## Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Selection 1

### Chapter 3. The Logic of Relatives

#### §1. Individual and Simple Terms

214.   Just as we had to begin the study of Logical Addition and Multiplication by considering $\infty$ and $0,$ terms which might have been introduced under the Algebra of the Copula, being defined in terms of the copula only, without the use of $+$ or $\times,$ but which had not been there introduced, because they had no application there, so we have to begin the study of relatives by considering the doctrine of individuals and simples,— a doctrine which makes use only of the conceptions of non-relative logic, but which is wholly without use in that part of the subject, while it is the very foundation of the conception of a relative, and the basis of the method of working with the algebra of relatives.

215.   The germ of the correct theory of individuals and simples is to be found in Kant’s Critic of the Pure Reason, “Appendix to the Transcendental Dialectic,” where he lays it down as a regulative principle, that, if

$\begin{array}{lll} a \,-\!\!\!< b & ~ & b \,\overline{-\!\!\!<}\, a, \end{array}$

then it is always possible to find a term $x,$ that

$\begin{array}{lll} a \,-\!\!\!< x & ~ & x \,-\!\!\!< b \\[8pt] x \,\overline{-\!\!\!<}\, a & ~ & b \,\overline{-\!\!\!<}\, x. \end{array}$

Kant’s distinction of regulative and constitutive principles is unsound, but this law of continuity, as he calls it, must be accepted as a fact.  The proof of it, which I have given elsewhere, depends on the continuity of space, time, and the intensities of the qualities which enter into the definition of any term.  If, for instance, we say that Europe, Asia, Africa, and North America are continents, but not all the continents, there remains over only South America.  But we may distinguish between South America as it now exists and South America in former geological times;  we may, therefore, take $x$ as including Europe, Asia, Africa, North America, and South America as it exists now, and every $x$ is a continent, but not every continent is $x.$

### 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.

### Resources

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