BOCHVAR S ALGEBRAS AND CORRESPONDING PROPOSITIONAL CALCULI

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1 Bulletin of the Section of Logic Volume 9/1 (1980), pp reedition 2010 [original edition, pp ] Viktor Finn Revas Grigolia BOCHVAR S ALGEBRAS AND CORRESPONDING PROPOSITIONAL CALCULI 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 (B n -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 B n is obtained on the basis of algebraic axiomatization. 1. A B n -algebra (2 < n < ℵ 0 ) is a universal algebra A = A,,,, J 0,..., J, 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, J 0,..., J 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

2 40 Viktor Finn and Revas Grigolia A7. (x y) = x y A8. 0 x = x A9. J J i x = J i x, 0 i n 1 A10. J 0 J i x = J i x, 0 i n 1 A11. J i J j x = 0, 0 < i < n 1, 0 j n 1 A12. J i ( x) = J i x A13. J i x = (J 0 x... J i 1 x J i+1 x... J x) A14. J i x J i x = 1, 0 i n 1 A15. (J i x J k x) J i x = J i x, 0 i, k n 1 A16. x Jix = x, n 1 i n 1 i A17. J k (x y) = k (J k x J j y) k (J i x J k y) 0 k < [ n 2 ] j=0 A18. J k (x y) = (J i x J k y) (J k x J i y) i=1 (J i x J k y) (J k x J i y) (J i x J k y) (J k x J i y), n 1 k [ n+1 A19. ( i)(0 i n 1)J i x = J i y x = y [m] is the largest integer k such that k m. B n -algebras are quasi-lattices in the sense of P lonka [2] with the operation of involution for which De Morgan axioms hold, and with unary J-operations J 0,..., J. 1 The algebra B n = R n,,,, J 0,..., J, 0, 1, where R n = {0,,..., n 2, 1}, x = 1 x, x y = min(max(x, y), max( x, x), max( { 1, x = 1 y, y)), x y = max(min(x, y), min(, x), min( y, y)), J i x = 0, x i, 0 i n 1, is an example of the B n -algebra. Proposition. The class of all B n -algebras is a quasi-variety but it is not a variety. x y iff J i (x y) = J i x [ n+1 2 ] i n 1, J j (x y) = J j y 0 j < [ n 2 ]. 2 ] Theorem 1.1. The relation is a partially ordered relation on A.

3 Bochvar s Algebras and Corresponding Propositional Calculi 41 A subset F of the set A is a filter of the B n -algebra A iff (i) 1 F, (ii) if x, y F then x y F, (iii) if x F and x y then y F, (iv) if x F then J x F. Theorem 1.2. If F is a filter, then the relation R on A defined by xry iff J i x J i y, J i x J i y F [ n+1 2 ] 1 n 1, J jx J j y, J j x J j y F, 0 j < [ n 2 ], is a congruence relation. Let us consider the algebra B m = R m,,,, J 0,..., J, 0, 1. Let f be a mapping of the set {0, 1,..., m 1} into the set {0, 1,..., } (m < n) such that (1) f(0) = 0, (2) f(m 1) = n 1, (3) x, y {0,..., m 1} x y = f(x) f(y), (4) f(m 1 i) = n 1 f(i) where 0 i m 1. From the definition of f it follows that such f does not exist if m is odd and n is odd and n is even. The algebra B f m = R m,,,, J 0,..., J f(1),..., J f(2),..., J f(m 2),..., J, 0, 1 where J f(i) = J i x for i {0,..., m 1} and J k x = 0 for k {0,..., n 1} {0,..., m 1} is a B n -algebra. Lemma 1.3. If the filter F is maximal, then A/F is isomorphic to B f m for suitable f and m, where 2 m n if n is odd, and m = 2k, 2 m n if n is even. Representation theorem. Every B n -algebra A is isomorphic to the subdirect product of algebras B f m, where 2 m n if n is odd, and m = 2k, 2 m n is n is even. The formulae of the logic B n are constructed by means of propositional variables and the connectives,,, J 0,..., J (where, are binary and, J 0,..., J are unary) in the usual way. We shall denote them by α, β, γ.... Formulae of the form J i α, J i α will be denoted by ξ, η, ζ,.... We introduce the following abbreviations: α β α β, 0 J α J α, 1 J α J α, α β ((J i α J i β) (J i β J i α)). Now we shall construct the calculi B n by giving a finite number of axiom schemes and inference rules of modus ponens:

4 42 Viktor Finn and Revas Grigolia B n 1. (α α) α B n 2. (α β) (β α) B n 3. (α (β γ) ((α β) γ) B n 4. (α (β α)) ((α β) (α γ)) B n 5. ( (α)) α B n 6. ( J α J α) ( J α J α) B n 7. (α β) ( α β) B n 8. (( J α J α) β) β B n 9. J ξ ξ B n 10. J 0 ξ ξ B n 11. J i ξ ( J α J α), 0 < i < n 1 B n 12. J i ( α) J i α, 0 i n 1 B n 13. J i α (J 0 α... J i 1 α J i+1 α... J α) 0 i n 1 B n 14. (J i α J i α) (J α J α), 0 i < n 1 B n 15. ((J i α J k β) J i α) J i α, 0 i, k n 1 B n 16. (α J i α) α, n 1 i n 1 i. B n 17. J k (α β) = k (J k α J j β) k (J i α J k β), 0 k < [ n 2 ] B n 18. j=0 J k (α β) = (J i α J k β) (J k α J i β) (J i α J k β) (J k α J i β) (J i α J k β) (J k α J i β), n 1 k [ n+1 2 ]. B n 19. ξ (η ζ) B n 20. (ξ (η ζ)) ((ξ η) (ξ ζ)) B n 21. (ξ η) ξ B n 22. (ξ η) η B n 23. (ξ η) ((ξ ζ) (ξ (η ζ))) B n 24. ξ (ξ η) B n 25. η (ξ η) B n 26. (ξ ζ) ((η ζ) ((ξ η) ζ))) B n 27. (ξ η) ( η ξ) B n 28. ξ ξ B n 29. ξ ξ

5 Bochvar s Algebras and Corresponding Propositional Calculi 43 Inference rule: α, α β β A formula α is said to be a tautology if α considered as an algebraic polinom has the value 1 for each assignment of variables by elements of the algebra B n. Completeness theorem. For each formula α, α is a theorem B n ( B) iff α is a tautology. Note that the Bochvar s 3-valued logic has already been formalized in several different ways [2,3,5]. Some authors treat this logic as the nonsenselogic or the logic of significance. References [1] D. A. Bochvar, Ob odnom trojokhznachnom ischislenii i ego primenenii k analizu paradoksov klassicheskogo rasshirennogo phunktisionalnego ischislenia, Matematicheski sbornik 4 (1938), no 2, pp [2] J. P lonka, On distributive quasi-lattices, Fundamenta Mathematicae LX (1967), pp [3] V. K. Finn, On the functional properties and axiomatization of D. A. Bochvar s three-valued logic, IV International Congress for Logic, Methodology and Philosophy of Science, Bucharest (1971), pp [4] Z. Goddard and R. Routley, The logic of significance and context, N. Y. (1973), v. 1. [5] K. Piróg-Rzepecka, Systems of nonsense-logics, Warszawa- Wroc law, Department of Semiotic Problems of Informatics of All-Union Institute of Scientific and Technical Information, the Academy of Science of USSR, Moscov Department of Logic The Institute of Cybernetics of the Academy of Sciences of the Georgian SSR, Tbilissi