Selection from C.S. Peirce’s “Logic Of Relatives” (1870)

Demonstration of the sort called mathematical is founded on suppositions of particular cases.  The geometrician draws a figure;  the algebraist assumes a letter to signify a single quantity fulfilling the required conditions.  But while the mathematician supposes an individual case, his hypothesis is yet perfectly general, because he considers no characters of the individual case but those which must belong to every such case.

The advantage of his procedure lies in the fact that the logical laws of individual terms are simpler than those which relate to general terms, because individuals are either identical or mutually exclusive, and cannot intersect or be subordinated to one another as classes can.

Mathematical demonstration is not, therefore, more restricted to matters of intuition than any other kind of reasoning.  Indeed, logical algebra conclusively proves that mathematics extends over the whole realm of formal logic;  and any theory of cognition which cannot be adjusted to this fact must be abandoned.  We may reap all the advantages which the mathematician is supposed to derive from intuition by simply making general suppositions of individual cases.  (CP 3.92)

References

• Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870.  Reprinted, Collected Papers 3.45–149, Chronological Edition 2, 359–429.  Online (1) (2) (3).
• 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.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.

Resources

• Peirce’s 1870 Logic Of Relatives