Peirce's 1870 “Logic Of Relatives ” • Comment 10.2

Post   : Peirce's 1870 “Logic Of Relatives” • Comment 10.2
http://inquiryintoinquiry.com/2014/03/14/peirces-1870-logic-of-relatives-%e2%80%a2-comment-10-2/
Posted : March 14, 2014 at 4:00 pm
Author : Jon Awbrey

To say that a relative term “imparts a relation” is to say that it conveys information
about the space of tuples in a cartesian product, that is, it determines a particular
subset of that space.  When we study the combinations of relative terms, from the most
elementary forms of composition to the most complex patterns of correlation, we are
considering the ways that these constraints, determinations, and informations, as
imparted by relative terms, are compounded in the formation of syntax.

Let us go back and look more carefully at just how it happens that Peirce’s adjacent terms
and subjacent indices manage to impart their respective measures of information about relations.
Consider the examples shown in Figures 7 and 8, where connecting lines of identity have been
drawn between the corresponding occurrences of the subjacent marks of reference:  † ‡ ∥ § ¶.

Figure 7.  Lover of a Servant of a Woman
☞http://inquiryintoinquiry.files.wordpress.com/2014/03/lor-1870-figure-71.jpg

Figure 8.  Giver of a Horse to a Lover of a Woman
☞http://inquiryintoinquiry.files.wordpress.com/2014/03/lor-1870-figure-8.jpg

One way to approach the problem of “information fusion” in Peirce’s syntax
is to soften the distinction between adjacent terms and subjacent signs and
treat the types of constraints they separately signify more on a par with
each other.  To that purpose, let us consider a way of thinking about
relational composition that emphasizes the set-theoretic constraints
involved in the construction of a composite relation.

For example, given the relations L ⊆ X × Y and M ⊆ Y × Z, Table 9 and Figure 10
present two ways of picturing the constraints that are involved in constructing
the relational composition L ∘ M ⊆ X × Z.

Table 9.  Relational Composition

..... | 1 | 1 | 1 |
===================
L ... | X | Y | . |
M ... | . | Y | Z |
L ◦ M | X | . | Z |

The way to read Table 9 is to imagine that you are playing a game that involves placing
tokens on the squares of a board that is marked in just this way.  The rules are that you
have to place a single token on each marked square in the middle of the board in such a way
that all the indicated constraints are satisfied.  That is, you have to place a token whose
denomination is a value in the set X on each of the squares marked X, and similarly for the
squares marked Y and Z, meanwhile leaving all the blank squares empty.  Moreover, the tokens
placed in each row and column have to obey the relational constraints that are indicated at
the heads of the corresponding row and column.  Thus, the two tokens from X have to denote
the very same value from X, and likewise for Y and Z, while the pairs of tokens on the rows
marked L and M are required to denote elements that are in the relations L and M, respectively.
The upshot is that when just this much is done, that is, when the L, M, and 1 relations are
satisfied, then the row marked L ◦ M will automatically bear the tokens of a pair of elements
in the composite relation L ◦ M.

Figure 10 shows a different way of viewing the same situation.

Figure 10.  Relational Composition
☞http://inquiryintoinquiry.files.wordpress.com/2014/03/lor-1870-figure-101.jpg

-- 

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