Writings of Charles S. Peirce: A Chronological Edition, Volume 2. Charles S. Peirce

      So

      But

      Let be the sign of logical subtraction; so defined that

      Here it will be observed that x is not completely determinate. It may vary from a to a with b taken away. This minimum may be denoted by a —b. It is also to be observed that if the sphere of b reaches at all beyond a, the expression is uninterpretable. If then we denote the contradictory negative of a class by the letter which denotes the class itself, with a line above it,¹ if we denote by v a wholly indeterminate class, and if we allow to be a wholly uninterpretable symbol, we have

      which is uninterpretable unless

      If we define zero by the following identities, in which x may be any class whatever,

      then, zero denotes the class which does not go beyond any class, that is nothing or nonentity.

      Let a ; b be read a logically divided by b, and be defined by the condition that

      x is not fully determined by this condition. It will vary from a to and will be uninterpretable if a is not wholly contained under b. Hence, allowing [1;0] to be some uninterpretable symbol,

      which is uninterpretable unless

      Unity may be defined by the following identities in which x may be any class whatever.

      Then unity denotes the class of which any class is a part; that is, what is or ens.

      It is plain that if for the moment we allow a : b to denote the maximum value of a ; b, then

      So that

      The rules for the transformation of expressions involving logical subtraction and division would be very complicated. The following method is, therefore, resorted to.

      It is plain that any operations consisting solely of logical addition and multiplication, being performed upon interpretable symbols, can result in nothing uninterpretable. Hence, if φ + × x signifies such an operation performed upon symbols of which x is one, we have

      where a and b are interpretable.

      It is plain, also, that all four operations being performed in any way upon any symbols, will, in general, give a result of which one term is interpretable and another not; although either of these terms may disappear. We have then

      We have seen that if either of these coefficients i and j is uninterpretable, the other factor of the same term is nothing, or else the whole expression is uninterpretable. But

      Hence

      Developing by (18) , we have,

      So that, by (11),

      Developing x; y in the same way, we have²

      So that, by (14),

      Boole gives (20), but not (19).

      In solving identities we must remember that

      From the value of b cannot be obtained.

      From a ; b the value of b cannot be determined.

      Required to eliminate x.

      Logically multiplying these identities, we get

      For two terms disappear because of (17).

      But we have, by (18),

      Multiplying logically by x we get

      Substituting these values above, we have

      Required to eliminate x.

      Now, developing as in (18), only in reference to φ(1) and φ(0) instead of to x and y,

      But by (18) we have also,