errata : p. 218, last line but one, for "A" read "A " [owing to brittleness of the type, the same error is liable to occur elsewhere].
218 MATHEMA TICAL LOGIC [PART I I a,. au b is in the class whenever a and b are in the class. I b. a n b is in the class whenever a and b are in the class. II a. There is an element A such that a v A = a for every element a, II b. There is an element V such that a n V = a for every element a. III a, a v b = b v a whenever a, b, a u band b u a are in the class. III b. an b = b n a whenever a, b, an band b n a are in the class. IVa. au (b n e) = (a u b)n(a u e) whenever a, b,e, au b, a u c, b n e, a v(bne), and (a u b) n (a v e) are in the class. IV b. an (b u e) = (a n b)u(a n e) whenever a, b, e, a n b, a n c, b u c, a n(bve), and (a n b) u (a n e) are in the class. V. If the elements A and V in postulates II a and II b exist and are uniq ue, then for every element a there is an element - a such that a u - a = V and a n - a = A. VI. ’fhere are at least two elements, {IJ and y, in the class, such that x 1= y, The form of the above postulates is such that they are mutually inde- pendent, i.e. any nine of them are satisfied by interpretations of the symbols which do not satisfy the remaining one. For our purposes, " K" must be replaced by "Cis," A and V will be the null-class and the universal class, which are defined in *24. Then the above ten postulates are proved below, as follows: I a, in *22.37, namely" f- . a v fj € Cis" I b, in *22’36, nalnely " f- . (.( n fj E Cis" ]1 a, in *24-24, namely" f- . a u A = (.( " II b, in *24’26, namely " f- . (.( n V = a " III a, in *22.57, namely" f- . a u = fj u a" III b, in *22.51, namely" f- . a n = n a" IVa, in *22.69, namely" f-. (a v) n (a v "I) = (.( v (13 n ",)" IV b, in *22.68, namely" f- . (a n) v (a n "I) = a n (fj v ",)" V, in *24.21’22, namely" f- . (.( n - a = A " and " f- . a u - (.( = V " VI, in *24.1, namely cc f- . A =4= V JJ Hence, assuming Huntington’s analysis of the postulates fol’ the formal algebra of logic, the propositions proved in what follows suffice to establish that this algebra holds for classes. The corresponding propositions of *2? and *25 prove that it holds for relations, substituting Rei, 0, n, A, V for Cis, v, n, A, V.
