Post : Peirce's 1870 “Logic Of Relatives” • Comment 12.3
http://inquiryintoinquiry.com/2014/06/12/peirces-1870-logic-of-relatives-%e2%80%a2-comment-12-3/
Posted : June 12, 2014 at 4:32 pm
Author : Jon Awbrey
Peircers,
We now have two ways of computing a logical involution that raises a dyadic
relative term to the
power of a monadic absolute term, for example, l^w for “lover of every woman”.
The first method operates in the medium of set theory, expressing the
denotation of the term l^w as
the intersection of a set of relational applications:
• l^w = ⋂_{x ∈ W} L ⋅ x
The second method operates in the matrix representation, expressing the value
of the matrix *L*^*W*
with respect to an argument u as a product of coefficient powers:
• (*L*^*W*)_u = ∏_{v ∈ X} *L*_{uv}^*W*_v
Abstract formulas like these are more easily grasped with the aid of a concrete
example and a
picture of the relations involved.
Example 6
=========
Consider a universe of discourse X that is subject to the following data:
• X = {a, b, c, d, e, f, g, h, i}
• W = {d, f}
• L = {b:a, b:c, c:b, c:d, e:d, e:e, e:f, g:f, g:h, h:g, h:i}
Figure 55 shows the placement of W within X and the placement of L within X × X.
Figure 55. Bigraph Representation of W ⊆ X and L ⊆ X × X
☞http://inquiryintoinquiry.files.wordpress.com/2014/06/lor-1870-figure-55.jpg
To highlight the role of W more clearly, the Figure represents the absolute
term “w” by means of the
relative term “w,” that conveys the same information.
Computing the denotation of l^w by way of the set-theoretic formula, we can
show our work as follows:
• l^w = ⋂_{x ∈ W} L ⋅ x = L ⋅ d ∩ L ⋅ f = {c, e} ∩ {e, g} = {e}
With the above Figure in mind, we can visualize the computation
of (*L*^*W*)_u = ∏_{v ∈ X} *L*_{uv}^*W*_v as follows:
1. Pick a specific u in the bottom row of the Figure.
2. Pan across the elements v in the middle row of the Figure.
3. If u links to v then *L*_{uv} = 1, otherwise *L*_{uv} = 0.
4. If v in the middle row links to v in the top row
then *W*_v = 1, otherwise *W*_v = 0.
5. Compute the value *L*_{uv}^*W*_v = (*L*_{uv} ⇐ *W*_v)
for each v in the middle row.
6. If any of the values *L*_{uv}^*W*_v is 0
then the product ∏_{v ∈ X} *L*_{uv}^*W*_v is 0,
otherwise it is 1.
As a general observation, we know that the value of (*L*^*W*)_u goes to 0 just
as soon as we find a
v ∈ X such that *L*_{uv} = 0 and *W*_v = 1, in other words, such that (u, v) ∉
L but v ∈ W. If
there is no such v then (*L*^*W*)_u = 1.
Running through the program for each u ∈ X, the only case that produces a
non-zero result is
(*L*^*W*)_e = 1. That portion of the work can be sketched as follows:
• (*L*^*W*)_e = ∏_{v ∈ X} *L*_{ev}^{*W*_v} = 0^0 ⋅ 0^0 ⋅ 0^0 ⋅ 1^1 ⋅ 1^0 ⋅ 1^1
⋅ 0^0 ⋅ 0^0 ⋅ 0^0 = 1
Regards,
Jon
--
academia: http://independent.academia.edu/JonAwbrey
my word press blog: http://inquiryintoinquiry.com/
inquiry list: http://stderr.org/pipermail/inquiry/
isw: http://intersci.ss.uci.edu/wiki/index.php/JLA
oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey
facebook page: https://www.facebook.com/JonnyCache
-----------------------------
PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L
to this message. PEIRCE-L posts should go to [email protected] . To
UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] with the
line "UNSubscribe PEIRCE-L" in the BODY of the message. More at
http://www.cspeirce.com/peirce-l/peirce-l.htm .
