Peirce’s 1870 “Logic of Relatives” • Comment 8.5

Posted on February 20, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 8.5

I continue with my commentary on CP 3.73, developing the Othello example as a way of illustrating Peirce’s formalism.

Since multiplication by a dyadic relative term is a logical analogue of matrix multiplication in linear algebra, all the products computed above can be represented by logical matrices, that is, by arrays of boolean \{ 0, 1 \} coordinate values.  Absolute terms and dyadic relatives are represented as 1-dimensional and 2-dimensional arrays, respectively.

The equations defining the absolute terms are given again below, first as logical sums of individual terms and then as n-tuples of boolean coordinates.

Othello Universe

Since we are going to be regarding these tuples as column arrays, it is convenient to arrange them in a table of the following form.

Othello Column Array

Here are the dyadic relative terms again, followed by their representation as coefficient matrices, in this case bordered by row and column labels to remind us what the coefficient values are meant to signify.

\begin{array}{*{13}{c}} \mathit{l} & = & \mathrm{B\!:\!C} & +\!\!, & \mathrm{C\!:\!B} & +\!\!, & \mathrm{D\!:\!O} & +\!\!, & \mathrm{E\!:\!I} & +\!\!, & \mathrm{I\!:\!E} & +\!\!, & \mathrm{O\!:\!D} \end{array}

Logical Matrix L

\begin{array}{*{13}{c}} \mathit{s} & = & \mathrm{C\!:\!O} & +\!\!, & \mathrm{E\!:\!D} & +\!\!, & \mathrm{I\!:\!O} & +\!\!, & \mathrm{J\!:\!D} & +\!\!, & \mathrm{J\!:\!O} \end{array}

Logical Matrix S

Here are the matrix representations of the products we calculated before.

Logical Matrix L1

Logical Matrix LO

Logical Matrix LM

Logical Matrix LW

Logical Matrix S1

Logical Matrix SO

Logical Matrix SM

Logical Matrix SW

Logical Matrix LS

Logical Matrix SL

Resources

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

Posted in C.S. Peirce, Logic, Logic of Relatives, Logical Matrices, Mathematics, Relation Theory, Visualization | Tagged , , , , , , | 10 Comments

Peirce’s 1870 “Logic of Relatives” • Comment 8.4

Posted on February 19, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 8.4

I continue with my commentary on CP 3.73, developing the Othello example as a way of illustrating Peirce’s formalism.

To familiarize ourselves with the forms of calculation available in Peirce’s notation, let us compute a few of the simplest products we find at hand in the Othello universe.

Here are the absolute terms:

\begin{array}{*{15}{c}} \mathbf{1} & = & \mathrm{B} & +\!\!, & \mathrm{C} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{b} & = & \mathrm{O} \\[6pt] \mathrm{m} & = & \mathrm{C} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{w} & = & \mathrm{B} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} \end{array}

Here are the dyadic relative terms:

\begin{array}{*{13}{c}} \mathit{l} & = & \mathrm{B} \!:\! \mathrm{C} & +\!\!, & \mathrm{C} \!:\! \mathrm{B} & +\!\!, & \mathrm{D} \!:\! \mathrm{O} & +\!\!, & \mathrm{E} \!:\! \mathrm{I} & +\!\!, & \mathrm{I} \!:\! \mathrm{E} & +\!\!, & \mathrm{O} \!:\! \mathrm{D} \\[6pt] \mathit{s} & = & \mathrm{C} \!:\! \mathrm{O} & +\!\!, & \mathrm{E} \!:\! \mathrm{D} & +\!\!, & \mathrm{I} \!:\! \mathrm{O} & +\!\!, & \mathrm{J} \!:\! \mathrm{D} & +\!\!, & \mathrm{J} \!:\! \mathrm{O} \end{array}

