Peirce’s 1870 “Logic of Relatives” • Comment 11.7

Posted on May 5, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.7

We come now to the special cases of dyadic relations known as functions.  It will serve a dual purpose in the present exposition to take the class of functions as a source of object examples for clarifying the more abstruse concepts of Relation Theory.

To begin, let us recall the definition of a local flag L_{a @ j} of a k-adic relation L.

Display 1

For a dyadic relation L \subseteq X \times Y the notation for local flags can be simplified in two ways.  First, the local flags L_{u @ 1} and L_{v @ 2} are often more conveniently notated as L_{u @ X} and L_{v @ Y}, respectively.  Second, the notation may be streamlined even further by making the following definitions.

Display 2

In light of these conventions, the local flags of a dyadic relation L \subseteq X \times Y may be comprehended under the following descriptions.

Display 3

The following definitions are also useful.

Display 4

A sufficient illustration is supplied by the earlier example E.

Dyadic Relation E
\text{Figure 35. Dyadic Relation}~ E

Figure 36 shows the local flag E_{3 @ X} of E.

Local Flag E_{3 @ X}
\text{Figure 36. Local Flag}~ E_{3 @ X}

Figure 37 shows the local flag E_{2 @ Y} of E.

Local Flag E_{2 @ Y}
\text{Figure 37. Local Flag}~ E_{2 @ Y}

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Peirce’s 1870 “Logic of Relatives” • Comment 11.6

Posted on May 4, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.6

Let’s continue working our way through the above definitions, constructing appropriate examples as we go.

Relation E_1 \subseteq X \times Y exemplifies the quality of totality at X.

Dyadic Relation E₁
\text{Dyadic Relation}~ E_1

Relation E_2 \subseteq X \times Y exemplifies the quality of totality at Y.

Dyadic Relation E₂
\text{Dyadic Relation}~ E_2

Relation E_3 \subseteq X \times Y exemplifies the quality of tubularity at X.

Dyadic Relation E₃
\text{Dyadic Relation}~ E_3

Relation E_4 \subseteq X \times Y exemplifies the quality of tubularity at Y.

Dyadic Relation E₄
\text{Dyadic Relation}~ E_4

So E_3 is a pre-function e_3 : X \rightharpoonup Y and E_4 is a pre-function e_4 : X \leftharpoonup Y.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.5

Posted on May 2, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.5

Everyone knows the right sort of diagram can be a great aid in rendering complex matters comprehensible.  With that in mind, let’s extract what we need from the Relation Theory article to illuminate Peirce’s 1870 Logic of Relatives and use it to fashion what icons we can within the current frame of discussion.

For the immediate present, we may begin with dyadic relations and describe the most frequently encountered species of relations and functions in terms of their local and numerical incidence properties.

Let P \subseteq X \times Y be an arbitrary dyadic relation.  The following properties of P can then be defined.

Display 1

If P \subseteq X \times Y is tubular at X, then P is known as a partial function or a pre-function from X to Y, frequently signalized by renaming P with an alternate lower case name, say {}^{\backprime\backprime} p {}^{\prime\prime}, and writing p : X \rightharpoonup Y.

Just by way of formalizing the definition:

Display 2

To illustrate these properties, let us fashion a generic enough example of a dyadic relation, E \subseteq X \times Y, where X = Y = \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \}, and where the bigraph picture of E is shown in Figure 30.

Dyadic Relation E
\text{Figure 30. Dyadic Relation}~ E

If we scan along the X dimension from 0 to 9 we see that the incidence degrees of the X nodes with the Y domain are 0, 1, 2, 3, 1, 1, 1, 2, 0, 0 in that order.

If we scan along the Y dimension from 0 to 9 we see that the incidence degrees of the Y nodes with the X domain are 0, 0, 3, 2, 1, 1, 2, 1, 1, 0 in that order.

Thus, E is not total at either X or Y since there are nodes in both X and Y having incidence degrees less than 1.

Also, E is not tubular at either X or Y since there are nodes in both X and Y having incidence degrees greater than 1.

Clearly then the relation E cannot qualify as a pre-function, much less as a function, on either of its relational domains.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.4

Posted on May 1, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.4

The task before us is to clarify the relationships among relative terms, relations, and the special cases of relations given by equivalence relations, functions, and so on.

