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{{DISPLAYTITLE:Differential Logic}}
{{DISPLAYTITLE:Differential Logic : Sketch 2}}
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
'''Note.''' ''The present Sketch is largely superseded by the article “[[Differential Logic : Introduction]]” but I have preserved it here for the sake of the remaining ideas that have yet to be absorbed elsewhere.''
'''Differential logic''' is the component of logic whose object is the description of variation — for example, the aspects of change, difference, distribution, and diversity — in [[universes of discourse]] that are subject to logical description. In formal logic, differential logic treats the principles that govern the use of a ''differential logical calculus'', that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.
'''Differential logic''' is the component of logic whose object is the description of variation — for example, the aspects of change, difference, distribution, and diversity — in [[universes of discourse]] that are subject to logical description. In formal logic, differential logic treats the principles that govern the use of a ''differential logical calculus'', that is, a formal system with the expressive capacity to describe change and diversity in logical universes of discourse.
Now there are many ways to dance around this idea, and I feel like I have tried them all, before one gets down to acting on it, and there many issues of interpretation and justification that we will have to clear up after the fact, that is, before we can be sure that it all really makes any sense, but I think this time I'll just jump in, and show you the form in which this idea first came to me.
Now there are many ways to dance around this idea, and I feel like I have tried them all, before one gets down to acting on it, and there many issues of interpretation and justification that we will have to clear up after the fact, that is, before we can be sure that it all really makes any sense, but I think this time I'll just jump in, and show you the form in which this idea first came to me.
Start with a proposition of the form <math>x ~\operatorname{and}~ y,</math> which is graphed as two labels attached to a root node:
Start with a proposition of the form <math>x ~\mathrm{and}~ y,\!</math> which is graphed as two labels attached to a root node:
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<pre>
<pre>
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Written as a string, this is just the concatenation "<math>x~y</math>".
Written as a string, this is just the concatenation "<math>x~y\!</math>".
The proposition <math>xy\!</math> may be taken as a boolean function <math>f(x, y)\!</math> having the abstract type <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},</math> where <math>\mathbb{B} = \{ 0, 1 \}</math> is read in such a way that <math>0\!</math> means <math>\operatorname{false}</math> and <math>1\!</math> means <math>\operatorname{true}.</math>
The proposition <math>xy\!</math> may be taken as a boolean function <math>f(x, y)\!</math> having the abstract type <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\!</math> where <math>\mathbb{B} = \{ 0, 1 \}~\!</math> is read in such a way that <math>0\!</math> means <math>\mathrm{false}\!</math> and <math>1\!</math> means <math>\mathrm{true}.\!</math>
In this style of graphical representation, the value <math>\operatorname{true}</math> looks like a blank label and the value <math>\operatorname{false}</math> looks like an edge.
In this style of graphical representation, the value <math>\mathrm{true}\!</math> looks like a blank label and the value <math>\mathrm{false}\!</math> looks like an edge.
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Back to the proposition <math>xy.~\!</math> Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>xy\!</math> is true, as shown here:
Back to the proposition <math>xy.~\!</math> Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>xy\!</math> is true, as shown here:
{| align="center" cellpadding="10"
{| align="center" cellspacing="10"
| [[Image:Venn Diagram X And Y.jpg|500px]]
| [[Image:Venn Diagram X And Y.jpg|500px]]
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Now ask yourself: What is the value of the proposition <math>xy\!</math> at a distance of <math>\operatorname{d}x</math> and <math>\operatorname{d}y</math> from the cell <math>xy\!</math> where you are standing?
Now ask yourself: What is the value of the proposition <math>xy\!</math> at a distance of <math>\mathrm{d}x\!</math> and <math>\mathrm{d}y\!</math> from the cell <math>xy\!</math> where you are standing?
Don't think about it — just compute:
Don't think about it — just compute:
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To make future graphs easier to draw in ASCII, I will use devices like '''<code>@=@=@</code>''' and '''<code>o=o=o</code>''' to identify several nodes into one, as in this next redrawing:
To make future graphs easier to draw in ASCII, I will use devices like '''<code>@=@=@</code>''' and '''<code>o=o=o</code>''' to identify several nodes into one, as in this next redrawing:
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<pre>
<pre>
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However you draw it, these expressions follow because the expression <math>x + \operatorname{d}x,</math> where the plus sign indicates addition in <math>\mathbb{B},</math> that is, addition modulo 2, and thus corresponds to the exclusive disjunction operation in logic, parses to a graph of the following form:
However you draw it, these expressions follow because the expression <math>x + \mathrm{d}x,\!</math> where the plus sign indicates addition in <math>\mathbb{B},\!</math> that is, addition modulo 2, and thus corresponds to the exclusive disjunction operation in logic, parses to a graph of the following form:
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<pre>
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Next question: What is the difference between the value of the proposition <math>xy\!</math> "over there" and the value of the proposition <math>xy\!</math> where you are, all expressed as general formula, of course? Here 'tis:
Next question: What is the difference between the value of the proposition <math>xy\!</math> "over there" and the value of the proposition <math>xy\!</math> where you are, all expressed as general formula, of course? Here 'tis:
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Oh, I forgot to mention: Computed over <math>\mathbb{B},</math> plus and minus are the very same operation. This will make the relationship between the differential and the integral parts of the resulting calculus slightly stranger than usual, but never mind that now.
Oh, I forgot to mention: Computed over <math>\mathbb{B},\!</math> plus and minus are the very same operation. This will make the relationship between the differential and the integral parts of the resulting calculus slightly stranger than usual, but never mind that now.
Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the point where <math>xy\!</math> is true? Well, substituting <math>1\!</math> for <math>x\!</math> and <math>1\!</math> for <math>y\!</math> in the graph amounts to the same thing as erasing those labels:
Last question, for now: What is the value of this expression from your current standpoint, that is, evaluated at the point where <math>xy\!</math> is true? Well, substituting <math>1\!</math> for <math>x\!</math> and <math>1\!</math> for <math>y\!</math> in the graph amounts to the same thing as erasing those labels:
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And this is equivalent to the following graph:
And this is equivalent to the following graph:
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We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.
We have just met with the fact that the differential of the '''''and''''' is the '''''or''''' of the differentials.
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<math>x ~\operatorname{and}~ y \quad \xrightarrow{~\operatorname{Diff}~} \quad \operatorname{d}x ~\operatorname{or}~ \operatorname{d}y</math>
<math>x ~\mathrm{and}~ y \quad \xrightarrow{~\mathrm{Diff}~} \quad \mathrm{d}x ~\mathrm{or}~ \mathrm{d}y\!</math>
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A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}</math> or <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.
A function like this has an abstract type and a concrete type. The abstract type is what we invoke when we write things like <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\!</math> or <math>f : \mathbb{B}^2 \to \mathbb{B}.\!</math> The concrete type takes into account the qualitative dimensions or the "units" of the case, which can be explained as follows.
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{| align="center" cellspacing="10"
| Let <math>X\!</math> be the set of values <math>\{ \texttt{(} x \texttt{)},~ x \} ~=~ \{ \operatorname{not}~ x,~ x \}.</math>
| Let <math>X\!</math> be the set of values <math>\{ \texttt{(} x \texttt{)},~ x \} ~=~ \{ \mathrm{not}~ x,~ x \}.\!</math>
|-
|-
| Let <math>Y\!</math> be the set of values <math>\{ \texttt{(} y \texttt{)},~ y \} ~=~ \{ \operatorname{not}~ y,~ y \}.</math>
| Let <math>Y\!</math> be the set of values <math>\{ \texttt{(} y \texttt{)},~ y \} ~=~ \{ \mathrm{not}~ y,~ y \}.\!</math>
|}
|}
Then interpret the usual propositions about <math>x, y\!</math> as functions of the concrete type <math>f : X \times Y \to \mathbb{B}.</math>
Then interpret the usual propositions about <math>x, y\!</math> as functions of the concrete type <math>f : X \times Y \to \mathbb{B}.\!</math>
We are going to consider various ''operators'' on these functions. Here, an operator <math>\operatorname{F}</math> is a function that takes one function <math>f\!</math> into another function <math>\operatorname{F}f.</math>
We are going to consider various ''operators'' on these functions. Here, an operator <math>\mathrm{F}\!</math> is a function that takes one function <math>f\!</math> into another function <math>\mathrm{F}f.\!</math>
The first couple of operators that we need to consider are logical analogues of those that occur in the classical ''finite difference calculus'', namely:
The first couple of operators that we need to consider are logical analogues of those that occur in the classical ''finite difference calculus'', namely:
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| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\operatorname{D}.</math>
| The ''difference operator'' <math>\Delta,\!</math> written here as <math>\mathrm{D}.\!</math>
|-
|-
| The ''enlargement" operator'' <math>\Epsilon,\!</math> written here as <math>\operatorname{E}.</math>
| The ''enlargement" operator'' <math>\Epsilon,\!</math> written here as <math>\mathrm{E}.\!</math>
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|}
These days, <math>\operatorname{E}</math> is more often called the ''shift operator''.
These days, <math>\mathrm{E}\!</math> is more often called the ''shift operator''.
In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse, passing from the space <math>U = X \times Y</math> to its ''differential extension'', <math>\operatorname{E}U,</math> that has the following description:
In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse, passing from the space <math>U = X \times Y\!</math> to its ''differential extension'', <math>\mathrm{E}U,\!</math> that has the following description:
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{| align="center" cellspacing="10"
| <math>\operatorname{E}U ~=~ U \times \operatorname{d}U ~=~ X \times Y \times \operatorname{d}X \times \operatorname{d}Y,</math>
| <math>\mathrm{E}U ~=~ U \times \mathrm{d}U ~=~ X \times Y \times \mathrm{d}X \times \mathrm{d}Y,\!</math>
|}
|}
with
with
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| <math>\operatorname{d}X = \{ \texttt{(} \operatorname{d}x \texttt{)}, \operatorname{d}x \}</math> and <math>\operatorname{d}Y = \{ \texttt{(} \operatorname{d}y \texttt{)}, \operatorname{d}y \}.</math>
| <math>\mathrm{d}X = \{ \texttt{(} \mathrm{d}x \texttt{)}, \mathrm{d}x \}\!</math> and <math>\mathrm{d}Y = \{ \texttt{(} \mathrm{d}y \texttt{)}, \mathrm{d}y \}.\!</math>
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The interpretations of these new symbols can be diverse, but the easiest
The interpretations of these new symbols can be diverse, but the easiest option for now is just to say that <math>\mathrm{d}x\!</math> means "change <math>x\!</math>" and <math>\mathrm{d}y\!</math> means "change <math>y\!</math>". To draw the differential extension <math>\mathrm{E}U~\!</math> of our present universe <math>U = X \times Y\!</math> as a venn diagram, it would take us four logical dimensions <math>X, Y, \mathrm{d}X, \mathrm{d}Y,\!</math> but we can project a suggestion of what it's about on the universe <math>X \times Y\!</math> by drawing arrows that cross designated borders, labeling the arrows as
option for now is just to say that <math>\operatorname{d}x</math> means "change <math>x\!</math>" and <math>\operatorname{d}y</math> means "change <math>y\!</math>". To draw the differential extension <math>\operatorname{E}U</math> of our present universe <math>U = X \times Y</math> as a venn diagram, it would take us four logical dimensions <math>X, Y, \operatorname{d}X, \operatorname{d}Y,</math> but we can project a suggestion of what it's about on the universe <math>X \times Y</math> by drawing arrows that cross designated borders, labeling the arrows as
<math>\mathrm{d}x\!</math> when crossing the border between <math>x\!</math> and <math>\texttt{(} x \texttt{)}\!</math> and as <math>\mathrm{d}y\!</math> when crossing the border between <math>y\!</math> and <math>\texttt{(} y \texttt{)},\!</math> in either direction, in either case.
<math>\operatorname{d}x</math> when crossing the border between <math>x\!</math> and <math>\texttt{(} x \texttt{)}</math> and as <math>\operatorname{d}y</math> when crossing the border between <math>y\!</math> and <math>\texttt{(} y \texttt{)},</math> in either direction, in either case.
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Propositions can be formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in all the same ways that propositions can be formed on ordinary logical variables alone. For instance, the proposition <math>\texttt{(} \operatorname{d}x \texttt{(} \operatorname{d}y \texttt{))}</math> may be read to say that <math>\operatorname{d}x \Rightarrow \operatorname{d}y,</math> in other words, there is "no change in <math>x\!</math> without a change in <math>y\!</math>".
Propositions can be formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in all the same ways that propositions can be formed on ordinary logical variables alone. For instance, the proposition <math>\texttt{(} \mathrm{d}x \texttt{(} \mathrm{d}y \texttt{))}\!</math> may be read to say that <math>\mathrm{d}x \Rightarrow \mathrm{d}y,\!</math> in other words, there is "no change in <math>x\!</math> without a change in <math>y\!</math>".
Given the proposition <math>f(x, y)\!</math> in <math>U = X \times Y,</math> the (first order) ''enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f</math> in <math>\operatorname{E}U</math> that is defined by the formula <math>\operatorname{E}f(x, y, \operatorname{d}x, \operatorname{d}y) = f(x + \operatorname{d}x, y + \operatorname{d}y).</math>
Given the proposition <math>f(x, y)\!</math> in <math>U = X \times Y,\!</math> the (first order) ''enlargement'' of <math>f\!</math> is the proposition <math>\mathrm{E}f\!</math> in <math>\mathrm{E}U~\!</math> that is defined by the formula <math>\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y) = f(x + \mathrm{d}x, y + \mathrm{d}y).\!</math>
Applying the enlargement operator <math>\operatorname{E}</math> to the present example, <math>f(x, y) = xy,\!</math> we may compute the result as follows:
Applying the enlargement operator <math>\mathrm{E}\!</math> to the present example, <math>f(x, y) = xy,\!</math> we may compute the result as follows:
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<math>\operatorname{E}f(x, y, \operatorname{d}x, \operatorname{d}y) \quad = \quad (x + \operatorname{d}x)(y + \operatorname{d}y).</math>
<math>\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y) \quad = \quad (x + \mathrm{d}x)(y + \mathrm{d}y).\!</math>
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Given the proposition <math>f(x, y)\!</math> in <math>U = X \times Y,</math> the (first order) ''difference'' of <math>f\!</math> is the proposition <math>\operatorname{D}f</math> in <math>\operatorname{E}U</math> that is defined by the formula <math>\operatorname{D}f = \operatorname{E}f - f,</math> that is, <math>\operatorname{D}f(x, y, \operatorname{d}x, \operatorname{d}y) = f(x + \operatorname{d}x, y + \operatorname{d}y) - f(x, y).</math>
Given the proposition <math>f(x, y)\!</math> in <math>U = X \times Y,\!</math> the (first order) ''difference'' of <math>f\!</math> is the proposition <math>\mathrm{D}f~\!</math> in <math>\mathrm{E}U~\!</math> that is defined by the formula <math>\mathrm{D}f = \mathrm{E}f - f,\!</math> that is, <math>\mathrm{D}f(x, y, \mathrm{d}x, \mathrm{d}y) = f(x + \mathrm{d}x, y + \mathrm{d}y) - f(x, y).\!</math>
In the example <math>f(x, y) = xy,\!</math> the result is:
In the example <math>f(x, y) = xy,\!</math> the result is:
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<math>\operatorname{D}f(x, y, \operatorname{d}x, \operatorname{d}y) \quad = \quad (x + \operatorname{d}x)(y + \operatorname{d}y) - xy.</math>
<math>\mathrm{D}f(x, y, \mathrm{d}x, \mathrm{d}y) \quad = \quad (x + \mathrm{d}x)(y + \mathrm{d}y) - xy.\!</math>
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We did not yet go through the trouble to interpret this (first order) ''difference of conjunction'' fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition <math>xy,\!</math> that is, at the place where <math>x = 1\!</math> and <math>y = 1.\!</math> This evaluation is written in the form <math>\operatorname{D}f|_{xy}</math> or <math>\operatorname{D}f|_{(1, 1)},</math> and we arrived at the locally applicable law that is stated and illustrated as follows:
We did not yet go through the trouble to interpret this (first order) ''difference of conjunction'' fully, but were happy simply to evaluate it with respect to a single location in the universe of discourse, namely, at the point picked out by the singular proposition <math>xy,\!</math> that is, at the place where <math>x = 1\!</math> and <math>y = 1.\!</math> This evaluation is written in the form <math>\mathrm{D}f|_{xy}\!</math> or <math>\mathrm{D}f|_{(1, 1)},\!</math> and we arrived at the locally applicable law that is stated and illustrated as follows:
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<math>f(x, y) ~=~ xy ~=~ x ~\operatorname{and}~ y \quad \Rightarrow \quad \operatorname{D}f|_{xy} ~=~ \texttt{((} \operatorname{dx} \texttt{)(} \operatorname{d}y \texttt{))} ~=~ \operatorname{d}x ~\operatorname{or}~ \operatorname{d}y.</math>
<math>f(x, y) ~=~ xy ~=~ x ~\mathrm{and}~ y \quad \Rightarrow \quad \mathrm{D}f|_{xy} ~=~ \texttt{((} \mathrm{dx} \texttt{)(} \mathrm{d}y \texttt{))} ~=~ \mathrm{d}x ~\mathrm{or}~ \mathrm{d}y.\!</math>
|-
|-
| align="center" | [[Image:Venn Diagram Difference Conj At Conj.jpg|500px]]
| align="center" | [[Image:Venn Diagram Difference Conj At Conj.jpg|500px]]
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The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \operatorname{d}x \texttt{)(} \operatorname{d}y \texttt{))}</math> into the following exclusive disjunction:
The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \mathrm{d}x \texttt{)(} \mathrm{d}y \texttt{))}\!</math> into the following exclusive disjunction:
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{| align="center" cellspacing="10"
| align="center" | <math>\operatorname{d}x ~\texttt{(} \operatorname{d}y \texttt{)} ~+~ \operatorname{d}y ~\texttt{(} \operatorname{d}x \texttt{)} ~+~ \operatorname{d}x ~\operatorname{d}y.</math>
| align="center" | <math>\mathrm{d}x ~\texttt{(} \mathrm{d}y \texttt{)} ~+~ \mathrm{d}y ~\texttt{(} \mathrm{d}x \texttt{)} ~+~ \mathrm{d}x ~\mathrm{d}y.\!</math>
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==Note 3==
==Note 3==
Last time we computed what will variously be called the ''difference map'', the ''difference proposition'', or the ''local proposition'' <math>\operatorname{D}f_p</math> for the proposition <math>f(x, y) = xy\!</math> at the point <math>p\!</math> where <math>x = 1\!</math> and <math>y = 1.\!</math>
Last time we computed what will variously be called the ''difference map'', the ''difference proposition'', or the ''local proposition'' <math>\mathrm{D}f_p\!</math> for the proposition <math>f(x, y) = xy\!</math> at the point <math>p\!</math> where <math>x = 1\!</math> and <math>y = 1.\!</math>
In the universe <math>U = X \times Y,</math> the four propositions <math>xy,~ x\texttt{(}y\texttt{)},~ \texttt{(}x\texttt{)}y,~ \texttt{(}x\texttt{)(}y\texttt{)}</math> that indicate the "cells", or the smallest regions of the venn diagram, are called ''singular propositions''. These serve as an alternative notation for naming the points <math>(1, 1),~ (1, 0),~ (0, 1),~ (0, 0),\!</math> respectively.
