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Directory:Jon Awbrey/Papers/Differential Logic : Introduction (view source)

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'''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. A definition that broad naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation that pervade or precede quantitative models. To the extent that a logical inquiry makes use of a formal system, its differential component 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 a logical universe of discourse.

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'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''

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'''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. A definition that broad naturally incorporates any study of variation by way of mathematical models, but differential logic is especially charged with the qualitative aspects of variation that pervade or precede quantitative models. To the extent that a logical inquiry makes use of a formal system, its differential component 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 a logical universe of discourse.

A simple example of a differential logical calculus is furnished by a ''[[differential propositional calculus]]''. A differential propositional calculus is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a universe of discourse or transformations that map a source universe into a target universe. This augments ordinary propositional calculus in the same way that the differential calculus of Leibniz and Newton augments the analytic geometry of Descartes.

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|- style="background:#f0f0ff"

| <math>\text{Graph}\!</math>

−

| <math>\text{Expression}\!</math>

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| <math>\text{Expression}~\!</math>

| <math>\text{Interpretation}\!</math>

| <math>\text{Other Notations}\!</math>

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| height="100px" | [[Image:Rooted Node.jpg|20px]]

| <math>~</math>

−

| <math>\~~operatorname~~{true}</math>

+

| <math>\mathrm{true}</math>

| <math>1\!</math>

|-

| height="100px" | [[Image:Rooted Edge.jpg|20px]]

| <math>\texttt{(~)}</math>

−

| <math>\~~operatorname~~{false}</math>

+

| <math>\mathrm{false}</math>

| <math>0\!</math>

|-

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|-

| height="120px" | [[Image:Cactus (A) Big.jpg|20px]]

−

| <math>\texttt{(} a \texttt{)}</math>

+

| <math>\texttt{(} a \texttt{)}\!</math>

−

| <math>\~~operatorname~~{not}~ a</math>

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| <math>\mathrm{not}~ a</math>

| <math>\lnot a \quad \bar{a} \quad \tilde{a} \quad a^\prime</math>

|-

| height="100px" | [[Image:Cactus ABC Big.jpg|50px]]

| <math>a ~ b ~ c</math>

−

| <math>a ~\~~operatorname~~{and}~ b ~\~~operatorname~~{and}~ c</math>

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| <math>a ~\mathrm{and}~ b ~\mathrm{and}~ c</math>

| <math>a \land b \land c</math>

|-

| height="160px" | [[Image:Cactus ((A)(B)(C)) Big.jpg|65px]]

| <math>\texttt{((} a \texttt{)(} b \texttt{)(} c \texttt{))}</math>

−

| <math>a ~\~~operatorname~~{or}~ b ~\~~operatorname~~{or}~ c</math>

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| <math>a ~\mathrm{or}~ b ~\mathrm{or}~ c</math>

| <math>a \lor b \lor c</math>

|-

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|

<math>\begin{matrix}

−

a ~\~~operatorname~~{implies}~ b

+

a ~\mathrm{implies}~ b

\\[6pt]

−

\~~operatorname~~{if}~ a ~\~~operatorname~~{then}~ b

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\mathrm{if}~ a ~\mathrm{then}~ b

\end{matrix}</math>

| <math>a \Rightarrow b</math>

|-

−

| height="120px" | [[Image:Cactus (A,B) Big.jpg|65px]]

+

| height="120px" | [[Image:Cactus (A,B) Big ISW.jpg|65px]]

| <math>\texttt{(} a \texttt{,} b \texttt{)}</math>

|

<math>\begin{matrix}

−

a ~\~~operatorname~~{not~equal~to}~ b

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a ~\mathrm{not~equal~to}~ b

\\[6pt]

−

a ~\~~operatorname~~{exclusive~or}~ b

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a ~\mathrm{exclusive~or}~ b

\end{matrix}</math>

|

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|

<math>\begin{matrix}

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a ~\~~operatorname~~{is~equal~to}~ b

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a ~\mathrm{is~equal~to}~ b

\\[6pt]

−

a ~\~~operatorname~~{if~and~only~if}~ b

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a ~\mathrm{if~and~only~if}~ b

\end{matrix}</math>

|

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|

<math>\begin{matrix}

−

\~~operatorname~~{just~one~of}

+

\mathrm{just~one~of}

\\

a, b, c

\\

−

\~~operatorname~~{is~false}.

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\mathrm{is~false}.

\end{matrix}</math>

|

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|

<math>\begin{matrix}

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\~~operatorname~~{just~one~of}

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\mathrm{just~one~of}

\\

a, b, c

\\

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\~~operatorname~~{is~true}.

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\mathrm{is~true}.

\\[6pt]

−

\~~operatorname~~{partition~all}

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\mathrm{partition~all}

\\

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\~~operatorname~~{into}~ a, b, c.

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\mathrm{into}~ a, b, c.

\end{matrix}</math>

|

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|

<math>\begin{matrix}

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\~~operatorname~~{oddly~many~of}

+

\mathrm{oddly~many~of}

\\

a, b, c

\\

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\~~operatorname~~{are~true}.

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\mathrm{are~true}.

\end{matrix}</math>

|

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|

<math>\begin{matrix}

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\~~operatorname~~{partition}~ x

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\mathrm{partition}~ x

\\

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\~~operatorname~~{into}~ a, b, c.

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\mathrm{into}~ a, b, c.

\\[6pt]

−

\~~operatorname~~{genus}~ x ~\~~operatorname~~{comprises}

+

\mathrm{genus}~ x ~\mathrm{comprises}

\\

−

\~~operatorname~~{species}~ a, b, c.

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\mathrm{species}~ a, b, c.

\end{matrix}</math>

|

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An efficient calculus for the realm of logic represented by boolean functions and elementary propositions makes it feasible to compute the finite differences and the differentials of those functions and propositions.

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For example, consider a proposition of the form <math>{}^{\backprime\backprime} \, p ~\~~operatorname~~{and}~ q \, {}^{\prime\prime}</math> that is graphed as two letters attached to a root node:

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For example, consider a proposition of the form <math>{}^{\backprime\backprime} \, p ~\mathrm{and}~ q \, {}^{\prime\prime}</math> that is graphed as two letters attached to a root node:

{| align="center" cellpadding="10"

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Written as a string, this is just the concatenation <math>p~q</math>.

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The proposition <math>pq\!</math> may be taken as a boolean function <math>f(p, q)\!</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>

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The proposition <math>pq\!</math> may be taken as a boolean function <math>f(p, q)\!</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>

Imagine yourself standing in a fixed cell of the corresponding venn diagram, say, the cell where the proposition <math>pq\!</math> is true, as shown in the following Figure:

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|}

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Now ask yourself: What is the value of the proposition <math>pq\!</math> at a distance of <math>\~~operatorname~~{d}p</math> and <math>\~~operatorname~~{d}q</math> from the cell <math>pq\!</math> where you are standing?

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Now ask yourself: What is the value of the proposition <math>pq\!</math> at a distance of <math>\mathrm{d}p</math> and <math>\mathrm{d}q</math> from the cell <math>pq\!</math> where you are standing?

Don't think about it — just compute:

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|}

−

The cactus formula <math>\texttt{(p, dp)(q, dq)}</math> and its corresponding graph arise by substituting <math>p + \~~operatorname~~{d}p</math> for <math>p\!</math> and <math>q + \~~operatorname~~{d}q</math> for <math>q\!</math> in the boolean product or logical conjunction <math>pq\!</math> and writing the result in the two dialects of cactus syntax. This follows from the fact that the boolean sum <math>p + \~~operatorname~~{d}p</math> is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:

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The cactus formula <math>\texttt{(p, dp)(q, dq)}</math> and its corresponding graph arise by substituting <math>p + \mathrm{d}p</math> for <math>p\!</math> and <math>q + \mathrm{d}q</math> for <math>q\!</math> in the boolean product or logical conjunction <math>pq\!</math> and writing the result in the two dialects of cactus syntax. This follows from the fact that the boolean sum <math>p + \mathrm{d}p</math> is equivalent to the logical operation of exclusive disjunction, which parses to a cactus graph of the following form:

{| align="center" cellpadding="10"

−

| [[Image:Cactus Graph (P,dP).jpg|500px]]

+

| [[Image:Cactus Graph (P,dP) ISW.jpg|500px]]

|}

Next question: What is the difference between the value of the

−

proposition <math>pq\!</math> over there, at a distance of <math>\~~operatorname~~{d}p</math> and <math>\~~operatorname~~{d}q,</math> and the value of the proposition <math>pq\!</math> where you are standing, all expressed in the form of a general formula, of course? Here is the appropriate formulation:

+

proposition <math>pq\!</math> over there, at a distance of <math>\mathrm{d}p</math> and <math>\mathrm{d}q,</math> and the value of the proposition <math>pq\!</math> where you are standing, all expressed in the form of a general formula, of course? Here is the appropriate formulation:

{| align="center" cellpadding="10"

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{| align="center" cellpadding="10"

−

| [[Image:Cactus Graph ((dP)(dQ)).jpg|500px]]

+

| [[Image:Cactus Graph ((dP)(dQ)) ISW.jpg|500px]]

|}

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|

<math>\begin{matrix}

−

p ~\~~operatorname~~{and}~ q

+

p ~\mathrm{and}~ q

& \quad &

−

\xrightarrow{\quad\~~operatorname~~{Diff}\quad}

+

\xrightarrow{\quad\mathrm{Diff}\quad}

& \quad &

−

\~~operatorname~~{d}p ~\~~operatorname~~{or}~ \~~operatorname~~{d}q

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\mathrm{d}p ~\mathrm{or}~ \mathrm{d}q

\end{matrix}</math>

|}

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{| align="center" cellpadding="10"

−

| [[Image:Venn Diagram F = P And Q.jpg|500px]]

+

| [[Image:Venn Diagram F = P And Q ISW.jpg|500px]]

|-

−

| [[Image:Cactus Graph F = P And Q.jpg|500px]]

+

| [[Image:Cactus Graph F = P And Q ISW.jpg|500px]]

|}

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{| align="center" cellpadding="10" width="90%"

−

| Let <math>P\!</math> be the set of values <math>\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \~~operatorname~~{not}~ p,~ p \} ~\cong~ \mathbb{B}.</math>

+

| Let <math>P\!</math> be the set of values <math>\{ \texttt{(} p \texttt{)},~ p \} ~=~ \{ \mathrm{not}~ p,~ p \} ~\cong~ \mathbb{B}.</math>

|-

−

| Let <math>Q\!</math> be the set of values <math>\{ \texttt{(} q \texttt{)},~ q \} ~=~ \{ \~~operatorname~~{not}~ q,~ q \} ~\cong~ \mathbb{B}.</math>

+

| Let <math>Q\!</math> be the set of values <math>\{ \texttt{(} q \texttt{)},~ q \} ~=~ \{ \mathrm{not}~ q,~ q \} ~\cong~ \mathbb{B}.</math>

|}

Then interpret the usual propositions about <math>p, q\!</math> as functions of the concrete type <math>f : P \times Q \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 the pair that play a founding role in the classical finite difference calculus, namely:

{| align="center" cellpadding="10" width="90%"

−

| 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''.