The first obstacle to get past is the order convention Peirce’s orientation to relative terms causes him to use for functions.  To focus on a concrete example of immediate use in this discussion, let’s take the “number of” function Peirce denotes by means of square brackets and re-formulate it as a dyadic relative term v in the following way.

v(t) ~:=~ [t] ~=~ \text{the number of the term}~ t.

To set the dyadic relative term v within a suitable context of interpretation, let’s suppose v corresponds to a relation V \subseteq \mathbb{R} \times S where \mathbb{R} is the set of real numbers and S is a suitable syntactic domain, here described as a set of terms.  The dyadic relation V is at first sight a function from S to \mathbb{R}.  It is, however, not always possible to assign a number to every term in whatever syntactic domain S we happen to pick, so we may eventually be forced to treat the dyadic relation V as a partial function from S to \mathbb{R}.  All things considered, then, let’s try the following budget of strategies and compromises.

First, let’s adapt the arrow notation for functions in such a way as to allow detaching the functional orientation from the order in which the names of domains are written on the page.  Second, let’s change the notation for partial functions, or pre-functions, to mark more clearly their distinction from functions.  This produces the following scheme.

q : X \to Y means q is functional at X.

q : X \leftarrow Y means q is functional at Y.

q : X \rightharpoonup Y means q is pre-functional at X.

q : X \leftharpoonup Y means q is pre-functional at Y.

Until it becomes necessary to stipulate otherwise, let’s assume v is a function in \mathbb{R} of S, written v : \mathbb{R} \leftarrow S, amounting to a functional alias of the dyadic relation V \subseteq \mathbb{R} \times S and associated with the dyadic relative term v whose rèlate lies in the set \mathbb{R} of real numbers and whose correlate lies in the set S of syntactic terms.

Note.  Please refer to the article on Relation Theory for the definitions of functions and pre‑functions used in the above discussion.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.3

Posted on April 30, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.3

Before I can discuss Peirce’s “number of” function in greater detail I will need to deal with an expositional difficulty I have been carefully dancing around all this time, but one which will no longer abide its assigned place under the rug.

Functions have long been understood, from well before Peirce’s time to ours, as special cases of dyadic relations, so the “number of” function is already to be numbered among the class of dyadic relatives we’ve been dealing with all this time.  But Peirce’s manner of representing a dyadic relative term mentions the “rèlate” first and the “correlate” second, a convention going over into functional terms as making the functional value first and the functional argument second.  The problem is, almost anyone brought up in our present time frame is accustomed to thinking of a function as a set of ordered pairs where the order in each pair lists the functional argument first and the functional value second.

Syntactic wrinkles of this sort can be ironed out smoothly enough in a framework of flexible interpretive conventions, but not without introducing an order of anachronism into Peirce’s text I want to avoid as much as possible.  This will require me to experiment with various styles of compromise.  Among other things, the interpretation of Peirce’s 1870 “Logic of Relatives” can be facilitated by introducing a few items of background material on relations in general, as regarded from a combinatorial point of view.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.2

Posted on April 30, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.2

NOF Said …

Let’s bring together the various things Peirce has said about the number of function up to this point in the paper.

NOF 1

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)

NOF 2

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)

NOF 3

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)

NOF 4

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 \times 3 is a pair of triplets, and 3 \times 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].

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

[\mathit{t}][\mathrm{f}] ~=~ [\mathit{t}\mathrm{f}]

holds arithmetically.

So if men are just as apt to be black as things in general,

[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}],

where the difference between [\mathrm{m}] and [\mathrm{m,}] must not be overlooked.

It is to be observed that

[\mathit{1}] ~=~ \mathfrak{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.

(Peirce, CP 3.76)

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Peirce’s 1870 “Logic of Relatives” • Comment 11.1

Posted on April 29, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.1

Dear Reader,

We have reached a suitable place to pause in our reading of Peirce’s text — actually, it’s more like a place to run as fast as we can along a parallel track — where I can pay off a few of the expository IOUs I’ve been using to pave the way to this point.

The more pressing debts that come to mind are concerned with Peirce’s “number of” function which maps a term t into a number [t] and with my justification for calling a certain style of illustration the hypergraph picture of relational composition.  As it happens, there is a thematic relation between these topics, and so I can make my way forward by addressing them together.