In the universe <math>U = X \times Y,\!</math> the four propositions <math>xy,~ x\texttt{(}y\texttt{)},~ \texttt{(}x\texttt{)}y,~ \texttt{(}x\texttt{)(}y\texttt{)}\!</math> that indicate the "cells", or the smallest regions of the venn diagram, are called ''singular propositions''. These serve as an alternative notation for naming the points <math>(1, 1),~ (1, 0),~ (0, 1),~ (0, 0),\!</math> respectively.
Thus we can write <math>\operatorname{D}f_p = \operatorname{D}f|p = \operatorname{D}f|(1, 1) = \operatorname{D}f|xy,</math> so long as we know the frame of reference in force.
Thus we can write <math>\mathrm{D}f_p = \mathrm{D}f|p = \mathrm{D}f|(1, 1) = \mathrm{D}f|xy,\!</math> so long as we know the frame of reference in force.
Sticking with the example <math>f(x, y) = xy,\!</math> let us compute the value of the difference proposition <math>\operatorname{D}f</math> at all 4 points.
Sticking with the example <math>f(x, y) = xy,\!</math> let us compute the value of the difference proposition <math>\mathrm{D}f~\!</math> at all 4 points.
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The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
The easy way to visualize the values of these graphical expressions is just to notice the following equivalents:
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The Figure shows the points of the extended universe <math>\operatorname{E}U = X \times Y \times \operatorname{d}X \times \operatorname{d}Y</math> that satisfy the difference proposition <math>\operatorname{D}f,</math> namely, these:
The Figure shows the points of the extended universe <math>\mathrm{E}U = X \times Y \times \mathrm{d}X \times \mathrm{d}Y\!</math> that satisfy the difference proposition <math>\mathrm{D}f,\!</math> namely, these:
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{| align="center" cellpadding="6"
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<math>\begin{array}{rcccc}
<math>\begin{array}{rcccc}
1. & x & y & \operatorname{d}x & \operatorname{d}y
1. & x & y & \mathrm{d}x & \mathrm{d}y
\\
\\
2. & x & y & \operatorname{d}x & (\operatorname{d}y)
2. & x & y & \mathrm{d}x & (\mathrm{d}y)
\\
\\
3. & x & y & (\operatorname{d}x) & \operatorname{d}y
3. & x & y & (\mathrm{d}x) & \mathrm{d}y
\\
\\
4. & x & (y) & (\operatorname{d}x) & \operatorname{d}y
4. & x & (y) & (\mathrm{d}x) & \mathrm{d}y
\\
\\
5. & (x) & y & \operatorname{d}x & (\operatorname{d}y)
5. & (x) & y & \mathrm{d}x & (\mathrm{d}y)
\\
\\
6. & (x) & (y) & \operatorname{d}x & \operatorname{d}y
6. & (x) & (y) & \mathrm{d}x & \mathrm{d}y
\end{array}</math>
\end{array}\!</math>
|}
|}
An inspection of these six points should make it easy to understand <math>\operatorname{D}f</math> as telling you what you have to do from each point of <math>U\!</math> in order to change the value borne by the proposition <math>f(x, y).\!</math>
An inspection of these six points should make it easy to understand <math>\mathrm{D}f~\!</math> as telling you what you have to do from each point of <math>U\!</math> in order to change the value borne by the proposition <math>f(x, y).\!</math>
==Note 4==
==Note 4==
We have been studying the action of the difference operator <math>\operatorname{D},</math> also known as the ''localization operator'', on the proposition <math>f : X \times Y \to \mathbb{B}</math> that is commonly known as the conjunction <math>x \cdot y.</math> We described <math>\operatorname{D}f</math> as a (first order) differential proposition, that is, a proposition of the type <math>\operatorname{D}f : X \times Y \times \operatorname{d}X \times \operatorname{d}Y \to \mathbb{B}.</math> Abstracting from the augmented venn diagram that illustrates how the ''models'' or ''satisfying interpretations'' of <math>\operatorname{D}f</math> distribute within the extended universe <math>\operatorname{E}U = X \times Y \times \operatorname{d}X \times \operatorname{d}Y,</math> we can depict <math>\operatorname{D}f</math> in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>U = X \times Y</math> and whose arrows are labeled with the elements of <math>\operatorname{d}U = \operatorname{d}X \times \operatorname{d}Y.\!</math>
We have been studying the action of the difference operator <math>\mathrm{D},\!</math> also known as the ''localization operator'', on the proposition <math>f : X \times Y \to \mathbb{B}\!</math> that is commonly known as the conjunction <math>x \cdot y.\!</math> We described <math>\mathrm{D}f~\!</math> as a (first order) differential proposition, that is, a proposition of the type <math>\mathrm{D}f : X \times Y \times \mathrm{d}X \times \mathrm{d}Y \to \mathbb{B}.\!</math> Abstracting from the augmented venn diagram that illustrates how the ''models'' or ''satisfying interpretations'' of <math>\mathrm{D}f~\!</math> distribute within the extended universe <math>\mathrm{E}U = X \times Y \times \mathrm{d}X \times \mathrm{d}Y,\!</math> we can depict <math>\mathrm{D}f~\!</math> in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>U = X \times Y\!</math> and whose arrows are labeled with the elements of <math>\mathrm{d}U = \mathrm{d}X \times \mathrm{d}Y.\!</math>
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
& = & x & \cdot & y
& = & x & \cdot & y
\\[4pt]
\\[4pt]
\operatorname{D}f
\mathrm{D}f
& = & x & \cdot & y & \cdot & ((\operatorname{d}x)(\operatorname{d}y))
& = & x & \cdot & y & \cdot & ((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
& + & x & \cdot & (y) & \cdot & ~(\operatorname{d}x)~\operatorname{d}y~~
& + & x & \cdot & (y) & \cdot & ~(\mathrm{d}x)~\mathrm{d}y~~
\\[4pt]
\\[4pt]
& + & (x) & \cdot & y & \cdot & ~~\operatorname{d}x~(\operatorname{d}y)~
& + & (x) & \cdot & y & \cdot & ~~\mathrm{d}x~(\mathrm{d}y)~
\\[4pt]
\\[4pt]
& + & (x) & \cdot & (y) & \cdot & ~~\operatorname{d}x~~\operatorname{d}y~~
& + & (x) & \cdot & (y) & \cdot & ~~\mathrm{d}x~~\mathrm{d}y~~
\end{array}</math>
\end{array}\!</math>
|}
|}
==Note 5==
==Note 5==
The ''enlargement'' or ''shift'' operator <math>\operatorname{E}</math> exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features that play out on the surface of our initial example, <math>f(x, y) = xy.\!</math>
The ''enlargement'' or ''shift'' operator <math>\mathrm{E}\!</math> exhibits a wealth of interesting and useful properties in its own right, so it pays to examine a few of the more salient features that play out on the surface of our initial example, <math>f(x, y) = xy.\!</math>
A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
A suitably generic definition of the extended universe of discourse is afforded by the following set-up:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
|
|
<math>\begin{array}{ccclll}
<math>\begin{array}{ccclll}
X_1 \times \ldots \times X_k.
X_1 \times \ldots \times X_k.
\\[6pt]
\\[6pt]
\text{Let} & \operatorname{d}U
\text{Let} & \mathrm{d}U
& = &
& = &
\operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k.
\mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k.
\\[6pt]
\\[6pt]
\text{Then} & \operatorname{E}U
\text{Then} & \mathrm{E}U
& = &
& = &
X_1 \times \ldots \times X_k ~\times~ \operatorname{d}X_1 \times \ldots \times \operatorname{d}X_k
X_1 \times \ldots \times X_k ~\times~ \mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k
& = &
& = &
U \times \operatorname{d}U.
U \times \mathrm{d}U.
\end{array}</math>
\end{array}\!</math>
|}
|}
For a proposition of the form <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},</math> the (first order) ''enlargement'' of <math>f\!</math> is the proposition <math>\operatorname{E}f : \operatorname{E}U \to \mathbb{B}</math> that is defined by the following equation:
For a proposition of the form <math>f : X_1 \times \ldots \times X_k \to \mathbb{B},\!</math> the (first order) ''enlargement'' of <math>f\!</math> is the proposition <math>\mathrm{E}f : \mathrm{E}U \to \mathbb{B}\!</math> that is defined by the following equation:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| <math>\operatorname{E}f(x_1, \ldots, x_k, \operatorname{d}x_1, \ldots, \operatorname{d}x_k) ~=~ f(x_1 + \operatorname{d}x_1, \ldots, x_k + \operatorname{d}x_k).</math>
| <math>\mathrm{E}f(x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k) ~=~ f(x_1 + \mathrm{d}x_1, \ldots, x_k + \mathrm{d}x_k).\!</math>
|}
|}
The ''differential variables'' <math>\operatorname{d}x_j</math> are boolean variables of the same basic type as the ordinary variables <math>x_j.\!</math> It is conventional to distinguish the (first order) differential variables with the operative prefix "<math>\operatorname{d}</math>", but this is purely optional. It is their existence in particular relations to the initial variables, not their names, that defines them as differential variables.
The ''differential variables'' <math>\mathrm{d}x_j\!</math> are boolean variables of the same basic type as the ordinary variables <math>x_j.\!</math> It is conventional to distinguish the (first order) differential variables with the operative prefix "<math>\mathrm{d}\!</math>", but this is purely optional. It is their existence in particular relations to the initial variables, not their names, that defines them as differential variables.
In the case of logical conjunction, <math>f(x, y) = xy,\!</math> the computation of the enlargement <math>\operatorname{E}f</math> begins as follows:
In the case of logical conjunction, <math>f(x, y) = xy,\!</math> the computation of the enlargement <math>\mathrm{E}f\!</math> begins as follows:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| <math>\operatorname{E}f(x, y, \operatorname{d}x, \operatorname{d}y) ~=~ (x + \operatorname{d}x)(y + \operatorname{d}y).</math>
| <math>\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y) ~=~ (x + \mathrm{d}x)(y + \mathrm{d}y).\!</math>
|}
|}
Given that this expression uses nothing more than the ''boolean ring'' operations of addition <math>(+)\!</math> and multiplication <math>(\cdot),</math> it is permissible to multiply things out in the usual manner to arrive at the following result:
Given that this expression uses nothing more than the ''boolean ring'' operations of addition <math>(+)\!</math> and multiplication <math>(\cdot),\!</math> it is permissible to multiply things out in the usual manner to arrive at the following result:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| <math>\operatorname{E}f(x, y, \operatorname{d}x, \operatorname{d}y) ~=~ x~y ~+~ x~\operatorname{d}y ~+~ y~\operatorname{d}x ~+~ \operatorname{d}x~\operatorname{d}y.</math>
| <math>\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y) ~=~ x~y ~+~ x~\mathrm{d}y ~+~ y~\mathrm{d}x ~+~ \mathrm{d}x~\mathrm{d}y.\!</math>
|}
|}
To understand what this means in logical terms, it is useful to go back and analyze the above expression for <math>\operatorname{E}f</math> in the same way that we did for <math>\operatorname{D}f.</math> Toward that end, the next set of Figures represent the computation of the ''enlarged'' or ''shifted'' proposition <math>\operatorname{E}f</math> at each of the 4 points in the universe of discourse <math>U = X \times Y.</math>
To understand what this means in logical terms, it is useful to go back and analyze the above expression for <math>\mathrm{E}f\!</math> in the same way that we did for <math>\mathrm{D}f.\!</math> Toward that end, the next set of Figures represent the computation of the ''enlarged'' or ''shifted'' proposition <math>\mathrm{E}f\!</math> at each of the 4 points in the universe of discourse <math>U = X \times Y.\!</math>
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<pre>
<pre>
|}
|}
Given the sort of data that arises from this form of analysis, we can now fold the disjoined ingredients back into a boolean expansion or a DNF that is equivalent to the proposition <math>\operatorname{E}f.</math>
Given the sort of data that arises from this form of analysis, we can now fold the disjoined ingredients back into a boolean expansion or a DNF that is equivalent to the proposition <math>\mathrm{E}f.\!</math>
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| <math>\operatorname{E}f ~~=~~ xy \cdot \operatorname{E}f_{xy} ~~+~~ x(y) \cdot \operatorname{E}f_{x(y)} ~~+~~ (x)y \cdot \operatorname{E}f_{(x)y} ~~+~~ (x)(y) \cdot \operatorname{E}f_{(x)(y)}.</math>
| <math>\mathrm{E}f ~~=~~ xy \cdot \mathrm{E}f_{xy} ~~+~~ x(y) \cdot \mathrm{E}f_{x(y)} ~~+~~ (x)y \cdot \mathrm{E}f_{(x)y} ~~+~~ (x)(y) \cdot \mathrm{E}f_{(x)(y)}.\!</math>
|}
|}
Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element <math>(\operatorname{d}x)(\operatorname{d}y)</math> is drawn as a loop at the point <math>x~y.</math>
Here is a summary of the result, illustrated by means of a digraph picture, where the "no change" element <math>(\mathrm{d}x)(\mathrm{d}y)\!</math> is drawn as a loop at the point <math>x~y.\!</math>
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
& = & x & \cdot & y
& = & x & \cdot & y
\\[4pt]
\\[4pt]
\operatorname{E}f
\mathrm{E}f
& = & x & \cdot & y & \cdot & (\operatorname{d}x)(\operatorname{d}y)
& = & x & \cdot & y & \cdot & (\mathrm{d}x)(\mathrm{d}y)
\\[4pt]
\\[4pt]
& + & x & \cdot & (y) & \cdot & (\operatorname{d}x)~\operatorname{d}y~
& + & x & \cdot & (y) & \cdot & (\mathrm{d}x)~\mathrm{d}y~
\\[4pt]
\\[4pt]
& + & (x) & \cdot & y & \cdot & ~\operatorname{d}x~(\operatorname{d}y)
& + & (x) & \cdot & y & \cdot & ~\mathrm{d}x~(\mathrm{d}y)
\\[4pt]
\\[4pt]
& + & (x) & \cdot & (y) & \cdot & ~\operatorname{d}x~~\operatorname{d}y~\end{array}</math>
& + & (x) & \cdot & (y) & \cdot & ~\mathrm{d}x~~\mathrm{d}y~\end{array}\!</math>
|}
|}
We may understand the enlarged proposition <math>\operatorname{E}f</math> as telling us all the different ways to reach a model of the proposition <math>f\!</math> from each point of the universe <math>U.\!</math>
We may understand the enlarged proposition <math>\mathrm{E}f\!</math> as telling us all the different ways to reach a model of the proposition <math>f\!</math> from each point of the universe <math>U.\!</math>
==Note 6==
==Note 6==
To broaden our experience with simple examples, let us examine the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.\!</math> A few Tables are set here that detail the actions of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> on each of these functions, allowing us to view the results in several different ways.
To broaden our experience with simple examples, let us examine the sixteen functions of concrete type <math>X \times Y \to \mathbb{B}\!</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.\!</math> A few Tables are set here that detail the actions of <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> on each of these functions, allowing us to view the results in several different ways.
Tables A1 and A2 show two ways of arranging the 16 boolean functions on two variables, giving equivalent expressions for each function in several different systems of notation.
