# The Philosophical Review/Volume 1/Summary: Johnson - The Logical Calculus - Part 1

This is the first part of a two-part article; the second part is Summary: Johnson - The Logical Calculus - Part 2
I. Begins by attempting to enumerate briefly the principles common to every species of symbolic calculus: (1) the representation of symbols, (2) the convention of permanence of import, (3) the possibility of equivalence, (4) the method of substitution, (5) the prepositional import of equivalence, (6) the inferential relation between equivalence, (7) the distinction between universal and particular symbols, (8) the applicational interpretation of universals, (9) the force of the bracket, and (10) the postulate of homogeneity. II. ($\S \;$2) The Synthesis of Propositions. A. The inferential mode of synthesis has been prominent in traditional synthesis, but a more general view of synthesis is here taken, in which inference will be shown to be dependent on and subordinate to pure synthesis. B. The conception of a proposition in general is indicated by the article not. III. ($\S \;$5) All that formal logic can do in the way of synthesis of propositions is contained in the laws regulating the use of the words and and not, i.e. logic is limited to a development of the conceptions of pure synthesis and pure negation. The fundamental laws or axioms that regulate these operations must now be given. IV. The Fundamental Laws of Prepositional Synthesis ($\S \;$6). 1. The Commutative Law: xy = yx. 2. The Associative Law: xy.z = x.yz. 3. The Law of Tautology: xx = x. 4. The Law of Reciprocity: x = x. 5. The Law of Dichotomy: x = xy xy. 1 says that the order of pure synthesis is indifferent; 2, that the mode of grouping in pure synthesis is indifferent; 3, that the mere repetition of a proposition does not in any way add to its force or alter it; 4, that the denial ​of the denial of a proposition is equivalent to its affirmation; 5, that the denial of any proposition is equivalent to the denial of its conjunction with any other proposition, together with the denial of its conjunction with the contradictory of that other proposition. V. ($\S \;$8) These results can be put into more familiar forms. We have two fundamental types of synthesis, — conjunctive and disjunctive. Each of these types has four varieties involving x, y, or their contradictories, viz. conjunctives xy, xy, xy, xy; disjunctives xy, xy, xy, xy. In our view there is no essential difference between the disjunctive, hypothetical, and alternative forms. This is shown symbolically. VI. ($\S \;$9) Other Interpretations of the Disjunctive, Hypothetical, and Alternative Mode of Synthesis. VII. Other Modes of Synthesis than not and and, e.g. if and or ($\S \;$10). VIII. ($\S \;$11) The Primary Analysis of Propositions. The molecular proposition contains two sorts of elements, a subjective and a predicative term. The usual analysis of the predicate term into copula and predicate term is a verbal device, devoid of logical significance. IX. ($\S \;$12) Synthesis of Propositions as modified by their Analysis: (1) of propositions containing a common individual derivative subject name; (2) of singular propositions containing a common predication, but different subjects; (3) of propositions which refer to the same aggregate subjects. X. ($\S \;$13) Foregoing Principles applied to the Logic of Relatives. XI. Criticism of the preceding analysis. If the above analysis is correct, it will establish the point that all the familiar methods of formal logic and the less familiar results of relative logic depend, not on the peculiar relation of subject and predication, but on the proportional synthesis involved in the quantitative element of the universal or particular judgment.