A set of five independent postulates for Boolean algebras, with application to logical constants
Henry Maurice Sheffer · Transactions of the American Mathematical Society · 1913
Postulate-sets for determining the class of Boolean algebrasf have been given by Schröder,^ Whttehead, § and Huntington.||Schroder's set of ten postulates assumes-in addition to an undefined class K, common to all these postulate-sets-an undefined dyadic relation, 4 > and Boole's 1f undefined binary AT-rules** of combination, + and X ; Whitehead's two sets, the first of thirteen, and the second of fifteen, postulates, and Huntington's first set, of ten postulates, assume the same undefined AT-rules of combination, + and X , which Huntington writes respectively © and o ; Huntington's second set, of nine postulates, assumes Schroder's undefined relation 4 > * Presented to the Society, December 31, 1912.f We employ the term Boolean algebras in its plural form for the following reasons: (1) none of the equivalent postulate-sets here referred to is in terms of its undefined entities onevalued ("categorical")-that is, each determines not a single algebra but a class of algebras; one should not speak, therefore, of "der identische Kalkül" (Schröder), "the Algebra of Symbolic Logic" (Whitehead), or "the algebra of logic" (Huntington); (2) Peano's Formulario Mathematico and Whitehead and Russell's Principia Mathematica, each of which includes, as a part, the algebras under consideration, have a far stronger title to the name "algebra of logic"; (3) "The Algebra of Symbolic Logic, viewed as a distinct algebra, is due to Boole" (Whitehead, loc.cit., p. 115)."This algebra in all its essential particulars was invented and perfected by Boole" (ib., p. 35, footnote).i Ernst Schröder: Vorlesungen über die Algebra der Logik (Exakte Logik), Erster Band, 1890.The postulates, under various names, are scattered throughout the volume; collected into one list by E. Müller: