Ivan Ivanovich Schegalkin

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Ivan Ivanovich Zhegalkin (Иван Иванович Жегалкин;. Scientific transliteration Ivan Ivanovich Žegalkin ) (born July 22 . Jul / 3. August  1869 greg. In Mtsensk ; † 28. March 1947 in Moscow ) was a Russian logician and mathematician.

Могила математика Ивана Жегалкина.JPG

Schegalkin was a lecturer from 1902 to 1911 and a professor at Moscow University from 1917 to 1947 . He was one of the founders of the Soviet school of mathematical logic. In 1927 he developed an algebraic logic calculus that optimized the logic of George Boole and was later referred to as the Boolean ring .

Propositional calculus

Schegalkin built his propositional calculus on the conjunction AND as multiplication and the exclusive disjunction EITHER-OR as addition and formulated the following system of axioms:

  • associative: (ab) c = a (bc)
  • commutative: ab = ba
  • neutral: 1a = a
  • nihilated: 0a = 0
  • idempotent: aa = a
  • associative: (a + b) + c = a + (b + c)
  • commutative: a + b = b + a
  • neutral: 0 + a = a
  • self-inverse: a + a = 0
  • distributive: (a + b) c = ac + bc

The rule "nihilated" is superfluous and provable: 0a = (a + a) a = aa + aa = 0. The "self-inverse" rule does not hold in Boole's original algebra ; he only named and used the remaining axioms. However, this rule is decisive, because it turns Boolean algebra, which also contains logically meaningless formulas, into an independent logic calculus in which all formulas have a logical meaning. One finds "self-inverse" as early as 1891 in Giuseppe Peano , in which, however, the second neutral law and the distributive law are missing. Schegalkin established the complete system of axioms for the first time. The definition –a = a creates a so-called Boolean ring with the subtraction ab = a + b. The definitions and and create a Boolean algebra in the modern sense.

Fonts

With French review:

  • О технике вычислений предложений в символической логике (Žegalkin, II: Sur le calcul des propositions dans la logique symbolique), in: Matematicheskij Sbornik 34 (1927), 9–28.27 [1]
  • Арифметизация Символической Логики (Žegalkin, II: L'arithmetisation de la logique symbolique), in: Matematicheskij Sbornik 35 (1928), 311–377. [2]

literature

  • DA Vladimirov: Boolean algebras . Edited in German by G. Eisenreich. Berlin 1972.

Individual evidence

  1. ↑ Schegalkin (source I see below) in: Matematicheskij Sbornik 34 (1927), 11f.
  2. ↑ Peano: Opere scelte II, rules p. 107 (3rd) (5th), p. 109 (2nd), p. 125 (25th) - (28th) notated with o instead of +; 1895 expanded with neutral law in: Peano: Opere scelte II, rules p. 177f (6.) (10.), P. 181f (2.) (4 '.) (25.) - (28.)
  3. ↑ Schegalkin (source II see below) in: Matematicheskij Sbornik 35 (1928), 322, definition of subtraction.
personal data
SURNAME Schegalkin, Ivan Ivanovich
ALTERNATIVE NAMES Жегалкин, Иван Иванович (Russian); Žegalkin, Ivan Ivanovič (scientific transliteration)
BRIEF DESCRIPTION Russian logician and mathematician
DATE OF BIRTH August 3, 1869
PLACE OF BIRTH Mtsensk
DATE OF DEATH March 28, 1947
Place of death Moscow