Peirce’s 1870 “Logic Of Relatives” • Comment 9.7

From this point forward we may think of idempotents, selectives, and zero-one diagonal matrices as being roughly equivalent notions. The only reason I say roughly is that we are comparing ideas at different levels of abstraction in proposing these connections.

We have covered the way that Peirce uses his invention of the comma modifier to assimilate boolean multiplication, logical conjunction, and what we may think of as serial selection under his more general account of relative multiplication.

But the comma functor has its application to relative terms of any arity, not just the zeroth arity of absolute terms, and so there will be a lot more to explore on this point. But now I must return to the anchorage of Peirce’s text and hopefully get a chance to revisit this topic later.

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Peirce’s 1870 “Logic Of Relatives” • Comment 9.6

By way of fixing the current array of relational concepts in our minds, let us work through a sample of products from our relational multiplication table that will serve to illustrate the application of a comma relative to an absolute term, presented in both matrix and bigraph pictures.

Example 1

Example 2

Example 3

Example 4

Example 5

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Peirce’s 1870 “Logic Of Relatives” • Comment 9.5

Peirce’s comma operation, in its application to an absolute term, is tantamount to the representation of that term’s denotation as an idempotent transformation, which is commonly represented as a diagonal matrix. Hence the alternate name, diagonal extension.

An idempotent element x is given by the abstract condition that xx = x, but elements like these are commonly encountered in more concrete circumstances, acting as operators or transformations on other sets or spaces, and in that action they will often be represented as matrices of coefficients.

Let’s see how this looks in the matrix and graph pictures of absolute and relative terms.

Absolute Terms

\begin{array}{*{17}{l}}  \mathbf{1} & = & \text{anything} & = &  \mathrm{B} & +\!\!, &  \mathrm{C} & +\!\!, &  \mathrm{D} & +\!\!, &  \mathrm{E} & +\!\!, &  \mathrm{I} & +\!\!, &  \mathrm{J} & +\!\!, &  \mathrm{O}  \\[6pt]  \mathrm{m} & = & \text{man} & = &  \mathrm{C} & +\!\!, &  \mathrm{I} & +\!\!, &  \mathrm{J} & +\!\!, &  \mathrm{O}  \\[6pt]  \mathrm{n} & = & \text{noble} & = &  \mathrm{C} & +\!\!, &  \mathrm{D} & +\!\!, &  \mathrm{O}  \\[6pt]  \mathrm{w} & = & \text{woman} & = &  \mathrm{B} & +\!\!, &  \mathrm{D} & +\!\!, &  \mathrm{E}  \end{array}

Previously, we represented absolute terms as column arrays. The above four terms are given by the columns of the following table:

\begin{array}{c|cccc}  & \mathbf{1} & \mathrm{m} & \mathrm{n} & \mathrm{w} \\  \hline  \mathrm{B} & 1 & 0 & 0 & 1 \\  \mathrm{C} & 1 & 1 & 1 & 0 \\  \mathrm{D} & 1 & 0 & 1 & 1 \\  \mathrm{E} & 1 & 0 & 0 & 1 \\  \mathrm{I} & 1 & 1 & 0 & 0 \\  \mathrm{J} & 1 & 1 & 0 & 0 \\  \mathrm{O} & 1 & 1 & 1 & 0  \end{array}

The types of graphs known as bigraphs or bipartite graphs can be used to picture simple relative terms, dyadic relations, and their corresponding logical matrices. One way to bring absolute terms and their corresponding sets of individuals into the bigraph picture is to mark the nodes in some way, for example, hollow nodes for non-members and filled nodes for members of the indicated set, as shown below:


LOR 1870 Figure 4.1
(4.1)

LOR 1870 Figure 4.2
(4.2)

LOR 1870 Figure 4.3
(4.3)

LOR 1870 Figure 4.4
(4.4)

