Peirce's 1870 “Logic Of Relatives ” • Selection 7

Post   : Peirce's 1870 “Logic Of Relatives” • Selection 7
http://inquiryintoinquiry.com/2014/02/07/peirces-1870-logic-of-relatives-%E2%80%A2-selection-7/
Posted : February 7, 2014 at 10:24 am
Author : Jon Awbrey
Tags   : Peirce, Logic, Logic of Relatives, Mathematics, Relation Theory, Semiotics

We continue with §3. Application of the Algebraic Signs to Logic.

<quote>

The Signs for Multiplication (cont.)
------------------------------------

The associative principle does not hold in this counting of factors.  Because it does not hold,
these subjacent numbers are frequently inconvenient in practice, and I therefore use also another
mode of showing where the correlate of a term is to be found.  This is by means of the marks of
reference, † ‡ ∥ § ¶, which are placed subjacent to the relative term and before and above the
correlate.  Thus, giver of a horse to a lover of a woman may be written:

g_†‡ †l_∥ ∥w ‡h.

http://s0.wp.com/latex.php?latex=\mathfrak{g}_{\dagger\ddagger}+{}^\dagger\mathit{l}_\parallel+{}^\parallel\mathrm{w}+{}^\ddagger\mathrm{h}.&bg=ffffff&fg=333333&s=0

The asterisk I use exclusively to refer to the last correlate of the last relative of the algebraic
term.

Now, considering the order of multiplication to be: — a term, a correlate of it, a correlate of that
correlate, etc. — there is no violation of the associative principle.  The only violations of it in
this mode of notation are that in thus passing from relative to correlate, we skip about among the
factors in an irregular manner, and that we cannot substitute in such an expression as goh a single
letter for oh.

I would suggest that such a notation may be found useful in treating other cases of non-associative
multiplication.  By comparing this with what was said above [CP 3.55] concerning functional
multiplication, it appears that multiplication by a conjugative term is functional, and that the
letter denoting such a term is a symbol of operation.  I am therefore using two alphabets, the Greek
and [Gothic], where only one was necessary.  But it is convenient to use both.

(Peirce, CP 3.71–72)

</quote>

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'' (CP 3.45–149),
''Chronological Edition'' (CE 2, 359–429).

Online:
1. http://www.jstor.org/stable/25058006
2. https://archive.org/details/jstor-25058006
3. http://books.google.com/books?id=fFnWmf5oLaoC

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.  Cited as (CP volume.paragraph).

Peirce, C.S., ''Writings of Charles S. Peirce : A Chronological Edition'', Peirce Edition Project
(eds.), Indiana University Press, Bloomington and Indianoplis, IN, 1981–.  Cited as (CE volume, page).

-- 

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