Peirce's 1870 Logic Of Relatives

Author's Note. The text that follows is a collection of notes that will eventually be developed into a paper on Charles Sanders Peirce's 1870 memoir on the logic of relative terms.

Preliminaries

Application of the Algebraic Signs to Logic

Peirce's text employs a number of different typefaces to denote different types of logical entities. The following Tables indicate the LaTeX typefaces that we will use for Peirce's stock examples.

\(\text{Absolute Terms (Monadic Relatives)}\!\)

\(\begin{array}{ll} \mathrm{a}. & \text{animal} \\ \mathrm{b}. & \text{black} \\ \mathrm{f}. & \text{Frenchman} \\ \mathrm{h}. & \text{horse} \\ \mathrm{m}. & \text{man} \\ \mathrm{p}. & \text{President of the United States Senate} \\ \mathrm{r}. & \text{rich person} \\ \mathrm{u}. & \text{violinist} \\ \mathrm{v}. & \text{Vice-President of the United States} \\ \mathrm{w}. & \text{woman} \end{array}\)

\(\text{Simple Relative Terms (Dyadic Relatives)}\!\)

\(\begin{array}{ll} \mathit{a}. & \text{enemy} \\ \mathit{b}. & \text{benefactor} \\ \mathit{c}. & \text{conqueror} \\ \mathit{e}. & \text{emperor} \\ \mathit{h}. & \text{husband} \\ \mathit{l}. & \text{lover} \\ \mathit{m}. & \text{mother} \\ \mathit{n}. & \text{not} \\ \mathit{o}. & \text{owner} \\ \mathit{s}. & \text{servant} \\ \mathit{w}. & \text{wife} \end{array}\)

\(\text{Conjugative Terms (Higher Adic Relatives)}\!\)

\(\begin{array}{ll} \mathfrak{b}. & \text{betrayer to ------ of ------} \\ \mathfrak{g}. & \text{giver to ------ of ------} \\ \mathfrak{t}. & \text{transferrer from ------ to ------} \\ \mathfrak{w}. & \text{winner over of ------ to ------ from ------} \end{array}\)

Selection 1

Use of the Letters

The letters of the alphabet will denote logical signs. Now logical terms are of three grand classes. The first embraces those whose logical form involves only the conception of quality, and which therefore represent a thing simply as "a ——". These discriminate objects in the most rudimentary way, which does not involve any consciousness of discrimination. They regard an object as it is in itself as such (quale); for example, as horse, tree, or man. These are absolute terms. The second class embraces terms whose logical form involves the conception of relation, and which require the addition of another term to complete the denotation. These discriminate objects with a distinct consciousness of discrimination. They regard an object as over against another, that is as relative; as father of, lover of, or servant of. These are simple relative terms. The third class embraces terms whose logical form involves the conception of bringing things into relation, and which require the addition of more than one term to complete the denotation. They discriminate not only with consciousness of discrimination, but with consciousness of its origin. They regard an object as medium or third between two others, that is as conjugative; as giver of —— to ——, or buyer of —— for —— from ——. These may be termed conjugative terms. The conjugative term involves the conception of third, the relative that of second or other, the absolute term simply considers an object. No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship. Whether this reason for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory of not, the fact itself is made perfectly evident by the study of the logic of relatives. (Peirce, CP 3.63).

I am going to experiment with an interlacing commentary on Peirce's 1870 "Logic of Relatives" paper, revisiting some critical transitions from several different angles and calling attention to a variety of puzzles, problems, and potentials that are not so often remarked or tapped.

What strikes me about the initial installment this time around is its use of a certain pattern of argument that I can recognize as invoking a "closure principle", and this is a figure of reasoning that Peirce uses in three other places: his discussion of "continuous predicates", his definition of sign relations, and in the pragmatic maxim itself.

One might also call attention to the following two statements:

Now logical terms are of three grand classes.

No fourth class of terms exists involving the conception of fourth, because when that of third is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship.

Selection 2

Numbers Corresponding to Letters

I propose to use the term "universe" to denote that class of individuals about which alone the whole discourse is understood to run. The universe, therefore, in this sense, as in Mr. De Morgan's, is different on different occasions. In this sense, moreover, discourse may run upon something which is not a subjective part of the universe; for instance, upon the qualities or collections of the individuals it contains. I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. Thus in a universe of perfect men (men), the number of "tooth of" would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus, \([t].\!\) (Peirce, CP 3.65).

Peirce's remarks at CP 3.65 are so replete with remarkable ideas, some of them so taken for granted in mathematical discourse that they usually escape explicit mention, and others so suggestive of things to come in a future remote from his time of writing, and yet so smoothly introduced in passing that it's all too easy to overlook their consequential significance, that I can do no better here than to highlight these ideas in other words, whose main advantage is to be a little more jarring to the mind's sensibilities.

• This mapping of letters to numbers, or logical terms to mathematical quantities, is the very core of what "quantification theory" is all about, and definitely more to the point than the mere "innovation" of using distinctive symbols for the so-called "quantifiers". We will speak of this more later on.

• The mapping of logical terms to numerical measures, to express it in current language, would probably be recognizable as some kind of "morphism" or "functor" from a logical domain to a quantitative co-domain.

• Notice that Peirce follows the mathematician's usual practice, then and now, of making the status of being an "individual" or a "universal" relative to a discourse in progress. I have come to appreciate more and more of late how radically different this "patchwork" or "piecewise" approach to things is from the way of some philosophers who seem to be content with nothing less than many worlds domination, which means that they are never content and rarely get started toward the solution of any real problem. Just my observation, I hope you understand.

• It is worth noting that Peirce takes the "plural denotation" of terms for granted, or what's the number of a term for, if it could not vary apart from being one or nil?

• I also observe that Peirce takes the individual objects of a particular universe of discourse in a "generative" way, not a "totalizing" way, and thus they afford us with the basis for talking freely about collections, constructions, properties, qualities, subsets, and "higher types", as the phrase is mint.

Selection 3

The Signs of Inclusion, Equality, Etc.

I shall follow Boole in taking the sign of equality to signify identity. Thus, if \(\mathrm{v}\!\) denotes the Vice-President of the United States, and \(\mathrm{p}\!\) the President of the Senate of the United States,

\(\mathrm{v} = \mathrm{p}\!\)

means that every Vice-President of the United States is President of the Senate, and every President of the United States Senate is Vice-President. The sign "less than" is to be so taken that

\(\mathrm{f} < \mathrm{m}\!\)

means that every Frenchman is a man, but there are men besides Frenchmen. Drobisch has used this sign in the same sense. It will follow from these significations of \(=\!\) and \(<\!\) that the sign \(-\!\!\!<\!\) (or \(\leqq\), "as small as") will mean "is". Thus,

\(\mathrm{f} -\!\!\!< \mathrm{m}\)

means "every Frenchman is a man", without saying whether there are any other men or not. So,

\(\mathit{m} -\!\!\!< \mathit{l}\)

will mean that every mother of anything is a lover of the same thing; although this interpretation in some degree anticipates a convention to be made further on. These significations of \(=\!\) and \(<\!\) plainly conform to the indispensable conditions. Upon the transitive character of these relations the syllogism depends, for by virtue of it, from

\(\mathrm{f} -\!\!\!< \mathrm{m}\)   and \(\mathrm{m} -\!\!\!< \mathrm{a}\)   we can infer that \(\mathrm{f} -\!\!\!< \mathrm{a}\)

that is, from every Frenchman being a man and every man being an animal, that every Frenchman is an animal. But not only do the significations of \(=\!\) and \(<\!\) here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes. So, to write \(5 < 7\!\) is to say that \(5\!\) is part of \(7\!\), just as to write \(\mathrm{f} < \mathrm{m}\!\) is to say that Frenchmen are part of men. Indeed, if \(\mathrm{f} < \mathrm{m}\!\), then the number of Frenchmen is less than the number of men, and if \(\mathrm{v} = \mathrm{p}\!\), then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities. (Peirce, CP 3.66).