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These days, <math>\mathrm{E}</math> is more often called the ''shift operator''.

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In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse. Starting from the initial space <math>X = P \times Q,</math> its ''(first order) differential extension'' <math>\~~operatorname~~{E}X</math> is constructed according to the following specifications:

+

In order to describe the universe in which these operators operate, it is necessary to enlarge the original universe of discourse. Starting from the initial space <math>X = P \times Q,</math> its ''(first order) differential extension'' <math>\mathrm{E}X</math> is constructed according to the following specifications:

{| align="center" cellpadding="10" width="90%"

|

<math>\begin{array}{rcc}

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\~~operatorname~~{E}X & = & X \times \~~operatorname~~{d}X

+

\mathrm{E}X & = & X \times \mathrm{d}X

\end{array}</math>

|}

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P \times Q

\\[4pt]

−

\~~operatorname~~{d}X

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\mathrm{d}X

& = &

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\~~operatorname~~{d}P \times \~~operatorname~~{d}Q

+

\mathrm{d}P \times \mathrm{d}Q

\\[4pt]

−

\~~operatorname~~{d}P

+

\mathrm{d}P

& = &

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\{ \texttt{(} \~~operatorname~~{d}p \texttt{)},~ \~~operatorname~~{d}p \}

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\{ \texttt{(} \mathrm{d}p \texttt{)},~ \mathrm{d}p \}

\\[4pt]

−

\~~operatorname~~{d}Q

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\mathrm{d}Q

& = &

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\{ \texttt{(} \~~operatorname~~{d}q \texttt{)},~ \~~operatorname~~{d}q \}

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\{ \texttt{(} \mathrm{d}q \texttt{)},~ \mathrm{d}q \}

\end{array}</math>

|}

The interpretations of these new symbols can be diverse, but the easiest

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option for now is just to say that <math>\~~operatorname~~{d}p</math> means “change <math>p\!</math>” and <math>\~~operatorname~~{d}q</math> means “change <math>q\!</math>”.

+

option for now is just to say that <math>\mathrm{d}p</math> means “change <math>p\!</math>” and <math>\mathrm{d}q</math> means “change <math>q\!</math>”.

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Drawing a venn diagram for the differential extension <math>\~~operatorname~~{E}X = X \times \~~operatorname~~{d}X</math> requires four logical dimensions, <math>P, Q, \~~operatorname~~{d}P, \~~operatorname~~{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\~~operatorname~~{d}p</math> and <math>\~~operatorname~~{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\~~operatorname~~{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\~~operatorname~~{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction.

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Drawing a venn diagram for the differential extension <math>\mathrm{E}X = X \times \mathrm{d}X</math> requires four logical dimensions, <math>P, Q, \mathrm{d}P, \mathrm{d}Q,</math> but it is possible to project a suggestion of what the differential features <math>\mathrm{d}p</math> and <math>\mathrm{d}q</math> are about on the 2-dimensional base space <math>X = P \times Q</math> by drawing arrows that cross the boundaries of the basic circles in the venn diagram for <math>X\!,</math> reading an arrow as <math>\mathrm{d}p</math> if it crosses the boundary between <math>p\!</math> and <math>\texttt{(} p \texttt{)}</math> in either direction and reading an arrow as <math>\mathrm{d}q</math> if it crosses the boundary between <math>q\!</math> and <math>\texttt{(} q \texttt{)}</math> in either direction.

{| align="center" cellpadding="10"

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|}

−

Propositions are formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in the same ways that propositions are formed on ordinary logical variables alone. For example, the proposition <math>\texttt{(} \~~operatorname~~{d}p \texttt{(} \~~operatorname~~{d}q \texttt{))}</math> says the same thing as <math>\~~operatorname~~{d}p \Rightarrow \~~operatorname~~{d}q,</math> in other words, that there is no change in <math>p\!</math> without a change in <math>q.\!</math>

+

Propositions are formed on differential variables, or any combination of ordinary logical variables and differential logical variables, in the same ways that propositions are formed on ordinary logical variables alone. For example, the proposition <math>\texttt{(} \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{))}</math> says the same thing as <math>\mathrm{d}p \Rightarrow \mathrm{d}q,</math> in other words, that there is no change in <math>p\!</math> without a change in <math>q.\!</math>

−

Given the proposition <math>f(p, q)\!</math> over the space <math>X = P \times Q,</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\~~operatorname~~{E}f</math> over the differential extension <math>\~~operatorname~~{E}X</math> that is defined by the

+

Given the proposition <math>f(p, q)\!</math> over the space <math>X = P \times Q,</math> the ''(first order) enlargement'' of <math>f\!</math> is the proposition <math>\mathrm{E}f</math> over the differential extension <math>\mathrm{E}X</math> that is defined by the

following formula:

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|

<math>\begin{matrix}

−

\~~operatorname~~{E}f(p, q, \~~operatorname~~{d}p, \~~operatorname~~{d}q)

+

\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)

& = &

−

f(p + \~~operatorname~~{d}p,~ q + \~~operatorname~~{d}q)

+

f(p + \mathrm{d}p,~ q + \mathrm{d}q)

& = &

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f( \texttt{(} p, \~~operatorname~~{d}p \texttt{)},~ \texttt{(} q, \~~operatorname~~{d}q \texttt{)} )

+

f( \texttt{(} p, \mathrm{d}p \texttt{)},~ \texttt{(} q, \mathrm{d}q \texttt{)} )

\end{matrix}</math>

|}

−

In the example <math>f(p, q) = pq,\!</math> the enlargement <math>\~~operatorname~~{E}f</math> is computed as follows:

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In the example <math>f(p, q) = pq,\!</math> the enlargement <math>\mathrm{E}f</math> is computed as follows:

{| align="center" cellpadding="10" style="text-align:center"

|

<math>\begin{matrix}

−

\~~operatorname~~{E}f(p, q, \~~operatorname~~{d}p, \~~operatorname~~{d}q)

+

\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)

& = &

−

(p + \~~operatorname~~{d}p)(q + \~~operatorname~~{d}q)

+

(p + \mathrm{d}p)(q + \mathrm{d}q)

& = &

−

\texttt{(} p, \~~operatorname~~{d}p \texttt{)(} q, \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} p, \mathrm{d}p \texttt{)(} q, \mathrm{d}q \texttt{)}

\end{matrix}</math>

|-

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|}

−

Given the proposition <math>f(p, q)\!</math> over <math>X = P \times Q,</math> the ''(first order) difference'' of <math>f\!</math> is the proposition <math>\~~operatorname~~{D}f</math> over <math>\~~operatorname~~{E}X</math> that is defined by the formula <math>\~~operatorname~~{D}f = \~~operatorname~~{E}f - f,</math> or, written out in full:

+

Given the proposition <math>f(p, q)\!</math> over <math>X = P \times Q,</math> the ''(first order) difference'' of <math>f\!</math> is the proposition <math>\mathrm{D}f</math> over <math>\mathrm{E}X</math> that is defined by the formula <math>\mathrm{D}f = \mathrm{E}f - f,</math> or, written out in full:

{| align="center" cellpadding="10" style="text-align:center"

|

<math>\begin{matrix}

−

\~~operatorname~~{D}f(p, q, \~~operatorname~~{d}p, \~~operatorname~~{d}q)

+

\mathrm{D}f(p, q, \mathrm{d}p, \mathrm{d}q)

& = &

−

f(p + \~~operatorname~~{d}p,~ q + \~~operatorname~~{d}q) - f(p, q)

+

f(p + \mathrm{d}p,~ q + \mathrm{d}q) - f(p, q)

& = &

−

\texttt{(} f( \texttt{(} p, \~~operatorname~~{d}p \texttt{)},~ \texttt{(} q, \~~operatorname~~{d}q \texttt{)} ),~ f(p, q) \texttt{)}

+

\texttt{(} f( \texttt{(} p, \mathrm{d}p \texttt{)},~ \texttt{(} q, \mathrm{d}q \texttt{)} ),~ f(p, q) \texttt{)}

\end{matrix}</math>

|}

−

In the example <math>f(p, q) = pq,\!</math> the difference <math>\~~operatorname~~{D}f</math> is computed as follows:

+

In the example <math>f(p, q) = pq,\!</math> the difference <math>\mathrm{D}f</math> is computed as follows:

{| align="center" cellpadding="10" style="text-align:center"

|

<math>\begin{matrix}

−

\~~operatorname~~{D}f(p, q, \~~operatorname~~{d}p, \~~operatorname~~{d}q)

+

\mathrm{D}f(p, q, \mathrm{d}p, \mathrm{d}q)

& = &

−

(p + \~~operatorname~~{d}p)(q + \~~operatorname~~{d}q) - pq

+

(p + \mathrm{d}p)(q + \mathrm{d}q) - pq

& = &

−

\texttt{((} p, \~~operatorname~~{d}p \texttt{)(} q, \~~operatorname~~{d}q \texttt{)}, pq \texttt{)}

+

\texttt{((} p, \mathrm{d}p \texttt{)(} q, \mathrm{d}q \texttt{)}, pq \texttt{)}

\end{matrix}</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>pq,\!</math> that is, at the place where <math>p = 1\!</math> and <math>q = 1.\!</math> This evaluation is written in the form <math>\~~operatorname~~{D}f|_{pq}</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>pq,\!</math> that is, at the place where <math>p = 1\!</math> and <math>q = 1.\!</math> This evaluation is written in the form <math>\mathrm{D}f|_{pq}</math> or <math>\mathrm{D}f|_{(1, 1)},</math> and we arrived at the locally applicable law that is stated and illustrated as follows:

{| align="center" cellpadding="10" style="text-align:center"

|

−

<math>f(p, q) ~=~ pq ~=~ p ~\~~operatorname~~{and}~ q \quad \Rightarrow \quad \~~operatorname~~{D}f|_{pq} ~=~ \texttt{((} \~~operatorname~~{dp} \texttt{)(} \~~operatorname~~{d}q \texttt{))} ~=~ \~~operatorname~~{d}p ~\~~operatorname~~{or}~ \~~operatorname~~{d}q</math>

+

<math>f(p, q) ~=~ pq ~=~ p ~\mathrm{and}~ q \quad \Rightarrow \quad \mathrm{D}f|_{pq} ~=~ \texttt{((} \mathrm{dp} \texttt{)(} \mathrm{d}q \texttt{))} ~=~ \mathrm{d}p ~\mathrm{or}~ \mathrm{d}q</math>

|-

| [[Image:Venn Diagram PQ Difference Conj At Conj.jpg|500px]]

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|}

−

The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \~~operatorname~~{d}p \texttt{)(} \~~operatorname~~{d}q \texttt{))}</math> into the following exclusive disjunction:

+

The picture shows the analysis of the inclusive disjunction <math>\texttt{((} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{))}</math> into the following exclusive disjunction:

{| align="center" cellpadding="10" style="text-align:center"

|

<math>\begin{matrix}

−

\~~operatorname~~{d}p ~\texttt{(} \~~operatorname~~{d}q \texttt{)}

+

\mathrm{d}p ~\texttt{(} \mathrm{d}q \texttt{)}

& + &

−

\texttt{(} \~~operatorname~~{d}p \texttt{)}~ \~~operatorname~~{d}q

+

\texttt{(} \mathrm{d}p \texttt{)}~ \mathrm{d}q

& + &

−

\~~operatorname~~{d}p ~\~~operatorname~~{d}q

+

\mathrm{d}p ~\mathrm{d}q

\end{matrix}</math>

|}

Line 459: Line 461:

The differential proposition that results may be interpreted to say “change <math>p\!</math> or change <math>q\!</math> or both”. And this can be recognized as just what you need to do if you happen to find yourself in the center cell and require a complete and detailed description of ways to escape it.