Diagonal Extensions

\begin{array}{lll}  \mathbf{1,} & = & \text{anything that is}\, \underline{~~~~}  \\[6pt]  & = & \mathrm{B\!:\!B} ~+\!\!,~ \mathrm{C\!:\!C} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{E\!:\!E} ~+\!\!,~ \mathrm{I\!:\!I} ~+\!\!,~ \mathrm{J\!:\!J} ~+\!\!,~ \mathrm{O\!:\!O}  \\[9pt]  \mathrm{m,} & = & \text{man that is}\, \underline{~~~~}  \\[6pt]  & = & \mathrm{C\!:\!C} ~+\!\!,~ \mathrm{I\!:\!I} ~+\!\!,~ \mathrm{J\!:\!J} ~+\!\!,~ \mathrm{O\!:\!O}  \\[9pt]  \mathrm{n,} & = & \text{noble that is}\, \underline{~~~~}  \\[6pt]  & = & \mathrm{C\!:\!C} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{O\!:\!O}  \\[9pt]  \mathrm{w,} & = & \text{woman that is}\, \underline{~~~~}  \\[6pt]  & = & \mathrm{B\!:\!B} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{E\!:\!E}  \end{array}

Naturally enough, the diagonal extensions are represented by diagonal matrices:

\begin{array}{c|*{7}{c}}  \mathbf{1,} & \mathrm{B} & \mathrm{C} & \mathrm{D} &  \mathrm{E}  & \mathrm{I} & \mathrm{J} & \mathrm{O} \\  \hline  \mathrm{B} & 1 &   &   &   &   &   &   \\  \mathrm{C} &   & 1 &   &   &   &   &   \\  \mathrm{D} &   &   & 1 &   &   &   &   \\  \mathrm{E} &   &   &   & 1 &   &   &   \\  \mathrm{I} &   &   &   &   & 1 &   &   \\  \mathrm{J} &   &   &   &   &   & 1 &   \\  \mathrm{O} &   &   &   &   &   &   & 1  \end{array}

\begin{array}{c|*{7}{c}}  \mathrm{m,} & \mathrm{B} & \mathrm{C} & \mathrm{D} &  \mathrm{E}  & \mathrm{I} & \mathrm{J} & \mathrm{O} \\  \hline  \mathrm{B} & 0 &   &   &   &   &   &   \\  \mathrm{C} &   & 1 &   &   &   &   &   \\  \mathrm{D} &   &   & 0 &   &   &   &   \\  \mathrm{E} &   &   &   & 0 &   &   &   \\  \mathrm{I} &   &   &   &   & 1 &   &   \\  \mathrm{J} &   &   &   &   &   & 1 &   \\  \mathrm{O} &   &   &   &   &   &   & 1  \end{array}

\begin{array}{c|*{7}{c}}  \mathrm{n,} & \mathrm{B} & \mathrm{C} & \mathrm{D} &  \mathrm{E}  & \mathrm{I} & \mathrm{J} & \mathrm{O} \\  \hline  \mathrm{B} & 0 &   &   &   &   &   &   \\  \mathrm{C} &   & 1 &   &   &   &   &   \\  \mathrm{D} &   &   & 1 &   &   &   &   \\  \mathrm{E} &   &   &   & 0 &   &   &   \\  \mathrm{I} &   &   &   &   & 0 &   &   \\  \mathrm{J} &   &   &   &   &   & 0 &   \\  \mathrm{O} &   &   &   &   &   &   & 1  \end{array}

\begin{array}{c|*{7}{c}}  \mathrm{w,} & \mathrm{B} & \mathrm{C} & \mathrm{D} &  \mathrm{E}  & \mathrm{I} & \mathrm{J} & \mathrm{O} \\  \hline  \mathrm{B} & 1 &   &   &   &   &   &   \\  \mathrm{C} &   & 0 &   &   &   &   &   \\  \mathrm{D} &   &   & 1 &   &   &   &   \\  \mathrm{E} &   &   &   & 1 &   &   &   \\  \mathrm{I} &   &   &   &   & 0 &   &   \\  \mathrm{J} &   &   &   &   &   & 0 &   \\  \mathrm{O} &   &   &   &   &   &   & 0  \end{array}

Cast into the bigraph picture of dyadic relations, the diagonal extension of an absolute term takes on a very distinctive sort of “straight-laced” character:


LOR 1870 Figure 5.1
(5.1)

LOR 1870 Figure 5.2
(5.2)

LOR 1870 Figure 5.3
(5.3)

LOR 1870 Figure 5.4
(5.4)
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Peirce’s 1870 “Logic Of Relatives” • Comment 9.4

Boole rationalized the properties of what we now call boolean multiplication, roughly equivalent to logical conjunction, in terms of the laws that apply to selective operations. Peirce, in his turn, taking a very significant step of analysis that has seldom been recognized for what it would lead to, does not consider this multiplication to be a fundamental operation, but derives it as a by-product of relative multiplication by a comma relative. Thus, Peirce makes logical conjunction a special case of relative composition.