The quantifier mapping from terms to their numbers that Peirce signifies by means of the square bracket notation \([t]\!\) has one of its principal uses in providing a basis for the computation of frequencies, probabilities, and all of the other statistical measures that can be constructed from these, and thus in affording what may be called a principle of correspondence between probability theory and its limiting case in the forms of logic.

This brings us once again to the relativity of contingency and necessity, as one way of approaching necessity is through the avenue of probability, describing necessity as a probability of 1, but the whole apparatus of probability theory only figures in if it is cast against the backdrop of probability space axioms, the reference class of distributions, and the sample space that we cannot help but to abdeuce upon the scene of observations. Aye, there's the snake eyes. And with them we can see that there is always an irreducible quantum of facticity to all our necessities. More plainly spoken, it takes a fairly complex conceptual infrastructure just to begin speaking of probabilities, and this setting can only be set up by means of abductive, fallible, hypothetical, and inherently risky mental acts.

Pragmatic thinking is the logic of abduction, which is just another way of saying that it addresses the question: "What may be hoped?" We have to face the possibility that it may be just as impossible to speak of "absolute identity" with any hope of making practical philosophical sense as it is to speak of "absolute simultaneity" with any hope of making operational physical sense.

Selection 4

The Signs for Addition

The sign of addition is taken by Boole so that

\(x + y\!\)

denotes everything denoted by \(x\!\), and, besides, everything denoted by \(y\!\). Thus

\(\mathrm{m} + \mathrm{w}\!\)

denotes all men, and, besides, all women. This signification for this sign is needed for connecting the notation of logic with that of the theory of probabilities. But if there is anything which is denoted by both terms of the sum, the latter no longer stands for any logical term on account of its implying that the objects denoted by one term are to be taken besides the objects denoted by the other. For example,

\(\mathrm{f} + \mathrm{u}\!\)

means all Frenchmen besides all violinists, and, therefore, considered as a logical term, implies that all French violinists are besides themselves. For this reason alone, in a paper which is published in the Proceedings of the Academy for March 17, 1867, I preferred to take as the regular addition of logic a non-invertible process, such that

\(\mathrm{m} ~+\!\!,~ \mathrm{b}\)

stands for all men and black things, without any implication that the black things are to be taken besides the men; and the study of the logic of relatives has supplied me with other weighty reasons for the same determination. Since the publication of that paper, I have found that Mr. W. Stanley Jevons, in a tract called Pure Logic, or the Logic of Quality [1864], had anticipated me in substituting the same operation for Boole's addition, although he rejects Boole's operation entirely and writes the new one with a  \(+\!\)  sign while withholding from it the name of addition. It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions. But the notation has other recommendations. The conception of taking together involved in these processes is strongly analogous to that of summation, the sum of 2 and 5, for example, being the number of a collection which consists of a collection of two and a collection of five. Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves — provided all the terms summed are mutually exclusive. Addition being taken in this sense, nothing is to be denoted by zero, for then

\(x ~+\!\!,~ 0 ~=~ x\)

whatever is denoted by \(x\!\); and this is the definition of zero. This interpretation is given by Boole, and is very neat, on account of the resemblance between the ordinary conception of zero and that of nothing, and because we shall thus have

\([0] ~=~ 0.\)

(Peirce, CP 3.67).

A wealth of issues arise here that I hope to take up in depth at a later point, but for the moment I shall be able to mention only the barest sample of them in passing.

The two papers that precede this one in CP 3 are Peirce's papers of March and September 1867 in the 'Proceedings of the American Academy of Arts and Sciences', titled "On an Improvement in Boole's Calculus of Logic" and "Upon the Logic of Mathematics", respectively. Among other things, these two papers provide us with further clues about the motivating considerations that brought Peirce to introduce the "number of a term" function, signified here by square brackets. I have already quoted from the "Logic of Mathematics" paper in a related connection. Here are the links to those excerpts:

• [1]
• [2]

In setting up a correspondence between "letters" and "numbers", my sense is that Peirce is "nocking an arrow", or constructing some kind of structure-preserving map from a logical domain to a numerical domain, and this interpretation is here reinforced by the careful attention that he gives to the conditions under which precisely which aspects of structure are preserved, plus his telling recognition of the criterial fact that zeroes are preserved by the mapping. But here's the catch, the arrow is from the qualitative domain to the quantitative domain, which is just the opposite of what I tend to expect, since I think of quantitative measures as preserving more information than qualitative measures. To curtail the story, it is possible to sort this all out, but that is a story for another day.

Other than that, I just want to red flag the beginnings of another one of those "failures to communicate" that so dogged the disciplines in the 20th Century, namely, the fact that Peirce seemed to have an inkling about the problems that would be caused by using the plus sign for inclusive disjunction, but, as it happens, his advice was overridden by the usages in various different communities, rendering the exchange of information among engineering, mathematical, and philosophical specialties a minefield in place of mindfield to this very day.

Selection 5

The Signs for Multiplication

I shall adopt for the conception of multiplication the application of a relation, in such a way that, for example, \(\mathit{l}\mathrm{w}\!\) shall denote whatever is lover of a woman. This notation is the same as that used by Mr. De Morgan, although he apears not to have had multiplication in his mind. \(\mathit{s}(\mathrm{m} ~+\!\!,~ \mathrm{w})\) will, then, denote whatever is servant of anything of the class composed of men and women taken together. So that:

\(\mathit{s}(\mathrm{m} ~+\!\!,~ \mathrm{w}) ~=~ \mathit{s}\mathrm{m} ~+\!\!,~ \mathit{s}\mathrm{w}\).

\((\mathit{l} ~+\!\!,~ \mathit{s})\mathrm{w}\) will denote whatever is lover or servant to a woman, and:

\((\mathit{l} ~+\!\!,~ \mathit{s})\mathrm{w} ~=~ \mathit{l}\mathrm{w} ~+\!\!,~ \mathit{s}\mathrm{w}\).

\((\mathit{s}\mathit{l})\mathrm{w}\!\) will denote whatever stands to a woman in the relation of servant of a lover, and:

\((\mathit{s}\mathit{l})\mathrm{w} ~=~ \mathit{s}(\mathit{l}\mathrm{w})\).

Thus all the absolute conditions of multiplication are satisfied. The term "identical with ——" is a unity for this multiplication. That is to say, if we denote "identical with ——" by \(\mathit{1}\!\) we have:

\(x \mathit{1} ~=~ x\),

whatever relative term \(x\!\) may be. For what is a lover of something identical with anything, is the same as a lover of that thing. (Peirce, CP 3.68).

Peirce in 1870 is five years down the road from the Peirce of 1865–1866 who lectured extensively on the role of sign relations in the logic of scientific inquiry, articulating their involvement in the three types of inference, and inventing the concept of "information" to explain what it is that signs convey in the process. By this time, then, the semiotic or sign relational approach to logic is so implicit in his way of working that he does not always take the trouble to point out its distinctive features at each and every turn. So let's take a moment to draw out a few of these characters.