−

Last time we computed what is variously called the ''difference map'', the ''difference proposition'', or the ''local proposition'' <math>\~~operatorname~~{D}f_x</math> of the proposition <math>f(p, q) = pq\!</math> at the point <math>x\!</math> where <math>p = 1\!</math> and <math>q = 1.\!</math>

+

Last time we computed what is variously called the ''difference map'', the ''difference proposition'', or the ''local proposition'' <math>\mathrm{D}f_x</math> of the proposition <math>f(p, q) = pq\!</math> at the point <math>x\!</math> where <math>p = 1\!</math> and <math>q = 1.\!</math>

In the universe <math>X = P \times Q,</math> the four propositions <math>pq,~ p \texttt{(} q \texttt{)},~ \texttt{(} p \texttt{)} q,~ \texttt{(} p \texttt{)(} q \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_x = \~~operatorname~~{D}f|x = \~~operatorname~~{D}f|(1, 1) = \~~operatorname~~{D}f|pq,</math> so long as we know the frame of reference in force.

+

Thus we can write <math>\mathrm{D}f_x = \mathrm{D}f|x = \mathrm{D}f|(1, 1) = \mathrm{D}f|pq,</math> so long as we know the frame of reference in force.

−

In the example <math>f(p, q) = pq,\!</math> the value of the difference proposition <math>\~~operatorname~~{D}f_x</math> at each of the four points in <math>x \in X\!</math> may be computed in graphical fashion as shown below:

+

In the example <math>f(p, q) = pq,\!</math> the value of the difference proposition <math>\mathrm{D}f_x</math> at each of the four points in <math>x \in X\!</math> may be computed in graphical fashion as shown below:

{| align="center" cellpadding="10"

Line 474: Line 476:

| [[Image:Cactus Graph Df@P(Q) = (dP)dQ.jpg|500px]]

|-

−

| [[Image:Cactus Graph Df@(P)Q = dP(dQ).jpg|500px]]

+

| [[Image:Cactus Graph Df@(P)Q = dP(dQ) ISW Alt.jpg|500px]]

|-

| [[Image:Cactus Graph Df@(P)(Q) = dP dQ.jpg|500px]]

Line 493: Line 495:

|}

−

The Figure shows the points of the extended universe <math>\~~operatorname~~{E}X = P \times Q \times \~~operatorname~~{d}P \times \~~operatorname~~{d}Q</math> that are indicated by the difference map <math>\~~operatorname~~{D}f : \~~operatorname~~{E}X \to \mathbb{B},</math> namely, the following six points or singular propositions::

+

The Figure shows the points of the extended universe <math>\mathrm{E}X = P \times Q \times \mathrm{d}P \times \mathrm{d}Q</math> that are indicated by the difference map <math>\mathrm{D}f : \mathrm{E}X \to \mathbb{B},</math> namely, the following six points or singular propositions::

{| align="center" cellpadding="10"

|

<math>\begin{array}{rcccc}

−

1. & p & q & \~~operatorname~~{d}p & \~~operatorname~~{d}q

+

1. & p & q & \mathrm{d}p & \mathrm{d}q

\\

−

2. & p & q & \~~operatorname~~{d}p & (\~~operatorname~~{d}q)

+

2. & p & q & \mathrm{d}p & (\mathrm{d}q)

\\

−

3. & p & q & (\~~operatorname~~{d}p) & \~~operatorname~~{d}q

+

3. & p & q & (\mathrm{d}p) & \mathrm{d}q

\\

−

4. & p & (q) & (\~~operatorname~~{d}p) & \~~operatorname~~{d}q

+

4. & p & (q) & (\mathrm{d}p) & \mathrm{d}q

\\

−

5. & (p) & q & \~~operatorname~~{d}p & (\~~operatorname~~{d}q)

+

5. & (p) & q & \mathrm{d}p & (\mathrm{d}q)

\\

−

6. & (p) & (q) & \~~operatorname~~{d}p & \~~operatorname~~{d}q

+

6. & (p) & (q) & \mathrm{d}p & \mathrm{d}q

\end{array}</math>

|}

−

The information borne by <math>\~~operatorname~~{D}f</math> should be clear enough from a survey of these six points — they tell you what you have to do from each point of <math>X\!</math> in order to change the value borne by <math>f(p, q),\!</math> that is, the move you have to make in order to reach a point where the value of the proposition <math>f(p, q)\!</math> is different from what it is where you started.

+

The information borne by <math>\mathrm{D}f</math> should be clear enough from a survey of these six points — they tell you what you have to do from each point of <math>X\!</math> in order to change the value borne by <math>f(p, q),\!</math> that is, the move you have to make in order to reach a point where the value of the proposition <math>f(p, q)\!</math> is different from what it is where you started.

−

We have been studying the action of the difference operator <math>\~~operatorname~~{D}</math> on propositions of the form <math>f : P \times Q \to \mathbb{B},</math> as illustrated by the example <math>f(p, q) = pq\!</math> that is known in logic as the conjunction of <math>p\!</math> and <math>q.\!</math> The resulting difference map <math>\~~operatorname~~{D}f</math> is a (first order) differential proposition, that is, a proposition of the form <math>\~~operatorname~~{D}f : P \times Q \times \~~operatorname~~{d}P \times \~~operatorname~~{d}Q \to \mathbb{B}.</math>

+

We have been studying the action of the difference operator <math>\mathrm{D}</math> on propositions of the form <math>f : P \times Q \to \mathbb{B},</math> as illustrated by the example <math>f(p, q) = pq\!</math> that is known in logic as the conjunction of <math>p\!</math> and <math>q.\!</math> The resulting difference map <math>\mathrm{D}f</math> is a (first order) differential proposition, that is, a proposition of the form <math>\mathrm{D}f : P \times Q \times \mathrm{d}P \times \mathrm{d}Q \to \mathbb{B}.</math>

−

Abstracting from the augmented venn diagram that shows how the ''models'' or ''satisfying interpretations'' of <math>\~~operatorname~~{D}f</math> distribute over the extended universe of discourse <math>\~~operatorname~~{E}X = P \times Q \times \~~operatorname~~{d}P \times \~~operatorname~~{d}Q,</math> the difference map <math>\~~operatorname~~{D}f</math> can be represented in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>X = P \times Q</math> and whose arrows are labeled with the elements of <math>\~~operatorname~~{d}X = \~~operatorname~~{d}P \times \~~operatorname~~{d}Q,</math> as shown in the following Figure.

+

Abstracting from the augmented venn diagram that shows how the ''models'' or ''satisfying interpretations'' of <math>\mathrm{D}f</math> distribute over the extended universe of discourse <math>\mathrm{E}X = P \times Q \times \mathrm{d}P \times \mathrm{d}Q,</math> the difference map <math>\mathrm{D}f</math> can be represented in the form of a ''digraph'' or ''directed graph'', one whose points are labeled with the elements of <math>X = P \times Q</math> and whose arrows are labeled with the elements of <math>\mathrm{d}X = \mathrm{d}P \times \mathrm{d}Q,</math> as shown in the following Figure.

{| align="center" cellpadding="10" style="text-align:center"

Line 526: Line 528:

& = & p & \cdot & q

\\[4pt]

−

\~~operatorname~~{D}f

+

\mathrm{D}f

−

& = & p & \cdot & q & \cdot & ((\~~operatorname~~{d}p)(\~~operatorname~~{d}q))

+

& = & p & \cdot & q & \cdot & ((\mathrm{d}p)(\mathrm{d}q))

\\[4pt]

−

& + & p & \cdot & (q) & \cdot & ~(\~~operatorname~~{d}p)~\~~operatorname~~{d}q~~

+

& + & p & \cdot & (q) & \cdot & ~(\mathrm{d}p)~\mathrm{d}q~~

\\[4pt]

−

& + & (p) & \cdot & q & \cdot & ~~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)~

+

& + & (p) & \cdot & q & \cdot & ~~\mathrm{d}p~(\mathrm{d}q)~

\\[4pt]

−

& + & (p) & \cdot & (q) & \cdot & ~~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~~

+

& + & (p) & \cdot & (q) & \cdot & ~~\mathrm{d}p~~\mathrm{d}q~~

−

\end{array}</math>

+

\end{array}\!</math>

|}

Any proposition worth its salt can be analyzed from many different points of view, any one of which has the potential to reveal an unsuspected aspect of the proposition's meaning. We will encounter more and more of these alternative readings as we go.

−

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(p, q) = pq.\!</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(p, q) = pq.\!</math>

A suitably generic definition of the extended universe of discourse is afforded by the following set-up:

Line 552: Line 554:

\\[6pt]

\text{Let} &

−

\~~operatorname~~{d}X

+

\mathrm{d}X

& = &

−

\~~operatorname~~{d}X_1 \times \ldots \times \~~operatorname~~{d}X_k.

+

\mathrm{d}X_1 \times \ldots \times \mathrm{d}X_k.