Tables A1 and A2 show two ways of arranging the 16 boolean functions on two variables, giving equivalent expressions for each function in several different systems of notation.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}</math>
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="15%" |
| width="15%" |
<p><math>\mathcal{L}_1</math></p>
<p><math>\mathcal{L}_1\!</math></p>
<p><math>\text{Decimal}</math></p>
<p><math>\text{Decimal}\!</math></p>
| width="15%" |
| width="15%" |
<p><math>\mathcal{L}_2</math></p>
<p><math>\mathcal{L}_2\!</math></p>
<p><math>\text{Binary}</math></p>
<p><math>\text{Binary}\!</math></p>
| width="15%" |
| width="15%" |
<p><math>\mathcal{L}_3</math></p>
<p><math>\mathcal{L}_3\!</math></p>
<p><math>\text{Vector}</math></p>
<p><math>\text{Vector}\!</math></p>
| width="15%" |
| width="15%" |
<p><math>\mathcal{L}_4</math></p>
<p><math>\mathcal{L}_4\!</math></p>
<p><math>\text{Cactus}</math></p>
<p><math>\text{Cactus}\!</math></p>
| width="25%" |
| width="25%" |
<p><math>\mathcal{L}_5</math></p>
<p><math>\mathcal{L}_5\!</math></p>
<p><math>\text{English}</math></p>
<p><math>\text{English}\!</math></p>
| width="15%" |
| width="15%" |
<p><math>\mathcal{L}_6</math></p>
<p><math>\mathcal{L}_6~\!</math></p>
<p><math>\text{Ordinary}</math></p>
<p><math>\text{Ordinary}\!</math></p>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
|
|
\\[4pt]
\\[4pt]
f_7
f_7
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
f_{0111}
f_{0111}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
0~1~1~1
0~1~1~1
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x~~y)
(x~~y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
\text{not both}~ x ~\text{and}~ y
\text{not both}~ x ~\text{and}~ y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
\lnot x \lor \lnot y
\lnot x \lor \lnot y
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{15}
f_{15}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
f_{1111}
f_{1111}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
1~1~1~1
1~1~1~1
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((x,~y))
((x,~y))
\\[4pt]
\\[4pt]
22:03, 8 December 2014 (UTC)y~~
\\[4pt]
\\[4pt]
~(x~(y))
~(x~(y))
\\[4pt]
\\[4pt]
~~x21:56, 7 December 2014 (UTC)
~~x22:03, 8 December 2014 (UTC)
\\[4pt]
\\[4pt]
((x)~y)~
((x)~y)~
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
\text{true}
\text{true}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
1
1
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}</math>
|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="15%" |
| width="15%" |
<p><math>\mathcal{L}_1</math></p>
<p><math>\mathcal{L}_1\!</math></p>
<p><math>\text{Decimal}</math></p>
<p><math>\text{Decimal}\!</math></p>
| width="15%" |
| width="15%" |
<p><math>\mathcal{L}_2</math></p>
<p><math>\mathcal{L}_2\!</math></p>
<p><math>\text{Binary}</math></p>
<p><math>\text{Binary}\!</math></p>
| width="15%" |
| width="15%" |
<p><math>\mathcal{L}_3</math></p>
<p><math>\mathcal{L}_3\!</math></p>
<p><math>\text{Vector}</math></p>
<p><math>\text{Vector}\!</math></p>
| width="15%" |
| width="15%" |
<p><math>\mathcal{L}_4</math></p>
<p><math>\mathcal{L}_4\!</math></p>
<p><math>\text{Cactus}</math></p>
<p><math>\text{Cactus}\!</math></p>
| width="25%" |
| width="25%" |
<p><math>\mathcal{L}_5</math></p>
<p><math>\mathcal{L}_5\!</math></p>
<p><math>\text{English}</math></p>
<p><math>\text{English}\!</math></p>
| width="15%" |
| width="15%" |
<p><math>\mathcal{L}_6</math></p>
<p><math>\mathcal{L}_6~\!</math></p>
<p><math>\text{Ordinary}</math></p>
<p><math>\text{Ordinary}\!</math></p>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
|
|
| <math>f_0\!</math>
| <math>f_0\!</math>
| <math>f_{0000}\!</math>
| <math>f_{0000}\!</math>
| <math>0~0~0~0</math>
| <math>0~0~0~0\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>\text{false}\!</math>
| <math>\text{false}\!</math>
| <math>0\!</math>
| <math>0\!</math>
\\[4pt]
\\[4pt]
f_8
f_8
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
f_{1000}
f_{1000}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
1~0~0~0
1~0~0~0
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x ~\text{and}~ y
x ~\text{and}~ y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x \land y
x \land y
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{12}
f_{12}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
f_{1100}
f_{1100}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
1~1~0~0
1~1~0~0
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x
x
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x
x
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_9
f_9
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
f_{1001}
f_{1001}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
1~0~0~1
1~0~0~1
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x ~\text{equal to}~ y
x ~\text{equal to}~ y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x = y
x = y
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{10}
f_{10}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
f_{1010}
f_{1010}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
1~0~1~0
1~0~1~0
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
y
y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
y
y
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{14}
f_{14}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
f_{1110}
f_{1110}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
1~1~1~0
1~1~1~0
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)(y))
((x)(y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x ~\text{or}~ y
x ~\text{or}~ y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x \lor y
x \lor y
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
| <math>f_{1111}\!</math>
| <math>f_{1111}\!</math>
| <math>1~1~1~1</math>
| <math>1~1~1~1\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>\text{true}\!</math>
| <math>\text{true}\!</math>
| <math>1\!</math>
| <math>1\!</math>
<br>
<br>
The next four Tables expand the expressions of <math>\operatorname{E}f</math> and <math>\operatorname{D}f</math> in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, here being collected into a set of seven natural classes.
The next four Tables expand the expressions of <math>\mathrm{E}f\!</math> and <math>\mathrm{D}f~\!</math> in two different ways, for each of the sixteen functions. Notice that the functions are given in a different order, here being collected into a set of seven natural classes.
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
|+ <math>\text{Table A3.}~~\mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="10%" |
| width="10%" |
| width="18%" | <math>f\!</math>
| width="18%" | <math>f\!</math>
| width="18%" |
| width="18%" |
<p><math>\operatorname{T}_{11} f</math></p>
<p><math>\mathrm{T}_{11} f\!</math></p>
<p><math>\operatorname{E}f|_{\operatorname{d}x~\operatorname{d}y}</math></p>
<p><math>\mathrm{E}f|_{\mathrm{d}x~\mathrm{d}y}\!</math></p>
| width="18%" |
| width="18%" |
<p><math>\operatorname{T}_{10} f</math></p>
<p><math>\mathrm{T}_{10} f\!</math></p>
<p><math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math></p>
<p><math>\mathrm{E}f|_{\mathrm{d}x(\mathrm{d}y)}\!</math></p>
| width="18%" |
| width="18%" |
<p><math>\operatorname{T}_{01} f</math></p>
<p><math>\mathrm{T}_{01} f\!</math></p>
<p><math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math></p>
<p><math>\mathrm{E}f|_{(\mathrm{d}x)\mathrm{d}y}\!</math></p>
| width="18%" |
| width="18%" |
<p><math>\operatorname{T}_{00} f</math></p>
<p><math>\mathrm{T}_{00} f\!</math></p>
<p><math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math></p>
<p><math>\mathrm{E}f|_{(\mathrm{d}x)(\mathrm{d}y)}\!</math></p>
|-
|-
| <math>f_0\!</math>
| <math>f_0\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_8
f_8
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)(y)
(x)(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)~y~
(x)~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~(y)
~x~(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{12}
f_{12}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)
(x)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)
(x)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_9
f_9
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~(x,~y)~
~(x,~y)~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~(x,~y)~
~(x,~y)~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{10}
f_{10}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(y)
(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(y)
(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{14}
f_{14}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)(y))
((x)(y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~x~~y~)
(~x~~y~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~x~(y))
(~x~(y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)~y~)
((x)~y~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)(y))
((x)(y))
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| colspan="2" | <math>\text{Fixed Point Total}\!</math>
| colspan="2" | <math>\text{Fixed Point Total}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
|+ <math>\text{Table A4.}~~\mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="10%" |
| width="10%" |
| width="18%" | <math>f\!</math>
| width="18%" | <math>f\!</math>
| width="18%" |
| width="18%" |
<math>\operatorname{D}f|_{\operatorname{d}x~\operatorname{d}y}</math>
<math>\mathrm{D}f|_{\mathrm{d}x~\mathrm{d}y}\!</math>
| width="18%" |
| width="18%" |
<math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
<math>\mathrm{D}f|_{\mathrm{d}x(\mathrm{d}y)}\!</math>
| width="18%" |
| width="18%" |
<math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
<math>\mathrm{D}f|_{(\mathrm{d}x)\mathrm{d}y}\!</math>
| width="18%" |
| width="18%" |
<math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
<math>\mathrm{D}f|_{(\mathrm{d}x)(\mathrm{d}y)}\!</math>
|-
|-
| <math>f_0\!</math>
| <math>f_0\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_8
f_8
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{12}
f_{12}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_9
f_9
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{10}
f_{10}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{14}
f_{14}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)(y))
((x)(y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(y)
(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)
(x)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
|}
|}
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|+ <math>\text{Table A5.}~~\mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="10%" |
| width="10%" |
| width="18%" | <math>f\!</math>
| width="18%" | <math>f\!</math>
| width="18%" | <math>\operatorname{E}f|_{xy}</math>
| width="18%" | <math>\mathrm{E}f|_{xy}\!</math>
| width="18%" | <math>\operatorname{E}f|_{x(y)}</math>
| width="18%" | <math>\mathrm{E}f|_{x(y)}\!</math>
| width="18%" | <math>\operatorname{E}f|_{(x)y}</math>
| width="18%" | <math>\mathrm{E}f|_{(x)y}\!</math>
| width="18%" | <math>\operatorname{E}f|_{(x)(y)}</math>
| width="18%" | <math>\mathrm{E}f|_{(x)(y)}\!</math>
|-
|-
| <math>f_0\!</math>
| <math>f_0\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_8
f_8
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}x~~\operatorname{d}y~
~\mathrm{d}x~~\mathrm{d}y~
\\[4pt]
\\[4pt]
~\operatorname{d}x~(\operatorname{d}y)
~\mathrm{d}x~(\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x)~\operatorname{d}y~
(\mathrm{d}x)~\mathrm{d}y~
\\[4pt]
\\[4pt]
(\operatorname{d}x)(\operatorname{d}y)
(\mathrm{d}x)(\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}x~(\operatorname{d}y)
~\mathrm{d}x~(\mathrm{d}y)
\\[4pt]
\\[4pt]
~\operatorname{d}x~~\operatorname{d}y~
~\mathrm{d}x~~\mathrm{d}y~
\\[4pt]
\\[4pt]
(\operatorname{d}x)(\operatorname{d}y)
(\mathrm{d}x)(\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x)~\operatorname{d}y~
(\mathrm{d}x)~\mathrm{d}y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x)~\operatorname{d}y~
(\mathrm{d}x)~\mathrm{d}y~
\\[4pt]
\\[4pt]
(\operatorname{d}x)(\operatorname{d}y)
(\mathrm{d}x)(\mathrm{d}y)
\\[4pt]
\\[4pt]
~\operatorname{d}x~~\operatorname{d}y~
~\mathrm{d}x~~\mathrm{d}y~
\\[4pt]
\\[4pt]
~\operatorname{d}x~(\operatorname{d}y)
~\mathrm{d}x~(\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x)(\operatorname{d}y)
(\mathrm{d}x)(\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x)~\operatorname{d}y~
(\mathrm{d}x)~\mathrm{d}y~
\\[4pt]
\\[4pt]
~\operatorname{d}x~(\operatorname{d}y)
~\mathrm{d}x~(\mathrm{d}y)
\\[4pt]
\\[4pt]
~\operatorname{d}x~~\operatorname{d}y~
~\mathrm{d}x~~\mathrm{d}y~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{12}
f_{12}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}x~
~\mathrm{d}x~
\\[4pt]
\\[4pt]
(\operatorname{d}x)
(\mathrm{d}x)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}x~
~\mathrm{d}x~
\\[4pt]
\\[4pt]
(\operatorname{d}x)
(\mathrm{d}x)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x)
(\mathrm{d}x)
\\[4pt]
\\[4pt]
~\operatorname{d}x~
~\mathrm{d}x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x)
(\mathrm{d}x)
\\[4pt]
\\[4pt]
~\operatorname{d}x~
~\mathrm{d}x~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_9
f_9
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~(\operatorname{d}x,~\operatorname{d}y)~
~(\mathrm{d}x,~\mathrm{d}y)~
\\[4pt]
\\[4pt]
((\operatorname{d}x,~\operatorname{d}y))
((\mathrm{d}x,~\mathrm{d}y))
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((\operatorname{d}x,~\operatorname{d}y))
((\mathrm{d}x,~\mathrm{d}y))
\\[4pt]
\\[4pt]
~(\operatorname{d}x,~\operatorname{d}y)~
~(\mathrm{d}x,~\mathrm{d}y)~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((\operatorname{d}x,~\operatorname{d}y))
((\mathrm{d}x,~\mathrm{d}y))
\\[4pt]
\\[4pt]
~(\operatorname{d}x,~\operatorname{d}y)~
~(\mathrm{d}x,~\mathrm{d}y)~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~(\operatorname{d}x,~\operatorname{d}y)~
~(\mathrm{d}x,~\mathrm{d}y)~
\\[4pt]
\\[4pt]
((\operatorname{d}x,~\operatorname{d}y))
((\mathrm{d}x,~\mathrm{d}y))
\end{matrix}</math>
\end{matrix}~\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{10}
f_{10}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}y~
~\mathrm{d}y~
\\[4pt]
\\[4pt]
(\operatorname{d}y)
(\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}y)
(\mathrm{d}y)
\\[4pt]
\\[4pt]
~\operatorname{d}y~
~\mathrm{d}y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}y~
~\mathrm{d}y~
\\[4pt]
\\[4pt]
(\operatorname{d}y)
(\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}y)
(\mathrm{d}y)
\\[4pt]
\\[4pt]
~\operatorname{d}y~
~\mathrm{d}y~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{14}
f_{14}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)(y))
((x)(y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
((\operatorname{d}x)~\operatorname{d}y~)
((\mathrm{d}x)~\mathrm{d}y~)
\\[4pt]
\\[4pt]
(~\operatorname{d}x~(\operatorname{d}y))
(~\mathrm{d}x~(\mathrm{d}y))
\\[4pt]
\\[4pt]
(~\operatorname{d}x~~\operatorname{d}y~)
(~\mathrm{d}x~~\mathrm{d}y~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((\operatorname{d}x)~\operatorname{d}y~)
((\mathrm{d}x)~\mathrm{d}y~)
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
(~\operatorname{d}x~~\operatorname{d}y~)
(~\mathrm{d}x~~\mathrm{d}y~)
\\[4pt]
\\[4pt]
(~\operatorname{d}x~(\operatorname{d}y))
(~\mathrm{d}x~(\mathrm{d}y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(~\operatorname{d}x~(\operatorname{d}y))
(~\mathrm{d}x~(\mathrm{d}y))
\\[4pt]
\\[4pt]
(~\operatorname{d}x~~\operatorname{d}y~)
(~\mathrm{d}x~~\mathrm{d}y~)
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
((\operatorname{d}x)~\operatorname{d}y~)
((\mathrm{d}x)~\mathrm{d}y~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(~\operatorname{d}x~~\operatorname{d}y~)
(~\mathrm{d}x~~\mathrm{d}y~)
\\[4pt]
\\[4pt]
(~\operatorname{d}x~(\operatorname{d}y))
(~\mathrm{d}x~(\mathrm{d}y))
\\[4pt]
\\[4pt]
((\operatorname{d}x)~\operatorname{d}y~)
((\mathrm{d}x)~\mathrm{d}y~)
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
|}
|}
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|+ <math>\text{Table A6.}~~\mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="10%" |
| width="10%" |
| width="18%" | <math>f\!</math>
| width="18%" | <math>f\!</math>
| width="18%" | <math>\operatorname{D}f|_{xy}</math>
| width="18%" | <math>\mathrm{D}f|_{xy}\!</math>
| width="18%" | <math>\operatorname{D}f|_{x(y)}</math>
| width="18%" | <math>\mathrm{D}f|_{x(y)}\!</math>
| width="18%" | <math>\operatorname{D}f|_{(x)y}</math>
| width="18%" | <math>\mathrm{D}f|_{(x)y}~\!</math>
| width="18%" | <math>\operatorname{D}f|_{(x)(y)}</math>
| width="18%" | <math>\mathrm{D}f|_{(x)(y)}\!</math>
|-
|-
| <math>f_0\!</math>
| <math>f_0\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_8
f_8
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\\[4pt]
\\[4pt]
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{12}
f_{12}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}x
\mathrm{d}x
\\[4pt]
\\[4pt]
\operatorname{d}x
\mathrm{d}x
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}x
\mathrm{d}x
\\[4pt]
\\[4pt]
\operatorname{d}x
\mathrm{d}x
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}x
\mathrm{d}x
\\[4pt]
\\[4pt]
\operatorname{d}x
\mathrm{d}x
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}x
\mathrm{d}x
\\[4pt]
\\[4pt]
\operatorname{d}x
\mathrm{d}x
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_9
f_9
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{10}
f_{10}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}y
\mathrm{d}y
\\[4pt]
\\[4pt]
\operatorname{d}y
\mathrm{d}y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}y
\mathrm{d}y
\\[4pt]
\\[4pt]
\operatorname{d}y
\mathrm{d}y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}y
\mathrm{d}y
\\[4pt]
\\[4pt]
\operatorname{d}y
\mathrm{d}y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}y
\mathrm{d}y
\\[4pt]
\\[4pt]
\operatorname{d}y
\mathrm{d}y
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{14}
f_{14}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)(y))
((x)(y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\\[4pt]
\\[4pt]
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
|}
|}
==Note 7==
==Note 7==
If you think that I linger in the realm of logical difference calculus out of sheer vacillation about getting down to the differential proper, it is probably out of a prior expectation that you derive from the art or the long-engrained practice of real analysis. But the fact is that ordinary calculus only rushes on to the sundry orders of approximation because the strain of comprehending the full import of <math>\operatorname{E}</math> and <math>\operatorname{D}</math> at once whelm over its discrete and finite powers to grasp them. But here, in the fully serene idylls of ZOL, we find ourselves fit with the compass of a wit that is all we'd ever wish to explore their effects with care.
