of time, place, size, causation, &c. in which one thing may stand to another. By A ᔕ B I mean, then, any two objects of thought related to each other in any conceivable manner. *General Formula of Logical Inference.* The one supreme rule of inference consists, as I have said, in the direction to affirm of anything whatever is known of its like, equal or equivalent. The *Substitution of Similars* is a phrase which seems aptly to express the capacity of mutual replacement existing in any two objects which are like or equivalent to a sufficient degree. It is matter for further investigation to ascertain when and for what purposes a degree of similarity less than complete identity is sufficient to warrant substitution. For the present we think only of the exact sameness expressed in the form A = B. Now if we take the letter C to denote any third conceivable object, and use the sign ᔕ in its stated meaning of *indefinite relation*, then the general formula of all inference may be thus exhibited:-- From A = B ᔕ C we may infer A ᔕ C or, in words--*In whatever relation a thing stands to a second thing, in the same relation it stands to the like or equivalent of that second thing.* The identity between A and B allows us indifferently to place A where B was, or B where A was; and there is no limit to the variety of special meanings which we can bestow upon the signs used in this formula consistently with its truth. Thus if we first specify only the meaning of the sign ᔕ, we may say that if *C is the weight of B*, then *C is also the weight of A*. Similarly If C is the father of B, C is the father of A; If C is a fragment of B, C is a fragment of A; If C is a quality of B, C is a quality of A; If C is a species of B, C is a species of A; If C is the equal of B, C is the equal of A; and so on *ad infinitum*. We may also endow with special meanings the letter-terms A, B, and C, and the process of inference will never be false. Thus let the sign ᔕ mean “is height of,” and let A = Snowdon, B = Highest mountain in England or Wales, C = 3,590 feet;