\\[6pt]

\text{Then} &

−

\~~operatorname~~{E}X

+

\mathrm{E}X

& = &

−

X \times \~~operatorname~~{d}X

+

X \times \mathrm{d}X

\\[6pt]

&

−

& = & 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

\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}X \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}X \to \mathbb{B}</math> that is defined by the following equation:

{| align="center" cellpadding="10" width="90%"

|

<math>\begin{array}{l}

−

\~~operatorname~~{E}f(x_1, \ldots, x_k, \~~operatorname~~{d}x_1, \ldots, \~~operatorname~~{d}x_k)

+

\mathrm{E}f(x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k)

\\[6pt]

−

= \quad f(x_1 + \~~operatorname~~{d}x_1, \ldots, x_k + \~~operatorname~~{d}x_k)

+

= \quad f(x_1 + \mathrm{d}x_1, \ldots, x_k + \mathrm{d}x_k)

\\[6pt]

−

= \quad f( \texttt{(} x_1, \~~operatorname~~{d}x_1 \texttt{)}, \ldots, \texttt{(} x_k, \~~operatorname~~{d}x_k \texttt{)} )

+

= \quad f( \texttt{(} x_1, \mathrm{d}x_1 \texttt{)}, \ldots, \texttt{(} x_k, \mathrm{d}x_k \texttt{)} )

\end{array}</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> Although it is conventional to distinguish the (first order) differential variables with the operative prefix “<math>\~~operatorname~~{d}</math>” this way of notating differential variables is entirely 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> Although it is conventional to distinguish the (first order) differential variables with the operative prefix “<math>\mathrm{d}</math>” this way of notating differential variables is entirely optional. It is their existence in particular relations to the initial variables, not their names, that defines them as differential variables.

−

In the example of logical conjunction, <math>f(p, q) = pq,\!</math> the enlargement <math>\~~operatorname~~{E}f</math> is formulated as follows:

+

In the example of logical conjunction, <math>f(p, q) = pq,\!</math> the enlargement <math>\mathrm{E}f</math> is formulated as follows:

{| align="center" cellpadding="10" width="90%"

|

<math>\begin{array}{l}

−

\~~operatorname~~{E}f(p, q, \~~operatorname~~{d}p, \~~operatorname~~{d}q)

+

\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)

\\[6pt]

−

= \quad (p + \~~operatorname~~{d}p)(q + \~~operatorname~~{d}q)

+

= \quad (p + \mathrm{d}p)(q + \mathrm{d}q)

\\[6pt]

−

= \quad \texttt{(} p, \~~operatorname~~{d}p \texttt{)(} q, \~~operatorname~~{d}q \texttt{)}

+

= \quad \texttt{(} p, \mathrm{d}p \texttt{)(} q, \mathrm{d}q \texttt{)}

\end{array}</math>

|}

Line 599: Line 601:

|

<math>\begin{matrix}

−

\~~operatorname~~{E}f(p, q, \~~operatorname~~{d}p, \~~operatorname~~{d}q)

+

\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)

& = &

p~q

& + &

−

p~\~~operatorname~~{d}q

+

p~\mathrm{d}q

& + &

−

q~\~~operatorname~~{d}p

+

q~\mathrm{d}p

& + &

−

\~~operatorname~~{d}p~\~~operatorname~~{d}q

+

\mathrm{d}p~\mathrm{d}q

\end{matrix}</math>

|}

−

To understand what the ''enlarged'' or ''shifted'' proposition means in logical terms, it serves 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 value of <math>\~~operatorname~~{E}f_x</math> at each <math>x \in X</math> may be computed in graphical fashion as shown below:

+

To understand what the ''enlarged'' or ''shifted'' proposition means in logical terms, it serves 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 value of <math>\mathrm{E}f_x</math> at each <math>x \in X</math> may be computed in graphical fashion as shown below:

{| align="center" cellpadding="10" style="text-align:center"

Line 625: Line 627:

|}

−

Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition <math>\~~operatorname~~{E}f.</math>

+

Given the data that develops in this form of analysis, the disjoined ingredients can now be folded back into a boolean expansion or a disjunctive normal form (DNF) that is equivalent to the enlarged proposition <math>\mathrm{E}f.</math>

{| align="center" cellpadding="10" width="90%"

|

<math>\begin{matrix}

−

\~~operatorname~~{E}f

+

\mathrm{E}f

& = &

−

pq \cdot \~~operatorname~~{E}f_{pq}

+

pq \cdot \mathrm{E}f_{pq}

& + &

−

p(q) \cdot \~~operatorname~~{E}f_{p(q)}

+

p(q) \cdot \mathrm{E}f_{p(q)}

& + &

−

(p)q \cdot \~~operatorname~~{E}f_{(p)q}

+

(p)q \cdot \mathrm{E}f_{(p)q}

& + &

−

(p)(q) \cdot \~~operatorname~~{E}f_{(p)(q)}

+

(p)(q) \cdot \mathrm{E}f_{(p)(q)}

\end{matrix}</math>

|}

−

Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element <math>(\~~operatorname~~{d}p)(\~~operatorname~~{d}q)</math> is drawn as a loop at the point <math>p~q.</math>

+

Here is a summary of the result, illustrated by means of a digraph picture, where the “no change” element <math>(\mathrm{d}p)(\mathrm{d}q)</math> is drawn as a loop at the point <math>p~q.</math>

{| align="center" cellpadding="10" style="text-align:center"

Line 652: Line 654:

& = & p & \cdot & q

\\[4pt]

−

\~~operatorname~~{E}f

+

\mathrm{E}f

−

& = & p & \cdot & q & \cdot & (\~~operatorname~~{d}p)(\~~operatorname~~{d}q)

+

& = & p & \cdot & q & \cdot & (\mathrm{d}p)(\mathrm{d}q)

\\[4pt]

−

& + & p & \cdot & (q) & \cdot & (\~~operatorname~~{d}p)~\~~operatorname~~{d}q~

+

& + & p & \cdot & (q) & \cdot & (\mathrm{d}p)~\mathrm{d}q~

\\[4pt]

−

& + & (p) & \cdot & q & \cdot & ~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)

+

& + & (p) & \cdot & q & \cdot & ~\mathrm{d}p~(\mathrm{d}q)

\\[4pt]

−

& + & (p) & \cdot & (q) & \cdot & ~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~\end{array}</math>

+

& + & (p) & \cdot & (q) & \cdot & ~\mathrm{d}p~~\mathrm{d}q~\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>X.\!</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>X.\!</math>

==Propositional Forms on Two Variables==

−

To broaden our experience with simple examples, let us examine the sixteen functions of concrete type <math>P \times Q \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>P \times Q \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.

Line 770: Line 772:

(p)~q~

\\[4pt]

−

(p)~~~~~~~

+

(p)[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])

\\[4pt]

~p~(q)

\\[4pt]

−

~~~~~~~(q)

+

[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])(q)

\\[4pt]

(p,~q)

Line 852: Line 854:

\\[4pt]

f_{1111}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 877: Line 879:

((p,~q))

\\[4pt]

−

~~~~~q~~

+

22:10, 8 December 2014 (UTC)q~~

\\[4pt]

~(p~(q))

\\[4pt]

