Bochvar's algebras and corresponding propositional calculi
Viktor Finn, Revaz Grigolia · 1980
This is an abstract of the paper which is to appear in “Disallowance po neoclassicists logikam i teorii mnozhestv ” (“Nauka”). In [1] D. A. Bochvar formulated a 3-valued logic. He analyzed the paradoxes of Russel and Weyl, and by means of the logic he proved that the paradox formulae were meaningless. In this paper the class of algebras (Bn-algebras) corresponding to n-valued generalizations of the Bochovar’s 3-valued logic is investigated. The class is defined axiomatically. The axiomatization for Bochovar’s n-valued logic Bn is obtained on the basis of algebraic axiomatization. 1. A Bn-algebra (2 < n < ℵ0) is a universal algebra A = 〈A,∪,∩,∼, J0,..., Jn−1, 0, 1〉, where A is a nonempty set of elements, 0 and 1 are constant elements of A, ∪ and ∩ are binary operations on elements of A, and ∼, J0,..., Jn−1 are unary operations on elements of A obeying the following axioms: A1. x ∪ x = x A2. x ∪ y = y ∪ x A3. x ∪ (y ∪ z) = (x ∪ y) ∪ z A4. x ∩ (y ∪ z) = (x ∩ z) ∪ (x ∩ y) A5. ∼ ∼ x = x A6. ∼ 1 = 0 40 Viktor Finn and Revas Grigolia A7. ∼ (x ∪ y) = ∼ x ∩ ∼ y A8. 0 ∪ x = x A9. Jn−1Jix = Jix, 0 ≤ i ≤ n − 1 A10. J0Jix = ∼ Jix, 0 ≤ i ≤ n − 1 A11. JiJjx = 0, 0 < i < n − 1, 0 ≤ j ≤ n − 1 A12. Ji( ∼ x) = Jn−1−ix