Peirce’s 1870 “Logic of Relatives” • Sets as Sums

Peirce’s 1870 “Logic of Relatives”Sets as Sums

Peirce’s way of representing sets as logical sums may seem arcane, but it’s quite often 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 classes and collections with explicit decrees of axioms.  And back in the days when a computer was simply a person who computed, well before the advent of electronic computers we take for granted today, mathematicians commonly used generating functions as a rough and ready sort of addressable memory to organize, store, and keep track of their accounts on a wide variety of formal objects.

Let’s 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 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 the problem setup as the problem of computing the following product:

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

The factor (1 + a) represents the option we have, in choosing a subset of \{ a, b, c \}, to exclude the element a (signified by the 1), or else to include it (signified by the a), proceeding in a similar fashion with 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 that being displayed in a vertical array:

\begin{matrix}  (1 ~+~ a) \\  (1 ~+~ b) \\  (1 ~+~ c)  \end{matrix}

I picture that as a playboard with six bumpers, the ball chuting down the board in such a way as to strike 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 the sum:

\begin{array}{*{15}{c}}  1 & + & a & + & b & + & c & + & ab & + & ac & + & bc & + & abc.  \end{array}

This informs us that the subsets of choice are:

\begin{matrix}  \varnothing, & \{a\}, & \{b\}, & \{c\}, & \{a, b\}, & \{a, c\}, & \{b, c\}, & \{a, b, c\}.  \end{matrix}

And so they are.


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6 Responses to Peirce’s 1870 “Logic of Relatives” • Sets as Sums

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