−

~~~~p~~~~~~~

+

~~p22:10, 8 December 2014 (UTC)

\\[4pt]

((p)~q)~

Line 1,038: Line 1,040:

\\[4pt]

f_{12}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 1,044: Line 1,046:

\\[4pt]

f_{1100}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 1,087: Line 1,089:

\\[4pt]

1~0~0~1

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 1,207: Line 1,209:

| <math>f_{15}\!</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>1\!</math>

Line 1,217: Line 1,219:

===Transforms Expanded over Differential Features===

−

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, partitioned into seven natural classes by a group action.

+

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, partitioned into seven natural classes by a group action.

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

−

|+ <math>\text{Table A3.}~~\~~operatorname~~{E}f ~\text{Expanded over Differential Features}~ \{ \~~operatorname~~{d}p, \~~operatorname~~{d}q \}</math>

+

|+ <math>\text{Table A3.}~~\mathrm{E}f ~\text{Expanded over Differential Features}~ \{ \mathrm{d}p, \mathrm{d}q \}</math>

|- style="background:#f0f0ff"

| width="10%" |  

| width="18%" | <math>f\!</math>

| width="18%" |

−

<math>\~~operatorname~~{T}_{11} f</math>

+

<math>\mathrm{T}_{11} f</math>

−

<math>\~~operatorname~~{E}f|_{\~~operatorname~~{d}p~\~~operatorname~~{d}q}</math>

+

<math>\mathrm{E}f|_{\mathrm{d}p~\mathrm{d}q}</math>

| width="18%" |

−

<math>\~~operatorname~~{T}_{10} f</math>

+

<math>\mathrm{T}_{10} f</math>

−

<math>\~~operatorname~~{E}f|_{\~~operatorname~~{d}p(\~~operatorname~~{d}q)}</math>

+

<math>\mathrm{E}f|_{\mathrm{d}p(\mathrm{d}q)}</math>

| width="18%" |

−

<math>\~~operatorname~~{T}_{01} f</math>

+

<math>\mathrm{T}_{01} f</math>

−

<math>\~~operatorname~~{E}f|_{(\~~operatorname~~{d}p)\~~operatorname~~{d}q}</math>

+

<math>\mathrm{E}f|_{(\mathrm{d}p)\mathrm{d}q}</math>

| width="18%" |

−

<math>\~~operatorname~~{T}_{00} f</math>

+

<math>\mathrm{T}_{00} f</math>

−

<math>\~~operatorname~~{E}f|_{(\~~operatorname~~{d}p)(\~~operatorname~~{d}q)}</math>

+

<math>\mathrm{E}f|_{(\mathrm{d}p)(\mathrm{d}q)}</math>

|-

| <math>f_0\!</math>

Line 1,312: Line 1,314:

\\[4pt]

f_{12}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 1,496: Line 1,498:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

−

|+ <math>\text{Table A4.}~~\~~operatorname~~{D}f ~\text{Expanded over Differential Features}~ \{ \~~operatorname~~{d}p, \~~operatorname~~{d}q \}</math>

+

|+ <math>\text{Table A4.}~~\mathrm{D}f ~\text{Expanded over Differential Features}~ \{ \mathrm{d}p, \mathrm{d}q \}</math>

|- style="background:#f0f0ff"

| width="10%" |  

| width="18%" | <math>f\!</math>

| width="18%" |

−

<math>\~~operatorname~~{D}f|_{\~~operatorname~~{d}p~\~~operatorname~~{d}q}</math>

+

<math>\mathrm{D}f|_{\mathrm{d}p~\mathrm{d}q}</math>

| width="18%" |

−

<math>\~~operatorname~~{D}f|_{\~~operatorname~~{d}p(\~~operatorname~~{d}q)}</math>

+

<math>\mathrm{D}f|_{\mathrm{d}p(\mathrm{d}q)}</math>

| width="18%" |

−

<math>\~~operatorname~~{D}f|_{(\~~operatorname~~{d}p)\~~operatorname~~{d}q}</math>

+

<math>\mathrm{D}f|_{(\mathrm{d}p)\mathrm{d}q}</math>

| width="18%" |

−

<math>\~~operatorname~~{D}f|_{(\~~operatorname~~{d}p)(\~~operatorname~~{d}q)}</math>

+

<math>\mathrm{D}f|_{(\mathrm{d}p)(\mathrm{d}q)}</math>

|-

| <math>f_0\!</math>

Line 1,582: Line 1,584:

\\[4pt]

f_{12}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 1,764: Line 1,766:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

−

|+ <math>\text{Table A5.}~~\~~operatorname~~{E}f ~\text{Expanded over Ordinary Features}~ \{ p, q \}</math>

+

|+ <math>\text{Table A5.}~~\mathrm{E}f ~\text{Expanded over Ordinary Features}~ \{ p, q \}</math>

|- style="background:#f0f0ff"

| width="10%" |  

| width="18%" | <math>f\!</math>

−

| width="18%" | <math>\~~operatorname~~{E}f|_{pq}</math>

+

| width="18%" | <math>\mathrm{E}f|_{pq}</math>

−

| width="18%" | <math>\~~operatorname~~{E}f|_{p(q)}</math>

+

| width="18%" | <math>\mathrm{E}f|_{p(q)}</math>

−

| width="18%" | <math>\~~operatorname~~{E}f|_{(p)q}</math>

+

| width="18%" | <math>\mathrm{E}f|_{(p)q}</math>

−

| width="18%" | <math>\~~operatorname~~{E}f|_{(p)(q)}</math>

+

| width="18%" | <math>\mathrm{E}f|_{(p)(q)}</math>

|-

| <math>f_0\!</math>

Line 1,802: Line 1,804:

|

<math>\begin{matrix}

−

~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~

+

~\mathrm{d}p~~\mathrm{d}q~

\\[4pt]

−

~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)

+

~\mathrm{d}p~(\mathrm{d}q)

\\[4pt]

−

(\~~operatorname~~{d}p)~\~~operatorname~~{d}q~

+

(\mathrm{d}p)~\mathrm{d}q~

\\[4pt]

−

(\~~operatorname~~{d}p)(\~~operatorname~~{d}q)

+

(\mathrm{d}p)(\mathrm{d}q)

\end{matrix}</math>

|

<math>\begin{matrix}

−

~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)

+

~\mathrm{d}p~(\mathrm{d}q)

\\[4pt]

−

~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~

+

~\mathrm{d}p~~\mathrm{d}q~

\\[4pt]

−

(\~~operatorname~~{d}p)(\~~operatorname~~{d}q)

+

(\mathrm{d}p)(\mathrm{d}q)

\\[4pt]

−

(\~~operatorname~~{d}p)~\~~operatorname~~{d}q~

+

(\mathrm{d}p)~\mathrm{d}q~

\end{matrix}</math>

|

<math>\begin{matrix}

−

(\~~operatorname~~{d}p)~\~~operatorname~~{d}q~

+

(\mathrm{d}p)~\mathrm{d}q~

\\[4pt]

−

(\~~operatorname~~{d}p)(\~~operatorname~~{d}q)

+

(\mathrm{d}p)(\mathrm{d}q)

\\[4pt]

−

~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~

+

~\mathrm{d}p~~\mathrm{d}q~

\\[4pt]

−

~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)

+

~\mathrm{d}p~(\mathrm{d}q)

\end{matrix}</math>

|

<math>\begin{matrix}

−

(\~~operatorname~~{d}p)(\~~operatorname~~{d}q)

+

(\mathrm{d}p)(\mathrm{d}q)

\\[4pt]

−

(\~~operatorname~~{d}p)~\~~operatorname~~{d}q~

+

(\mathrm{d}p)~\mathrm{d}q~

\\[4pt]

−

~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)

+

~\mathrm{d}p~(\mathrm{d}q)

\\[4pt]

−

~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~

+

~\mathrm{d}p~~\mathrm{d}q~

\end{matrix}</math>

|-

Line 1,846: Line 1,848:

\\[4pt]

f_{12}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 1,855: Line 1,857:

|

<math>\begin{matrix}

−

~\~~operatorname~~{d}p~

+

~\mathrm{d}p~

\\[4pt]

−

(\~~operatorname~~{d}p)

+

(\mathrm{d}p)

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

−

~\~~operatorname~~{d}p~

+

~\mathrm{d}p~

\\[4pt]

−

(\~~operatorname~~{d}p)

+

(\mathrm{d}p)

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

−

(\~~operatorname~~{d}p)

+

(\mathrm{d}p)

\\[4pt]

−

~\~~operatorname~~{d}p~

+

~\mathrm{d}p~

\end{matrix}</math>

|

<math>\begin{matrix}

−

(\~~operatorname~~{d}p)

+

(\mathrm{d}p)

\\[4pt]

−

~\~~operatorname~~{d}p~

+

~\mathrm{d}p~

\end{matrix}</math>

|-

Line 1,892: Line 1,894:

|

<math>\begin{matrix}

−

~(\~~operatorname~~{d}p,~\~~operatorname~~{d}q)~

+

~(\mathrm{d}p,~\mathrm{d}q)~

\\[4pt]

−

((\~~operatorname~~{d}p,~\~~operatorname~~{d}q))

+

((\mathrm{d}p,~\mathrm{d}q))

\end{matrix}</math>

|

<math>\begin{matrix}

−

((\~~operatorname~~{d}p,~\~~operatorname~~{d}q))

+

((\mathrm{d}p,~\mathrm{d}q))

\\[4pt]

−

~(\~~operatorname~~{d}p,~\~~operatorname~~{d}q)~

+

~(\mathrm{d}p,~\mathrm{d}q)~

\end{matrix}</math>

|

<math>\begin{matrix}

−

((\~~operatorname~~{d}p,~\~~operatorname~~{d}q))

+

((\mathrm{d}p,~\mathrm{d}q))

\\[4pt]

−

~(\~~operatorname~~{d}p,~\~~operatorname~~{d}q)~

+

~(\mathrm{d}p,~\mathrm{d}q)~

\end{matrix}</math>

|

<math>\begin{matrix}

−

~(\~~operatorname~~{d}p,~\~~operatorname~~{d}q)~

+

~(\mathrm{d}p,~\mathrm{d}q)~

\\[4pt]

−

((\~~operatorname~~{d}p,~\~~operatorname~~{d}q))

+

((\mathrm{d}p,~\mathrm{d}q))

\end{matrix}</math>

|-

Line 1,929: Line 1,931:

|

<math>\begin{matrix}

−

~\~~operatorname~~{d}q~

+

~\mathrm{d}q~

\\[4pt]

−

(\~~operatorname~~{d}q)

+

(\mathrm{d}q)

\end{matrix}</math>

|

<math>\begin{matrix}

−

(\~~operatorname~~{d}q)

+

(\mathrm{d}q)

\\[4pt]

−

~\~~operatorname~~{d}q~

+

~\mathrm{d}q~

\end{matrix}</math>

|

<math>\begin{matrix}

−

~\~~operatorname~~{d}q~

+

~\mathrm{d}q~

\\[4pt]

−

(\~~operatorname~~{d}q)

+

(\mathrm{d}q)

\end{matrix}</math>

|

<math>\begin{matrix}

−

(\~~operatorname~~{d}q)

+

(\mathrm{d}q)

\\[4pt]

−

~\~~operatorname~~{d}q~

+

~\mathrm{d}q~

\end{matrix}</math>

|-

Line 1,974: Line 1,976:

|

<math>\begin{matrix}

−

((\~~operatorname~~{d}p)(\~~operatorname~~{d}q))

+

((\mathrm{d}p)(\mathrm{d}q))

\\[4pt]

−

((\~~operatorname~~{d}p)~\~~operatorname~~{d}q~)

+

((\mathrm{d}p)~\mathrm{d}q~)

\\[4pt]

−

(~\~~operatorname~~{d}p~(\~~operatorname~~{d}q))

+

(~\mathrm{d}p~(\mathrm{d}q))

\\[4pt]

−

(~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~)

+

(~\mathrm{d}p~~\mathrm{d}q~)

\end{matrix}</math>

|

<math>\begin{matrix}

−

((\~~operatorname~~{d}p)~\~~operatorname~~{d}q~)

+

((\mathrm{d}p)~\mathrm{d}q~)

\\[4pt]

−

((\~~operatorname~~{d}p)(\~~operatorname~~{d}q))

+

((\mathrm{d}p)(\mathrm{d}q))

\\[4pt]

−

(~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~)

+

(~\mathrm{d}p~~\mathrm{d}q~)

\\[4pt]

−

(~\~~operatorname~~{d}p~(\~~operatorname~~{d}q))

+

(~\mathrm{d}p~(\mathrm{d}q))

\end{matrix}</math>

|

<math>\begin{matrix}

−

(~\~~operatorname~~{d}p~(\~~operatorname~~{d}q))

+

(~\mathrm{d}p~(\mathrm{d}q))

\\[4pt]

−

(~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~)

+

(~\mathrm{d}p~~\mathrm{d}q~)

\\[4pt]

−

((\~~operatorname~~{d}p)(\~~operatorname~~{d}q))

+

((\mathrm{d}p)(\mathrm{d}q))

\\[4pt]

−

((\~~operatorname~~{d}p)~\~~operatorname~~{d}q~)

+

((\mathrm{d}p)~\mathrm{d}q~)

\end{matrix}</math>

|

<math>\begin{matrix}

−

(~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~)

+

(~\mathrm{d}p~~\mathrm{d}q~)

\\[4pt]

−

(~\~~operatorname~~{d}p~(\~~operatorname~~{d}q))

+

(~\mathrm{d}p~(\mathrm{d}q))

\\[4pt]

−

((\~~operatorname~~{d}p)~\~~operatorname~~{d}q~)

+

((\mathrm{d}p)~\mathrm{d}q~)

\\[4pt]

−

((\~~operatorname~~{d}p)(\~~operatorname~~{d}q))

+

((\mathrm{d}p)(\mathrm{d}q))

\end{matrix}</math>

|-

Line 2,024: Line 2,026:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

−

|+ <math>\text{Table A6.}~~\~~operatorname~~{D}f ~\text{Expanded over Ordinary Features}~ \{ p, q \}</math>

+

|+ <math>\text{Table A6.}~~\mathrm{D}f ~\text{Expanded over Ordinary Features}~ \{ p, q \}</math>

|- style="background:#f0f0ff"

| width="10%" |  

| width="18%" | <math>f\!</math>