If you think that I linger in the realm of logical difference calculus out of sheer vacillation about getting down to the differential proper, it is probably out of a prior expectation that you derive from the art or the long-engrained practice of real analysis. But the fact is that ordinary calculus only rushes on to the sundry orders of approximation because the strain of comprehending the full import of <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> at once whelm over its discrete and finite powers to grasp them. But here, in the fully serene idylls of ZOL, we find ourselves fit with the compass of a wit that is all we'd ever wish to explore their effects with care.
So let us do just that.
So let us do just that.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
|+ <math>\text{Table A3.}~~\mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="10%" |
| width="10%" |
| width="18%" | <math>f\!</math>
| width="18%" | <math>f\!</math>
| width="18%" |
| width="18%" |
<p><math>\operatorname{T}_{11} f</math></p>
<p><math>\mathrm{T}_{11} f\!</math></p>
<p><math>\operatorname{E}f|_{\operatorname{d}x~\operatorname{d}y}</math></p>
<p><math>\mathrm{E}f|_{\mathrm{d}x~\mathrm{d}y}\!</math></p>
| width="18%" |
| width="18%" |
<p><math>\operatorname{T}_{10} f</math></p>
<p><math>\mathrm{T}_{10} f\!</math></p>
<p><math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math></p>
<p><math>\mathrm{E}f|_{\mathrm{d}x(\mathrm{d}y)}\!</math></p>
| width="18%" |
| width="18%" |
<p><math>\operatorname{T}_{01} f</math></p>
<p><math>\mathrm{T}_{01} f\!</math></p>
<p><math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math></p>
<p><math>\mathrm{E}f|_{(\mathrm{d}x)\mathrm{d}y}\!</math></p>
| width="18%" |
| width="18%" |
<p><math>\operatorname{T}_{00} f</math></p>
<p><math>\mathrm{T}_{00} f\!</math></p>
<p><math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math></p>
<p><math>\mathrm{E}f|_{(\mathrm{d}x)(\mathrm{d}y)}\!</math></p>
|-
|-
| <math>f_0\!</math>
| <math>f_0\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
| <math>(~)</math>
| <math>(~)\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_8
f_8
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)(y)
(x)(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)~y~
(x)~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~(y)
~x~(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{12}
f_{12}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)
(x)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)
(x)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_9
f_9
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~(x,~y)~
~(x,~y)~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~(x,~y)~
~(x,~y)~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{10}
f_{10}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(y)
(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(y)
(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{14}
f_{14}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)(y))
((x)(y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~x~~y~)
(~x~~y~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~x~(y))
(~x~(y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)~y~)
((x)~y~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)(y))
((x)(y))
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
| <math>((~))</math>
| <math>((~))\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| colspan="2" | <math>\text{Fixed Point Total}\!</math>
| colspan="2" | <math>\text{Fixed Point Total}\!</math>
<br>
<br>
The shift operator <math>\operatorname{E}</math> can be understood as enacting a ''substitution operation'' on the proposition that is given as its argument.
The shift operator <math>\mathrm{E}\!</math> can be understood as enacting a ''substitution operation'' on the proposition that is given as its argument.
For example, the action of <math>\operatorname{E}</math> on the conjunction <math>f(x, y) = xy\!</math> is defined as follows:
For example, the action of <math>\mathrm{E}\!</math> on the conjunction <math>f(x, y) = xy\!</math> is defined as follows:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
|
|
<math>\begin{array}{lcl}
<math>\begin{array}{lcl}
\operatorname{E} ~:~ (U \to \mathbb{B})
\mathrm{E} ~:~ (U \to \mathbb{B})
& \to &
& \to &
(\operatorname{E}U \to \mathbb{B}),
(\mathrm{E}U \to \mathbb{B}),
\\[6pt]
\\[6pt]
\operatorname{E} ~:~ f(x, y)
\mathrm{E} ~:~ f(x, y)
& \mapsto &
& \mapsto &
\operatorname{E}f(x, y, \operatorname{d}x, \operatorname{d}y),
\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y),
\\[6pt]
\\[6pt]
\operatorname{E}f(x, y, \operatorname{d}x, \operatorname{d}y)
\mathrm{E}f(x, y, \mathrm{d}x, \mathrm{d}y)
& = &
& = &
f(x + \operatorname{d}x, y + \operatorname{d}y).
f(x + \mathrm{d}x, y + \mathrm{d}y).
\end{array}</math>
\end{array}\!</math>
|}
|}
Evaluating <math>\operatorname{E}f</math> at particular values of <math>\operatorname{d}x</math> and <math>\operatorname{d}y,</math> for example, <math>\operatorname{d}x = i</math> and <math>\operatorname{d}y = j,</math> where <math>i\!</math> and <math>j\!</math> are values in <math>\mathbb{B},</math> produces the following result:
Evaluating <math>\mathrm{E}f\!</math> at particular values of <math>\mathrm{d}x\!</math> and <math>\mathrm{d}y,\!</math> for example, <math>\mathrm{d}x = i\!</math> and <math>\mathrm{d}y = j,\!</math> where <math>i\!</math> and <math>j\!</math> are values in <math>\mathbb{B},\!</math> produces the following result:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
|
|
<math>\begin{array}{lclcl}
<math>\begin{array}{lclcl}
\operatorname{E}_{ij}
\mathrm{E}_{ij}
& : &
& : &
(U \to \mathbb{B})
(U \to \mathbb{B})
(U \to \mathbb{B}),
(U \to \mathbb{B}),
\\[6pt]
\\[6pt]
\operatorname{E}_{ij}
\mathrm{E}_{ij}
& : &
& : &
f
f
& \mapsto &
& \mapsto &
\operatorname{E}_{ij}f,
\mathrm{E}_{ij}f,
\\[6pt]
\\[6pt]
\operatorname{E}_{ij}f
\mathrm{E}_{ij}f
& = &
& = &
\operatorname{E}f|_{\operatorname{d}x = i, \operatorname{d}y = j}
\mathrm{E}f|_{\mathrm{d}x = i, \mathrm{d}y = j}
& = &
& = &
f(x + i, y + j).
f(x + i, y + j).
\end{array}</math>
\end{array}\!</math>
|}
|}
The notation is a little awkward, but the data of Table A3 should make the sense clear. The important thing to observe is that <math>\operatorname{E}_{ij}</math> has the effect of transforming each proposition <math>f : U \to \mathbb{B}</math> into a proposition <math>f^\prime : U \to \mathbb{B}.</math> As it happens, the action of each <math>\operatorname{E}_{ij}</math> is one-to-one and onto, so the gang of four operators <math>\{ \operatorname{E}_{ij} : i, j \in \mathbb{B} \}</math> is an example of what is called a ''transformation group'' on the set of sixteen propositions. Bowing to a longstanding local and linear tradition, I will therefore redub the four elements of this group as <math>\operatorname{T}_{00}, \operatorname{T}_{01}, \operatorname{T}_{10}, \operatorname{T}_{11},</math> to bear in mind their transformative character, or nature, as the case may be. Abstractly viewed, this group of order four has the following operation table:
The notation is a little awkward, but the data of Table A3 should make the sense clear. The important thing to observe is that <math>\mathrm{E}_{ij}\!</math> has the effect of transforming each proposition <math>f : U \to \mathbb{B}\!</math> into a proposition <math>f^\prime : U \to \mathbb{B}.\!</math> As it happens, the action of each <math>\mathrm{E}_{ij}\!</math> is one-to-one and onto, so the gang of four operators <math>\{ \mathrm{E}_{ij} : i, j \in \mathbb{B} \}\!</math> is an example of what is called a ''transformation group'' on the set of sixteen propositions. Bowing to a longstanding local and linear tradition, I will therefore redub the four elements of this group as <math>\mathrm{T}_{00}, \mathrm{T}_{01}, \mathrm{T}_{10}, \mathrm{T}_{11},\!</math> to bear in mind their transformative character, or nature, as the case may be. Abstractly viewed, this group of order four has the following operation table:
<br>
<br>
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|- style="height:50px"
|- style="height:50px"
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{T}_{00}</math>
<math>\mathrm{T}_{00}\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{T}_{01}</math>
<math>\mathrm{T}_{01}\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{T}_{10}</math>
<math>\mathrm{T}_{10}\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{T}_{11}</math>
<math>\mathrm{T}_{11}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{T}_{00}</math>
| style="border-right:1px solid black" | <math>\mathrm{T}_{00}\!</math>
| <math>\operatorname{T}_{00}</math>
| <math>\mathrm{T}_{00}\!</math>
| <math>\operatorname{T}_{01}</math>
| <math>\mathrm{T}_{01}\!</math>
| <math>\operatorname{T}_{10}</math>
| <math>\mathrm{T}_{10}\!</math>
| <math>\operatorname{T}_{11}</math>
| <math>\mathrm{T}_{11}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{T}_{01}</math>
| style="border-right:1px solid black" | <math>\mathrm{T}_{01}\!</math>
| <math>\operatorname{T}_{01}</math>
| <math>\mathrm{T}_{01}\!</math>
| <math>\operatorname{T}_{00}</math>
| <math>\mathrm{T}_{00}\!</math>
| <math>\operatorname{T}_{11}</math>
| <math>\mathrm{T}_{11}\!</math>
| <math>\operatorname{T}_{10}</math>
| <math>\mathrm{T}_{10}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{T}_{10}</math>
| style="border-right:1px solid black" | <math>\mathrm{T}_{10}\!</math>
| <math>\operatorname{T}_{10}</math>
| <math>\mathrm{T}_{10}\!</math>
| <math>\operatorname{T}_{11}</math>
| <math>\mathrm{T}_{11}\!</math>
| <math>\operatorname{T}_{00}</math>
| <math>\mathrm{T}_{00}\!</math>
| <math>\operatorname{T}_{01}</math>
| <math>\mathrm{T}_{01}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{T}_{11}</math>
| style="border-right:1px solid black" | <math>\mathrm{T}_{11}\!</math>
| <math>\operatorname{T}_{11}</math>
| <math>\mathrm{T}_{11}\!</math>
| <math>\operatorname{T}_{10}</math>
| <math>\mathrm{T}_{10}\!</math>
| <math>\operatorname{T}_{01}</math>
| <math>\mathrm{T}_{01}\!</math>
| <math>\operatorname{T}_{00}</math>
| <math>\mathrm{T}_{00}\!</math>
|}
|}
It happens that there are just two possible groups of 4 elements. One is the cyclic group <math>Z_4\!</math> (from German ''Zyklus''), which this is not. The other is the Klein four-group <math>V_4\!</math> (from German ''Vier''), which this is.
It happens that there are just two possible groups of 4 elements. One is the cyclic group <math>Z_4\!</math> (from German ''Zyklus''), which this is not. The other is the Klein four-group <math>V_4\!</math> (from German ''Vier''), which this is.
More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called ''orbits''. One says that the orbits are preserved by the action of the group. There is an ''Orbit Lemma'' of immense utility to "those who count" which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group. In this instance, <math>\operatorname{T}_{00}</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get: <math>\text{Number of orbits}~ = (4 + 4 + 4 + 16) \div 4 = 7.</math> Amazing!
More concretely viewed, the group as a whole pushes the set of sixteen propositions around in such a way that they fall into seven natural classes, called ''orbits''. One says that the orbits are preserved by the action of the group. There is an ''Orbit Lemma'' of immense utility to "those who count" which, depending on your upbringing, you may associate with the names of Burnside, Cauchy, Frobenius, or some subset or superset of these three, vouching that the number of orbits is equal to the mean number of fixed points, in other words, the total number of points (in our case, propositions) that are left unmoved by the separate operations, divided by the order of the group. In this instance, <math>\mathrm{T}_{00}\!</math> operates as the group identity, fixing all 16 propositions, while the other three group elements fix 4 propositions each, and so we get: <math>\text{Number of orbits}~ = (4 + 4 + 4 + 16) \div 4 = 7.\!</math> Amazing!
==Note 8==
==Note 8==
We have been contemplating functions of the type <math>f : U \to \mathbb{B}</math> and studying the action of the operators <math>\operatorname{E}</math> and <math>\operatorname{D}</math> on this family. These functions, that we may identify for our present aims with propositions, inasmuch as they capture their abstract forms, are logical analogues of ''scalar potential fields''. These are the sorts of fields that are so picturesquely presented in elementary calculus and physics textbooks by images of snow-covered hills and parties of skiers who trek down their slopes like least action heroes. The analogous scene in propositional logic presents us with forms more reminiscent of plateaunic idylls, being all plains at one of two levels, the mesas of verity and falsity, as it were, with nary a niche to inhabit between them, restricting our options for a sporting gradient of downhill dynamics to just one of two: standing still on level ground or falling off a bluff.
We have been contemplating functions of the type <math>f : U \to \mathbb{B}\!</math> and studying the action of the operators <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> on this family. These functions, that we may identify for our present aims with propositions, inasmuch as they capture their abstract forms, are logical analogues of ''scalar potential fields''. These are the sorts of fields that are so picturesquely presented in elementary calculus and physics textbooks by images of snow-covered hills and parties of skiers who trek down their slopes like least action heroes. The analogous scene in propositional logic presents us with forms more reminiscent of plateaunic idylls, being all plains at one of two levels, the mesas of verity and falsity, as it were, with nary a niche to inhabit between them, restricting our options for a sporting gradient of downhill dynamics to just one of two: standing still on level ground or falling off a bluff.
We are still working well within the logical analogue of the classical finite difference calculus, taking in the novelties that the logical transmutation of familiar elements is able to bring to light. Soon we will take up several different notions of approximation relationships that may be seen to organize the space of propositions, and these will allow us to define several different forms of differential analysis applying to propositions. In time we will find reason to consider more general types of maps, having concrete types of the form <math>X_1 \times \ldots \times X_k \to Y_1 \times \ldots \times Y_n</math> and abstract types <math>\mathbb{B}^k \to \mathbb{B}^n.</math> We will think of these mappings as transforming universes of discourse into themselves or into others, in short, as ''transformations of discourse''.
We are still working well within the logical analogue of the classical finite difference calculus, taking in the novelties that the logical transmutation of familiar elements is able to bring to light. Soon we will take up several different notions of approximation relationships that may be seen to organize the space of propositions, and these will allow us to define several different forms of differential analysis applying to propositions. In time we will find reason to consider more general types of maps, having concrete types of the form <math>X_1 \times \ldots \times X_k \to Y_1 \times \ldots \times Y_n\!</math> and abstract types <math>\mathbb{B}^k \to \mathbb{B}^n.\!</math> We will think of these mappings as transforming universes of discourse into themselves or into others, in short, as ''transformations of discourse''.
Before we continue with this intinerary, however, I would like to highlight another sort of differential aspect that concerns the ''boundary operator'' or the ''marked connective'' that serves as one of the two basic connectives in the cactus language for ZOL.
Before we continue with this intinerary, however, I would like to highlight another sort of differential aspect that concerns the ''boundary operator'' or the ''marked connective'' that serves as one of the two basic connectives in the cactus language for ZOL.
For example, consider the proposition <math>f\!</math> of concrete type <math>f : X \times Y \times Z \to \mathbb{B}</math> and abstract type <math>f : \mathbb{B}^3 \to \mathbb{B}</math> that is written <math>\texttt{(} x, y, z \texttt{)}</math> in cactus syntax. Taken as an assertion in what Peirce called the ''existential interpretation'', <math>\texttt{(} x, y, z \texttt{)}</math> says that just one of <math>x, y, z\!</math> is false. It is useful to consider this assertion in relation to the conjunction <math>xyz\!</math> of the features that are engaged as its arguments. A venn diagram of <math>\texttt{(} x, y, z \texttt{)}</math> looks like this:
For example, consider the proposition <math>f\!</math> of concrete type <math>f : X \times Y \times Z \to \mathbb{B}\!</math> and abstract type <math>f : \mathbb{B}^3 \to \mathbb{B}\!</math> that is written <math>\texttt{(} x, y, z \texttt{)}\!</math> in cactus syntax. Taken as an assertion in what Peirce called the ''existential interpretation'', <math>\texttt{(} x, y, z \texttt{)}\!</math> says that just one of <math>x, y, z\!</math> is false. It is useful to consider this assertion in relation to the conjunction <math>xyz\!</math> of the features that are engaged as its arguments. A venn diagram of <math>\texttt{(} x, y, z \texttt{)}\!</math> looks like this:
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
|}
|}
In relation to the center cell indicated by the conjunction <math>xyz,\!</math> the region indicated by <math>\texttt{(} x, y, z \texttt{)}</math> is comprised of the adjacent or bordering cells. Thus they are the cells that are just across the boundary of the center cell, as if reached by way of Leibniz's ''minimal changes'' from the point of origin, here, <math>xyz.\!</math>
In relation to the center cell indicated by the conjunction <math>xyz,\!</math> the region indicated by <math>\texttt{(} x, y, z \texttt{)}\!</math> is comprised of the adjacent or bordering cells. Thus they are the cells that are just across the boundary of the center cell, as if reached by way of Leibniz's ''minimal changes'' from the point of origin, here, <math>xyz.\!</math>
The same sort of boundary relationship holds for any cell of origin that one chooses to indicate. One way to indicate a cell is by forming a logical conjunction of positive and negative basis features, that is, by constructing an expression of the form <math>e_1 \cdot \ldots \cdot e_k,</math> where <math>e_j = x_j ~\text{or}~ e_j = \texttt{(} x_j \texttt{)},</math> for <math>j = 1 ~\text{to}~ k.</math> The proposition <math>\texttt{(} e_1, \ldots, e_k \texttt{)}</math> indicates the disjunctive region consisting of the cells that are just next door to <math>e_1 \cdot \ldots \cdot e_k.</math>
The same sort of boundary relationship holds for any cell of origin that one chooses to indicate. One way to indicate a cell is by forming a logical conjunction of positive and negative basis features, that is, by constructing an expression of the form <math>e_1 \cdot \ldots \cdot e_k,\!</math> where <math>e_j = x_j ~\text{or}~ e_j = \texttt{(} x_j \texttt{)},\!</math> for <math>j = 1 ~\text{to}~ k.\!</math> The proposition <math>\texttt{(} e_1, \ldots, e_k \texttt{)}\!</math> indicates the disjunctive region consisting of the cells that are just next door to <math>e_1 \cdot \ldots \cdot e_k.\!</math>
==Note 9==
==Note 9==
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|- style="height:50px"
|- style="height:50px"
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{e}</math>
<math>\mathrm{e}\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{f}</math>
<math>\mathrm{f}\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{g}</math>
<math>\mathrm{g}\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{h}</math>
<math>\mathrm{h}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
| style="border-right:1px solid black" | <math>\mathrm{e}\!</math>
| <math>\operatorname{e}</math>
| <math>\mathrm{e}\!</math>
| <math>\operatorname{f}</math>
| <math>\mathrm{f}\!</math>
| <math>\operatorname{g}</math>
| <math>\mathrm{g}\!</math>
| <math>\operatorname{h}</math>
| <math>\mathrm{h}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
| style="border-right:1px solid black" | <math>\mathrm{f}\!</math>
| <math>\operatorname{f}</math>
| <math>\mathrm{f}\!</math>
| <math>\operatorname{e}</math>
| <math>\mathrm{e}\!</math>
| <math>\operatorname{h}</math>
| <math>\mathrm{h}\!</math>
| <math>\operatorname{g}</math>
| <math>\mathrm{g}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
| style="border-right:1px solid black" | <math>\mathrm{g}\!</math>
| <math>\operatorname{g}</math>
| <math>\mathrm{g}\!</math>
| <math>\operatorname{h}</math>
| <math>\mathrm{h}\!</math>
| <math>\operatorname{e}</math>
| <math>\mathrm{e}\!</math>
| <math>\operatorname{f}</math>
| <math>\mathrm{f}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
| style="border-right:1px solid black" | <math>\mathrm{h}\!</math>
| <math>\operatorname{h}</math>
| <math>\mathrm{h}\!</math>
| <math>\operatorname{g}</math>
| <math>\mathrm{g}\!</math>
| <math>\operatorname{f}</math>
| <math>\mathrm{f}\!</math>
| <math>\operatorname{e}</math>
| <math>\mathrm{e}\!</math>
|}
|}
<br>
<br>
This operation table is abstractly the same as, or isomorphic to, the versions with the <math>\operatorname{E}_{ij}</math> operators and the <math>\operatorname{T}_{ij}</math> transformations that we discussed earlier. That is to say, the story is the same — only the names have been changed. An abstract group can have a multitude of significantly and superficially different representations. Even after we have long forgotten the details of the particular representation that we may have come in with, there are species of concrete representations, called the ''regular representations'', that are always readily available, as they can be generated from the mere data of the abstract operation table itself.
