Ivan Ivanovich Zhegalkin

Ivan Ivanovich Schegalkin ( Иван Иванович Жегалкин; scientific transliteration Ivan Ivanovich Žegalkin ) (born 22 Julijul / August 3 1869greg in Mtsensk, .. † March 28, 1947 in Moscow) was a Russian logician and mathematician.

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

Propositional calculus

Schegalkins built his propositional calculus on the conjunction ' as multiplication and exclusive disjunction EITHER - OR as addition and formulated the following axiom system:

  • Associative: (ab) c = a ( bc)
  • Commutative: ab = ba
  • Neutral: 1a = a
  • Nihil: 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 " nihil " is superfluous and provable: 0a = ( a a ) a = aa aa = 0 The rule " self-inverse " does not apply in Boole's original algebra; he called and used only the remaining axioms. This rule, however, is crucial because it makes Boole's calculus, which also contains logically meaningless formulas, an independent logical calculus in which bear a logical meaning all formulas. One can find " self-inverse " in 1891 with Giuseppe Peano, but in which the second neutral law and the distributive missing. The complete system of axioms presented Schegalkin for the first time. By definition a = a results in a Boolean ring, the Stone 1934/1936 introduced, inspired by Schegalkin. By the definitions and results in a Boolean algebra, and in the modern sense.

Writings

With French review:

  • О технике вычислений предложений в символической логике ( Žegalkin, II: Sur le calcul of propositions dans la logique symbolique ), in: Matematicheskij Sbornik 34 (1927 ), 9-28.
  • Арифметизация Символической Логики ( Žegalkin, II: L' arithmetisation de la logique symbolique ), in: Matematicheskij Sbornik 35 (1928 ), 311-377.
Logical conjunction Marshall Harvey Stone
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