At this point we have two good pictures of how to compute the relational compositions of dyadic relations, namely, the bigraph representation and the matrix representation, each of which has its differential advantages in different types of situations.

But we lack a comparable picture of how to compute the richer variety of relational compositions involving triadic or higher adicity relations.  As a matter of fact, we run into a non-trivial classification problem simply to enumerate the different types of compositions arising in those cases.

Therefore let us inaugurate a systematic study of relational composition, general enough to articulate the “generative potency” of Peirce’s 1870 Logic of Relatives.

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Peirce’s 1870 “Logic of Relatives” • Selection 11

Posted on April 29, 2014 by Jon Awbrey

We continue with §3. Application of the Algebraic Signs to Logic.

Peirce’s 1870 “Logic of Relatives” • Selection 11

The Signs for Multiplication (concl.)

The conception of multiplication we have adopted is that of the application of one relation to another.  So, a quaternion being the relation of one vector to another, the multiplication of quaternions is the application of one such relation to a second.

Even ordinary numerical multiplication involves the same idea, for 2 \times 3 is a pair of triplets, and 3 \times 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].

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

[\mathit{t}][\mathrm{f}] ~=~ [\mathit{t}\mathrm{f}]

holds arithmetically.

So if men are just as apt to be black as things in general,

[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}],

where the difference between [\mathrm{m}] and [\mathrm{m,}] must not be overlooked.

It is to be observed that

[\mathit{1}] ~=~ \mathfrak{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.

(Peirce, CP 3.76)

References

  • 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 3.45–149, Chronological Edition 2, 359–429.  Online (1) (2) (3).
  • 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.
  • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.

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Peirce’s 1870 “Logic of Relatives” • Comment 10.12

Posted on April 26, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 10.12

Potential ambiguities in Peirce’s two versions of the “rich black man” example can be resolved by providing them with explicit graphical markups, as shown in Figures 28 and 29.

Man that is Black that is Rich

\text{Figure 28. Man that is Black that is Rich}

Man that is a Rich Individual and is a Black Person that is that Rich Individual

\text{Figure 29. Man that is a Rich Individual and is}
\text{a Black Person that is that Rich Individual}

On the other hand, as the forms of relational composition become more complex, the corresponding algebraic products of elementary relatives, for example, \mathrm{(x\!:\!y\!:\!z)(y\!:\!z)(z)}, will not always determine unique results without the addition of more information about the intended linkings of terms.

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Peirce’s 1870 “Logic of Relatives” • Comment 10.11

Posted on April 25, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 10.11

Let us return to the point where we left off unpacking the contents of CP 3.73.  Here Peirce remarks that the comma operator can be iterated at will.

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,\!,\!b,\!r} interpreted like \mathfrak{g}\mathit{o}\mathrm{h} means a man that is a rich individual and is a black [person] that is that rich individual.  But this has no other meaning than \mathrm{m,\!b,\!r} or a man that is a black [person] 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.

(Peirce, CP 3.73)

Again, let’s check whether this makes sense on the stage of our small but dramatic model.  Let’s say Desdemona and Othello are rich and, among the persons of the play, only they.  On this premiss we obtain a sample of absolute terms sufficiently ample to work through Peirce’s example.

Display 1

One application of the comma operator yields the following dyadic relatives.

Display 2

Another application of the comma operator generates the following triadic relatives.

Display 3

Assuming the associativity of multiplication among dyadic relatives, the product \mathrm{m,\!b,\!r} may be computed by a brute force method to yield the following result.

Display 4

This says that a man that is black that is rich is Othello, which is true on the premisses of our present universe of discourse.

Following the standard associative combinations of \mathfrak{g}\mathit{o}\mathrm{h}, the product \mathrm{m,\!,\!b,\!r} is multiplied out along the following lines, where the trinomials of the form \mathrm{(X\!:\!Y\!:\!Z)(Y\!:\!Z)(Z)} are the only ones producing a non‑null result, namely, \mathrm{(X\!:\!Y\!:\!Z)(Y\!:\!Z)(Z) = X}.

Display 5

So we have that \mathrm{m,\!,\!b,\!r} ~=~ \mathrm{m,\!b,\!r}.

In closing, observe how the teridentity relation has turned up again in this context, as the second comma‑ing of the universal term itself.

Display 6

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