This operation table is abstractly the same as, or isomorphic to, the versions with the <math>\mathrm{E}_{ij}\!</math> operators and the <math>\mathrm{T}_{ij}\!</math> transformations that we discussed earlier. That is to say, the story is the same — only the names have been changed. An abstract group can have a multitude of significantly and superficially different representations. Even after we have long forgotten the details of the particular representation that we may have come in with, there are species of concrete representations, called the ''regular representations'', that are always readily available, as they can be generated from the mere data of the abstract operation table itself.
To see how a regular representation is constructed from the abstract operation table, pick a group element at the top of the table and "consider its effects" on each of the group elements listed on the left. These effects may be recorded in one of the ways that Peirce often used, as a ''logical aggregate'' of elementary dyadic relatives, that is, as a logical disjunction or sum whose terms represent the <math>\operatorname{input} : \operatorname{output}</math> pairs that are produced by each group element in turn. This forms one of the two possible ''regular representations'' of the group, specifically, the one that is called the ''post-regular representation'' or the ''right regular representation''. It has long been conventional to organize the terms of this logical sum in the form of a matrix:
To see how a regular representation is constructed from the abstract operation table, pick a group element at the top of the table and "consider its effects" on each of the group elements listed on the left. These effects may be recorded in one of the ways that Peirce often used, as a ''logical aggregate'' of elementary dyadic relatives, that is, as a logical disjunction or sum whose terms represent the <math>\mathrm{input} : \mathrm{output}\!</math> pairs that are produced by each group element in turn. This forms one of the two possible ''regular representations'' of the group, specifically, the one that is called the ''post-regular representation'' or the ''right regular representation''. It has long been conventional to organize the terms of this logical sum in the form of a matrix:
Reading "<math>+\!</math>" as a logical disjunction:
Reading "<math>+\!</math>" as a logical disjunction:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{G}
\mathrm{G}
& = & \operatorname{e}
& = & \mathrm{e}
& + & \operatorname{f}
& + & \mathrm{f}
& + & \operatorname{g}
& + & \mathrm{g}
& + & \operatorname{h}
& + & \mathrm{h}
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
And so, by expanding effects, we get:
And so, by expanding effects, we get:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{G}
\mathrm{G}
& = & \operatorname{e}:\operatorname{e}
& = & \mathrm{e}:\mathrm{e}
& + & \operatorname{f}:\operatorname{f}
& + & \mathrm{f}:\mathrm{f}
& + & \operatorname{g}:\operatorname{g}
& + & \mathrm{g}:\mathrm{g}
& + & \operatorname{h}:\operatorname{h}
& + & \mathrm{h}:\mathrm{h}
\\[4pt]
\\[4pt]
& + & \operatorname{e}:\operatorname{f}
& + & \mathrm{e}:\mathrm{f}
& + & \operatorname{f}:\operatorname{e}
& + & \mathrm{f}:\mathrm{e}
& + & \operatorname{g}:\operatorname{h}
& + & \mathrm{g}:\mathrm{h}
& + & \mathrm{h}:\mathrm{g}
& + & \mathrm{h}:\mathrm{g}
\\[4pt]
\\[4pt]
& + & \operatorname{e}:\operatorname{g}
& + & \mathrm{e}:\mathrm{g}
& + & \operatorname{f}:\operatorname{h}
& + & \mathrm{f}:\mathrm{h}
& + & \operatorname{g}:\operatorname{e}
& + & \mathrm{g}:\mathrm{e}
& + & \operatorname{h}:\operatorname{f}
& + & \mathrm{h}:\mathrm{f}
\\[4pt]
\\[4pt]
& + & \operatorname{e}:\operatorname{h}
& + & \mathrm{e}:\mathrm{h}
& + & \operatorname{f}:\operatorname{g}
& + & \mathrm{f}:\mathrm{g}
& + & \operatorname{g}:\operatorname{f}
& + & \mathrm{g}:\mathrm{f}
& + & \operatorname{h}:\operatorname{e}
& + & \mathrm{h}:\mathrm{e}
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
The idea about the regular representations of a group is universally known as Cayley's Theorem, typically stated in the following form:
The idea about the regular representations of a group is universally known as Cayley's Theorem, typically stated in the following form:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| Every group is isomorphic to a subgroup of <math>\operatorname{Aut}(S),</math> the group of automorphisms of a suitable set <math>S\!</math>.
| Every group is isomorphic to a subgroup of <math>\mathrm{Aut}(S),\!</math> the group of automorphisms of a suitable set <math>S\!</math>.
|}
|}
There is a considerable generalization of these regular representations to a broad class of relational algebraic systems in Peirce's earliest papers. The crux of the whole idea is this:
There is a considerable generalization of these regular representations to a broad class of relational algebraic systems in Peirce's earliest papers. The crux of the whole idea is this:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| Consider the effects of the symbol, whose meaning you wish to investigate, as they play out on all the stages of context where you can imagine that symbol playing a role.
| Consider the effects of the symbol, whose meaning you wish to investigate, as they play out on all the stages of context where you can imagine that symbol playing a role.
|}
|}
This idea of contextual definition is basically the same as Jeremy Bentham's notion of ''paraphrasis'', a "method of accounting for fictions by explaining various purported terms away" (Quine, in Van Heijenoort, p. 216). Today we'd call these constructions ''term models''. This, again, is the big idea behind Schönfinkel's combinators <math>\operatorname{S}, \operatorname{K}, \operatorname{I},</math> and hence of lambda calculus, and I reckon you know where that leads.
This idea of contextual definition is basically the same as Jeremy Bentham's notion of ''paraphrasis'', a "method of accounting for fictions by explaining various purported terms away" (Quine, in Van Heijenoort, p. 216). Today we'd call these constructions ''term models''. This, again, is the big idea behind Schönfinkel's combinators <math>\mathrm{S}, \mathrm{K}, \mathrm{I},\!</math> and hence of lambda calculus, and I reckon you know where that leads.
==Note 11==
==Note 11==
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|+ <math>\text{Klein Four-Group}~ V_4</math>
|+ <math>\text{Klein Four-Group}~ V_4\!</math>
|- style="height:50px"
|- style="height:50px"
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{e}</math>
<math>\mathrm{e}\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{f}</math>
<math>\mathrm{f}\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{g}</math>
<math>\mathrm{g}\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{h}</math>
<math>\mathrm{h}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
| style="border-right:1px solid black" | <math>\mathrm{e}\!</math>
| <math>\operatorname{e}</math>
| <math>\mathrm{e}\!</math>
| <math>\operatorname{f}</math>
| <math>\mathrm{f}\!</math>
| <math>\operatorname{g}</math>
| <math>\mathrm{g}\!</math>
| <math>\operatorname{h}</math>
| <math>\mathrm{h}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
| style="border-right:1px solid black" | <math>\mathrm{f}\!</math>
| <math>\operatorname{f}</math>
| <math>\mathrm{f}\!</math>
| <math>\operatorname{e}</math>
| <math>\mathrm{e}\!</math>
| <math>\operatorname{h}</math>
| <math>\mathrm{h}\!</math>
| <math>\operatorname{g}</math>
| <math>\mathrm{g}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
| style="border-right:1px solid black" | <math>\mathrm{g}\!</math>
| <math>\operatorname{g}</math>
| <math>\mathrm{g}\!</math>
| <math>\operatorname{h}</math>
| <math>\mathrm{h}\!</math>
| <math>\operatorname{e}</math>
| <math>\mathrm{e}\!</math>
| <math>\operatorname{f}</math>
| <math>\mathrm{f}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
| style="border-right:1px solid black" | <math>\mathrm{h}\!</math>
| <math>\operatorname{h}</math>
| <math>\mathrm{h}\!</math>
| <math>\operatorname{g}</math>
| <math>\mathrm{g}\!</math>
| <math>\operatorname{f}</math>
| <math>\mathrm{f}\!</math>
| <math>\operatorname{e}</math>
| <math>\mathrm{e}\!</math>
|}
|}
Reading "<math>+\!</math>" as a logical disjunction:
Reading "<math>+\!</math>" as a logical disjunction:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| <math>\mathrm{G} ~=~ \mathrm{e} ~+~ \mathrm{f} ~+~ \mathrm{g} ~+~ \mathrm{h}~\!</math>
|}
|}
Expanding effects, we get:
Expanding effects, we get:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{G}
\mathrm{G}
& + & \mathrm{g}:\mathrm{f}
& + & \mathrm{g}:\mathrm{f}
& + & \mathrm{h}:\mathrm{e}
& + & \mathrm{h}:\mathrm{e}
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
Working through the construction for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular post-representations:
Working through the construction for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular post-representations:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{e}
\mathrm{e}
& + & \mathrm{g}:\mathrm{f}
& + & \mathrm{g}:\mathrm{f}
& + & \mathrm{h}:\mathrm{e}
& + & \mathrm{h}:\mathrm{e}
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
So if somebody asks you, say, "What is <math>\operatorname{g}</math>?", you can say, "I don't know for certain, but in practice its effects go a bit like this:
So if somebody asks you, say, "What is <math>\mathrm{g}\!</math>?", you can say, "I don't know for certain, but in practice its effects go a bit like this:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| <math>\mathrm{g} \quad \text{converts} \quad \mathrm{e} ~\text{to}~ \mathrm{g}, \quad \mathrm{f} ~\text{to}~ \mathrm{h}, \quad \mathrm{g} ~\text{to}~ \mathrm{e}, \quad \mathrm{h} ~\text{to}~ \mathrm{f}.\!</math>
|}
|}
Working through this alternative for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular ante-representations:
Working through this alternative for each one of the four group elements, we arrive at the following exegeses of their senses, giving their regular ante-representations:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
|
<math>\begin{matrix}
<math>\begin{matrix}
\mathrm{e}
\mathrm{e}
& + & \mathrm{f}:\mathrm{g}
& + & \mathrm{f}:\mathrm{g}
& + & \mathrm{e}:\mathrm{h}
& + & \mathrm{e}:\mathrm{h}
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
Peirce describes the action of an "elementary dual relative" in this way:
Peirce describes the action of an "elementary dual relative" in this way:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| Elementary simple relatives are connected together in systems of four. For if <math>\mathrm{A}\!:\!\mathrm{B}</math> be taken to denote the elementary relative which multiplied into <math>\mathrm{B}\!</math> gives <math>\mathrm{A},\!</math> then this relation existing as elementary, we have the four elementary relatives
| Elementary simple relatives are connected together in systems of four. For if <math>\mathrm{A}\!:\!\mathrm{B}\!</math> be taken to denote the elementary relative which multiplied into <math>\mathrm{B}\!</math> gives <math>\mathrm{A},\!</math> then this relation existing as elementary, we have the four elementary relatives
|-
|-
| align="center" | <math>\mathrm{A}\!:\!\mathrm{A} \qquad \mathrm{A}\!:\!\mathrm{B} \qquad \mathrm{B}\!:\!\mathrm{A} \qquad \mathrm{B}\!:\!\mathrm{B}.</math>
| align="center" | <math>\mathrm{A}\!:\!\mathrm{A} \qquad \mathrm{A}\!:\!\mathrm{B} \qquad \mathrm{B}\!:\!\mathrm{A} \qquad \mathrm{B}\!:\!\mathrm{B}.\!</math>
|-
|-
| C.S. Peirce, ''Collected Papers'', CP 3.123.
| C.S. Peirce, ''Collected Papers'', CP 3.123.
And though he is well aware that it is not at all necessary to arrange elementary relatives into arrays, matrices, or tables, when he does so he tends to prefer organizing dyadic relations in the following manner:
And though he is well aware that it is not at all necessary to arrange elementary relatives into arrays, matrices, or tables, when he does so he tends to prefer organizing dyadic relations in the following manner:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{bmatrix}
<math>\begin{bmatrix}
\\
\\
\mathrm{C}\!:\!\mathrm{A} & \mathrm{C}\!:\!\mathrm{B} & \mathrm{C}\!:\!\mathrm{C}
\mathrm{C}\!:\!\mathrm{A} & \mathrm{C}\!:\!\mathrm{B} & \mathrm{C}\!:\!\mathrm{C}
\end{bmatrix}</math>
\end{bmatrix}\!</math>
|}
|}
That conforms to the way that the last school of thought I matriculated into stipulated that we tabulate material:
That conforms to the way that the last school of thought I matriculated into stipulated that we tabulate material:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{bmatrix}
<math>\begin{bmatrix}
\\
\\
e_{31} & e_{32} & e_{33}
e_{31} & e_{32} & e_{33}
\end{bmatrix}</math>
\end{bmatrix}\!</math>
|}
|}
So, for example, let us suppose that we have the small universe <math>\{ \mathrm{A}, \mathrm{B}, \mathrm{C} \},\!</math> and the 2-adic relation <math>\mathit{m} = {}^{\backprime\backprime}\, \text{mover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:56, 7 December 2014 (UTC)}\, {}^{\prime\prime}</math> that is represented by the following matrix:
So, for example, let us suppose that we have the small universe <math>\{ \mathrm{A}, \mathrm{B}, \mathrm{C} \},\!</math> and the 2-adic relation <math>\mathit{m} = {}^{\backprime\backprime}\, \text{mover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 22:03, 8 December 2014 (UTC)}\, {}^{\prime\prime}\!</math> that is represented by the following matrix:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{bmatrix}
<math>\begin{bmatrix}
m_\mathrm{CB} (\mathrm{C}\!:\!\mathrm{B}) &
m_\mathrm{CB} (\mathrm{C}\!:\!\mathrm{B}) &
m_\mathrm{CC} (\mathrm{C}\!:\!\mathrm{C})
m_\mathrm{CC} (\mathrm{C}\!:\!\mathrm{C})
\end{bmatrix}</math>
\end{bmatrix}\!</math>
|}
|}
Also, let <math>\mathit{m}\!</math> be such that:
Also, let <math>\mathit{m}\!</math> be such that:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{array}{l}
<math>\begin{array}{l}
\\
\\
\mathrm{C} ~\text{is a mover of}~ \mathrm{C} ~\text{and}~ \mathrm{A}.
\mathrm{C} ~\text{is a mover of}~ \mathrm{C} ~\text{and}~ \mathrm{A}.