Sign relations, like any non-trivial brand of 3-adic relations, can become overwhelming to think about once the cardinality of the object, sign, and interpretant domains or the complexity of the relation itself ascends beyond the simplest examples. Furthermore, most of the strategies that we would normally use to control the complexity, like neglecting one of the domains, in effect, projecting the 3-adic sign relation onto one of its 2-adic faces, or focusing on a single ordered triple of the form \((o, s, i)\!\) at a time, can result in our receiving a distorted impression of the sign relation's true nature and structure.

I find that it helps me to draw, or at least to imagine drawing, diagrams of the following form, where I can keep tabs on what's an object, what's a sign, and what's an interpretant sign, for a selected set of sign-relational triples.

Here is how I would picture Peirce's example of equivalent terms, \(\mathrm{v} = \mathrm{p}\!\), where \(^{\backprime\backprime} \mathrm{v} ^{\prime\prime}\) denotes the Vice-President of the United States, and \(^{\backprime\backprime} \mathrm{p} ^{\prime\prime}\) denotes the President of the Senate of the United States.

o-----------------------------o-----------------------------o | Objective Framework (OF) | Interpretive Framework (IF) | o-----------------------------o-----------------------------o | Objects | Signs | o-----------------------------o-----------------------------o | | | o "v" | | / | | / | | / | | o ... o-----------@ | | \ | | \ | | \ | | o "p" | | | o-----------------------------o-----------------------------o

Depending on whether we interpret the terms \(^{\backprime\backprime} \mathrm{v} ^{\prime\prime}\) and \(^{\backprime\backprime} \mathrm{p} ^{\prime\prime}\) as applying to persons who hold these offices at one particular time or as applying to all those persons who have held these offices over an extended period of history, their denotations may be either singular of plural, respectively.

As a shortcut technique for indicating general denotations or plural referents, I will use the elliptic convention that represents these by means of figures like "o o o" or "o … o", placed at the object ends of sign relational triads.

For a more complex example, here is how I would picture Peirce's example of an equivalence between terms that comes about by applying one of the distributive laws, for relative multiplication over absolute summation.

o-----------------------------o-----------------------------o | Objective Framework (OF) | Interpretive Framework (IF) | o-----------------------------o-----------------------------o | Objects | Signs | o-----------------------------o-----------------------------o | | | o "'s'(m +, w)" | | / | | / | | / | | o ... o-----------@ | | \ | | \ | | \ | | o "'s'm +, 's'w" | | | o-----------------------------o-----------------------------o

Selection 6

The Signs for Multiplication (cont.)

A conjugative term like giver naturally requires two correlates, one denoting the thing given, the other the recipient of the gift. We must be able to distinguish, in our notation, the giver of \(\mathrm{A}\!\) to \(\mathrm{B}\!\) from the giver to \(\mathrm{A}\!\) of \(\mathrm{B}\!\), and, therefore, I suppose the signification of the letter equivalent to such a relative to distinguish the correlates as first, second, third, etc., so that "giver of —— to ——" and "giver to —— of ——" will be expressed by different letters. Let \(\mathfrak{g}\) denote the latter of these conjugative terms. Then, the correlates or multiplicands of this multiplier cannot all stand directly after it, as is usual in multiplication, but may be ranged after it in regular order, so that:

\(\mathfrak{g}\mathit{x}\mathit{y}\)

will denote a giver to \(\mathit{x}\!\) of \(\mathit{y}\!\). But according to the notation, \(\mathit{x}\!\) here multiplies \(\mathit{y}\!\), so that if we put for \(\mathit{x}\!\) owner (\(\mathit{o}\!\)), and for \(\mathit{y}\!\) horse (\(\mathrm{h}\!\)),

\(\mathfrak{g}\mathit{o}\mathrm{h}\)

appears to denote the giver of a horse to an owner of a horse. But let the individual horses be \(\mathrm{H}, \mathrm{H}^{\prime}, \mathrm{H}^{\prime\prime}\), etc. Then:

\(\mathrm{h} ~=~ \mathrm{H} ~+\!\!,~ \mathrm{H}^{\prime} ~+\!\!,~ \mathrm{H}^{\prime\prime} ~+\!\!,~ \text{etc.}\)

\(\mathfrak{g}\mathit{o}\mathrm{h} ~=~ \mathfrak{g}\mathit{o}(\mathrm{H} ~+\!\!,~ \mathrm{H}^{\prime} ~+\!\!,~ \mathrm{H}^{\prime\prime} ~+\!\!,~ \text{etc.}) ~=~ \mathfrak{g}\mathit{o}\mathrm{H} ~+\!\!,~ \mathfrak{g}\mathit{o}\mathrm{H}^{\prime} ~+\!\!,~ \mathfrak{g}\mathit{o}\mathrm{H}^{\prime\prime} ~+\!\!,~ \text{etc.}\)

Now this last member must be interpreted as a giver of a horse to the owner of that horse, and this, therefore must be the interpretation of \(\mathfrak{g}\mathit{o}\mathrm{h}\). This is always very important. A term multiplied by two relatives shows that the same individual is in the two relations. If we attempt to express the giver of a horse to a lover of a woman, and for that purpose write:

\(\mathfrak{g}\mathit{l}\mathrm{w}\mathrm{h}\),

we have written giver of a woman to a lover of her, and if we add brackets, thus,

\(\mathfrak{g}(\mathit{l}\mathrm{w})\mathrm{h}\),

we abandon the associative principle of multiplication. A little reflection will show that the associative principle must in some form or other be abandoned at this point. But while this principle is sometimes falsified, it oftener holds, and a notation must be adopted which will show of itself when it holds. We already see that we cannot express multiplication by writing the multiplicand directly after the multiplier; let us then affix subjacent numbers after letters to show where their correlates are to be found. The first number shall denote how many factors must be counted from left to right to reach the first correlate, the second how many 'more' must be counted to reach the second, and so on. Then, the giver of a horse to a lover of a woman may be written:

\(\mathfrak{g}_{12} \mathit{l}_1 \mathrm{w} \mathrm{h} ~=~ \mathfrak{g}_{11} \mathit{l}_2 \mathrm{h} \mathrm{w} ~=~ \mathfrak{g}_{2(-1)} \mathrm{h} \mathit{l}_1 \mathrm{w}\).

Of course a negative number indicates that the former correlate follows the latter by the corresponding positive number. A subjacent zero makes the term itself the correlate. Thus,

\(\mathit{l}_0\!\)

denotes the lover of that lover or the lover of himself, just as \(\mathfrak{g}\mathit{o}\mathrm{h}\) denotes that the horse is given to the owner of itself, for to make a term doubly a correlate is, by the distributive principle, to make each individual doubly a correlate, so that:

\(\mathit{l}_0 ~=~ \mathit{L}_0 ~+\!\!,~ \mathit{L}_0^{\prime} ~+\!\!,~ \mathit{L}_0^{\prime\prime} ~+\!\!,~ \text{etc.}\)

A subjacent sign of infinity may indicate that the correlate is indeterminate, so that:

\(\mathit{l}_\infty\)

will denote a lover of something. We shall have some confirmation of this presently. If the last subjacent number is a one it may be omitted. Thus we shall have:

\(\mathit{l}_1 ~=~ \mathit{l}\),

\(\mathfrak{g}_{11} ~=~ \mathfrak{g}_1 ~=~ \mathfrak{g}\).

