Post : Peirce's 1880 “Algebra Of Logic” Chapter 3 • Selection 4
http://inquiryintoinquiry.com/2015/02/12/peirces-1880-algebra-of-logic-chapter-3-%e2%80%a2-selection-4/
Posted : February 12, 2015 at 4:00 pm
<blockquote>
Chapter 3. The Logic of Relatives (cont.)
§2. Relatives (cont.)
220. Every relative, like every term of singular reference, is general;
its definition describes a system in general terms; and, as general, it
may be conceived either as a logical sum of individual relatives, or as
a logical product of simple relatives. An individual relative refers
to a system all the members of which are individual. The expressions
(A : B)
(A : B : C)
may denote individual relatives. Taking dual individual relatives,
for instance, we may arrange them all in an infinite block, thus,
A:A A:B A:C A:D A:E etc.
B:A B:B B:C B:D B:E etc.
C:A C:B C:C C:D C:E etc.
D:A D:B D:C D:D D:E etc.
E:A E:B E:C E:D E:E etc.
etc. etc. etc. etc. etc. etc.
In the same way, triple individual relatives may be arranged
in a cube, and so forth. The logical sum of all the relatives
in this infinite block will be the relative universe, ∞, where
x ─< ∞,
whatever dual relative x may be. It is needless to distinguish
the dual universe, the triple universe, etc., because, by adding
a perfectly indefinite additional member to the system, a dual
relative may be converted into a triple relative, etc. Thus, for
''lover of a woman'', we may write ''lover of a woman coexisting
with anything''. In the same way, a term of single reference is
equivalent to a relative with an indefinite correlate; thus,
''woman'' is equivalent to ''woman coexisting with anything''.
Thus, we shall have
A = A:A + A:B + A:C + A:D + A:E + etc.
A:B = A:B:A + A:B:B + A:B:C + A:B:D + etc.
</blockquote>
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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