# The Definition of Equality in Mathematics

A definition is simply the convention that, for the sake of brevity or some other convenience, a certain new sign is to be used instead of a group of signs whose meaning is already known. Thus, it is the sign of _equality_ that forms the principal copula. The theory of definition has been most minutely studied, in modern times by Frege and Peano.Philip E. B. Jourdain.Girton, Cambridge. England. Contents* 1 Preface * 1 Bibliography * 1.1 Introduction * 1.2 The Two Interpretations of the Logical Calculus * 1.3 Relation of Inclusion * 1.4 Definition of Equality * 1.5 Principle of Identity * 1.6 Principle of the Syllogism * 1.7 Multiplication and Addition * 1.8 Principles of Simplification and Composition * 1.9 The Laws of Tautology and of Absorption * 1.10 Theorems on Multiplication and Addition * 1.11 The First Formula for Transforming Inclusions into Equalities * 1.12 The Distributive Law * 1.13 Definition of 0 and 1 * 1.14 The Law of Duality * 1.15 Definition of Negation * 1.16 The Principles of Contradiction and of Excluded Middle * 1.17 Law of Double Negation * 1.18 Second Formulas for Transforming Inclusions into Equalities * 1.19 The Law of Contraposition * 1.20 Postulate of Existence * 1.21 The Development of 0 and of 1 * 1.22 Properties of the Constituents * 1.23 Logical Functions * 1.24 The Law of Development * 1.25 The Formulas of De Morgan * 1.26 Disjunctive Sums * 1.27 Properties of Developed Functions * 1.28 The Limits of a Function * 1.29 Formula of Poretsky. * 1.30 Schroder’s Theorem. * 1.31 The Resultant of Elimination * 1.32 The Case of Indetermination * 1.33 Sums and Products of Functions * 0.34 The Expression of an Inclusion by Means of an Indeterminate * 0.35 The Expression of a Double Inclusion by Means of an Indeterminate * 0.36 Solution of an Equation Involving One Unknown Quantity * 0.37 Elimination of Several Unknown Quantities * 0.38 Theorem Concerning the Values of a Function * 0.39 Conditions of Impossibility and Indetermination * 0.40 Solution of Equations Containing Several Unknown Quantities * 0.41 The Problem of Boole * 0.42 The Method of Poretsky * 0.43 The Law of Forms * 0.44 The Law of Consequences * 0.45 The Law of Causes * 0.46 Forms of Consequences and Causes * 0.47 * 0.48 The Geometrical Diagrams of Venn * 0.49 The Logical Machine of Jevons * 0.50 Table of Consequences * 0.51 Table of Causes * 0.52 The Number of Possible Assertions * 0.53 Particular Propositions * 0.54 Solution of an Inequation with One Unknown * 0.55 System of an Equation and an Inequation * 0.56 Formulas Peculiar to the Calculus of Propositions * 0.57 Equivalence of an Implication and an Alternative * 0.58 Law of Importation and Exportation * 0.59 Reduction of Inequalities to Equalities * 0