This enables us to retain our former expressions \(\mathit{l}\mathrm{w}\!\), \(\mathfrak{g}\mathit{o}\mathrm{h}\), etc. (Peirce, CP 3.69–70).

Comment : Sets as Logical Sums

Peirce's way of representing sets as logical 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 a computer was just a person who computed, 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 element \(a\!\) 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:

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

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 the product \((1 + a)(1 + b)(1 + c)\!\), one obtains:

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

And this informs us that the subsets of choice are:

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

Selection 7

The Signs for Multiplication (cont.)

The associative principle does not hold in this counting of factors. Because it does not hold, these subjacent numbers are frequently inconvenient in practice, and I therefore use also another mode of showing where the correlate of a term is to be found. This is by means of the marks of reference, \(\dagger ~ \ddagger ~ \parallel ~ \S ~ \P\), which are placed subjacent to the relative term and before and above the correlate. Thus, giver of a horse to a lover of a woman may be written:

\(\mathfrak{g}_{\dagger\ddagger} \, ^\dagger\mathit{l}_\parallel \, ^\parallel\mathrm{w} \, ^\ddagger\mathrm{h}\)

The asterisk I use exclusively to refer to the last correlate of the last relative of the algebraic term. Now, considering the order of multiplication to be: — a term, a correlate of it, a correlate of that correlate, etc. — there is no violation of the associative principle. The only violations of it in this mode of notation are that in thus passing from relative to correlate, we skip about among the factors in an irregular manner, and that we cannot substitute in such an expression as \(\mathfrak{g}\mathit{o}\mathrm{h}\) a single letter for \(\mathit{o}\mathrm{h}.\!\) I would suggest that such a notation may be found useful in treating other cases of non-associative multiplication. By comparing this with what was said above [in CP 3.55] concerning functional multiplication, it appears that multiplication by a conjugative term is functional, and that the letter denoting such a term is a symbol of operation. I am therefore using two alphabets, the Greek and Kennerly, where only one was necessary. But it is convenient to use both. (Peirce, CP 3.71–72).

Comment : Proto-Graphical Syntax

It is clear from our last excerpt that Peirce is already on the verge of a graphical syntax for the logic of relatives. Indeed, it seems likely that he had already reached this point in his own thinking.

For instance, it seems quite impossible to read his last variation on the theme of a "giver of a horse to a lover of a woman" without drawing lines of identity to connect up the corresponding marks of reference, like this:

o---------------------------------------o | | | ! # | | / \ / \ | | o o o o | | `g`_!@ !'l'_# #w @h | | o o | | \_____________/ | | @ | | | o---------------------------------------o Giver of a Horse to a Lover of a Woman

Selection 8

The Signs for Multiplication (cont.)

Thus far, we have considered the multiplication of relative terms only. Since our conception of multiplication is the application of a relation, we can only multiply absolute terms by considering them as relatives. Now the absolute term "man" is really exactly equivalent to the relative term "man that is ——", and so with any other. I shall write a comma after any absolute term to show that it is so regarded as a relative term. Then "man that is black" will be written:

\(\mathrm{m},\!\mathrm{b}\!\)

But not only may any absolute term be thus regarded as a relative term, but any relative term may in the same way be regarded as a relative with one correlate more. It is convenient to take this additional correlate as the first one. Then:

\(\mathit{l},\!\mathit{s}\mathrm{w}\)

will denote a lover of a woman that is a servant of that woman. The comma here after \(\mathit{l}\!\) should not be considered as altering at all the meaning of \(\mathit{l}\!\), but as only a subjacent sign, serving to alter the arrangement of the correlates. In point of fact, since a comma may be added in this way to any relative term, it may be added to one of these very relatives formed by a comma, and thus by the addition of two commas an absolute term becomes a relative of two correlates. So:

\(\mathrm{m},\!,\!\mathrm{b},\!\mathrm{r}\)

interpreted like

\(\mathfrak{g}\mathit{o}\mathrm{h}\)

means a man that is a rich individual and is a black that is that rich individual. But this has no other meaning than:

\(\mathrm{m},\!\mathrm{b},\!\mathrm{r}\)

or a man that is a black that is rich. Thus we see that, after one comma is added, the addition of another does not change the meaning at all, so that whatever has one comma after it must be regarded as having an infinite number. If, therefore, \(\mathit{l},\!,\!\mathit{s}\mathrm{w}\) is not the same as \(\mathit{l},\!\mathit{s}\mathrm{w}\) (as it plainly is not, because the latter means a lover and servant of a woman, and the former a lover of and servant of and same as a woman), this is simply because the writing of the comma alters the arrangement of the correlates. And if we are to suppose that absolute terms are multipliers at all (as mathematical generality demands that we should}, we must regard every term as being a relative requiring an infinite number of correlates to its virtual infinite series "that is —— and is —— and is —— etc." Now a relative formed by a comma of course receives its subjacent numbers like any relative, but the question is, What are to be the implied subjacent numbers for these implied correlates? Any term may be regarded as having an infinite number of factors, those at the end being ones, thus:

\(\mathit{l},\!\mathit{s}\mathrm{w} ~=~ \mathit{l},\!\mathit{s}\mathit{w},\!\mathit{1},\!\mathit{1},\!\mathit{1},\!\mathit{1},\!\mathit{1},\!\mathit{1},\!\mathit{1}, ~\text{etc.}\)

A subjacent number may therefore be as great as we please. But all these ones denote the same identical individual denoted by \(\mathrm{w}\!\); what then can be the subjacent numbers to be applied to \(\mathit{s}\!\), for instance, on account of its infinite "that is"'s? What numbers can separate it from being identical with \(\mathrm{w}\!\)? There are only two. The first is zero, which plainly neutralizes a comma completely, since

\(\mathit{s},_0\!\mathrm{w} ~=~ \mathit{s}\mathrm{w}\)

and the other is infinity; for as \(1^\infty\) is indeterminate in ordinary algbra, so it will be shown hereafter to be here, so that to remove the correlate by the product of an infinite series of ones is to leave it indeterminate. Accordingly,

\(\mathrm{m},_\infty\)

should be regarded as expressing some man. Any term, then, is properly to be regarded as having an infinite number of commas, all or some of which are neutralized by zeros. "Something" may then be expressed by:

\(\mathit{1}_\infty\!\)

I shall for brevity frequently express this by an antique figure one \((\mathfrak{1}).\) "Anything" by:

\(\mathit{1}_0\!\)

I shall often also write a straight \(1\!\) for anything. (Peirce, CP 3.73).

Commentary Note 8.1

To my way of thinking, CP 3.73 is one of the most remarkable passages in the history of logic. In this first pass over its deeper contents I won't be able to accord it much more than a superficial dusting off.

Let us imagine a concrete example that will serve in developing the uses of Peirce's notation. Entertain a discourse whose universe \(X\!\) will remind us a little of the cast of characters in Shakespeare's Othello.

\(X ~=~ \{ \mathrm{Bianca}, \mathrm{Cassio}, \mathrm{Clown}, \mathrm{Desdemona}, \mathrm{Emilia}, \mathrm{Iago}, \mathrm{Othello} \}.\)

The universe \(X\!\) is "that class of individuals about which alone the whole discourse is understood to run" but its marking out for special recognition as a universe of discourse in no way rules out the possibility that "discourse may run upon something which is not a subjective part of the universe; for instance, upon the qualities or collections of the individuals it contains" (CP 3.65).

