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August 17 , 2008

The Logic of Algebra


The algebra of logic was founded by George Boole (1815-1864); it was developed and perfected by Ernst Schroder (1841-1902) The fundamental laws of this calculus were devised to express the principles of reasoning, the laws of thought But this calculus may be considered from the purely formal point of view, which is that of mathematics, as an algebra based upon certain principles arbitrarily laid down It belongs to the realm of philosophy to decide whether, and in what measure, this calculus corresponds to the actual operations of the mind, and is adapted to translate or even to replace argument; we cannot discuss this point here The formal value of this calculus and its interest for the mathematician are absolutely independent of the interpretation given it and of the application which can be made of it to logical problems In short, we shall discuss it not as logic but as algebra.

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  • Preface
  • Bibliography
  • Introduction
  • The Two Interpretations of the Logical Calculus
  • Relation of Inclusion
  • Definition of Equality
  • Principle of Identity
  • Principle of the Syllogism
  • Multiplication and Addition
  • Principles of Simplification and Composition
  • The Laws of Tautology and of Absorption
  • Theorems on Multiplication and Addition
  • The First Formula for Transforming Inclusions into Equalities
  • The Distributive Law
  • Definition of 0 and 1
  • The Law of Duality
  • Definition of Negation
  • The Principles of Contradiction and of Excluded Middle
  • Law of Double Negation
  • Second Formulas for Transforming Inclusions into Equalities
  • The Law of Couturat 
  • Postulate of Existence
  • The Development of 0 and of 1
  • Properties of the Constituents
  • Logical Functions
  • The Law of Development  
  • The Formulas of De Morgan
  • Disjunctive Sums
  • Properties of Developed Functions
  • The Limits of a Function
  • Formula of Poretsky     
  • Schroder's Theorem
  • The Resultant of Elimination 
  • The Case of Indetermination
  • Sums and Products of Functions
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