Here are a few of the simplest products among those terms:

\begin{array}{lll} \mathit{l}\mathbf{1} & = & \text{lover of anything} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{O} \\[6pt] & = & \text{anything except} ~ \mathrm{J} \end{array}

\begin{array}{lll} \mathit{l}\mathrm{O} & = & \text{lover of Othello} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & \mathrm{O} \\[6pt] & = & \mathrm{D} \end{array}

\begin{array}{lll} \mathit{l}\mathrm{m} & = & \text{lover of a man} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & (\mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} \end{array}

\begin{array}{lll} \mathit{l}\mathrm{w} & = & \text{lover of a woman} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E}) \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{O} \end{array}

\begin{array}{lll} \mathit{s}\mathbf{1} & = & \text{servant of anything} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{C} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{E} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} \end{array}

\begin{array}{lll} \mathit{s}\mathrm{O} & = & \text{servant of Othello} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & \mathrm{O} \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} \end{array}

\begin{array}{lll} \mathit{s}\mathrm{m} & = & \text{servant of a man} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & (\mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} ~+\!\!,~ \mathrm{O}) \\[6pt] & = & \mathrm{C} ~+\!\!,~ \mathrm{I} ~+\!\!,~ \mathrm{J} \end{array}

\begin{array}{lll} \mathit{s}\mathrm{w} & = & \text{servant of a woman} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E}) \\[6pt] & = & \mathrm{E} ~+\!\!,~ \mathrm{J} \end{array}

\begin{array}{lll} \mathit{l}\mathit{s} & = & \text{lover of a servant of}\, \underline{~~~~} \\[6pt] & = & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\ & & \times \\ & & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\[6pt] & = & \mathrm{B} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{D} \end{array}

\begin{array}{lll} \mathit{s}\mathit{l} & = & \text{servant of a lover of}\, \underline{~~~~} \\[6pt] & = & (\mathrm{C} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O}) \\ & & \times \\ & & (\mathrm{B} \!:\! \mathrm{C} ~+\!\!,~ \mathrm{C} \!:\! \mathrm{B} ~+\!\!,~ \mathrm{D} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{I} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{E} ~+\!\!,~ \mathrm{O} \!:\! \mathrm{D}) \\[6pt] & = & \mathrm{C} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{E} \!:\! \mathrm{O} ~+\!\!,~ \mathrm{I} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{D} ~+\!\!,~ \mathrm{J} \!:\! \mathrm{O} \end{array}

Among other things, one observes that the relative terms \mathit{l} and \mathit{s} do not commute, that is, \mathit{l}\mathit{s} is not equal to \mathit{s}\mathit{l}.

Resources

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

Posted in C.S. Peirce, Logic, Logic of Relatives, Logical Graphs, Mathematics, Relation Theory, Visualization | Tagged , , , , , , | 10 Comments

Peirce’s 1870 “Logic of Relatives” • Comment 8.3

Posted on February 18, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 8.3

I continue with my commentary on CP 3.73, developing the Othello example as a way of illustrating Peirce’s formalism.

It is critically important to distinguish a relation from a relative term.

  • The relation is an object of thought which may be regarded in extension as a set of ordered tuples known as its elementary relations.
  • The relative term is a sign which denotes certain objects, called its relates, as these are determined in relation to certain other objects, called its correlates.  Under most circumstances the relative term may be taken to denote the corresponding relation.

Returning to the Othello example, let us consider the dyadic relatives ^{\backprime\backprime} \text{lover of}\, \underline{~~~~}\, ^{\prime\prime} and ^{\backprime\backprime} \text{servant of}\, \underline{~~~~}\, ^{\prime\prime}.