In order to provide ourselves with the convenience of abbreviated terms, while preserving Peirce's conventions about capitalization, we may use the alternate names \(^{\backprime\backprime}\mathrm{u}^{\prime\prime}\) for the universe \(X\!\) and \(^{\backprime\backprime}\mathrm{Jeste}^{\prime\prime}\) for the character \(\mathrm{Clown}.\!\) This permits the above description of the universe of discourse to be rewritten in the following fashion:

\(\mathrm{u} ~=~ \{ \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}, \mathrm{I}, \mathrm{J}, \mathrm{O} \}\)

This specification of the universe of discourse could be summed up in Peirce's notation by the following equation:

\(\begin{array}{*{15}{c}} \mathbf{1} & = & \mathrm{B} & +\!\!, & \mathrm{C} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \end{array}\)

Within this discussion, then, the individual terms are as follows:

\(\begin{matrix} ^{\backprime\backprime}\mathrm{B}^{\prime\prime}, & ^{\backprime\backprime}\mathrm{C}^{\prime\prime}, & ^{\backprime\backprime}\mathrm{D}^{\prime\prime}, & ^{\backprime\backprime}\mathrm{E}^{\prime\prime}, & ^{\backprime\backprime}\mathrm{I}^{\prime\prime}, & ^{\backprime\backprime}\mathrm{J}^{\prime\prime}, & ^{\backprime\backprime}\mathrm{O}^{\prime\prime} \end{matrix}\)

Each of these terms denotes in a singular fashion the corresponding individual in \(X.\!\)

By way of general terms in this discussion, we may begin with the following set:

\(\begin{array}{ccl} ^{\backprime\backprime}\mathrm{b}^{\prime\prime} & = & ^{\backprime\backprime}\mathrm{black}^{\prime\prime} \'"`UNIQ-MathJax1-QINU`"' ==='"`UNIQ--h-72--QINU`"'Commentary Note 11.14=== First, a correction. Ignore for now the gloss that I gave in regard to Figure 19: {| align="center" cellspacing="6" width="90%" | <p>Here, I have used arrowheads to indicate the relational domains at which each of the relations \(J, K, L\!\) happens to be functional.

It is more like the feathers of the arrows that serve to mark the relational domains at which the relations \(J, K, L\!\) are functional, but it would take yet another construction to make this precise, as the feathers are not uniquely appointed but many splintered.

Now, as promised, let's look at a more homely example of a morphism, say, any one of the mappings \(J : \mathbb{R} \to \mathbb{R}\) (roughly speaking) that are commonly known as logarithm functions, where you get to pick your favorite base. In this case, \(K(r, s) = r + s\!\) and \(L(u, v) = u \cdot v,\) and the defining formula \(J(L(u, v)) = K(Ju, Jv)\!\) comes out looking like \(J(u \cdot v) = J(u) + J(v),\) writing a dot (\(\cdot\!\)) and a plus sign (\(+\!\)) for the ordinary 2-ary operations of arithmetical multiplication and arithmetical summation, respectively.

o-----------------------------------------------------------o | | | {+} {.} | | O O | | /|\ /|\ | | / | \ / | \ | | v | \ v | \ | | o o o o o o | | X X X Y Y Y | | o o o o o o | | ^ ^ ^ / / / | | \ \ \ / / | | \ \ / \ / / | | \ \ \ / | | \ / \ / \ / | | O O O | | J J J | | | o-----------------------------------------------------------o Figure 21. Logarithm Arrow J : {+} <- {.}

Thus, where the image \(J\!\) is the logarithm map, the compound \(K\!\) is the numerical sum, and the the ligature \(L\!\) is the numerical product, one obtains the immemorial mnemonic motto:

The image of the product is the sum of the images.

\(\begin{array}{lll} J(u \cdot v) & = & J(u) + J(v) \\[12pt] J(L(u, v)) & = & K(Ju, Jv) \end{array}\)

Commentary Note 11.15

I'm going to elaborate a little further on the subject of arrows, morphisms, or structure-preserving maps, as a modest amount of extra work at this point will repay ample dividends when it comes time to revisit Peirce's "number of" function on logical terms.

The structure that is preserved by a structure-preserving map is just the structure that we all know and love as a 3-adic relation. Very typically, it will be the type of 3-adic relation that defines the type of 2-ary operation that obeys the rules of a mathematical structure that is known as a group, that is, a structure that satisfies the axioms for closure, associativity, identities, and inverses.

For example, in the previous case of the logarithm map \(J,\!\) we have the data:

\(\begin{array}{lcccll} J & : & \mathbb{R} & \leftarrow & \mathbb{R} & \text{(properly restricted)} \\[6pt] K & : & \mathbb{R} & \leftarrow & \mathbb{R} \times \mathbb{R} & \text{where}~ K(r, s) = r + s \\[6pt] L & : & \mathbb{R} & \leftarrow & \mathbb{R} \times \mathbb{R} & \text{where}~ L(u, v) = u \cdot v \end{array}\)

Real number addition and real number multiplication (suitably restricted) are examples of group operations. If we write the sign of each operation in braces as a name for the 3-adic relation that constitutes or defines the corresponding group, then we have the following set-up:

\(\begin{matrix} J & : & [+] \leftarrow [\,\cdot\,] \\[6pt] [+] & \subseteq & \mathbb{R} \times \mathbb{R} \times \mathbb{R} \\[6pt] [\,\cdot\,] & \subseteq & \mathbb{R} \times \mathbb{R} \times \mathbb{R} \end{matrix}\)

In many cases, one finds that both group operations are indicated by the same sign, typically  \(\cdot\!\) ,  \(*\!\) ,  \(+\!\) , or simple concatenation, but they remain in general distinct whether considered as operations or as relations, no matter what signs of operation are used. In such a setting, our chiasmatic theme may run a bit like these two variants:

The image of the sum is the sum of the images.

The image of the product is the sum of the images.

Figure 22 presents a generic picture for groups \(G\!\) and \(H.\!\)

o-----------------------------------------------------------o | | | G H | | O O | | /|\ /|\ | | / | \ / | \ | | v | \ v | \ | | o o o o o o | | X X X Y Y Y | | o o o o o o | | ^ ^ ^ / / / | | \ \ \ / / | | \ \ / \ / / | | \ \ \ / | | \ / \ / \ / | | O O O | | J J J | | | o-----------------------------------------------------------o Figure 22. Group Homomorphism J : G <- H

In a setting where both groups are written with a plus sign, perhaps even constituting the very same group, the defining formula of a morphism, \(J(L(u, v)) = K(Ju, Jv),\!\) takes on the shape \(J(u + v) = Ju + Jv,\!\) which looks very analogous to the distributive multiplication of a sum \((u + v)\!\) by a factor \(J.\!\) Hence another popular name for a morphism: a linear map.

Commentary Note 11.16

We have enough material on morphisms now to go back and cast a more studied eye on what Peirce is doing with that "number of" function, the one that we apply to a logical term \(t\!\) by writing it in square brackets, as \([t].\!\) It is convenient to have a prefix notation for this function, and since Peirce reserves \(\mathit{n}\!\) for \(\operatorname{not},\!\) let's use \(v(t)\!\) as a variant for \([t].\!\)

My plan will be nothing less plodding than to work through all of the principal statements that Peirce has made about the "number of" function up to our present stopping place in the paper, namely, those that I collected once before and placed at this location:

• Commentary Note 11.2

I propose to assign to all logical terms, numbers; to an absolute term, the number of individuals it denotes; to a relative term, the average number of things so related to one individual. Thus in a universe of perfect men (men), the number of "tooth of" would be 32. The number of a relative with two correlates would be the average number of things so related to a pair of individuals; and so on for relatives of higher numbers of correlates. I propose to denote the number of a logical term by enclosing the term in square brackets, thus, \([t].\!\) (Peirce, CP 3.65).