\end{array}</math>
\end{array}\!</math>
|}
|}
In sum, <math>\mathit{m}\!</math> is represented by the following matrix:
In sum, <math>\mathit{m}\!</math> is represented by the following matrix:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{bmatrix}
<math>\begin{bmatrix}
0 \cdot (\mathrm{C}\!:\!\mathrm{B}) &
0 \cdot (\mathrm{C}\!:\!\mathrm{B}) &
1 \cdot (\mathrm{C}\!:\!\mathrm{C})
1 \cdot (\mathrm{C}\!:\!\mathrm{C})
\end{bmatrix}</math>
\end{bmatrix}\!</math>
|}
|}
In the paper "On the Relative Forms of Quaternions" (CP 3.323), we observe Peirce providing the following sorts of explanation:
In the paper "On the Relative Forms of Quaternions" (CP 3.323), we observe Peirce providing the following sorts of explanation:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
|
|
<p>If <math>X, Y, Z\!</math> denote the three rectangular components of a vector, and <math>W\!</math> denote numerical unity (or a fourth rectangular component, involving space of four dimensions), and <math>(Y:Z)\!</math> denote the operation of converting the <math>Y\!</math> component of a vector into its <math>Z\!</math> component, then</p>
<p>If <math>X, Y, Z\!</math> denote the three rectangular components of a vector, and <math>W\!</math> denote numerical unity (or a fourth rectangular component, involving space of four dimensions), and <math>(Y:Z)\!</math> denote the operation of converting the <math>Y\!</math> component of a vector into its <math>Z\!</math> component, then</p>
\\
\\
k & = & (Z:W) & - & (W:Z) & - & (X:Y) & + & (Y:X)
k & = & (Z:W) & - & (W:Z) & - & (X:Y) & + & (Y:X)
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
|-
|-
| align="center" |
| align="center" |
<math>i^2 = j^2 = k^2 = -1, \quad ijk = -1, \quad \text{etc}.</math>
<math>i^2 = j^2 = k^2 = -1, \quad ijk = -1, \quad \text{etc}.\!</math>
|-
|-
|
|
\\
\\
Z:W && Z:X && Z:Y && Z:Z
Z:W && Z:X && Z:Y && Z:Z
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\
\\
z && -y && x && w
z && -y && x && w
\end{array}</math>
\end{array}\!</math>
|-
|-
|
|
<p>The multiplication of such matrices follows the same laws as the multiplication of quaternions. The determinant of the matrix = the fourth power of the tensor of the quaternion.</p>
<p>The multiplication of such matrices follows the same laws as the multiplication of quaternions. The determinant of the matrix = the fourth power of the tensor of the quaternion.</p>
<p>The imaginary <math>x + y \sqrt{-1}</math> may likewise be represented by the matrix</p>
<p>The imaginary <math>x + y \sqrt{-1}\!</math> may likewise be represented by the matrix</p>
|-
|-
| align="center" |
| align="center" |
\\
\\
-y & x
-y & x
\end{array}</math>
\end{array}\!</math>
|-
|-
|
|
Back to our current subinstance, the example in support of our first example. I will try to reconstruct it in a less confusing way.
Back to our current subinstance, the example in support of our first example. I will try to reconstruct it in a less confusing way.
Consider the universe of discourse <math>\mathbf{1} = \mathrm{A} + \mathrm{B} + \mathrm{C}</math> and the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:56, 7 December 2014 (UTC)}\, {}^{\prime\prime},</math> as when "<math>X\!</math> is a data record that contains a pointer to <math>Y\!</math>". That interpretation is not important, it's just for the sake of intuition. In general terms, the 2-adic relation <math>n\!</math> can be represented by this matrix:
Consider the universe of discourse <math>\mathbf{1} = \mathrm{A} + \mathrm{B} + \mathrm{C}\!</math> and the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 22:03, 8 December 2014 (UTC)}\, {}^{\prime\prime},\!</math> as when "<math>X\!</math> is a data record that contains a pointer to <math>Y\!</math>". That interpretation is not important, it's just for the sake of intuition. In general terms, the 2-adic relation <math>n\!</math> can be represented by this matrix:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{bmatrix}
<math>\begin{bmatrix}
\mathit{n}_\mathrm{CB} (\mathrm{C}\!:\!\mathrm{B}) &
\mathit{n}_\mathrm{CB} (\mathrm{C}\!:\!\mathrm{B}) &
\mathit{n}_\mathrm{CC} (\mathrm{C}\!:\!\mathrm{C})
\mathit{n}_\mathrm{CC} (\mathrm{C}\!:\!\mathrm{C})
\end{bmatrix}</math>
\end{bmatrix}\!</math>
|}
|}
More specifically, let <math>\mathit{n}\!</math> be such that:
More specifically, let <math>\mathit{n}\!</math> be such that:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{array}{l}
<math>\begin{array}{l}
\\
\\
\mathrm{C} ~\text{is a noder of}~ \mathrm{C} ~\text{and}~ \mathrm{A}.
\mathrm{C} ~\text{is a noder of}~ \mathrm{C} ~\text{and}~ \mathrm{A}.
\end{array}</math>
\end{array}\!</math>
|}
|}
Filling in the instantial values of the coefficients <math>\mathit{n}_{ij},\!</math> as the indices <math>i\!</math> and <math>j\!</math> range over the universe of discourse, the relation <math>\mathit{n}\!</math> is represented by the following matrix:
Filling in the instantial values of the coefficients <math>\mathit{n}_{ij},\!</math> as the indices <math>i\!</math> and <math>j\!</math> range over the universe of discourse, the relation <math>\mathit{n}\!</math> is represented by the following matrix:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{bmatrix}
<math>\begin{bmatrix}
0 \cdot (\mathrm{C}\!:\!\mathrm{B}) &
0 \cdot (\mathrm{C}\!:\!\mathrm{B}) &
1 \cdot (\mathrm{C}\!:\!\mathrm{C})
1 \cdot (\mathrm{C}\!:\!\mathrm{C})
\end{bmatrix}</math>
\end{bmatrix}\!</math>
|}
|}
In Peirce's time, and even in some circles of mathematics today, the information indicated by the elementary relatives <math>(I\!:\!J),</math> as <math>I, J\!</math> range over the universe of discourse, would be referred to as the ''umbral elements'' of the algebraic operation represented by the matrix, though I seem to recall that Peirce preferred to call these terms the "ingredients". When this ordered basis is understood well enough, one will tend to drop any mention of it from the matrix itself, leaving us nothing but these bare bones:
In Peirce's time, and even in some circles of mathematics today, the information indicated by the elementary relatives <math>(I\!:\!J),\!</math> as <math>I, J\!</math> range over the universe of discourse, would be referred to as the ''umbral elements'' of the algebraic operation represented by the matrix, though I seem to recall that Peirce preferred to call these terms the "ingredients". When this ordered basis is understood well enough, one will tend to drop any mention of it from the matrix itself, leaving us nothing but these bare bones:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{bmatrix}
<math>\begin{bmatrix}
\\
\\
1 & 0 & 1
1 & 0 & 1
\end{bmatrix}</math>
\end{bmatrix}\!</math>
|}
|}
The various representations of <math>n\!</math> are nothing more than alternative ways of specifying its basic ingredients, namely, the following logical sum of elementary relatives:
The various representations of <math>n\!</math> are nothing more than alternative ways of specifying its basic ingredients, namely, the following logical sum of elementary relatives:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{array}{*{13}{c}}
<math>\begin{array}{*{13}{c}}
& + & \mathrm{B}\!:\!\mathrm{C}
& + & \mathrm{B}\!:\!\mathrm{C}
& + & \mathrm{C}\!:\!\mathrm{A}
& + & \mathrm{C}\!:\!\mathrm{A}
\end{array}</math>
\end{array}\!</math>
|}
|}
Recognizing <math>\mathit{1} = \mathrm{A}\!:\!\mathrm{A} + \mathrm{B}\!:\!\mathrm{B} + \mathrm{C}\!:\!\mathrm{C}</math> to be the identity transformation, the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:56, 7 December 2014 (UTC)}\, {}^{\prime\prime}</math> may be represented by an element <math>\mathit{1} + \mathrm{A}\!:\!\mathrm{B} + \mathrm{B}\!:\!\mathrm{C} + \mathrm{C}\!:\!\mathrm{A}</math> of the so-called ''group ring'', all of which just makes this element a special sort of linear transformation.
Recognizing <math>\mathit{1} = \mathrm{A}\!:\!\mathrm{A} + \mathrm{B}\!:\!\mathrm{B} + \mathrm{C}\!:\!\mathrm{C}\!</math> to be the identity transformation, the 2-adic relation <math>\mathit{n} = {}^{\backprime\backprime}\, \text{noder of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 22:03, 8 December 2014 (UTC)}\, {}^{\prime\prime}\!</math> may be represented by an element <math>\mathit{1} + \mathrm{A}\!:\!\mathrm{B} + \mathrm{B}\!:\!\mathrm{C} + \mathrm{C}\!:\!\mathrm{A}\!</math> of the so-called ''group ring'', all of which just makes this element a special sort of linear transformation.
Up to this point, we're still reading the elementary relatives of the form <math>I:J\!</math> in the way that Peirce reads them in logical contexts: <math>I\!</math> is the relate, <math>J\!</math> is the correlate, and in our current example we read <math>I:J,\!</math> or more exactly, <math>\mathit{n}_{ij} = 1,\!</math> to say that <math>I\!</math> is a noder of <math>J.\!</math> This is the mode of reading that we call ''multiplying on the left''.
Up to this point, we're still reading the elementary relatives of the form <math>I:J\!</math> in the way that Peirce reads them in logical contexts: <math>I\!</math> is the relate, <math>J\!</math> is the correlate, and in our current example we read <math>I:J,\!</math> or more exactly, <math>\mathit{n}_{ij} = 1,\!</math> to say that <math>I\!</math> is a noder of <math>J.\!</math> This is the mode of reading that we call ''multiplying on the left''.
In the algebraic, permutational, or transformational contexts of application, however, Peirce converts to the alternative mode of reading, although still calling <math>I\!</math> the relate and <math>J\!</math> the correlate, the elementary relative <math>I:J\!</math> now means that <math>I\!</math> gets changed into <math>J.\!</math> In this scheme of reading, the transformation <math>\mathrm{A}\!:\!\mathrm{B} + \mathrm{B}\!:\!\mathrm{C} + \mathrm{C}\!:\!\mathrm{A}</math> is a permutation of the aggregate <math>\mathbf{1} = \mathrm{A} + \mathrm{B} + \mathrm{C},</math> or what we would now call the set <math>\{ \mathrm{A}, \mathrm{B}, \mathrm{C} \},\!</math> in particular, it is the permutation that is otherwise notated as follows:
In the algebraic, permutational, or transformational contexts of application, however, Peirce converts to the alternative mode of reading, although still calling <math>I\!</math> the relate and <math>J\!</math> the correlate, the elementary relative <math>I:J\!</math> now means that <math>I\!</math> gets changed into <math>J.\!</math> In this scheme of reading, the transformation <math>\mathrm{A}\!:\!\mathrm{B} + \mathrm{B}\!:\!\mathrm{C} + \mathrm{C}\!:\!\mathrm{A}\!</math> is a permutation of the aggregate <math>\mathbf{1} = \mathrm{A} + \mathrm{B} + \mathrm{C},\!</math> or what we would now call the set <math>\{ \mathrm{A}, \mathrm{B}, \mathrm{C} \},\!</math> in particular, it is the permutation that is otherwise notated as follows:
{| align="center" cellpadding="6"
{| align="center" cellpadding="6"
\\
\\
\mathrm{B} & \mathrm{C} & \mathrm{A}
\mathrm{B} & \mathrm{C} & \mathrm{A}
\end{Bmatrix}</math>
\end{Bmatrix}\!</math>
|}
|}
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black; text-align:center; width:60%"
|+ <math>\text{Klein Four-Group}~ V_4</math>
|+ <math>\text{Klein Four-Group}~ V_4\!</math>
|- style="height:50px"
|- style="height:50px"
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot</math>
| width="12%" style="border-bottom:1px solid black; border-right:1px solid black" | <math>\cdot\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{e}</math>
<math>\mathrm{e}\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{f}</math>
<math>\mathrm{f}\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{g}</math>
<math>\mathrm{g}\!</math>
| width="22%" style="border-bottom:1px solid black" |
| width="22%" style="border-bottom:1px solid black" |
<math>\operatorname{h}</math>
<math>\mathrm{h}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{e}</math>
| style="border-right:1px solid black" | <math>\mathrm{e}\!</math>
| <math>\operatorname{e}</math>
| <math>\mathrm{e}\!</math>
| <math>\operatorname{f}</math>
| <math>\mathrm{f}\!</math>
| <math>\operatorname{g}</math>
| <math>\mathrm{g}\!</math>
| <math>\operatorname{h}</math>
| <math>\mathrm{h}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{f}</math>
| style="border-right:1px solid black" | <math>\mathrm{f}\!</math>
| <math>\operatorname{f}</math>
| <math>\mathrm{f}\!</math>
| <math>\operatorname{e}</math>
| <math>\mathrm{e}\!</math>
| <math>\operatorname{h}</math>
| <math>\mathrm{h}\!</math>
| <math>\operatorname{g}</math>
| <math>\mathrm{g}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{g}</math>
| style="border-right:1px solid black" | <math>\mathrm{g}\!</math>
| <math>\operatorname{g}</math>
| <math>\mathrm{g}\!</math>
| <math>\operatorname{h}</math>
| <math>\mathrm{h}\!</math>
| <math>\operatorname{e}</math>
| <math>\mathrm{e}\!</math>
| <math>\operatorname{f}</math>
| <math>\mathrm{f}\!</math>
|- style="height:50px"
|- style="height:50px"
| style="border-right:1px solid black" | <math>\operatorname{h}</math>
| style="border-right:1px solid black" | <math>\mathrm{h}\!</math>
| <math>\operatorname{h}</math>
| <math>\mathrm{h}\!</math>
| <math>\operatorname{g}</math>
| <math>\mathrm{g}\!</math>
| <math>\operatorname{f}</math>
| <math>\mathrm{f}\!</math>
| <math>\operatorname{e}</math>
| <math>\mathrm{e}\!</math>
|}
|}
<br>
<br>
A group operation table is really just a device for recording a certain 3-adic relation, to be specific, the set of triples of the form <math>(x, y, z)\!</math> satisfying the equation <math>x \cdot y = z.</math>
A group operation table is really just a device for recording a certain 3-adic relation, to be specific, the set of triples of the form <math>(x, y, z)\!</math> satisfying the equation <math>x \cdot y = z.\!</math>
In the case of <math>V_4 = (G, \cdot),</math> where <math>G\!</math> is the ''underlying set'' <math>\{ \operatorname{e}, \operatorname{f}, \operatorname{g}, \operatorname{h} \},</math> we have the 3-adic relation <math>L(V_4) \subseteq G \times G \times G</math> whose triples are listed below:
In the case of <math>V_4 = (G, \cdot),\!</math> where <math>G\!</math> is the ''underlying set'' <math>\{ \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h} \},\!</math> we have the 3-adic relation <math>L(V_4) \subseteq G \times G \times G\!</math> whose triples are listed below:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{matrix}
<math>\begin{matrix}
(\mathrm{h}, \mathrm{g}, \mathrm{f}) &
(\mathrm{h}, \mathrm{g}, \mathrm{f}) &
(\mathrm{h}, \mathrm{h}, \mathrm{e})
(\mathrm{h}, \mathrm{h}, \mathrm{e})
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3</math> is actually a function <math>L : G \times G \to G.</math> It is from this functional perspective that we can see an easy way to derive the two regular representations. Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators:
It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3\!</math> is actually a function <math>L : G \times G \to G.\!</math> It is from this functional perspective that we can see an easy way to derive the two regular representations. Since we have a function of the type <math>L : G \times G \to G,\!</math> we can define a couple of substitution operators:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| valign="top" | 1.
| valign="top" | 1.
| <math>\operatorname{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>x \cdot y.</math>
| <math>\mathrm{Sub}(x, (\underline{~~}, y))\!</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),\!</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>x \cdot y.\!</math>
|-
|-
| valign="top" | 2.
| valign="top" | 2.