−

| width="18%" | <math>\~~operatorname~~{D}f|_{pq}</math>

+

| width="18%" | <math>\mathrm{D}f|_{pq}</math>

−

| width="18%" | <math>\~~operatorname~~{D}f|_{p(q)}</math>

+

| width="18%" | <math>\mathrm{D}f|_{p(q)}</math>

−

| width="18%" | <math>\~~operatorname~~{D}f|_{(p)q}</math>

+

| width="18%" | <math>\mathrm{D}f|_{(p)q}</math>

−

| width="18%" | <math>\~~operatorname~~{D}f|_{(p)(q)}</math>

+

| width="18%" | <math>\mathrm{D}f|_{(p)(q)}</math>

|-

| <math>f_0\!</math>

Line 2,062: Line 2,064:

|

<math>\begin{matrix}

−

~~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~~

+

~~\mathrm{d}p~~\mathrm{d}q~~

\\[4pt]

−

~~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)~

+

~~\mathrm{d}p~(\mathrm{d}q)~

\\[4pt]

−

~(\~~operatorname~~{d}p)~\~~operatorname~~{d}q~~

+

~(\mathrm{d}p)~\mathrm{d}q~~

\\[4pt]

−

((\~~operatorname~~{d}p)(\~~operatorname~~{d}q))

+

((\mathrm{d}p)(\mathrm{d}q))

\end{matrix}</math>

|

<math>\begin{matrix}

−

~~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)~

+

~~\mathrm{d}p~(\mathrm{d}q)~

\\[4pt]

−

~~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~~

+

~~\mathrm{d}p~~\mathrm{d}q~~

\\[4pt]

−

((\~~operatorname~~{d}p)(\~~operatorname~~{d}q))

+

((\mathrm{d}p)(\mathrm{d}q))

\\[4pt]

−

~(\~~operatorname~~{d}p)~\~~operatorname~~{d}q~~

+

~(\mathrm{d}p)~\mathrm{d}q~~

\end{matrix}</math>

|

<math>\begin{matrix}

−

~(\~~operatorname~~{d}p)~\~~operatorname~~{d}q~~

+

~(\mathrm{d}p)~\mathrm{d}q~~

\\[4pt]

−

((\~~operatorname~~{d}p)(\~~operatorname~~{d}q))

+

((\mathrm{d}p)(\mathrm{d}q))

\\[4pt]

−

~~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~~

+

~~\mathrm{d}p~~\mathrm{d}q~~

\\[4pt]

−

~~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)~

+

~~\mathrm{d}p~(\mathrm{d}q)~

\end{matrix}</math>

|

<math>\begin{matrix}

−

((\~~operatorname~~{d}p)(\~~operatorname~~{d}q))

+

((\mathrm{d}p)(\mathrm{d}q))

\\[4pt]

−

~(\~~operatorname~~{d}p)~\~~operatorname~~{d}q~~

+

~(\mathrm{d}p)~\mathrm{d}q~~

\\[4pt]

−

~~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)~

+

~~\mathrm{d}p~(\mathrm{d}q)~

\\[4pt]

−

~~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~~

+

~~\mathrm{d}p~~\mathrm{d}q~~

\end{matrix}</math>

|-

Line 2,106: Line 2,108:

\\[4pt]

f_{12}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 2,115: Line 2,117:

|

<math>\begin{matrix}

−

\~~operatorname~~{d}p

+

\mathrm{d}p

\\[4pt]

−

\~~operatorname~~{d}p

+

\mathrm{d}p

\end{matrix}</math>

|

<math>\begin{matrix}

−

\~~operatorname~~{d}p

+

\mathrm{d}p

\\[4pt]

−

\~~operatorname~~{d}p

+

\mathrm{d}p

\end{matrix}</math>

|

<math>\begin{matrix}

−

\~~operatorname~~{d}p

+

\mathrm{d}p

\\[4pt]

−

\~~operatorname~~{d}p

+

\mathrm{d}p

\end{matrix}</math>

|

<math>\begin{matrix}

−

\~~operatorname~~{d}p

+

\mathrm{d}p

\\[4pt]

−

\~~operatorname~~{d}p

+

\mathrm{d}p

\end{matrix}</math>

|-

Line 2,152: Line 2,154:

|

<math>\begin{matrix}

−

(\~~operatorname~~{d}p,~\~~operatorname~~{d}q)

+

(\mathrm{d}p,~\mathrm{d}q)

\\[4pt]

−

(\~~operatorname~~{d}p,~\~~operatorname~~{d}q)

+

(\mathrm{d}p,~\mathrm{d}q)

\end{matrix}</math>

|

<math>\begin{matrix}

−

(\~~operatorname~~{d}p,~\~~operatorname~~{d}q)

+

(\mathrm{d}p,~\mathrm{d}q)

\\[4pt]

−

(\~~operatorname~~{d}p,~\~~operatorname~~{d}q)

+

(\mathrm{d}p,~\mathrm{d}q)

\end{matrix}</math>

|

<math>\begin{matrix}

−

(\~~operatorname~~{d}p,~\~~operatorname~~{d}q)

+

(\mathrm{d}p,~\mathrm{d}q)

\\[4pt]

−

(\~~operatorname~~{d}p,~\~~operatorname~~{d}q)

+

(\mathrm{d}p,~\mathrm{d}q)

\end{matrix}</math>

|

<math>\begin{matrix}

−

(\~~operatorname~~{d}p,~\~~operatorname~~{d}q)

+

(\mathrm{d}p,~\mathrm{d}q)

\\[4pt]

−

(\~~operatorname~~{d}p,~\~~operatorname~~{d}q)

+

(\mathrm{d}p,~\mathrm{d}q)

\end{matrix}</math>

|-

Line 2,189: Line 2,191:

|

<math>\begin{matrix}

−

\~~operatorname~~{d}q

+

\mathrm{d}q

\\[4pt]

−

\~~operatorname~~{d}q

+

\mathrm{d}q

\end{matrix}</math>

|

<math>\begin{matrix}

−

\~~operatorname~~{d}q

+

\mathrm{d}q

\\[4pt]

−

\~~operatorname~~{d}q

+

\mathrm{d}q

\end{matrix}</math>

|

<math>\begin{matrix}

−

\~~operatorname~~{d}q

+

\mathrm{d}q

\\[4pt]

−

\~~operatorname~~{d}q

+

\mathrm{d}q

\end{matrix}</math>

|

<math>\begin{matrix}

−

\~~operatorname~~{d}q

+

\mathrm{d}q

\\[4pt]

−

\~~operatorname~~{d}q

+

\mathrm{d}q

\end{matrix}</math>

|-

Line 2,234: Line 2,236:

|

<math>\begin{matrix}

−

((\~~operatorname~~{d}p)(\~~operatorname~~{d}q))

+

((\mathrm{d}p)(\mathrm{d}q))

\\[4pt]

−

~(\~~operatorname~~{d}p)~\~~operatorname~~{d}q~~

+

~(\mathrm{d}p)~\mathrm{d}q~~

\\[4pt]

−

~~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)~

+

~~\mathrm{d}p~(\mathrm{d}q)~

\\[4pt]

−

~~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~~

+

~~\mathrm{d}p~~\mathrm{d}q~~

\end{matrix}</math>

|

<math>\begin{matrix}

−

~(\~~operatorname~~{d}p)~\~~operatorname~~{d}q~~

+

~(\mathrm{d}p)~\mathrm{d}q~~

\\[4pt]

−

((\~~operatorname~~{d}p)(\~~operatorname~~{d}q))

+

((\mathrm{d}p)(\mathrm{d}q))

\\[4pt]

−

~~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~~

+

~~\mathrm{d}p~~\mathrm{d}q~~

\\[4pt]

−

~~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)~

+

~~\mathrm{d}p~(\mathrm{d}q)~

\end{matrix}</math>

|

<math>\begin{matrix}

−

~~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)~

+

~~\mathrm{d}p~(\mathrm{d}q)~

\\[4pt]

−

~~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~~

+

~~\mathrm{d}p~~\mathrm{d}q~~

\\[4pt]

−

((\~~operatorname~~{d}p)(\~~operatorname~~{d}q))

+

((\mathrm{d}p)(\mathrm{d}q))

\\[4pt]

−

~(\~~operatorname~~{d}p)~\~~operatorname~~{d}q~~

+

~(\mathrm{d}p)~\mathrm{d}q~~

\end{matrix}</math>

|

<math>\begin{matrix}

−

~~\~~operatorname~~{d}p~~\~~operatorname~~{d}q~~

+

~~\mathrm{d}p~~\mathrm{d}q~~

\\[4pt]

−

~~\~~operatorname~~{d}p~(\~~operatorname~~{d}q)~

+

~~\mathrm{d}p~(\mathrm{d}q)~

\\[4pt]

−

~(\~~operatorname~~{d}p)~\~~operatorname~~{d}q~~

+

~(\mathrm{d}p)~\mathrm{d}q~~

\\[4pt]

−

((\~~operatorname~~{d}p)(\~~operatorname~~{d}q))

+

((\mathrm{d}p)(\mathrm{d}q))

\end{matrix}</math>

|-

Line 2,285: Line 2,287:

==Operational Representation==

−

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 overwhelms its discrete and finite powers to grasp them. But here, in the fully serene idylls of [[zeroth order logic]], we find ourselves fit with the compass of a wit that is all we'd ever need 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 overwhelms its discrete and finite powers to grasp them. But here, in the fully serene idylls of [[zeroth order logic]], we find ourselves fit with the compass of a wit that is all we'd ever need to explore their effects with care.

So let us do just that.

Line 2,294: Line 2,296:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

−

|+ <math>\text{Table A3.}~~\~~operatorname~~{E}f ~\text{Expanded Over Differential Features}~ \{ \~~operatorname~~{d}p, \~~operatorname~~{d}q \}</math>

+

|+ <math>\text{Table A3.}~~\mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}p, \mathrm{d}q \}</math>

|- style="background:#f0f0ff"

| width="10%" |  

| width="18%" | <math>f\!</math>

| width="18%" |

−

<math>\~~operatorname~~{T}_{11} f</math>

+

<math>\mathrm{T}_{11} f</math>

−

<math>\~~operatorname~~{E}f|_{\~~operatorname~~{d}p~\~~operatorname~~{d}q}</math>

+

<math>\mathrm{E}f|_{\mathrm{d}p~\mathrm{d}q}</math>

| width="18%" |

−

<math>\~~operatorname~~{T}_{10} f</math>

+

<math>\mathrm{T}_{10} f</math>

−

<math>\~~operatorname~~{E}f|_{\~~operatorname~~{d}p(\~~operatorname~~{d}q)}</math>

+

<math>\mathrm{E}f|_{\mathrm{d}p(\mathrm{d}q)}</math>

| width="18%" |

−

<math>\~~operatorname~~{T}_{01} f</math>

+

<math>\mathrm{T}_{01} f</math>

−

<math>\~~operatorname~~{E}f|_{(\~~operatorname~~{d}p)\~~operatorname~~{d}q}</math>

+

<math>\mathrm{E}f|_{(\mathrm{d}p)\mathrm{d}q}</math>

| width="18%" |

−

<math>\~~operatorname~~{T}_{00} f</math>

+

<math>\mathrm{T}_{00} f</math>

−

<math>\~~operatorname~~{E}f|_{(\~~operatorname~~{d}p)(\~~operatorname~~{d}q)}</math>

+

<math>\mathrm{E}f|_{(\mathrm{d}p)(\mathrm{d}q)}</math>

|-

| <math>f_0\!</math>

Line 2,384: Line 2,386:

\\[4pt]

f_{12}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 2,567: Line 2,569:

−

The shift operator <math>\~~operatorname~~{E}</math> can be understood as enacting a substitution operation on the propositional form <math>f(p, q)\!</math> that is given as its argument. In our present focus on propositional forms that involve two variables, we have the following type specifications and definitions:

+

The shift operator <math>\mathrm{E}</math> can be understood as enacting a substitution operation on the propositional form <math>f(p, q)\!</math> that is given as its argument. In our present focus on propositional forms that involve two variables, we have the following type specifications and definitions:

{| align="center" cellpadding="6" width="90%"

|

<math>\begin{array}{lcl}

−

\~~operatorname~~{E} ~:~ (X \to \mathbb{B})

+

\mathrm{E} ~:~ (X \to \mathbb{B})

& \to &

−

(\~~operatorname~~{E}X \to \mathbb{B})

+

(\mathrm{E}X \to \mathbb{B})

\\[6pt]

−

\~~operatorname~~{E} ~:~ f(p, q)

+

\mathrm{E} ~:~ f(p, q)

& \mapsto &

−

\~~operatorname~~{E}f(p, q, \~~operatorname~~{d}p, \~~operatorname~~{d}q)

+

\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)

\\[6pt]

−

\~~operatorname~~{E}f(p, q, \~~operatorname~~{d}p, \~~operatorname~~{d}q)

+

\mathrm{E}f(p, q, \mathrm{d}p, \mathrm{d}q)

& = &

−

f(p + \~~operatorname~~{d}p, q + \~~operatorname~~{d}q)

+

f(p + \mathrm{d}p, q + \mathrm{d}q)

\\[6pt]

& = &

−

f( \texttt{(} p, \~~operatorname~~{d}p \texttt{)}, \texttt{(} q, \~~operatorname~~{d}q \texttt{)} )

+

f( \texttt{(} p, \mathrm{d}p \texttt{)}, \texttt{(} q, \mathrm{d}q \texttt{)} )

\end{array}</math>

|}

−

Evaluating <math>\~~operatorname~~{E}f</math> at particular values of <math>\~~operatorname~~{d}p</math> and <math>\~~operatorname~~{d}q,</math> for example, <math>\~~operatorname~~{d}p = i</math> and <math>\~~operatorname~~{d}q = 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}p</math> and <math>\mathrm{d}q,</math> for example, <math>\mathrm{d}p = i</math> and <math>\mathrm{d}q = 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%"

|

<math>\begin{array}{lclcl}

−