The relative term \mathit{l} equivalent to the rhematic expression ^{\backprime\backprime} \text{lover of}\, \underline{~~~~}\, ^{\prime\prime} is given by the following equation.

\begin{array}{*{13}{c}} \mathit{l} & = & \mathrm{B} \!:\! \mathrm{C} & +\!\!, & \mathrm{C} \!:\! \mathrm{B} & +\!\!, & \mathrm{D} \!:\! \mathrm{O} & +\!\!, & \mathrm{E} \!:\! \mathrm{I} & +\!\!, & \mathrm{I} \!:\! \mathrm{E} & +\!\!, & \mathrm{O} \!:\! \mathrm{D} \end{array}

In the interests of simplicity, let’s put aside all distinctions of rank and fealty, collapsing the motley crews of servant and subordinate under the heading of a single service, denoted by the relative term \mathit{s} for ^{\backprime\backprime} \text{servant of}\, \underline{~~~~}\, ^{\prime\prime}.  The terms of this unified service are given by the following equation.

\begin{array}{*{11}{c}} \mathit{s} & = & \mathrm{C} \!:\! \mathrm{O} & +\!\!, & \mathrm{E} \!:\! \mathrm{D} & +\!\!, & \mathrm{I} \!:\! \mathrm{O} & +\!\!, & \mathrm{J} \!:\! \mathrm{D} & +\!\!, & \mathrm{J} \!:\! \mathrm{O} \end{array}

The inclusion of \mathrm{I} \!:\! \mathrm{C} under \mathit{s} might be implied by the plot of the play but since it is so hotly arguable I will leave it out of the toll.

One thing more we need to watch out for:  There are different conventions in the field regarding the ordering of terms in their applications and different conventions are more convenient under different circumstances, so there’s little chance any one of them can be canonized once and for all.  In our current reading we apply relative terms from right to left and our conception of relative multiplication, or relational composition, needs to be adjusted accordingly.

Resources

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

Posted in C.S. Peirce, Logic, Logic of Relatives, Logical Graphs, Mathematics, Relation Theory, Visualization | Tagged , , , , , , | 10 Comments

Peirce’s 1870 “Logic of Relatives” • Comment 8.2

Posted on February 18, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 8.2

I continue with my commentary on CP 3.73, developing the Othello example as a way of illustrating Peirce’s formalism.

In the development of the story so far, we have a universe of discourse characterized by the following equations:

\begin{array}{*{15}{c}} \mathbf{1} & = & \mathrm{B} & +\!\!, & \mathrm{C} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{b} & = & \mathrm{O} \\[6pt] \mathrm{m} & = & \mathrm{C} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{w} & = & \mathrm{B} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} \end{array}

This much forms a basis for the collection of absolute terms to be used in this example.  Let us now consider how we might represent an exemplary collection of relative terms.

Consider the genesis of relative terms, for example:

\begin{array}{l} ^{\backprime\backprime}\, \text{lover of}\, \underline{~~~~}\, ^{\prime\prime} \\[6pt] ^{\backprime\backprime}\, \text{betrayer to}\, \underline{~~~~}\, \text{of}\, \underline{~~~~}\, ^{\prime\prime} \\[6pt] ^{\backprime\backprime}\, \text{winner over of}\, \underline{~~~~}\, \text{to}\, \underline{~~~~}\, \text{from}\, \underline{~~~~}\, ^{\prime\prime} \end{array}

We may regard these fill-in-the-blank forms as being derived by a kind of rhematic abstraction from the corresponding instances of absolute terms.

The following examples illustrate the relationships that exist among absolute terms, relative terms, relations, and elementary relations.

  • The relative term ^{\backprime\backprime} \text{lover of}\, \underline{~~~~}\, ^{\prime\prime} can be derived from the absolute term ^{\backprime\backprime} \text{lover of Emilia} ^{\prime\prime} by removing the absolute term ^{\backprime\backprime} \text{Emilia} ^{\prime\prime}.

    Iago is a lover of Emilia, so the relate-correlate pair \mathrm{I} \!:\! \mathrm{E} is an element of the dyadic relation associated with the relative term ^{\backprime\backprime} \text{lover of}\, \underline{~~~~}\, ^{\prime\prime}.