We may formalize the role of the "number of" function by assigning it a name and a type as \(v : S \to \mathbb{R},\) where \(S\!\) is a suitable set of signs, a so-called syntactic domain, that is ample enough to hold all of the terms that we might wish to number in a given discussion, and where \(\mathbb{R}\) is the real number domain.

Transcribing Peirce's example, we may let m = "man" and t = "tooth of ---". Then v(t) = [t] = [tm]÷[m], that is to say, in a universe of perfect human dentition, the number of the relative term "tooth of ---" is equal to the number of teeth of humans divided by the number of humans, that is, 32.

The 2-adic relative term t determines a 2-adic relation T ⊆ U × V, where U and V are two universes of discourse, possibly the same one, that hold among other things all of the teeth and all of the people that happen to be under discussion, respectively.

A rough indication of the bigraph for T might be drawn as follows, where I have tried to sketch in just the toothy part of U and the peoply part of V.

t_1     t_32  t_33    t_64  t_65    t_96  ...     ...
 o  ...  o     o  ...  o     o  ...  o     o  ...  o     U
  \  |  /       \  |  /       \  |  /       \  |  /
   \ | /         \ | /         \ | /         \ | /       T
    \|/           \|/           \|/           \|/
     o             o             o             o         V
    m_1           m_2           m_3           ...

Notice that the "number of" function v : S → R needs the data that is represented by this entire bigraph for T in order to compute the value [t].

Finally, one observes that this component of T is a function in the direction T : U → V, since we are counting only those teeth that ideally occupy one and only one mouth of a creature.

Commentary Note 11.17

I think that the reader is beginning to get an inkling of the crucial importance of the "number of" map in Peirce's way of looking at logic, for it's one of the plancks in the bridge from logic to the theories of probability, statistics, and information, in which logic forms but a limiting case at one scenic turnout on the expanding vista. It is, as a matter of necessity and a matter of fact, practically speaking, at any rate, one way that Peirce forges a link between the "eternal", logical, or rational realm and the "secular", empirical, or real domain.

With that little bit of encouragement and exhortation, let us return to the nitty gritty details of the text.

But not only do the significations of "=" and "<" here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations. Equality is, in fact, nothing but the identity of two numbers; numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes. So, to write 5 < 7 is to say that 5 is part of 7, just as to write f < m is to say that Frenchmen are part of men. Indeed, if f < m, then the number of Frenchmen is less than the number of men, and if v = p, then the number of Vice-Presidents is equal to the number of Presidents of the Senate; so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities. (Peirce, CP 3.66).

Peirce is here remarking on the principle that the measure v on terms "preserves" or "respects" the prevailing implication, inclusion, or subsumption relations that impose an ordering on those terms.

In these initiatory passages of the text, Peirce is using a single symbol "<" to denote the usual linear ordering on numbers, but also what amounts to the implication ordering on logical terms and the inclusion ordering on classes. Later, of course, he will introduce distinctive symbols for logical orders.

Now, the links among terms, sets, and numbers can be pursued in all directions, and Peirce has already indicated in an earlier paper how he would "construct" the integers from sets, that is, from the aggregate denotations of terms.

We will get back to that at another time.

In the immediate example, we have this sort of statement:

"if f < m, then the number of Frenchmen is less than the number of men"

In symbolic form, this would be written:

f < m ⇒ [f] < [m]

Here, the "<" on the left is a logical ordering on syntactic terms while the "<" on the right is an arithmetic ordering on real numbers.

The type of principle that comes up here is usually discussed under the question of whether a map between two ordered sets is "order-preserving" or not. The general type of question may be formalized in the following way.

Let X₁ be a set with an ordering denoted by "<₁".

Let X₂ be a set with an ordering denoted by "<₂".

What makes an ordering what it is will commonly be a set of axioms that defines the properties of the order relation in question. Since one frequently has occasion to view the same set in the light of several different order relations, one will often resort to explicit forms like (X, <₁), (X, <₂), and so on, to invoke a set with a given ordering.

A map F : (X₁, <₁) → (X₂, <₂) is order-preserving if and only if a statement of a particular form holds for all x and y in (X₁, <₁), specifically, this:

x <₁ y ⇒ Fx <₂ Fy

The action of the "number of" map v : (S, <₁) → (R, <₂) has just this character, as exemplified by its application to the case where x = f = "frenchman" and y = m = "man", like so:

f < m ⇒ [f] < [m]

f < m ⇒ vf < vm

Here, to be more exacting, we may interpret the "<" on the left as "proper subsumption", that is, excluding the equality case, while we read the "<" on the right as the usual "less than".

Commentary Note 11.18

There is a comment that I ought to make on the concept of a structure preserving map, including as a special case the idea of an order-preserving map. It seems to be a peculiarity of mathematical usage in general — at least, I don't think it's just me — that "preserving structure" always means "preserving some, not of necessity all of the structure in question". People sometimes express this by speaking of structure preservation in measure, the implication being that any property that is amenable to being qualified in manner is potentially amenable to being quantified in degree, perhaps in such a way as to answer questions like "How structure-preserving is it?".

Let's see how this remark applies to the order-preserving property of the "number of" mapping v : S → R. For any pair of absolute terms x and y in the syntactic domain S, we have the following implications, where "–<" denotes the logical subsumption relation on terms and "=<" is the "less than or equal to" relation on the real number domain R.

x –< y ⇒ vx =< vy

Equivalently:

x –< y ⇒ [x] =< [y]

It is easy to see that nowhere near all of the distinctions that make up the structure of the ordering on the left hand side will be preserved as one passes to the right hand side of these implication statements, but that is not required in order to call the map v "order-preserving", or what is also known as an "order morphism".

Commentary Note 11.19

Up to this point in the LOR of 1870, Peirce has introduced the "number of" measure on logical terms and discussed the extent to which this measure, v : S → R such that v : s ~> [s], exhibits a couple of important measure-theoretic principles:

1. The "number of" map exhibits a certain type of "uniformity property", whereby the value of the measure on a uniformly qualified population is in fact actualized by each member of the population.
2. The "number of" map satisfies an "order morphism principle", whereby the illative partial ordering of logical terms is reflected up to a partial extent by the arithmetical linear ordering of their measures.

Peirce next takes up the action of the "number of" map on the two types of, loosely speaking, "additive" operations that we normally consider in logic.

It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions. (CP 3.67).

The "regular non-invertible addition" is signified by "+,", corresponding to what we'd call the inclusive disjunction of logical terms or the union of their extensions as sets.

The "invertible addition" is signified in algebra by "+", corresponding to what we'd call the exclusive disjunction of logical terms or the symmetric difference of their sets, ignoring many details and nuances that are often important, of course.

But the notation has other recommendations. The conception of taking together involved in these processes is strongly analogous to that of summation, the sum of 2 and 5, for example, being the number of a collection which consists of a collection of two and a collection of five. (CP 3.67).

A full interpretation of this remark will require us to pick up the precise technical sense in which Peirce is using the word "collection", and that will take us back to his logical reconstruction of certain aspects of number theory, all of which I am putting off to another time, but it is still possible to get a rough sense of what he's saying relative to the present frame of discussion.

