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

Note 2.

      Identity (18′) is reducible to (18) by developement of the second member by (18).

       Note 3.

      It should be added that

      For

      where cannot be less than nothing. These limits have been treated both by Boole and by De Morgan.

       Note 4.

      The mode proposed for the expression of particular propositions is weak. What is really wanted is something much more fundamental. Another idea has since occurred to me which I have never worked out but which I can here briefly explain.

      If w denotes wise, and s denotes Solomon, then the expression ws cannot be interpreted by (18) or any principle of Boole’s calculus. It might then be used to denote wiser than Solomon. Thus, relative terms would be brought into the domain of the calculus. But if we are to have symbols for relative terms, we must have a symbol for not or other than. Let this be n. Then, ns will denote not Solomon.

      But Solomon is a singular. Let m denote man. Then shall wm denote wiser than every man or wiser than some man? In either case, nwm will be ambiguous. For it may be taken as n(wm) or as (nw)m.

      Now, if we simply adopt the formula

      which is imitated from algebra, then wm must be wiser than every man, and (nw)m must be other than wiser (no more wise) than any man, or other than wiser-than-some-man. So that wiser than some man will be n^((nw)m). So that both conceptions will be susceptible of expression.

      Then, we have the familiar-looking formulæ

      But we do not have

      That which we have expressed as nm can be expressed as 1 – m or 0/m without any special letter m. Can we not determine the value of n, then? Now n is fully defined by the condition that the logical product of the two expressions x and nx (whatever x may be) is zero; or

      That is to say, if either x or nx is zero the other may be of any value. That is, if need not be zero; or n⁰ is not necessarily zero or unity. And, if nx is zero, x may have any value; or, in general, when x is not zero nx may be zero. Algebraical ideas at once, therefore, suggest that ; so that non-man is expressed by 0^m.

      0⁰ must be taken as unity.

      The difference in meaning between h(km) and (hk)^m is this: if k means king and h hater and m as before man, the former is hater of every king over all men and the latter is the hater of every king of any man.

      Hence we have

      Some examples of the method in which these exponents may be made use of may now be shown. Let m be man, a animal. Then, every man is an animal; or

      or

      or

      Then,

      or

      And in the same way, if h denote head,

      and then

      That is, any man’s head is an animal’s head. This result cannot be reached by any ordinary forms of Logic or by Boole’s Calculus.

      The next step requires us to notice an operation to which no specific name nor symbol has been given in algebra. We have the series of quantities

      Let the nth of this series be denoted by the symbol k–n. Then, k₀ = 1 = k⁰ k₁ = 0.

      Then, recurring to our calculus let k denote one foot longer. Then

      km will be one foot longer than m

      kkm or will be two feet longer than m

       will be n feet longer than m

       will be no longer than m

       will be one foot shorter than m

      Now, let k denote killer. Then,

      km is killer of every m

      kkm is killer of every killer of every m

       is Every m

       is killed by every m.

      Then, we have, in general,

      If, then,

      The most curious symbol of this system must now be noticed. k_–1 and k₁ correspond to the active and passive voices respectively. But now it is also necessary to have a species of non-relative terms derived from relatives, which correspond to the middle voice. This may be indicated by a comma as exponent. Thus, if k denotes killer, k’ will be killer of himself. Then we shall have

      Let p denote parent, l lover,