Peirce's 1870 “Logic Of Relatives ” • Comment 9.4

Post   : Peirce's 1870 “Logic Of Relatives” • Comment 9.4
http://inquiryintoinquiry.com/2014/02/27/peirces-1870-logic-of-relatives-%e2%80%a2-comment-9-4/
Posted : February 27, 2014 at 7:48 pm
Author : Jon Awbrey

Peircers,

Boole rationalized the properties of what we now call ''boolean multiplication'', roughly equivalent 
to logical conjunction, in terms of the laws that apply to selective operations.  Peirce, in his 
turn, taking a very significant step of analysis that has seldom been recognized for what it would 
lead to, does not consider this multiplication to be a fundamental operation, but derives it as a 
by-product of relative multiplication by a comma relative.  Thus, Peirce makes logical conjunction a 
special case of relative composition.

This opens up a very wide field of inquiry, ''the operational significance of logical terms'', but 
it will be best to advance bit by bit and to lean on simple examples.

Back to Venice, and the close-knit party of absolutes and relatives that we were entertaining when 
last we were there.

Here is the list of absolute terms we had been considering before:

1  =  anything  =  B +, C +, D +, E +, I +, J +, O
m  =  man       =  C +, I +, J +, O
n  =  noble     =  C +, D +, O
w  =  woman     =  B +, D +, E

Here is the list of ''comma inflexions'' or ''diagonal extensions'' of these terms:

1, =  anything that is ────
    =  B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O

m, =  man that is ────
    =  C:C +, I:I +, J:J +, O:O

n, =  noble that is ────
    =  C:C +, D:D +, O:O

w, =  woman that is ────
    =  B:B +, D:D +, E:E

One observes that the diagonal extension of *1* is the same thing as the identity relation _1_.

Working with our smaller sample of absolute terms, we have already computed the sorts of products 
that apply the diagonal extension of an absolute term to another absolute term, for instance, these 
products:

m,n  =  man that is noble    =  C +, O
n,m  =  noble that is a man  =  C +, O

w,n  =  woman that is noble    =  D
n,w  =  noble that is a woman  =  D

This exercise gave us a bit of practical insight into why the commutative law holds for logical 
conjunction.

Further insight into the laws that govern this realm of logic, and the underlying reasons why they 
apply, might be gained by systematically working through the whole variety of different products 
that are generated by the operational means in sight, namely, the products obtained by appending a 
comma to each of the terms in {1, m, n, w} and then applying the relative term that results to each 
of those same terms in {1, m, n, w}.

But before we try to explore this territory more systematically, let us equip our intuitions with 
the forms of graphical and matrical representation that served us so well in our previous adventures.

Regards,

Jon

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