Post : Peirce's 1870 “Logic Of Relatives” • Comment 12.2
http://inquiryintoinquiry.com/2014/06/11/peirces-1870-logic-of-relatives-%e2%80%a2-comment-12-2/
Posted : June 11, 2014 at 12:36 am
Author : Jon Awbrey
Peircers,
Let us make a few preliminary observations about the operation of ''logical involution'', as Peirce
introduces it here:
“I shall take involution in such a sense that x^y will denote everything which is an x for every
individual of y. Thus l^w will be a lover of every woman.” (CP 3.77)
In ordinary arithmetic the ''involution'' x^y, or the ''exponentiation'' of x
to the power y,
is the repeated application of the multiplier x for as many times as there are
ones making up
the exponent y.
In analogous fashion, the logical involution l^w is the repeated application of the term l for as
many times as there are individuals under the term w. According to Peirce's interpretive rules, the
repeated applications of the base term l are distributed across the individuals of the exponent term
w. In particular, the base term l is not applied successively in the manner that would give
something like “a lover of a lover of … a lover of a woman”.
For example, suppose that a universe of discourse numbers among its elements just three women, W',
W'', W'''. This could be expressed in Peirce's notation by writing:
• w = W' +, W'' +, W'''
Under these circumstances the following equation would hold:
• l^w = l^(W' +, W'' +, W''') = (lW'),(lW''),(lW''')
This says that a lover of every woman in the given universe of discourse is a lover of W' that is a
lover of W'' that is a lover of W'''. In other words, a lover of every woman in this context is a
lover of W' and a lover of W'' and a lover of W'''.
The denotation of the term l^w is a subset of X that can be obtained as follows: For each flag of
the form L ⋆ x with x in W, collect the elements proj₁(L ⋆ x) that appear as the first components of
these ordered pairs, and then take the intersection of all these subsets. Putting it all together:
• l^w = ⋂_{x ∈ W} proj₁(L ⋆ x) = ⋂_{x ∈ W} L ∙ x
It is very instructive to examine the matrix representation of l^w at this point, not the least
because it effectively dispels the mystery of the name ''involution''. First, let us make the
following observation. To say that j is a lover of every woman is to say that j loves k if k is a
woman. This can be rendered in symbols as follows:
• j loves k ⇐ k is a woman
Reading the formula l^w as “j loves k if k is a woman” highlights the operation of converse
implication inherent in it, and this in turn reveals the analogy between implication and involution
that accounts for the aptness of the latter name.
The operations defined by the formulas x^y = z and (x⇐y) = z
for x, y, z in the boolean domain B = {0, 1} are tabulated below:
x^y = z
--------
0^0 = 1
0^1 = 0
1^0 = 1
1^1 = 1
x⇐y = z
---------
0⇐0 = 1
0⇐1 = 0
1⇐0 = 1
1⇐1 = 1
It is clear that these operations are isomorphic, amounting to the same operation of type B × B × B.
All that remains is to see how this operation on coefficient values in B induces the corresponding
operations on sets and terms.
The term l^w determines a selection of individuals from the universe of discourse X that may be
computed by means of the corresponding operation on coefficient matrices. If the terms l and w are
represented by the matrices *L* = Mat(l) and *W* = Mat(w), respectively, then the operation on terms
that produces the term l^w must be represented by a corresponding operation on matrices, say,
*L*^*W* = Mat(l)^Mat(w), that produces the matrix Mat(l^w). In other words, the involution
operation on matrices must be defined in such a way that the following equations hold:
• *L*^*W* = Mat(l)^Mat(w) = Mat(l^w)
The fact that l^w denotes the elements of a subset of X means that the matrix *L*^*W* is a
1-dimensional array of coefficients in B that is indexed by the elements of X. The value of the
matrix *L*^*W* at the index x in X is written (*L*^*W*)_x and computed as follows:
• (*L*^*W*)_x = ∏_{p ∈ X} (*L*_{x,p})^(*W*_p)
Regards,
Jon
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