  • The relative term ^{\backprime\backprime} \text{betrayer to}\, \underline{~~~~}\, \text{of}\, \underline{~~~~}\, ^{\prime\prime} can be derived from the absolute term ^{\backprime\backprime} \text{betrayer to Othello of Desdemona} ^{\prime\prime} by removing the absolute terms ^{\backprime\backprime} \text{Othello} ^{\prime\prime} and ^{\backprime\backprime} \text{Desdemona} ^{\prime\prime}.

    Iago is a betrayer to Othello of Desdemona, so the relate-correlate-correlate triple \mathrm{I} \!:\! \mathrm{O} \!:\! \mathrm{D} is an element of the triadic relation associated with the relative term ^{\backprime\backprime} \text{betrayer to}\, \underline{~~~~}\, \text{of}\, \underline{~~~~}\, ^{\prime\prime}.

  • The relative term ^{\backprime\backprime} \text{winner over of}\, \underline{~~~~}\, \text{to}\, \underline{~~~~}\, \text{from}\, \underline{~~~~}\, ^{\prime\prime} can be derived from the absolute term ^{\backprime\backprime} \text{winner over of Othello to Iago from Cassio} ^{\prime\prime} by removing the absolute terms ^{\backprime\backprime} \text{Othello} ^{\prime\prime}, ^{\backprime\backprime} \text{Iago} ^{\prime\prime}, and ^{\backprime\backprime} \text{Cassio} ^{\prime\prime}.

    Iago is a winner over of Othello to Iago from Cassio, so the elementary relative term \mathrm{I} \!:\! \mathrm{O} \!:\! \mathrm{I} \!:\! \mathrm{C} is an element of the tetradic relation associated with the relative term ^{\backprime\backprime} \text{winner over of}\, \underline{~~~~}\, \text{to}\, \underline{~~~~}\, \text{from}\, \underline{~~~~}\, ^{\prime\prime}.

Resources

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

Posted in C.S. Peirce, Logic, Logic of Relatives, Logical Graphs, Mathematics, Relation Theory, Visualization | Tagged , , , , , , | 10 Comments

Peirce’s 1870 “Logic of Relatives” • Comment 8.1

Posted on February 18, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 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 invent a concrete example to illustrate the use of Peirce’s notation.  Imagine a discourse whose universe X will remind us of the cast of characters in Shakespeare’s Othello.

X ~=~ \{ \text{Bianca}, \text{Cassio}, \text{Clown}, \text{Desdemona}, \text{Emilia}, \text{Iago}, \text{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 afford ourselves the convenience of abbreviated terms while preserving Peirce’s conventions about capitalization, we may use the alternate terms {}^{\backprime\backprime}\mathrm{u} {}^{\prime\prime} for the universe X and {}^{\backprime\backprime} \mathrm{Jeste} {}^{\prime\prime} for the character \text{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{array}{*{7}{c}} ^{\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{array}

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} \\[6pt] ^{\backprime\backprime}\mathrm{m}^{\prime\prime} & = & ^{\backprime\backprime}\mathrm{man}^{\prime\prime} \\[6pt] ^{\backprime\backprime}\mathrm{w}^{\prime\prime} & = & ^{\backprime\backprime}\mathrm{woman}^{\prime\prime} \end{array}

The denotation of a general term may be given by means of an equation between terms.

\begin{array}{*{15}{c}} \mathrm{b} & = & \mathrm{O} \\[6pt] \mathrm{m} & = & \mathrm{C} & +\!\!, & \mathrm{I} & +\!\!, & \mathrm{J} & +\!\!, & \mathrm{O} \\[6pt] \mathrm{w} & = & \mathrm{B} & +\!\!, & \mathrm{D} & +\!\!, & \mathrm{E} \end{array}

Resources

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

Posted in C.S. Peirce, Logic, Logic of Relatives, Logical Graphs, Mathematics, Relation Theory, Visualization | Tagged , , , , , , | 10 Comments

Peirce’s 1870 “Logic of Relatives” • Selection 8

Posted on February 17, 2014 by Jon Awbrey

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

Peirce’s 1870 “Logic of Relatives” • 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}\mathrm{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 algebra, 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)

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–.

