The Philosophical Review/Volume 1/Summary: Johnson - The Logical Calculus - Part 2

by Anonymous

The Logical Calculus. II. W. E. Johnson. Mind, New Series, I, 2, pp. 235-250.

The aim of the paper is not to add one more to the numerous systems of notation and symbolic method that have been already worked out more or less independently, but rather to bring out some underlying principles and assumptions which belong equally to the ordinary Formal Logic, to Symbolic Logic, and to the so-called Logic of Relatives. At the same time, it is hoped to present the work of different writers or different branches in a more systematic and comprehensive form than has hitherto been done. The Principle of Formal Inference. The principle of formal inference is expressed in the formula "(b and c) implies c"; i.e. a conclusion formally reached is simply a determinant of the data expressed in the premises. Notation for Propositional Synthesis. As determinative synthesis has always been represented by multiplication, we shall use the symbol a.b to stand for "a and b" The symbol a$\cdot$b may stand for "a or b." Notation for the Molecular Analyzed Proposition. J. adopts a notation suggested by Mr. Peirce's paper on the "Logic of Relatives " in the Johns Hopkins Studies in Logic: "We may express molecular propositions of any order as follows: p_x, p_xy, p_xyz, p_xyz0, etc., — representing such propositions as 'Coal is produced,' 'Coal is produced in England,' 'Coal was produced in England in 1890, etc" Notation for the Synthesis of Molecular Propositions. p_x$\cdot$q_x = (p$\cdot$q)_x : p_x$\cdot$q_x = (p$\cdot$q)_x. Then follow symbols for The Synthesis of Singly-Quantitative Propositions, for the Transition from the Synthesis of Unanalyzed Propositions to Multiple Quantification, for Complex Combinations involving Multiple Quantifications. The final outcome of this method of notation, according to J., is the same as that adopted at the end of Mr. Peirce's paper. The chief object of the paper is to exhibit the unity of the whole Logical Calculus, by showing its dependence on the single group of fundamental laws regulating the pure synthesis and pure negation of propositions.