# The Development of Symbolic Logic: Ideography and Calculus

The objects of a complete logical symbolism are: firstly, to avoid this disadvantage by providing an _ideography_, in which the signs represent ideas and the relations between them _directly_ (without the intermediary of words), and secondly, so to manage that, from given premises, we can, in this ideography, draw all the logical conclusions which they imply by means of rules of transformation of formulas analogous to those of algebra,–in fact, in which we can replace reasoning by the almost mechanical process of calculation. This second requirement is the requirement of a _calculus__ntiocinator_. It is essential that the ideography should be complete, that only symbols with a well-defined meaning should be used–to avoid the same sort of ambiguities that words have–and, consequently,–that no suppositions should be introduced implicitly, as is commonly the case if the meaning of signs is not well defined. Whatever premises are necessary and sufficient for a conclusion should be stated explicitly.Besides this, it is of practical importance,–though it is theoretically irrelevant,–that the ideography should be concise, so that it is a sort of stenography.The merits of such an ideography are obvious: rigor of reasoning is ensured by the calculus character; we are sure of not introducing unintentionally any premise; and we can see exactly on what propositions any demonstration depends.We can shortly, but very fairly accurately, characterize the dual development of the theory of symbolic logic during the last sixty years as follows: The _calculus__ntiocinator_ aspect of symbolic logic was developed by Boole, De Morgan, Jevons, Venn, C. S. Peirce, Schroder, Mrs. Ladd-Franklin and others; the _lingua__characteristic_ aspect was developed by Frege, Peano and Russell. Of course there is no hard and fast boundary-line between the domains of these two parties. Thus Peirce and Schroder early began to work at the foundations of arithmetic with the help of the calculus of relations; and thus they did not consider the logical calculus merely as an interesting branch of algebra. Then Peano paid particular attention to the calculus aspect of his symbolism. Frege has remarked that his own symbolism is meant to be a _calculus__ntiocinator_ as well as a _lingua__characteristic_, but the using of Frege’s symbolism as a calculus would be rather like using a three-legged stand-camera for what is called “snap-shot” photography, and one of the outwardly most noticeable things about Russell’s work is his combination of the symbolisms of Frege and Peano in such a way as to preserve nearly all of the merits of each.The present work is concerned with the _calculus__ntiocinator_ aspect, and shows, in an admirably succinct form, the beauty, symmetry and simplicity of the calculus of logic regarded as an algebra. In fact, it can hardly be doubted that some such form as the one in which Schroder left it is by far the best for exhibiting it from this point of view.8 The content of the present volume corresponds to the two first volumes of Schroder’s great but rather prolix treatise.9 Principally owing to the influence of C. S. Peirce, Schroderdeparted from the custom of Boole, Jevons, and himself (1877), which consisted in the making fundamental of the notion of _equality_, and adopted the notion of _subordination_ or _inclusion_ as a primitive notion. A more orthodox Boolian exposition is that of Venn, 10 which also contains many valuable historical notes.Symbolic Logic, London, 1881; 2nd ed., 1894.We will finally make two remarks.When Boole (cf. SS0.2 below) spoke of propositions determining a class of moments at which they are true, he really (as did MacColl ) used the word “proposition” for what we now call a “propositional function”. A “proposition” is a thing expressed by such a phrase as “twice two are four” or “twice two are five”, and is always true or always false. But we might seem to be stating a proposition when we say: “Mr. William Jennings Bryan is Candidate for the Presidency of the United States”, a statement which is sometimes true and sometimes false. But such a statement is like a mathematical _function_ in so far as it depends on a _variable_–the time. Functions of this kind are conveniently distinguished from such entities as that expressed by the phrase “twice two are four” by calling the latter entities “propositions” and the former entities “propositional functions”: when the variable in a propositional function is fixed, the function becomes a proposition. There is, of course, no sort of necessity why these special names should be used; the use of them is merely a question of convenience and convention.In the second place, it must be carefully observed that, in SS0.13, 0 and 1 are not _defined_ by expressions whose principal copulas are relations of inclusion