The "number of" map v : S → R evidently induces some sort of morphism with respect to logical sums. If this were straightforwardly true, we could write:

(?) v(x +, y) = vx + vy

Equivalently:

(?) [x +, y] = [x] + [y]

Of course, things are just not that simple in the case of inclusive disjunction and set-theoretic unions, so we'd "probably" invent a word like "sub-additive" to describe the principle that does hold here, namely:

v(x +, y) =< vx + vy

Equivalently:

[x +, y] =< [x] + [y]

This is why Peirce trims his discussion of this point with the following hedge:

Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves — provided all the terms summed are mutually exclusive. (CP 3.67).

Finally, a morphism with respect to addition, even a contingently qualified one, must do the right stuff on behalf of the additive identity:

Addition being taken in this sense, nothing is to be denoted by zero, for then: x +, 0 = x whatever is denoted by x; and this is the definition of zero. This interpretation is given by Boole, and is very neat, on account of the resemblance between the ordinary conception of zero and that of nothing, and because we shall thus have [0] = 0. (Peirce, CP 3.67).

With respect to the nullity 0 in S and the number 0 in R, we have:

v0 = [0] = 0.

In sum, therefor, it also serves that only preserves a due respect for the function of a vacuum in nature.

Commentary Note 11.20

We arrive at the last, for the time being, of Peirce's statements about the "number of" map.

The conception of multiplication we have adopted is that of the application of one relation to another. … Even ordinary numerical multiplication involves the same idea, for 2 × 3 is a pair of triplets, and 3 × 2 is a triplet of pairs, where "triplet of" and "pair of" are evidently relatives. If we have an equation of the form: xy = z and there are just as many x's per y as there are per things, things of the universe, then we have also the arithmetical equation: [x][y] = [z]. (Peirce, CP 3.76).

Peirce is here observing what we might dub a "contingent morphism" or a "skeptraphotic arrow", if you will. Provided that a certain condition, to be named and, what is more hopeful, to be clarified in short order, happens to be satisfied, we would find it holding that the "number of" map v : S → R such that vs = [s] serves to preserve the multiplication of relative terms, that is as much to say, the composition of relations, in the form: [xy] = [x][y].

So let us try to uncross Peirce's manifestly chiasmatic encryption of the condition that is called on in support of this preservation.

Proviso for [xy] = [x][y] —

there are just as many x’s per y as there are per things[,] things of the universe …

I have placed angle brackets around a comma that CP shows but CE omits, not that it helps much either way. So let us resort to the example:

For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then: [t][f] = [tf] holds arithmetically. (CP 3.76).

Now that is something that we can sink our teeth into, and trace the bigraph representation of the situation. In order to do this, it will help to recall our first examination of the "tooth of" relation, and to adjust the picture that we sketched of it on that occasion.

Transcribing Peirce's example, we may let m = "man" and t = "tooth of ---". Then v(t) = [t] = [tm]/[m], that is to say, in a universe of perfect human dentition, the number of the relative term "tooth of ---" is equal to the number of teeth of humans divided by the number of humans, that is, 32.

The 2-adic relative term t determines a 2-adic relation T ⊆ U × V, where U and V are two universes of discourse, possibly the same one, that hold among other things all of the teeth and all of the people that happen to be under discussion, respectively. To make the case as simple as we can and still cover the point, let's say that there are just four people in our initial universe of discourse, and that just two of them are French. The bigraphic composition below shows all of the pertinent facts of the case.

T_1     T_32  T_33    T_64  T_65    T_96  T_97    T_128
 o  ...  o     o  ...  o     o  ...  o     o  ...  o      U
  \  |  /       \  |  /       \  |  /       \  |  /
   \ | /         \ | /         \ | /         \ | /       't'
    \|/           \|/           \|/           \|/
     o             o             o             o          V = m = 1
                   |                           |
                   |                           |         'f'
                   |                           |
     o             o             o             o          V = m = 1
     J             K             L             M

Here, the order of relational composition flows up the page. For convenience, the absolute term f = "frenchman" has been converted by using the comma functor to give the idempotent representation ‘f’ = f, = "frenchman that is ---", and thus it can be taken as a selective from the universe of mankind.

By way of a legend for the figure, we have the following data:

m	=	J +, K +, L +, M	=	1
f	=	K +, M
f	=	K:K +, M:M
t	=	(T001 +, … +, T032):J		+,
		(T033 +, … +, T064):K		+,
		(T065 +, … +, T096):L		+,
		(T097 +, … +, T128):M

Now let's see if we can use this picture to make sense of the following statement:

For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then: [t][f] = [tf] holds arithmetically. (CP 3.76).

In the lingua franca of statistics, Peirce is saying this: That if the population of Frenchmen is a "fair sample" of the general population with regard to dentition, then the morphic equation [tf] = [t][f], whose transpose gives [t] = [tf]/[f], is every bite as true as the defining equation in this circumstance, namely, [t] = [tm]/[m].

Commentary Note 11.21

One more example and one more general observation, and then we will be all caught up with our homework on Peirce's "number of" function.

So if men are just as apt to be black as things in general: [m,][b] = [m,b] where the difference between [m] and [m,] must not be overlooked. (Peirce, CP 3.76).

The protasis, "men are just as apt to be black as things in general", is elliptic in structure, and presents us with a potential ambiguity. If we had no further clue to its meaning, it might be read as either of the following:

Men are just as apt to be black as things in general are apt to be black.

Men are just as apt to be black as men are apt to be things in general.

The second interpretation, if grammatical, is pointless to state, since it equates a proper contingency with an absolute certainty.

So I think it is safe to assume this paraphrase of what Peirce intends:

Men are just as likely to be black as things in general are likely to be black.

Stated in terms of the conditional probability:

P(b|m) = P(b)

From the definition of conditional probability:

P(b|m) = P(b & m)/P(m)

Equivalently:

P(b & m) = P(b|m)P(m)

Thus we may derive the equivalent statement:

P(b & m) = P(b|m)P(m) = P(b)P(m)

And this, of course, is the definition of independent events, as applied to the event of being Black and the event of being a Man.

It seems like a likely guess, then, that this is the content of Peirce's statement about frequencies, [m,b] = [m,][b], in this case normalized to produce the equivalent statement about probabilities: P(m & b) = P(m)P(b).

Let's see if this checks out.

Let n be the number of things in general, in Peirce's lingo, n = [1]. On the assumption that m and b are associated with independent events, we get [m,b] = P(m & b)n = P(m)P(b)n = P(m)[b] = [m,][b], so we have to interpret [m,] = "the average number of men per things in general" as P(m) = the probability of a thing in general being a man. Seems okay.

Commentary Note 11.22

Let's look at that last example from a different angle.

So if men are just as apt to be black as things in general: [m,][b] = [m,b] where the difference between [m] and [m,] must not be overlooked. (Peirce, CP 3.76).

In different lights the formula [m,b] = [m,][b] presents itself as an "aimed arrow", "fair sample", or "independence" condition.

The example apparently assumes a universe of "things in general", encompassing among other things the denotations of the absolute terms m = "man" and b = "black". That suggests to me that we might well illustrate this case in relief, by returning to our earlier staging of 'Othello' and seeing how well that universe of dramatic discourse observes the premiss that "men are just as apt to be black as things in general".