| <math>\operatorname{Sub}(x, (y, \underline{~~}))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>y \cdot x.</math>
| <math>\mathrm{Sub}(x, (y, \underline{~~}))\!</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),\!</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>y \cdot x.\!</math>
|}
|}
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>x \cdot y,</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : x \cdot y) ~|~ y \in G \}.</math> The pairs <math>(y : x \cdot y)</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run across the top margin. This aspect of pragmatic definition we recognize as the regular ante-representation:
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),\!</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)\!</math> into <math>x \cdot y,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : x \cdot y) ~|~ y \in G \}.\!</math> The pairs <math>(y : x \cdot y)\!</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run across the top margin. This aspect of pragmatic definition we recognize as the regular ante-representation:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{matrix}
<math>\begin{matrix}
& + & \mathrm{g}:\mathrm{f}
& + & \mathrm{g}:\mathrm{f}
& + & \mathrm{h}:\mathrm{e}
& + & \mathrm{h}:\mathrm{e}
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})</math> into <math>y \cdot x,</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : y \cdot x) ~|~ y \in G \}.</math> The pairs <math>(y : y \cdot x)</math> can be found by picking an <math>x\!</math> from the top margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run down the left margin. This aspect of pragmatic definition we recognize as the regular post-representation:
In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),\!</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})\!</math> into <math>y \cdot x,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : y \cdot x) ~|~ y \in G \}.\!</math> The pairs <math>(y : y \cdot x)~\!</math> can be found by picking an <math>x\!</math> from the top margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run down the left margin. This aspect of pragmatic definition we recognize as the regular post-representation:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<math>\begin{matrix}
<math>\begin{matrix}
& + & \mathrm{g}:\mathrm{f}
& + & \mathrm{g}:\mathrm{f}
& + & \mathrm{h}:\mathrm{e}
& + & \mathrm{h}:\mathrm{e}
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
==Note 19==
==Note 19==
So long as we're in the neighborhood, we might as well take in some more of the sights, for instance, the smallest example of a non-abelian (non-commutative) group. This is a group of six elements, say, <math>G = \{ \operatorname{e}, \operatorname{f}, \operatorname{g}, \operatorname{h}, \operatorname{i}, \operatorname{j} \},\!</math> with no relation to any other employment of these six symbols being implied, of course, and it can be most easily represented as the permutation group on a set of three letters, say, <math>X = \{ A, B, C \},\!</math> usually notated as <math>G = \operatorname{Sym}(X)</math> or more abstractly and briefly, as <math>\operatorname{Sym}(3)</math> or <math>S_3.\!</math> The next Table shows the intended correspondence between abstract group elements and the permutation or substitution operations in <math>\operatorname{Sym}(X).</math>
So long as we're in the neighborhood, we might as well take in some more of the sights, for instance, the smallest example of a non-abelian (non-commutative) group. This is a group of six elements, say, <math>G = \{ \mathrm{e}, \mathrm{f}, \mathrm{g}, \mathrm{h}, \mathrm{i}, \mathrm{j} \},\!</math> with no relation to any other employment of these six symbols being implied, of course, and it can be most easily represented as the permutation group on a set of three letters, say, <math>X = \{ A, B, C \},\!</math> usually notated as <math>G = \mathrm{Sym}(X)\!</math> or more abstractly and briefly, as <math>\mathrm{Sym}(3)\!</math> or <math>S_3.\!</math> The next Table shows the intended correspondence between abstract group elements and the permutation or substitution operations in <math>\mathrm{Sym}(X).\!</math>
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>
|+ <math>\text{Permutation Substitutions in}~ \mathrm{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="16%" | <math>\operatorname{e}</math>
| width="16%" | <math>\mathrm{e}\!</math>
| width="16%" | <math>\operatorname{f}</math>
| width="16%" | <math>\mathrm{f}\!</math>
| width="16%" | <math>\operatorname{g}</math>
| width="16%" | <math>\mathrm{g}\!</math>
| width="16%" | <math>\operatorname{h}</math>
| width="16%" | <math>\mathrm{h}\!</math>
| width="16%" | <math>\operatorname{i}</math>
| width="16%" | <math>\mathrm{i}~\!</math>
| width="16%" | <math>\operatorname{j}</math>
| width="16%" | <math>\mathrm{j}\!</math>
|-
|-
|
|
\\[6pt]
\\[6pt]
\mathrm{A} & \mathrm{B} & \mathrm{C}
\mathrm{A} & \mathrm{B} & \mathrm{C}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{C} & \mathrm{A} & \mathrm{B}
\mathrm{C} & \mathrm{A} & \mathrm{B}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{B} & \mathrm{C} & \mathrm{A}
\mathrm{B} & \mathrm{C} & \mathrm{A}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{A} & \mathrm{C} & \mathrm{B}
\mathrm{A} & \mathrm{C} & \mathrm{B}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{C} & \mathrm{B} & \mathrm{A}
\mathrm{C} & \mathrm{B} & \mathrm{A}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{B} & \mathrm{A} & \mathrm{C}
\mathrm{B} & \mathrm{A} & \mathrm{C}
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
Here is the operation table for <math>S_3,\!</math> given in abstract fashion:
Here is the operation table for <math>S_3,\!</math> given in abstract fashion:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| align="center" |
| align="center" |
<pre>
<pre>
|}
|}
By the way, we will meet with the symmetric group <math>S_3\!</math> again when we return to take up the study of Peirce's early paper "On a Class of Multiple Algebras" (CP 3.324–327), and also his late unpublished work "The Simplest Mathematics" (1902) (CP 4.227–323), with particular reference to the section that treats of "Trichotomic Mathematics" (CP 4.307–323).
By the way, we will meet with the symmetric group <math>S_3~\!</math> again when we return to take up the study of Peirce's early paper "On a Class of Multiple Algebras" (CP 3.324–327), and also his late unpublished work "The Simplest Mathematics" (1902) (CP 4.227–323), with particular reference to the section that treats of "Trichotomic Mathematics" (CP 4.307–323).
==Note 20==
==Note 20==
By way of collecting a short-term pay-off for all the work — not to mention all the peirce-spiration — that we sweated out over the regular representations of the Klein 4-group <math>V_4,\!</math> let us write out as quickly as possible in ''relative form'' a minimal budget of representations of the symmetric group on three letters, <math>S_3 = \operatorname{Sym}(3).</math> After doing the usual bit of compare and contrast among these divers representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscur'd details of Peirce's early "Algebra + Logic" papers.
By way of collecting a short-term pay-off for all the work — not to mention all the peirce-spiration — that we sweated out over the regular representations of the Klein 4-group <math>V_4,\!</math> let us write out as quickly as possible in ''relative form'' a minimal budget of representations of the symmetric group on three letters, <math>S_3 = \mathrm{Sym}(3).\!</math> After doing the usual bit of compare and contrast among these divers representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscur'd details of Peirce's early "Algebra + Logic" papers.
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>
|+ <math>\text{Permutation Substitutions in}~ \mathrm{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="16%" | <math>\operatorname{e}</math>
| width="16%" | <math>\mathrm{e}\!</math>
| width="16%" | <math>\operatorname{f}</math>
| width="16%" | <math>\mathrm{f}\!</math>
| width="16%" | <math>\operatorname{g}</math>
| width="16%" | <math>\mathrm{g}\!</math>
| width="16%" | <math>\operatorname{h}</math>
| width="16%" | <math>\mathrm{h}\!</math>
| width="16%" | <math>\operatorname{i}</math>
| width="16%" | <math>\mathrm{i}~\!</math>
| width="16%" | <math>\operatorname{j}</math>
| width="16%" | <math>\mathrm{j}\!</math>
|-
|-
|
|
\\[6pt]
\\[6pt]
\mathrm{A} & \mathrm{B} & \mathrm{C}
\mathrm{A} & \mathrm{B} & \mathrm{C}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{C} & \mathrm{A} & \mathrm{B}
\mathrm{C} & \mathrm{A} & \mathrm{B}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{B} & \mathrm{C} & \mathrm{A}
\mathrm{B} & \mathrm{C} & \mathrm{A}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{A} & \mathrm{C} & \mathrm{B}
\mathrm{A} & \mathrm{C} & \mathrm{B}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{C} & \mathrm{B} & \mathrm{A}
\mathrm{C} & \mathrm{B} & \mathrm{A}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{B} & \mathrm{A} & \mathrm{C}
\mathrm{B} & \mathrm{A} & \mathrm{C}
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
| align="center" |
| align="center" |
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{e}
\mathrm{e}
& = & \operatorname{A}\!:\!\operatorname{A}
& = & \mathrm{A}\!:\!\mathrm{A}
& + & \operatorname{B}\!:\!\operatorname{B}
& + & \mathrm{B}\!:\!\mathrm{B}
& + & \operatorname{C}\!:\!\operatorname{C}
& + & \mathrm{C}\!:\!\mathrm{C}
\\[4pt]
\\[4pt]
\operatorname{f}
\mathrm{f}
& = & \operatorname{A}\!:\!\operatorname{C}
& = & \mathrm{A}\!:\!\mathrm{C}
& + & \operatorname{B}\!:\!\operatorname{A}
& + & \mathrm{B}\!:\!\mathrm{A}
& + & \operatorname{C}\!:\!\operatorname{B}
& + & \mathrm{C}\!:\!\mathrm{B}
\\[4pt]
\\[4pt]
\operatorname{g}
\mathrm{g}
& = & \operatorname{A}\!:\!\operatorname{B}
& = & \mathrm{A}\!:\!\mathrm{B}
& + & \operatorname{B}\!:\!\operatorname{C}
& + & \mathrm{B}\!:\!\mathrm{C}
& + & \operatorname{C}\!:\!\operatorname{A}
& + & \mathrm{C}\!:\!\mathrm{A}
\\[4pt]
\\[4pt]
\operatorname{h}
\mathrm{h}
& = & \operatorname{A}\!:\!\operatorname{A}
& = & \mathrm{A}\!:\!\mathrm{A}
& + & \operatorname{B}\!:\!\operatorname{C}
& + & \mathrm{B}\!:\!\mathrm{C}
& + & \operatorname{C}\!:\!\operatorname{B}
& + & \mathrm{C}\!:\!\mathrm{B}
\\[4pt]
\\[4pt]
\operatorname{i}
\mathrm{i}
& = & \operatorname{A}\!:\!\operatorname{C}
& = & \mathrm{A}\!:\!\mathrm{C}
& + & \operatorname{B}\!:\!\operatorname{B}
& + & \mathrm{B}\!:\!\mathrm{B}
& + & \operatorname{C}\!:\!\operatorname{A}
& + & \mathrm{C}\!:\!\mathrm{A}
\\[4pt]
\\[4pt]
\operatorname{j}
\mathrm{j}
& = & \operatorname{A}\!:\!\operatorname{B}
& = & \mathrm{A}\!:\!\mathrm{B}
& + & \operatorname{B}\!:\!\operatorname{A}
& + & \mathrm{B}\!:\!\mathrm{A}
& + & \operatorname{C}\!:\!\operatorname{C}
& + & \mathrm{C}\!:\!\mathrm{C}
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
I have without stopping to think about it written out this natural representation of <math>S_3\!</math> in the style that comes most naturally to me, to wit, the "right" way, whereby an ordered pair configured as <math>X:Y\!</math> constitutes the turning of <math>X\!</math> into <math>Y.\!</math> It is possible that the next time we check in with CSP that we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.
I have without stopping to think about it written out this natural representation of <math>S_3~\!</math> in the style that comes most naturally to me, to wit, the "right" way, whereby an ordered pair configured as <math>X:Y\!</math> constitutes the turning of <math>X\!</math> into <math>Y.\!</math> It is possible that the next time we check in with CSP that we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.
==Note 21==
==Note 21==
To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table:
To construct the regular representations of <math>S_3,\!</math> we begin with the data of its operation table:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
|
<pre>
<pre>
Symmetric Group S_3
Symmetric Group S_3
Just by way of staying clear about what we are doing, let's return to the recipe that we worked out before:
Just by way of staying clear about what we are doing, let's return to the recipe that we worked out before:
It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3</math> is actually a function <math>L : G \times G \to G.</math> It is from this functional perspective that we can see an easy way to derive the two regular representations.
It is part of the definition of a group that the 3-adic relation <math>L \subseteq G^3\!</math> is actually a function <math>L : G \times G \to G.\!</math> It is from this functional perspective that we can see an easy way to derive the two regular representations.
Since we have a function of the type <math>L : G \times G \to G,</math> we can define a couple of substitution operators:
Since we have a function of the type <math>L : G \times G \to G,\!</math> we can define a couple of substitution operators:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| valign="top" | 1.
| valign="top" | 1.
| <math>\operatorname{Sub}(x, (\underline{~~}, y))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>x \cdot y.</math>
| <math>\mathrm{Sub}(x, (\underline{~~}, y))\!</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(\underline{~~}, y),\!</math> with the effect of producing the saturated rheme <math>(x, y)\!</math> that evaluates to <math>x \cdot y.\!</math>
|-
|-
| valign="top" | 2.
| valign="top" | 2.
| <math>\operatorname{Sub}(x, (y, \underline{~~}))</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>y \cdot x.</math>
| <math>\mathrm{Sub}(x, (y, \underline{~~}))\!</math> puts any specified <math>x\!</math> into the empty slot of the rheme <math>(y, \underline{~~}),\!</math> with the effect of producing the saturated rheme <math>(y, x)\!</math> that evaluates to <math>y \cdot x.\!</math>
|}
|}
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)</math> into <math>x \cdot y,</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : x \cdot y) ~|~ y \in G \}.</math> The pairs <math>(y : x \cdot y)</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the right margin. This produces the ''regular ante-representation'' of <math>S_3,\!</math> like so:
In (1) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(\underline{~~}, y),\!</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(\underline{~~}, y)\!</math> into <math>x \cdot y,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : x \cdot y) ~|~ y \in G \}.\!</math> The pairs <math>(y : x \cdot y)\!</math> can be found by picking an <math>x\!</math> from the left margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the right margin. This produces the ''regular ante-representation'' of <math>S_3,\!</math> like so:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
|
<math>\begin{array}{*{13}{c}}
<math>\begin{array}{*{13}{c}}
\operatorname{e}
\mathrm{e}
& = & \operatorname{e}:\operatorname{e}
& = & \mathrm{e}:\mathrm{e}
& + & \operatorname{f}:\operatorname{f}
& + & \mathrm{f}:\mathrm{f}
& + & \operatorname{g}:\operatorname{g}
& + & \mathrm{g}:\mathrm{g}
& + & \operatorname{h}:\operatorname{h}
& + & \mathrm{h}:\mathrm{h}
& + & \operatorname{i}:\operatorname{i}
& + & \mathrm{i}:\mathrm{i}
& + & \operatorname{j}:\operatorname{j}
& + & \mathrm{j}:\mathrm{j}
\\[4pt]
\\[4pt]
\operatorname{f}
\mathrm{f}
& = & \operatorname{e}:\operatorname{f}
& = & \mathrm{e}:\mathrm{f}
& + & \operatorname{f}:\operatorname{g}
& + & \mathrm{f}:\mathrm{g}
& + & \operatorname{g}:\operatorname{e}
& + & \mathrm{g}:\mathrm{e}
& + & \operatorname{h}:\operatorname{j}
& + & \mathrm{h}:\mathrm{j}
& + & \operatorname{i}:\operatorname{h}
& + & \mathrm{i}:\mathrm{h}
& + & \operatorname{j}:\operatorname{i}
& + & \mathrm{j}:\mathrm{i}
\\[4pt]
\\[4pt]
\operatorname{g}
\mathrm{g}
& = & \operatorname{e}:\operatorname{g}
& = & \mathrm{e}:\mathrm{g}
& + & \operatorname{f}:\operatorname{e}
& + & \mathrm{f}:\mathrm{e}
& + & \operatorname{g}:\operatorname{f}
& + & \mathrm{g}:\mathrm{f}
& + & \operatorname{h}:\operatorname{i}
& + & \mathrm{h}:\mathrm{i}
& + & \operatorname{i}:\operatorname{j}
& + & \mathrm{i}:\mathrm{j}
& + & \operatorname{j}:\operatorname{h}
& + & \mathrm{j}:\mathrm{h}
\\[4pt]
\\[4pt]
\operatorname{h}
\mathrm{h}
& = & \operatorname{e}:\operatorname{h}
& = & \mathrm{e}:\mathrm{h}
& + & \operatorname{f}:\operatorname{i}
& + & \mathrm{f}:\mathrm{i}
& + & \operatorname{g}:\operatorname{j}
& + & \mathrm{g}:\mathrm{j}
& + & \operatorname{h}:\operatorname{e}
& + & \mathrm{h}:\mathrm{e}
& + & \operatorname{i}:\operatorname{f}
& + & \mathrm{i}:\mathrm{f}
& + & \operatorname{j}:\operatorname{g}
& + & \mathrm{j}:\mathrm{g}
\\[4pt]
\\[4pt]
\operatorname{i}
\mathrm{i}
& = & \operatorname{e}:\operatorname{i}
& = & \mathrm{e}:\mathrm{i}
& + & \operatorname{f}:\operatorname{j}
& + & \mathrm{f}:\mathrm{j}
& + & \operatorname{g}:\operatorname{h}
& + & \mathrm{g}:\mathrm{h}
& + & \operatorname{h}:\operatorname{g}
& + & \mathrm{h}:\mathrm{g}
& + & \operatorname{i}:\operatorname{e}
& + & \mathrm{i}:\mathrm{e}
& + & \operatorname{j}:\operatorname{f}
& + & \mathrm{j}:\mathrm{f}
\\[4pt]
\\[4pt]
\operatorname{j}
\mathrm{j}
& = & \operatorname{e}:\operatorname{j}
& = & \mathrm{e}:\mathrm{j}
& + & \operatorname{f}:\operatorname{h}
& + & \mathrm{f}:\mathrm{h}
& + & \operatorname{g}:\operatorname{i}
& + & \mathrm{g}:\mathrm{i}
& + & \operatorname{h}:\operatorname{f}
& + & \mathrm{h}:\mathrm{f}
& + & \operatorname{i}:\operatorname{g}
& + & \mathrm{i}:\mathrm{g}
& + & \operatorname{j}:\operatorname{e}
& + & \mathrm{j}:\mathrm{e}
\end{array}</math>
\end{array}\!</math>
|}
|}
In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})</math> into <math>y \cdot x,</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : y \cdot x) ~|~ y \in G \}.</math> The pairs <math>(y : y \cdot x)</math> can be found by picking an <math>x\!</math> on the right margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the left margin. This generates the ''regular post-representation'' of <math>S_3,\!</math> like so:
In (2) we consider the effects of each <math>x\!</math> in its practical bearing on contexts of the form <math>(y, \underline{~~}),\!</math> as <math>y\!</math> ranges over <math>G,\!</math> and the effects are such that <math>x\!</math> takes <math>(y, \underline{~~})\!</math> into <math>y \cdot x,\!</math> for <math>y\!</math> in <math>G,\!</math> all of which is notated as <math>x = \{ (y : y \cdot x) ~|~ y \in G \}.\!</math> The pairs <math>(y : y \cdot x)~\!</math> can be found by picking an <math>x\!</math> on the right margin of the group operation table and considering its effects on each <math>y\!</math> in turn as these run along the left margin. This generates the ''regular post-representation'' of <math>S_3,\!</math> like so:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
|
<math>\begin{array}{*{13}{c}}
<math>\begin{array}{*{13}{c}}
\operatorname{e}
\mathrm{e}
& = & \operatorname{e}:\operatorname{e}
& = & \mathrm{e}:\mathrm{e}
& + & \operatorname{f}:\operatorname{f}
& + & \mathrm{f}:\mathrm{f}
& + & \operatorname{g}:\operatorname{g}
& + & \mathrm{g}:\mathrm{g}
& + & \operatorname{h}:\operatorname{h}
& + & \mathrm{h}:\mathrm{h}
& + & \operatorname{i}:\operatorname{i}
& + & \mathrm{i}:\mathrm{i}
& + & \operatorname{j}:\operatorname{j}
& + & \mathrm{j}:\mathrm{j}
\\[4pt]
\\[4pt]
\operatorname{f}
\mathrm{f}
& = & \operatorname{e}:\operatorname{f}
& = & \mathrm{e}:\mathrm{f}
& + & \operatorname{f}:\operatorname{g}
& + & \mathrm{f}:\mathrm{g}
& + & \operatorname{g}:\operatorname{e}
& + & \mathrm{g}:\mathrm{e}
& + & \operatorname{h}:\operatorname{i}
& + & \mathrm{h}:\mathrm{i}
& + & \operatorname{i}:\operatorname{j}
& + & \mathrm{i}:\mathrm{j}
& + & \operatorname{j}:\operatorname{h}
& + & \mathrm{j}:\mathrm{h}
\\[4pt]
\\[4pt]
\operatorname{g}
\mathrm{g}
& = & \operatorname{e}:\operatorname{g}
& = & \mathrm{e}:\mathrm{g}
& + & \operatorname{f}:\operatorname{e}
& + & \mathrm{f}:\mathrm{e}
& + & \operatorname{g}:\operatorname{f}
& + & \mathrm{g}:\mathrm{f}
& + & \operatorname{h}:\operatorname{j}
& + & \mathrm{h}:\mathrm{j}
& + & \operatorname{i}:\operatorname{h}
& + & \mathrm{i}:\mathrm{h}
& + & \operatorname{j}:\operatorname{i}
& + & \mathrm{j}:\mathrm{i}
\\[4pt]
\\[4pt]
\operatorname{h}
\mathrm{h}
& = & \operatorname{e}:\operatorname{h}
& = & \mathrm{e}:\mathrm{h}
& + & \operatorname{f}:\operatorname{j}
& + & \mathrm{f}:\mathrm{j}
& + & \operatorname{g}:\operatorname{i}
& + & \mathrm{g}:\mathrm{i}
& + & \operatorname{h}:\operatorname{e}
& + & \mathrm{h}:\mathrm{e}
& + & \operatorname{i}:\operatorname{g}
& + & \mathrm{i}:\mathrm{g}
& + & \operatorname{j}:\operatorname{f}
& + & \mathrm{j}:\mathrm{f}
\\[4pt]
\\[4pt]
\operatorname{i}
\mathrm{i}
& = & \operatorname{e}:\operatorname{i}
& = & \mathrm{e}:\mathrm{i}
& + & \operatorname{f}:\operatorname{h}
& + & \mathrm{f}:\mathrm{h}
& + & \operatorname{g}:\operatorname{j}
& + & \mathrm{g}:\mathrm{j}
& + & \operatorname{h}:\operatorname{f}
& + & \mathrm{h}:\mathrm{f}
& + & \operatorname{i}:\operatorname{e}
& + & \mathrm{i}:\mathrm{e}
& + & \operatorname{j}:\operatorname{g}
& + & \mathrm{j}:\mathrm{g}
\\[4pt]
\\[4pt]
\operatorname{j}
\mathrm{j}
& = & \operatorname{e}:\operatorname{j}
& = & \mathrm{e}:\mathrm{j}
& + & \operatorname{f}:\operatorname{i}
& + & \mathrm{f}:\mathrm{i}
& + & \operatorname{g}:\operatorname{h}
& + & \mathrm{g}:\mathrm{h}
& + & \operatorname{h}:\operatorname{g}
& + & \mathrm{h}:\mathrm{g}
& + & \operatorname{i}:\operatorname{f}
& + & \mathrm{i}:\mathrm{f}
& + & \operatorname{j}:\operatorname{e}
& + & \mathrm{j}:\mathrm{e}
\end{array}</math>
\end{array}\!</math>
|}
|}
If the ante-rep looks different from the post-rep, it is just as it should be, as <math>S_3\!</math> is non-abelian (non-commutative), and so the two representations differ in the details of their practical effects, though, of course, being representations of the same abstract group, they must be isomorphic.