\~~operatorname~~{E}_{ij}

+

\mathrm{E}_{ij}

& : &

(X \to \mathbb{B})

Line 2,600: Line 2,602:

(X \to \mathbb{B})

\\[6pt]

−

\~~operatorname~~{E}_{ij}

+

\mathrm{E}_{ij}

& : &

f

& \mapsto &

−

\~~operatorname~~{E}_{ij}f

+

\mathrm{E}_{ij}f

\\[6pt]

−

\~~operatorname~~{E}_{ij}f

+

\mathrm{E}_{ij}f

& = &

−

\~~operatorname~~{E}f|_{\~~operatorname~~{d}p = i, \~~operatorname~~{d}q = j}

+

\mathrm{E}f|_{\mathrm{d}p = i, \mathrm{d}q = j}

& = &

f(p + i, q + j)

Line 2,615: Line 2,617:

& = &

f( \texttt{(} p, i \texttt{)}, \texttt{(} q, j \texttt{)} )

−

\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 : X \to \mathbb{B}</math> into a proposition <math>f^\prime : X \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 : X \to \mathbb{B}</math> into a proposition <math>f^\prime : X \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:

Line 2,624: Line 2,626:

{| 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"

−

| 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" |

+

| 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" |

−

<math>\~~operatorname~~{T}_{01}</math>

+

<math>\mathrm{T}_{01}\!</math>

| 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" |

−

<math>\~~operatorname~~{T}_{11}</math>

+

<math>\mathrm{T}_{11}\!</math>

|- 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="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="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="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>

|}

Line 2,663: Line 2,666:

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!

{| align="center" cellpadding="0" cellspacing="0" width="90%"

Line 2,680: Line 2,683:

{| 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"

−

| 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" |

+

| width="22%" style="border-bottom:1px solid black" |

−

<math>\~~operatorname~~{e}</math>

+

<math>\mathrm{e}\!</math>

| width="22%" style="border-bottom:1px solid black" |

−

<math>\~~operatorname~~{f}</math>

+

<math>\mathrm{f}\!</math>

| width="22%" style="border-bottom:1px solid black" |

−

<math>\~~operatorname~~{g}</math>

+

<math>\mathrm{g}\!</math>

| width="22%" style="border-bottom:1px solid black" |

−

<math>\~~operatorname~~{h}</math>

+

<math>\mathrm{h}\!</math>

|- 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="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="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="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>

|}

−

This 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 took up earlier. That is to say, the story is the same, only the names have been changed. An abstract group can have a variety of significantly and superficially different representations. But even after we have long forgotten the details of any particular representation there is a type of concrete representations, called ''regular representations'', that are always readily available, as they can be generated from the mere data of the abstract operation table itself.

+

This 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 took up earlier. That is to say, the story is the same, only the names have been changed. An abstract group can have a variety of significantly and superficially different representations. But even after we have long forgotten the details of any particular representation there is a type of concrete representations, called ''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, select a group element from the top margin of the Table, and "consider its effects" on each of the group elements as they are listed along the left margin. We may record these effects as Peirce usually did, as a ''logical aggregate'' of elementary dyadic relatives, that is, as a logical disjunction or boolean sum whose terms represent the ordered pairs of <math>\~~operatorname~~{input} : \~~operatorname~~{output}</math> transactions that are produced by each group element in turn. This forms one of the two possible ''regular representations'' of the group, in this case 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 aggregate in the form of a matrix:

+

To see how a regular representation is constructed from the abstract operation table, select a group element from the top margin of the Table, and "consider its effects" on each of the group elements as they are listed along the left margin. We may record these effects as Peirce usually did, as a ''logical aggregate'' of elementary dyadic relatives, that is, as a logical disjunction or boolean sum whose terms represent the ordered pairs of <math>\mathrm{input} : \mathrm{output}</math> transactions that are produced by each group element in turn. This forms one of the two possible ''regular representations'' of the group, in this case 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 aggregate in the form of a matrix:

Reading "<math>+\!</math>" as a logical disjunction:

Line 2,726: Line 2,730:

| align="center" |

<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>

|}

Line 2,739: Line 2,743:

| align="center" |

<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]

−

& + & \~~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}

\\[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]

−

& + & \~~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>

|}

Line 2,767: Line 2,771:

{| align="center" cellpadding="6" width="90%"

−

| Every group is isomorphic to a subgroup of <math>\~~operatorname~~{Aut}(X),</math> the group of automorphisms of a suitably chosen set <math>X\!</math>.

+

| Every group is isomorphic to a subgroup of <math>\mathrm{Aut}(X),</math> the group of automorphisms of a suitably chosen set <math>X\!</math>.

|}

Line 2,776: Line 2,780:

|}

−

This idea of contextual definition by way of conduct transforming operators 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, ''From Frege to Gödel'', 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 by way of conduct transforming operators 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, ''From Frege to Gödel'', 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.

The next few excursions in this series will provide a scenic tour of various ideas in group theory that will turn out to be of constant guidance in several of the settings that are associated with our topic.

Line 2,805: Line 2,809:

|}

−

For example, given the set <math>X = \{ a, b, c \},\!</math> suppose that we have the 2-adic relative term <math>\mathit{m} = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~~~~~~~~}\, {}^{\prime\prime}</math> and

+

For example, given the set <math>X = \{ a, b, c \},\!</math> suppose that we have the 2-adic relative term <math>\mathit{m} = {}^{\backprime\backprime}\, \text{marker for}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 22:10, 8 December 2014 (UTC)}\, {}^{\prime\prime}</math> and

the associated 2-adic relation <math>M \subseteq X \times X,</math> the general pattern of whose common structure is represented by the following matrix:

Line 2,903: Line 2,907:

|}

−

Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{~~~~~~~~}\, {}^{\prime\prime}</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation.

+

Recognizing that <math>a\!:\!a + b\!:\!b + c\!:\!c</math> is the identity transformation otherwise known as <math>\mathit{1},\!</math> the 2-adic relative term <math>m = {}^{\backprime\backprime}\, \text{marker for}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 22:10, 8 December 2014 (UTC)}\, {}^{\prime\prime}</math> can be parsed as an element <math>\mathit{1} + a\!:\!b + b\!:\!c + c\!:\!a</math> of the so-called ''group ring'', all of which makes this element just a special sort of linear transformation.

Up to this point, we are still reading the elementary relatives of the form <math>i\!:\!j</math> in the way that Peirce read them in logical contexts: <math>i\!</math> is the relate, <math>j\!</math> is the correlate, and in our current example <math>i\!:\!j,</math> or more exactly, <math>m_{ij} = 1,\!</math> is taken to say that <math>i\!</math> is a marker for <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>a\!:\!b + b\!:\!c + c\!:\!a</math> is a permutation of the aggregate <math>\mathbf{1} = a + b + c,</math> or what we would now call the set <math>\{ a, b, 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>a\!:\!b + b\!:\!c + c\!:\!a\!</math> is a permutation of the aggregate <math>\mathbf{1} = a + b + c,\!</math> or what we would now call the set <math>\{ a, b, c \},\!</math> in particular, it is the permutation that is otherwise notated as follows:

{| align="center" cellpadding="6"

Line 2,929: Line 2,933:

{| 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"

−

| 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" |

+

| width="22%" style="border-bottom:1px solid black" |

−

<math>\~~operatorname~~{e}</math>

+

<math>\mathrm{e}\!</math>

| width="22%" style="border-bottom:1px solid black" |

−

<math>\~~operatorname~~{f}</math>

+

<math>\mathrm{f}\!</math>

| width="22%" style="border-bottom:1px solid black" |

−

<math>\~~operatorname~~{g}</math>

+

<math>\mathrm{g}\!</math>

| width="22%" style="border-bottom:1px solid black" |

−

<math>\~~operatorname~~{h}</math>

+

<math>\mathrm{h}\!</math>

|- 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="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="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="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>

|}

Line 2,970: Line 2,975:

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" |

<math>\begin{matrix}

−

(\~~operatorname~~{e}, \~~operatorname~~{e}, \~~operatorname~~{e}) &

+

(\mathrm{e}, \mathrm{e}, \mathrm{e}) &

−

(\~~operatorname~~{e}, \~~operatorname~~{f}, \~~operatorname~~{f}) &

+

(\mathrm{e}, \mathrm{f}, \mathrm{f}) &

−

(\~~operatorname~~{e}, \~~operatorname~~{g}, \~~operatorname~~{g}) &

+

(\mathrm{e}, \mathrm{g}, \mathrm{g}) &

−

(\~~operatorname~~{e}, \~~operatorname~~{h}, \~~operatorname~~{h})

+

(\mathrm{e}, \mathrm{h}, \mathrm{h})

\\[6pt]

−

(\~~operatorname~~{f}, \~~operatorname~~{e}, \~~operatorname~~{f}) &

+

(\mathrm{f}, \mathrm{e}, \mathrm{f}) &

−

(\~~operatorname~~{f}, \~~operatorname~~{f}, \~~operatorname~~{e}) &

+

(\mathrm{f}, \mathrm{f}, \mathrm{e}) &

−

(\~~operatorname~~{f}, \~~operatorname~~{g}, \~~operatorname~~{h}) &