Here are the relevant data:

1	=	B +, C +, D +, E +, I +, J +, O
b	=	O
m	=	C +, I +, J +, O
1,	=	B:B +, C:C +, D:D +, E:E +, I:I +, J:J +, O:O	b,	=	O:O
m,	=	C:C +, I:I +, J:J +, O:O

The "fair sampling" or "episkeptral arrow" condition is tantamount to this: "Men are just as apt to be black as things in general are apt to be black". In other words, men are a fair sample of things in general with respect to the factor of being black.

Should this hold, the consequence would be:

[m,b] = [m,][b].

When [b] is not zero, we obtain the result:

[m,] = [m,b]/[b].

Once again, the absolute term b = "black" is most felicitously depicted by way of its idempotent representation ‘b’ = b, = "black that is ---", and thus it can be taken as a selective from the universe of discourse.

Here is the bigraph for the composition:

m,b = "man that is black",

here represented in the equivalent form:

m,b, = "man that is black that is ---".

B   C   D   E   I   J   O
o   o   o   o   o   o   o   1
    |           |   |   |
    |           |   |   |   m,
    |           |   |   |
o   o   o   o   o   o   o   1
                        |
                        |   b,
                        |
o   o   o   o   o   o   o   1
B   C   D   E   I   J   O

Thus we observe one of the more factitious facts that hold in this universe of discourse, namely:

m,b = b.

Another way of saying that is:

b –< m.

That in itself is enough to puncture any notion that b and m are statistically independent, but let us continue to develop the plot a bit more.

Putting all of the general formulas and particular facts together, we arrive at following summation of situation in the Othello case:

If the fair sampling condition holds:

[m,] = [m,b]/[b] = [b]/[b] = `1`,

In fact, however, it is the case that:

[m,] = [m,1]/[1] = [m]/[1] = 4/7.

In sum, it is not the case in the Othello example that "men are just as apt to be black as things in general".

Expressed in terms of probabilities: P(m) = 4/7 and P(b) = 1/7.

If these were independent we'd have: P(mb) = 4/49.

On the contrary, P(mb) = P(b) = 1/7.

Another way to see it is as follows: P(b|m) = 1/4 while P(b) = 1/7.

Commentary Note 11.23

Let me try to sum up as succinctly as possible the lesson that we ought to take away from Peirce's last "number of" example, since I know that the account that I have given of it so far may appear to have wandered rather widely.

So if men are just as apt to be black as things in general: [m,][b] = [m,b] where the difference between [m] and [m,] must not be overlooked. C.S. Peirce, CP 3.76

In different lights the formula [m,b] = [m,][b] presents itself as an "aimed arrow", "fair sample", or "independence" condition. I had taken the tack of illustrating this polymorphous theme in bas relief, that is, via detour through a universe of discourse where it fails. Here's a brief reminder of the Othello example:

B   C   D   E   I   J   O
o   o   o   o   o   o   o   1
    |           |   |   |
    |           |   |   |   m,
    |           |   |   |
o   o   o   o   o   o   o   1
                        |
                        |   b,
                        |
o   o   o   o   o   o   o   1
B   C   D   E   I   J   O

The condition, "men are just as apt to be black as things in general", is expressible in terms of conditional probabilities as P(b|m) = P(b), written out, the probability of the event Black given the event Male is exactly equal to the unconditional probability of the event Black.

Thus, for example, it is sufficient to observe in the Othello setting that P(b|m) = 1/4 while P(b) = 1/7 in order to cognize the dependency, and thereby to tell that the ostensible arrow is anaclinically biased.

This reduction of a conditional probability to an absolute probability, in the form P(A|Z) = P(A), is a familiar disguise, and yet in practice one of the ways that we most commonly come to recognize the condition of independence P(AZ) = P(A)P(Z), via the definition of a conditional probability according to the rule P(A|Z) = P(AZ)/P(Z). To recall the familiar consequences, the definition of conditional probability plus the independence condition yields P(A|Z) = P(AZ)/P(Z) = P(A)P(Z)/P(Z), to wit, P(A|Z) = P(A).

As Hamlet discovered, there's a lot to be learned from turning a crank.

Commentary Note 11.24

And so we come to the end of the "number of" examples that we found on our agenda at this point in the text:

It is to be observed that: [!1!] = `1`. Boole was the first to show this connection between logic and probabilities. He was restricted, however, to absolute terms. I do not remember having seen any extension of probability to relatives, except the ordinary theory of expectation. Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it. C.S. Peirce, CP 3.76

There appears to be a problem with the printing of the text at this point. Let us first recall the conventions that I am using in this transcription: `1` for the "antique 1" that Peirce defines as !1!_∞ = "something", and !1! for the "bold 1" that signifies the ordinary 2-identity relation.

CP 3 gives [!1!] = `1`, which I cannot make any sense of. CE 2 gives [!1!] = 1 , which makes sense on the reading of "1" as denoting the natural number 1, and not as the absolute term "1" that denotes the universe of discourse. On this reading, [!1!] is the average number of things related by the identity relation !1! to one individual, and so it makes sense that [!1!] = 1 : N, where N is the set or the type of the natural numbers {0, 1, 2, …}.

With respect to the 2-identity !1! in the syntactic domain S and the number 1 in the non-negative integers N ⊂ R, we have:

v!1! = [!1!] = 1.

And so the "number of" mapping v : S → R has another one of the properties that would be required of an arrow S → R.

The manner in which these arrows and qualified arrows help us to construct a suspension bridge that unifies logic, semiotics, statistics, stochastics, and information theory will be one of the main themes that I aim to elaborate throughout the rest of this inquiry.

Selection 12

The Sign of Involution I shall take involution in such a sense that xy will denote everything which is an x for every individual of y. Thus 'l'w will be a lover of every woman. Then ('s''l')w will denote whatever stands to every woman in the relation of servant of every lover of hers; and 's'('l'w) will denote whatever is a servant of everything that is lover of a woman. So that ('s''l')w = 's'('l'w). (C.S. Peirce, CP 3.77).

Commentary Note 12

Let us make a few preliminary observations about the "logical sign of involution", as Peirce uses it here:

The Sign of Involution I shall take involution in such a sense that xy will denote everything which is an x for every individual of y. Thus 'l'w will be a lover of every woman. (C.S. Peirce, CP 3.77).

In arithmetic, the "involution" x^y, or the "exponentiation" of x to the power of y, is the iterated multiplication of the factor x, repeated as many times as there are ones making up the exponent y.

In analogous fashion, 'l'^w is the iterated multiplication of 'l', repeated as many times as there are individuals under the term w.

For example, suppose that the universe of discourse has, among other things, just the three women, W₁, W₂, W₃. This could be expressed in Peirce's notation by writing:

w = W₁ +, W₂ +, W₃.

In this setting, we would have:

'l'^w = 'l'^{(W₁ +, W₂ +, W₃)} = 'l'W₁ , 'l'W₂ , 'l'W₃.

That is, a lover of every woman in the universe of discourse would be a lover of W₁ and a lover of W₂ and lover of W₃.

References

• Boole, George (1854), An Investigation of the Laws of Thought, On Which are Founded the Mathematical Theories of Logic and Probabilities, Macmillan, 1854. Reprinted, Dover Publications, New York, NY, 1958.

• Peirce, C.S. (1870), "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic", Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870. Reprinted, Collected Papers (CP 3.45–149), Chronological Edition (CE 2, 359–429).

Bibliography

• Peirce, C.S., Bibliography.

• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958. Cited as (CP volume.paragraph).

• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianoplis, IN, 1981–. Cited as (CE volume, page).

See also

• Charles Sanders Peirce
• Logic of relatives
• Logic of Relatives (1870)
• Logic of Relatives (1883)

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