If the ante-rep looks different from the post-rep, it is just as it should be, as <math>S_3~\!</math> is non-abelian (non-commutative), and so the two representations differ in the details of their practical effects, though, of course, being representations of the same abstract group, they must be isomorphic.
==Note 22==
==Note 22==
|}
|}
You may be wondering what happened to the announced subject of ''Differential Logic'', and if you think that we have been taking a slight ''excursion'' — to use my favorite euphemism for ''digression'' — my reply to the charge of a scenic rout would need to be both "yes and no". What happened was this. At the sign-post marked by Sigil 7, we made the observation that the shift operators <math>\operatorname{E}_{ij}\!</math> form a transformation group that acts on the propositions of the form <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> Now group theory is a very attractive subject, but it did not really have the effect of drawing us so far off our initial course as you may at first think. For one thing, groups, in particular, the groups that have come to be named after the Norwegian mathematician Marius Sophus Lie, have turned out to be of critical utility in the solution of differential equations. For another thing, group operations afford us examples of triadic relations that have been extremely well-studied over the years, and this provides us with quite a bit of guidance in the study of sign relations, another class of triadic relations of significance for logical studies, in our brief acquaintance with which we have scarcely even started to break the ice. Finally, I could hardly avoid taking up the connection between group representations, a very generic class of logical models, and the all-important pragmatic maxim.
You may be wondering what happened to the announced subject of ''Differential Logic'', and if you think that we have been taking a slight ''excursion'' — to use my favorite euphemism for ''digression'' — my reply to the charge of a scenic rout would need to be both "yes and no". What happened was this. At the sign-post marked by Sigil 7, we made the observation that the shift operators <math>\mathrm{E}_{ij}\!</math> form a transformation group that acts on the propositions of the form <math>f : \mathbb{B}^2 \to \mathbb{B}.\!</math> Now group theory is a very attractive subject, but it did not really have the effect of drawing us so far off our initial course as you may at first think. For one thing, groups, in particular, the groups that have come to be named after the Norwegian mathematician Marius Sophus Lie, have turned out to be of critical utility in the solution of differential equations. For another thing, group operations afford us examples of triadic relations that have been extremely well-studied over the years, and this provides us with quite a bit of guidance in the study of sign relations, another class of triadic relations of significance for logical studies, in our brief acquaintance with which we have scarcely even started to break the ice. Finally, I could hardly avoid taking up the connection between group representations, a very generic class of logical models, and the all-important pragmatic maxim.
* [http://www-history.mcs.st-andrews.ac.uk/Biographies/Lie.html Biographical Data for Marius Sophus Lie (1842–1899)]
* [http://www-history.mcs.st-andrews.ac.uk/Biographies/Lie.html Biographical Data for Marius Sophus Lie (1842–1899)]
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Permutation Substitutions in}~ \operatorname{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}</math>
|+ <math>\text{Permutation Substitutions in}~ \mathrm{Sym} \{ \mathrm{A}, \mathrm{B}, \mathrm{C} \}\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="16%" | <math>\operatorname{e}</math>
| width="16%" | <math>\mathrm{e}\!</math>
| width="16%" | <math>\operatorname{f}</math>
| width="16%" | <math>\mathrm{f}\!</math>
| width="16%" | <math>\operatorname{g}</math>
| width="16%" | <math>\mathrm{g}\!</math>
| width="16%" | <math>\operatorname{h}</math>
| width="16%" | <math>\mathrm{h}\!</math>
| width="16%" | <math>\operatorname{i}</math>
| width="16%" | <math>\mathrm{i}~\!</math>
| width="16%" | <math>\operatorname{j}</math>
| width="16%" | <math>\mathrm{j}\!</math>
|-
|-
|
|
\\[6pt]
\\[6pt]
\mathrm{A} & \mathrm{B} & \mathrm{C}
\mathrm{A} & \mathrm{B} & \mathrm{C}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{C} & \mathrm{A} & \mathrm{B}
\mathrm{C} & \mathrm{A} & \mathrm{B}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{B} & \mathrm{C} & \mathrm{A}
\mathrm{B} & \mathrm{C} & \mathrm{A}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{A} & \mathrm{C} & \mathrm{B}
\mathrm{A} & \mathrm{C} & \mathrm{B}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{C} & \mathrm{B} & \mathrm{A}
\mathrm{C} & \mathrm{B} & \mathrm{A}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[6pt]
\\[6pt]
\mathrm{B} & \mathrm{A} & \mathrm{C}
\mathrm{B} & \mathrm{A} & \mathrm{C}
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
<br>
<br>
Then we rewrote these permutations — being functions <math>f : X \to X</math> they can also be recognized as being 2-adic relations <math>f \subseteq X \times X</math> — in ''relative form'', in effect, in the manner to which Peirce would have made us accustomed had he been given a relative half-a-chance:
Then we rewrote these permutations — being functions <math>f : X \to X\!</math> they can also be recognized as being 2-adic relations <math>f \subseteq X \times X\!</math> — in ''relative form'', in effect, in the manner to which Peirce would have made us accustomed had he been given a relative half-a-chance:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{e}
\mathrm{e}
& = & \mathrm{A}\!:\!\mathrm{A}
& = & \mathrm{A}\!:\!\mathrm{A}
& + & \mathrm{B}\!:\!\mathrm{B}
& + & \mathrm{B}\!:\!\mathrm{B}
& + & \mathrm{C}\!:\!\mathrm{C}
& + & \mathrm{C}\!:\!\mathrm{C}
\\[4pt]
\\[4pt]
\operatorname{f}
\mathrm{f}
& = & \mathrm{A}\!:\!\mathrm{C}
& = & \mathrm{A}\!:\!\mathrm{C}
& + & \mathrm{B}\!:\!\mathrm{A}
& + & \mathrm{B}\!:\!\mathrm{A}
& + & \mathrm{C}\!:\!\mathrm{B}
& + & \mathrm{C}\!:\!\mathrm{B}
\\[4pt]
\\[4pt]
\operatorname{g}
\mathrm{g}
& = & \mathrm{A}\!:\!\mathrm{B}
& = & \mathrm{A}\!:\!\mathrm{B}
& + & \mathrm{B}\!:\!\mathrm{C}
& + & \mathrm{B}\!:\!\mathrm{C}
& + & \mathrm{C}\!:\!\mathrm{A}
& + & \mathrm{C}\!:\!\mathrm{A}
\\[4pt]
\\[4pt]
\operatorname{h}
\mathrm{h}
& = & \mathrm{A}\!:\!\mathrm{A}
& = & \mathrm{A}\!:\!\mathrm{A}
& + & \mathrm{B}\!:\!\mathrm{C}
& + & \mathrm{B}\!:\!\mathrm{C}
& + & \mathrm{C}\!:\!\mathrm{B}
& + & \mathrm{C}\!:\!\mathrm{B}
\\[4pt]
\\[4pt]
\operatorname{i}
\mathrm{i}
& = & \mathrm{A}\!:\!\mathrm{C}
& = & \mathrm{A}\!:\!\mathrm{C}
& + & \mathrm{B}\!:\!\mathrm{B}
& + & \mathrm{B}\!:\!\mathrm{B}
& + & \mathrm{C}\!:\!\mathrm{A}
& + & \mathrm{C}\!:\!\mathrm{A}
\\[4pt]
\\[4pt]
\operatorname{j}
\mathrm{j}
& = & \mathrm{A}\!:\!\mathrm{B}
& = & \mathrm{A}\!:\!\mathrm{B}
& + & \mathrm{B}\!:\!\mathrm{A}
& + & \mathrm{B}\!:\!\mathrm{A}
& + & \mathrm{C}\!:\!\mathrm{C}
& + & \mathrm{C}\!:\!\mathrm{C}
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
These days one is much more likely to encounter the natural representation of <math>S_3\!</math> in the form of a ''linear representation'', that is, as a family of linear transformations that map the elements of a suitable vector space into each other, all of which would in turn usually be represented by a set of matrices like these:
These days one is much more likely to encounter the natural representation of <math>S_3~\!</math> in the form of a ''linear representation'', that is, as a family of linear transformations that map the elements of a suitable vector space into each other, all of which would in turn usually be represented by a set of matrices like these:
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Matrix Representations of Permutations in}~ \operatorname{Sym}(3)</math>
|+ <math>\text{Matrix Representations of Permutations in}~ \mathrm{Sym}(3)\!</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="16%" | <math>\operatorname{e}</math>
| width="16%" | <math>\mathrm{e}\!</math>
| width="16%" | <math>\operatorname{f}</math>
| width="16%" | <math>\mathrm{f}\!</math>
| width="16%" | <math>\operatorname{g}</math>
| width="16%" | <math>\mathrm{g}\!</math>
| width="16%" | <math>\operatorname{h}</math>
| width="16%" | <math>\mathrm{h}\!</math>
| width="16%" | <math>\operatorname{i}</math>
| width="16%" | <math>\mathrm{i}~\!</math>
| width="16%" | <math>\operatorname{j}</math>
| width="16%" | <math>\mathrm{j}\!</math>
|-
|-
|
|
\\
\\
0 & 0 & 1
0 & 0 & 1
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\
\\
0 & 1 & 0
0 & 1 & 0
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\
\\
1 & 0 & 0
1 & 0 & 0
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\
\\
0 & 1 & 0
0 & 1 & 0
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\
\\
1 & 0 & 0
1 & 0 & 0
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\
\\
0 & 0 & 1
0 & 0 & 1
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
The key to the mysteries of these matrices is revealed by noting that their coefficient entries are arrayed and overlaid on a place-mat marked like so:
The key to the mysteries of these matrices is revealed by noting that their coefficient entries are arrayed and overlaid on a place-mat marked like so:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
|
<math>\begin{bmatrix}
<math>\begin{bmatrix}
\mathrm{A}\!:\!\mathrm{A} &
\mathrm{A}\!:\!\mathrm{A} &
\mathrm{C}\!:\!\mathrm{B} &
\mathrm{C}\!:\!\mathrm{B} &
\mathrm{C}\!:\!\mathrm{C}
\mathrm{C}\!:\!\mathrm{C}
\end{bmatrix}</math>
\end{bmatrix}\!</math>
|}
|}
Here's one picture of how it begins, one angle on the point of departure:
Here's one picture of how it begins, one angle on the point of departure:
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
The least disturbance, it being provident and prudent both to take that first up, will arise in just one of three ways, in accord with the mode of discord that importunes on our equanimity, whether it is Expectation, Intention, Observation that incipiently incites the riot, departing as it will from congruence with the other two modes of being.
The least disturbance, it being provident and prudent both to take that first up, will arise in just one of three ways, in accord with the mode of discord that importunes on our equanimity, whether it is Expectation, Intention, Observation that incipiently incites the riot, departing as it will from congruence with the other two modes of being.
In short, we cross just one of the three lines that border on the center, or perhaps it is better to say that the objective situation transits one of the chordal bounds of harmony, for the moment marked as <math>\operatorname{d}_E, \operatorname{d}_I, \operatorname{d}_O</math> to note the fact one's Expectation, Intention, Observation, respectively, is the mode that we duly indite as the one that's sounding the sour note.
In short, we cross just one of the three lines that border on the center, or perhaps it is better to say that the objective situation transits one of the chordal bounds of harmony, for the moment marked as <math>\mathrm{d}_E, \mathrm{d}_I, \mathrm{d}_O\!</math> to note the fact one's Expectation, Intention, Observation, respectively, is the mode that we duly indite as the one that's sounding the sour note.
{| align="center" cellpadding="6" width="90%"
{| align="center" cellspacing="10"
| A difference between Expectation and Observation is experienced as a ''Surprise'', a phenomenon that calls for an Explanation.
| A difference between Expectation and Observation is experienced as a ''Surprise'', a phenomenon that calls for an Explanation.
|-
|-
|}
|}
At any rate, the modes of experiencing a surprising phenomenon or a problematic situation, as described just now, are already complex modalities, and will need to be analyzed further if we want to relate them to the minimal changes <math>\operatorname{d}_E, \operatorname{d}_I, \operatorname{d}_O.</math> Let me think about that for a little while and see what transpires.
At any rate, the modes of experiencing a surprising phenomenon or a problematic situation, as described just now, are already complex modalities, and will need to be analyzed further if we want to relate them to the minimal changes <math>\mathrm{d}_E, \mathrm{d}_I, \mathrm{d}_O.\!</math> Let me think about that for a little while and see what transpires.
==Note 25==
==Note 25==
==Document History==
==Document History==
===Ontology List (Apr–Jul 2002)===
===Differential Logic • Ontology List 2002===
* http://web.archive.org/web/20110612002240/http://suo.ieee.org/ontology/thrd28.html#04040
* http://web.archive.org/web/20110612002240/http://suo.ieee.org/ontology/thrd28.html#04040
# http://web.archive.org/web/20110612002235/http://suo.ieee.org/ontology/msg04290.html
# http://web.archive.org/web/20110612002235/http://suo.ieee.org/ontology/msg04290.html
===Inquiry List (May–Jul 2004)===
===Dynamics And Logic • Inquiry List 2004===
* http://stderr.org/pipermail/inquiry/2004-May/thread.html#1400
* http://stderr.org/pipermail/inquiry/2004-May/thread.html#1400
# http://stderr.org/pipermail/inquiry/2004-July/001688.html
# http://stderr.org/pipermail/inquiry/2004-July/001688.html
===NKS Forum (May–Jul 2004)===
===Dynamics And Logic • NKS Forum 2004===
* http://forum.wolframscience.com/archive/topic/420.html
* http://forum.wolframscience.com/archive/topic/420.html
# http://forum.wolframscience.com/showthread.php?postid=1602#post1602
# http://forum.wolframscience.com/showthread.php?postid=1602#post1602
# http://forum.wolframscience.com/showthread.php?postid=1603#post1603
# http://forum.wolframscience.com/showthread.php?postid=1603#post1603
[[Category:Artificial Intelligence]]
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[[Category:Charles Sanders Peirce]]
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[[Category:Equational Reasoning]]
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[[Category:Propositional Calculus]]
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