This opens up a very wide field of inquiry, the operational significance of logical terms, but it will be best to advance bit by bit and to lean on simple examples.

Back to Venice and the close-knit party of absolutes and relatives that we entertained when last stopping there.

Here is the list of absolute terms we had been considering before:

\begin{array}{*{17}{l}}  \mathbf{1} & = & \text{anything} & = &  \mathrm{B} & +\!\!, &  \mathrm{C} & +\!\!, &  \mathrm{D} & +\!\!, &  \mathrm{E} & +\!\!, &  \mathrm{I} & +\!\!, &  \mathrm{J} & +\!\!, &  \mathrm{O}  \\[6pt]  \mathrm{m} & = & \text{man} & = &  \mathrm{C} & +\!\!, &  \mathrm{I} & +\!\!, &  \mathrm{J} & +\!\!, &  \mathrm{O}  \\[6pt]  \mathrm{n} & = & \text{noble} & =  &  \mathrm{C} & +\!\!, &  \mathrm{D} & +\!\!, &  \mathrm{O}  \\[6pt]  \mathrm{w} & = & \text{woman} & = &  \mathrm{B} & +\!\!, &  \mathrm{D} & +\!\!, &  \mathrm{E}  \end{array}

Here is the list of comma inflexions or diagonal extensions of these terms:

\begin{array}{lll}  \mathbf{1,} & = & \text{anything that is}\, \underline{~~~~}  \\[6pt]  & = & \mathrm{B\!:\!B} ~+\!\!,~ \mathrm{C\!:\!C} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{E\!:\!E} ~+\!\!,~ \mathrm{I\!:\!I} ~+\!\!,~ \mathrm{J\!:\!J} ~+\!\!,~ \mathrm{O\!:\!O}  \\[9pt]  \mathrm{m,} & = & \text{man that is}\, \underline{~~~~}  \\[6pt]  & = & \mathrm{C\!:\!C} ~+\!\!,~ \mathrm{I\!:\!I} ~+\!\!,~ \mathrm{J\!:\!J} ~+\!\!,~ \mathrm{O\!:\!O}  \\[9pt]  \mathrm{n,} & = & \text{noble that is}\, \underline{~~~~}  \\[6pt]  & = & \mathrm{C\!:\!C} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{O\!:\!O}  \\[9pt]  \mathrm{w,} & = & \text{woman that is}\, \underline{~~~~}  \\[6pt]  & = & \mathrm{B\!:\!B} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{E\!:\!E}  \end{array}

One observes that the diagonal extension of \mathbf{1} is the same thing as the identity relation \mathit{1}.

Working with our smaller sample of absolute terms, we have already computed the sorts of products that apply the diagonal extension of an absolute term to another absolute term, for instance, these products:

\begin{array}{lllll}  \mathrm{m},\!\mathrm{n}  & = & \text{man that is noble}  & = & \mathrm{C} ~+\!\!,~ \mathrm{O}  \\[6pt]  \mathrm{n},\!\mathrm{m}  & = & \text{noble that is a man}  & = & \mathrm{C} ~+\!\!,~ \mathrm{O}  \\[6pt]  \mathrm{w},\!\mathrm{n}  & = & \text{woman that is noble}  & = & \mathrm{D}  \\[6pt]  \mathrm{n},\!\mathrm{w}  & = & \text{noble that is a woman}  & = & \mathrm{D}  \end{array}

This exercise gave us a bit of practical insight into why the commutative law holds for logical conjunction.

Further insight into the laws that govern this realm of logic, and the underlying reasons why they apply, might be gained by systematically working through the whole variety of different products that are generated by the operational means in sight, namely, the products obtained by appending a comma to each of the terms in \{ \mathbf{1}, \mathrm{m}, \mathrm{n}, \mathrm{w} \} and then applying the relative term that results to each of those same terms in \{ \mathbf{1}, \mathrm{m}, \mathrm{n}, \mathrm{w} \}.

