Minimal Logic

Minimal Logic 

a logical system that is a weakening of intuitionistic logic and constructive logic as a result of the exclusion from the postulates of the formula ┐A⊃(A ⊃ B ), which can be interpreted as “from a contradiction follows anything.” In spite of the unprovability of this logical principle and, moreover, of the formula ┐┐A ⊃ A (”law of removal of double negation”), in the minimal prepositional calculus (A. N. Kolmogorov, 1925; Norwegian logician I. Johansson, 1936) negations can be proved by contradiction on the basis of the “law of reduction to absurdity”: (A ⊃ B) ⊃ (A ⊃ ┐B) ⊃ ┐A).

This system may be expanded in the usual way to the minimal predicate calculus, which plays an important role in studies on the foundations of mathematics. Its logical methods (although not explicitly stated) are used, for example, in proofs of consistency of classical arithmetic, which were presented by the German logicians G. Gentzen (1936, 1938) and K. Schutte (1951) and by P. S. Novikov (1943). This calculus is also used as a logical foundation for metatheory in works on the ultraintuitionistic foundations of mathematics. A weakening (restriction) of minimal logic by excluding from the axioms the law of reduction to absurdity leads to positive logic.

REFERENCES

Kolmogorov, A. N. “O printsipe ’tertium non datur.’ “Matematicheskii sbornik, 1925, vol. 32, issue 4, pp. 646–67.
Kleene, S. C. Vvedenie v metamatematiku. Moscow, 1957. Pages 94, 490–91. (Translated from English).
Johansson, I. “Der Minimalkalkiil, ein reduzierter Formalismus.” Compositio mathematica, 1937, vol. 4, fasc. 1.
Wajsberg, M. “Untersuchungen liber den Aussagenkalkiil von A. Heyting.” Wiadomosci Mathematyczne, 1939, vol. 46.

IU. A. GASTEV