Quasi‐Boolean Algebras, Empirical Continuity and Three‐Valued Logic J. P. Cleave in Bristol (Great Britain)

Abstract

Z&chr. f. matb, Lol?i& und (hundlagcn d. M d h . Bd.82, S. 481-600 (1976) QUASI-B0OI;EAN ALGEBRAS, EMPIRICAL CONTINUITY AND THREE-VALUED LOGIC by J. P. CLEAVE in Bristol (Great Britain) rlnl Klg. 1We introduce QBA's by means of KORNER'S theory of inexact classes ([7] and [2]). Let A be a fixed domain of objects. By identifying classes with characteristic functions, a subclass of A is usually conceived as a function q~: A 4 (0, 1}, an element x of A being "in" the class if y(x) = 1 and excluded from the class if y(x) = 0. For the study of inexactness a third entity n can be introduced and an inexact class (or, rather, its characteristic function) conceived as a function q~: A --f (0, n, 1); x is in the class if ~( x ) = 1, excluded from the class if ~( x ) = 0, and is a neutral c m e if ~( x ) = 11. The subclasses of A form a Boolean algebra, C ( A ) , under the operat,ions of union, intersection and complementation : these operations have also been defined for inexact classes ([2], [7]) and their definitions can be rephrased in the following way. Let K, = ((0, n, 1); 0, 1, W, n, ') where the functions W, n, ' are defined by the tables shown in Fig. 1. Then the operations of union W, intersection n, and complementation -, on (characteristic functions of) inexact classes 9, p are defined byfor all a E A . (In [2] we used the integers -1 , 0 , + 1 instead of 0, n, 1 ; f , u , t are used in [7].) As a technical convenience we introduce here the universal class, 1, and the empty class 0 defined by l(a) = 1 and O(a) = 0 for all a E