But before we try to explore this territory more systematically, let us equip our intuitions with the forms of graphical and matrical representation that served us so well in our previous adventures.

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Peirce’s 1870 “Logic Of Relatives” • Comment 9.3

An idempotent element x in an algebraic system is one that obeys the idempotent law, that is, it satisfies the equation xx = x. Under most circumstances it is usual to write this as x^2 = x.

If the algebraic system in question falls under the additional laws that are necessary to carry out the required transformations then x^2 = x is convertible into x - x^2 = 0, and this into x(1 - x) = 0.

If the algebraic system in question happens to be a boolean algebra then the equation x(1 - x) = 0 says that x \land \lnot x is identically false, in effect, a statement of the classical principle of non-contradiction.

We have already seen how Boole found rationales for the commutative law and the idempotent law by contemplating the properties of selective operations.

It is time to bring these threads together, which we can do by considering the so-called idempotent representation of sets. This will give us one of the best ways to understand the significance that Boole attached to selective operations. It will also link up with the statements that Peirce makes regarding his dimension-augmenting comma operation.

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Peirce’s 1870 “Logic Of Relatives” • Comment 9.2

In setting up his discussion of selective operations and their corresponding selective symbols, Boole writes the following:

The operation which we really perform is one of selection according to a prescribed principle or idea.  To what faculties of the mind such an operation would be referred, according to the received classification of its powers, it is not important to inquire, but I suppose that it would be considered as dependent upon the two faculties of Conception or Imagination, and Attention.  To the one of these faculties might be referred the formation of the general conception;  to the other the fixing of the mental regard upon those individuals within the prescribed universe of discourse which answer to the conception.  If, however, as seems not improbable, the power of Attention is nothing more than the power of continuing the exercise of any other faculty of the mind, we might properly regard the whole of the mental process above described as referrible [sic] to the mental faculty of Imagination or Conception, the first step of the process being the conception of the Universe itself, and each succeeding step limiting in a definite manner the conception thus formed.  Adopting this view, I shall describe each such step, or any definite combination of such steps, as a definite act of conception.

(Boole, Laws of Thought, 43)

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Peirce’s 1870 “Logic Of Relatives” • Comment 9.1

Let us backtrack a few years and consider how George Boole explained his twin conceptions of selective operations and selective symbols.

Let us then suppose that the universe of our discourse is the actual universe, so that words are to be used in the full extent of their meaning, and let us consider the two mental operations implied by the words “white” and “men”.  The word “men” implies the operation of selecting in thought from its subject, the universe, all men;  and the resulting conception, men, becomes the subject of the next operation.  The operation implied by the word “white” is that of selecting from its subject, “men”, all of that class which are white.  The final resulting conception is that of “white men”.

Now it is perfectly apparent that if the operations above described had been performed in a converse order, the result would have been the same.  Whether we begin by forming the conception of “men”, and then by a second intellectual act limit that conception to “white men”, or whether we begin by forming the conception of “white objects”, and then limit it to such of that class as are “men”, is perfectly indifferent so far as the result is concerned.  It is obvious that the order of the mental processes would be equally indifferent if for the words “white” and “men” we substituted any other descriptive or appellative terms whatever, provided only that their meaning was fixed and absolute.  And thus the indifference of the order of two successive acts of the faculty of Conception, the one of which furnishes the subject upon which the other is supposed to operate, is a general condition of the exercise of that faculty.  It is a law of the mind, and it is the real origin of that law of the literal symbols of Logic which constitutes its formal expression (1) Chap. II [xy = yx]. 

It is equally clear that the mental operation above described is of such a nature that its effect is not altered by repetition.  Suppose that by a definite act of conception the attention has been fixed upon men, and that by another exercise of the same faculty we limit it to those of the race who are white.  Then any further repetition of the latter mental act, by which the attention is limited to white objects, does not in any way modify the conception arrived at, viz., that of white men.  This is also an example of a general law of the mind, and it has its formal expression in the law ((2) Chap. II) [x^2 = x] of the literal symbols.

(Boole, Laws of Thought, 44–45)

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