Resources

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

Posted in C.S. Peirce, Logic, Logic of Relatives, Logical Graphs, Mathematics, Relation Theory, Visualization | Tagged , , , , , , | 21 Comments

Peirce’s 1870 “Logic of Relatives” • Proto-Graphical Syntax

Posted on February 12, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Proto-Graphical Syntax

It is clear from our last Selection that Peirce is already on the verge of a graphical syntax for the logic of relative terms.  Indeed, it is likely he had already reached that point in his own thinking some time before.

For instance, it seems quite impossible for a person with any graphical sensitivity whatever to scan that last variation on “giver of a horse to a lover of a woman” without drawing or at least imagining lines of identity to connect the corresponding marks of reference, as shown in the following Figure.

Giver of a Horse to a Lover of a Woman

\text{Figure 3. Giver of a Horse to a Lover of a Woman}

Resources

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

Posted in C.S. Peirce, Logic, Logic of Relatives, Logical Graphs, Mathematics, Relation Theory, Visualization | Tagged , , , , , , | 12 Comments

Peirce’s 1870 “Logic of Relatives” • Selection 7

Posted on February 7, 2014 by Jon Awbrey

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

Peirce’s 1870 “Logic of Relatives” • 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:

Giver of a Horse to a Lover of a Woman

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 [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 [Gothic], where only one was necessary.  But it is convenient to use both.

(Peirce, CP 3.71–72)

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–.

Resources

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

Posted in C.S. Peirce, Logic, Logic of Relatives, Mathematics, Relation Theory, Semiotics, Triadicity, Visualization | Tagged , , , , , , , | 11 Comments

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

Posted on February 6, 2014 by Jon Awbrey

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

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

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

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

Suppose we are given a set of three elements, say, \{ a, b, c \}, and we are asked to find all the ways of choosing a subset from this collection.

We can represent the problem setup as the problem of computing the following product:

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

The factor (1 + a) represents the option we have, in choosing a subset of \{ a, b, c \}, to exclude the element a (signified by the 1), or else to include it (signified by the a), proceeding in a similar fashion with the other elements in their turn.

Probably on account of all those years I flippered away playing the oldtime pinball machines, I tend to imagine a product like that being displayed in a vertical array:

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

I picture that as a playboard with six bumpers, the ball chuting down the board in such a way as to strike exactly one of the two bumpers on each of the three levels.

So a trajectory of the ball where it hits the a bumper on the 1st level, hits the 1 bumper on the 2nd level, hits the c bumper on the 3rd level, and then exits the board, represents a single term in the desired product and corresponds to the subset \{ a, c \}.

Multiplying out the product (1 + a)(1 + b)(1 + c), one obtains the sum:

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

This informs us that the subsets of choice are:

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

And so they are.

Resources

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

Posted in C.S. Peirce, Logic, Logic of Relatives, Mathematics, Relation Theory, Semiotics, Triadicity, Visualization | Tagged , , , , , , , | 10 Comments

Peirce’s 1870 “Logic of Relatives” • Selection 6

Posted on February 5, 2014 by Jon Awbrey

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

Peirce’s 1870 “Logic of Relatives” • Selection 6

The application of a relation is one of the most basic operations in Peirce’s logic.  Because relation applications are so pervasive and because Peirce treats them on the pattern of algebraic multiplication, the part of §3 concerned with “The Signs for Multiplication” will occupy our attention for many days to come.

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}, ~\text{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}, ~\text{etc.}

(Peirce, CP 3.69–70)

Resources

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

Posted in C.S. Peirce, Logic, Logic of Relatives, Mathematics, Relation Theory, Semiotics, Triadicity, Visualization | Tagged , , , , , , , | 10 Comments