Peirce's 1870 “Logic Of Relatives ” • Sets as Sums

Post   : Peirce's 1870 “Logic Of Relatives” • Sets as Sums
http://inquiryintoinquiry.com/2014/02/06/peirces-1870-logic-of-relatives-%E2%80%A2-sets-as-sums/
Posted : February 6, 2014 at 11:00 am
Author : Jon Awbrey

Peircers,

The following comment is probably pertinent at this point.

Peirce's way of representing sets as logical sums may seem archaic, but it's long been used in
mathematics and remains the tool of choice in many branches of algebra, combinatorics, computing,
and statistics to this day.

Peirce applied this genre of representation to logic in fairly novel ways and the degree to which he
elaborated its use in the logic of relative terms is certainly original with him, but this
particular device, going under the handle of ''generating functions'', goes way back, well before
anyone thought of sticking a flag in set theory as a separate territory or of trying to fence off
our native possessions of sets with explicitly decreed axioms.  And back in the days when a
''computer'' was merely a person who computed, well before we had the sorts of electronic register
machines that we take so much for granted today, mathematicians commonly used generating functions
as a rough and ready kind of addressable memory to sort, store, and keep track of their accounts on
a wide variety of formal objects.

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

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 element “a” out (signified by the “1”), or else to include it (signified by the “a”), proceeding
in a similar fashion for the other elements in their turn.

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 way that
it strikes exactly one of the two bumpers on each 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 the product (1 + a)(1 + b)(1 + c), one obtains:

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

This informs us that the subsets of choice are:

∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}

And so they are.

-- 

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