Algebraic postulates and a geometric interpretation for the Lewis calculus of strict implication
Tang Tsao-Chen · Bulletin of the American Mathematical Society · 1938
Two further postulates for a Boolean ring with a unit element.If addition, subtraction, and multiplication are properly defined in logic, it may be shown* that the postulates for these operations are identical with those in a ring, in which every element is idempotent, satisfying the postulate XX ~-X .Such a ring is called a Boolean ring.The postulates for a Boolean ring with a unit element are therefore the following : A. Addition is always possible, commutative, and associative.B. Multiplication is always possible, associative, and both left-and right-distributive with respect to addition.C. Subtraction is always possible. D. xx = x.E. There exists an element 1 such that xl = x for every element x in the ring.Here we shall introduce a new operation, represented by x°°, which satisfies the following two further postulates : Fi.For every element x there exists an element x 00 such that X X "•"•" X • F2.For any two elements x and y we have (xy) 00 = x^y 00 .The postulates A-F 2 , obtained above, may be called the algebraic postulates for the Lewis calculus of strict implication.