[Inquiry] Re: Logic Of Relatives -- Commentary

[Jon Awbrey]

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LOR.  Commentary Note 6

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Peirce's way of representing sets as sums may seem archaic, but it is
quite often used, and is actually the tool of choice in many branches
of algebra, combinatorics, computing, and statistics to this very day.

Peirce's application to logic is fairly novel, and the degree of his
elaboration of the logic of relative terms is certainly original with
him, but this particular genre of representation, commonly going under
the handle of "generating functions", goes way back, well before anyone
thought to stick a flag in set theory as a separate territory or to try
to fence off our native possessions of it with expressly decreed axioms.
And back in the days when computers were people, before we had the sorts
of "electronic register machines" that we take so much for granted today,
mathematicians were constantly using generating functions as a rough and
ready type of addressable memory to sort, store, and keep track of their
accounts of a wide variety of formal objects of thought.

Let us look at a few simple examples of generating functions,
much as I encountered them during my own first adventures in
the Fair Land Of Combinatoria.

Suppose that we are given a set of three elements,
say, {a, b, c}, and we are asked to find all the
ways of choosing a subset from this collection.

We can represent this problem setup as the
problem of computing the following product:

(1 + a)(1 + b)(1 + c).

The factor (1 + a) represents the option that we have, in choosing
a subset of {a, b, c}, to leave the 'a' out (signified by the "1"),
or else to include it (signified by the "a"), and likewise for the
other elements 'b' and 'c' in their turns.

Probably on account of all those years I flippered away
playing the oldtime pinball machines, I tend to imagine
a product like this being displayed in a vertical array:

(1 + a)
(1 + b)
(1 + c)

I picture this as a playboard with six "bumpers",
the ball chuting down the board in such a career
that it strikes exactly one of the two bumpers
on each and every one of the three levels.

So a trajectory of the ball where it
hits the "a" bumper on the 1st level,
hits the "1" bumper on the 2nd level,
hits the "c" bumper on the 3rd level,
and then exits the board, represents
a single term in the desired product
and corresponds to the subset {a, c}.

Multiplying out (1 + a)(1 + b)(1 + c), one obtains:

1 + a + b + c + ab + ac + bc + abc.

And this informs us that the subsets of choice are:

{}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}.

Jon Awbrey

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