Chapter 3. The Logic of Relatives (cont.)
§1. Individual and Simple Terms (concl.)
216. Just as in mathematics we speak of infinitesimals and infinites, which are fictitious limits of continuous quantity, and every statement involving these expressions has its interpretation in the doctrine of limits, so in logic we may define an individual, 
but such that if
then
And in the same way, we may define the simple, 
but such that if
then
The individual and the simple, as here defined, are ideal limits, and every statement about either is to be interpreted by the doctrine of limits.
217. Every term may be conceived as a limitless logical sum of individuals, or as a limitless logical product of simples; thus,
![\begin{array}{lll} a & = & \mathrm{A}_1 + \mathrm{A}_2 + \mathrm{A}_3 + \mathrm{A}_4 + \mathrm{A}_5 + \text{etc.} \\[8pt] \overline{a} & = & \overline{\mathrm{A}}_1 \times \overline{\mathrm{A}}_2 \times \overline{\mathrm{A}}_3 \times \overline{\mathrm{A}}_4 \times \overline{\mathrm{A}}_5 \times \text{etc.} \end{array}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Barray%7D%7Blll%7D++a+%26+%3D+%26+%5Cmathrm%7BA%7D_1+%2B+%5Cmathrm%7BA%7D_2+%2B+%5Cmathrm%7BA%7D_3+%2B+%5Cmathrm%7BA%7D_4+%2B+%5Cmathrm%7BA%7D_5+%2B+%5Ctext%7Betc.%7D++%5C%5C%5B8pt%5D++%5Coverline%7Ba%7D+%26+%3D+%26+%5Coverline%7B%5Cmathrm%7BA%7D%7D_1+%5Ctimes+%5Coverline%7B%5Cmathrm%7BA%7D%7D_2+%5Ctimes+%5Coverline%7B%5Cmathrm%7BA%7D%7D_3+%5Ctimes+%5Coverline%7B%5Cmathrm%7BA%7D%7D_4+%5Ctimes+%5Coverline%7B%5Cmathrm%7BA%7D%7D_5+%5Ctimes+%5Ctext%7Betc.%7D++%5Cend%7Barray%7D&bg=ffffff&fg=000000&s=1&c=20201002)
It will be seen that a simple is the negative of an individual.
References
- Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57. Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
- Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Volume 3 : Exact Logic, 1933.
- Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–. Volume 4 (1879–1884), 1986.
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