+

(\mathrm{f}, \mathrm{g}, \mathrm{h}) &

−

(\~~operatorname~~{f}, \~~operatorname~~{h}, \~~operatorname~~{g})

+

(\mathrm{f}, \mathrm{h}, \mathrm{g})

\\[6pt]

−

(\~~operatorname~~{g}, \~~operatorname~~{e}, \~~operatorname~~{g}) &

+

(\mathrm{g}, \mathrm{e}, \mathrm{g}) &

−

(\~~operatorname~~{g}, \~~operatorname~~{f}, \~~operatorname~~{h}) &

+

(\mathrm{g}, \mathrm{f}, \mathrm{h}) &

−

(\~~operatorname~~{g}, \~~operatorname~~{g}, \~~operatorname~~{e}) &

+

(\mathrm{g}, \mathrm{g}, \mathrm{e}) &

−

(\~~operatorname~~{g}, \~~operatorname~~{h}, \~~operatorname~~{f})

+

(\mathrm{g}, \mathrm{h}, \mathrm{f})

\\[6pt]

−

(\~~operatorname~~{h}, \~~operatorname~~{e}, \~~operatorname~~{h}) &

+

(\mathrm{h}, \mathrm{e}, \mathrm{h}) &

−

(\~~operatorname~~{h}, \~~operatorname~~{f}, \~~operatorname~~{g}) &

+

(\mathrm{h}, \mathrm{f}, \mathrm{g}) &

−

(\~~operatorname~~{h}, \~~operatorname~~{g}, \~~operatorname~~{f}) &

+

(\mathrm{h}, \mathrm{g}, \mathrm{f}) &

−

(\~~operatorname~~{h}, \~~operatorname~~{h}, \~~operatorname~~{e})

+

(\mathrm{h}, \mathrm{h}, \mathrm{e})

\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%"

| 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>xy.\!</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>xy.~\!</math>

|-

| 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>yx.\!</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>yx.~\!</math>

|}

Line 3,012: Line 3,017:

| align="center" |

<math>\begin{matrix}

−

\~~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}

\\[4pt]

−

\~~operatorname~~{f}

+

\mathrm{f}

−

& = & \~~operatorname~~{e}\!:\!\~~operatorname~~{f}

+

& = & \mathrm{e}\!:\!\mathrm{f}

−

& + & \~~operatorname~~{f}\!:\!\~~operatorname~~{e}

+

& + & \mathrm{f}\!:\!\mathrm{e}

−

& + & \~~operatorname~~{g}\!:\!\~~operatorname~~{h}

+

& + & \mathrm{g}\!:\!\mathrm{h}

−

& + & \~~operatorname~~{h}\!:\!\~~operatorname~~{g}

+

& + & \mathrm{h}\!:\!\mathrm{g}

\\[4pt]

−

\~~operatorname~~{g}

+

\mathrm{g}

−

& = & \~~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]

−

\~~operatorname~~{h}

+

\mathrm{h}

−

& = & \~~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>

|}

Line 3,043: Line 3,048:

| align="center" |

<math>\begin{matrix}

−

\~~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}

\\[4pt]

−

\~~operatorname~~{f}

+

\mathrm{f}

−

& = & \~~operatorname~~{e}\!:\!\~~operatorname~~{f}

+

& = & \mathrm{e}\!:\!\mathrm{f}

−

& + & \~~operatorname~~{f}\!:\!\~~operatorname~~{e}

+

& + & \mathrm{f}\!:\!\mathrm{e}

−

& + & \~~operatorname~~{g}\!:\!\~~operatorname~~{h}

+

& + & \mathrm{g}\!:\!\mathrm{h}

−

& + & \~~operatorname~~{h}\!:\!\~~operatorname~~{g}

+

& + & \mathrm{h}\!:\!\mathrm{g}

\\[4pt]

−

\~~operatorname~~{g}

+

\mathrm{g}

−

& = & \~~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]

−

\~~operatorname~~{h}

+

\mathrm{h}

−

& = & \~~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>

|}

If the ante-rep looks the same as the post-rep, now that I'm writing them in the same dialect, that is because <math>V_4\!</math> is abelian (commutative), and so the two representations have the very same effects on each point of their bearing.

−

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>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

−

|+ <math>\text{Permutation Substitutions in}~ \~~operatorname~~{Sym} \{ a, b, c \}</math>

+

|+ <math>\text{Permutation Substitutions in}~ \mathrm{Sym} \{ a, b, c \}\!</math>

|- 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>

|-

|

Line 3,092: Line 3,097:

\\[6pt]

a & b & c

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 3,100: Line 3,105:

\\[6pt]

c & a & b

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 3,108: Line 3,113:

\\[6pt]

b & c & a

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 3,116: Line 3,121:

\\[6pt]

a & c & b

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 3,124: Line 3,129:

\\[6pt]

c & b & a

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 3,132: Line 3,137:

\\[6pt]

b & a & c

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|}

Line 3,145: Line 3,150:

|}

−

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).

−

By way of collecting a short-term pay-off for all the work that we did on 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 for the symmetric group on three letters, <math>\~~operatorname~~{Sym}(3).</math> After doing the usual bit of compare and contrast among the various representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscured details of Peirce's early "Algebra + Logic" papers.

+

By way of collecting a short-term pay-off for all the work that we did on 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 for the symmetric group on three letters, <math>\mathrm{Sym}(3).</math> After doing the usual bit of compare and contrast among the various representations, we will have enough concrete material beneath our abstract belts to tackle a few of the presently obscured details of Peirce's early “Algebra + Logic” papers.

−

Writing the permutations or substitutions of <math>\~~operatorname~~{Sym} \{ a, b, c \}</math> in relative form generates what is generally thought of as a ''natural representation'' of <math>S_3.\!</math>

+

Writing the permutations or substitutions of <math>\mathrm{Sym} \{ a, b, c \}\!</math> in relative form generates what is generally thought of as a ''natural representation'' of <math>S_3.~\!</math>

{| align="center" cellpadding="10" width="90%"

| align="center" |

<math>\begin{matrix}

−

\~~operatorname~~{e}

+

\mathrm{e}

& = & a\!:\!a

& + & b\!:\!b

& + & c\!:\!c

\\[4pt]

−

\~~operatorname~~{f}

+

\mathrm{f}

& = & a\!:\!c

& + & b\!:\!a

& + & c\!:\!b

\\[4pt]

−

\~~operatorname~~{g}

+

\mathrm{g}

& = & a\!:\!b

& + & b\!:\!c

& + & c\!:\!a

\\[4pt]

−

\~~operatorname~~{h}

+

\mathrm{h}

& = & a\!:\!a

& + & b\!:\!c

& + & c\!:\!b

\\[4pt]

−

\~~operatorname~~{i}

+

\mathrm{i}

& = & a\!:\!c

& + & b\!:\!b

& + & c\!:\!a

\\[4pt]

−

\~~operatorname~~{j}

+

\mathrm{j}

& = & a\!:\!b

& + & b\!:\!a

Line 3,186: Line 3,191:

|}

−

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 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 we will have to adjust our sense of direction, but that will be an easy enough bridge to cross when we come to it.

−

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="10" style="text-align:center"

−

| <math>\text{Symmetric Group}~ S_3</math>

+

| <math>\text{Symmetric Group}~ S_3\!</math>

|-

| [[Image:Symmetric Group S(3).jpg|500px]]

Line 3,198: Line 3,203:

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="10" width="90%"

| 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>xy.\!</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>xy.~\!</math>

|-

| 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>yx.\!</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>yx.~\!</math>

|}

Line 3,215: Line 3,220:

|

<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]

−

\~~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]

−

\~~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]

−

\~~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]

−

\~~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]

−

\~~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>

|}

Line 3,270: Line 3,275:

|

<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]

−

\~~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]

−

\~~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]

−

\~~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]

−

\~~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]

−

\~~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>

|}

−

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.

{| cellpadding="2" cellspacing="2" width="100%"

Line 3,336: Line 3,341:

The Reader may be wondering what happened to the announced subject of ''Dynamics And Logic''. What happened was a bit like this:

−

We made the observation that the shift operators <math>\{ \~~operatorname~~{E}_{ij} \}</math> form a transformation group that acts on the set of propositions of the form <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math> Group theory is a very attractive subject, but it did not draw us so far from our intended course as one might initially think. For one thing, groups, especially the groups that are named after the Norwegian mathematician [http://www-history.mcs.st-andrews.ac.uk/Biographies/Lie.html Marius Sophus Lie (1842–1899)], have turned out to be of critical utility in the solution of differential equations. For another thing, group operations provide us with an ample supply of triadic relations that have been extremely well-studied over the years, and thus they give us no small measure of useful guidance in the study of sign relations, another brand of 3-adic relations that have significance for logical studies, and in our acquaintance with which we have barely begun to break the ice. Finally, I couldn't resist taking up the links between group representations, amounting to the very archetypes of logical models, and the pragmatic maxim.

+

We made the observation that the shift operators <math>\{ \mathrm{E}_{ij} \}</math> form a transformation group that acts on the set of propositions of the form <math>f : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math> Group theory is a very attractive subject, but it did not draw us so far from our intended course as one might initially think. For one thing, groups, especially the groups that are named after the Norwegian mathematician [http://www-history.mcs.st-andrews.ac.uk/Biographies/Lie.html Marius Sophus Lie (1842–1899)], have turned out to be of critical utility in the solution of differential equations. For another thing, group operations provide us with an ample supply of triadic relations that have been extremely well-studied over the years, and thus they give us no small measure of useful guidance in the study of sign relations, another brand of 3-adic relations that have significance for logical studies, and in our acquaintance with which we have barely begun to break the ice. Finally, I couldn't resist taking up the links between group representations, amounting to the very archetypes of logical models, and the pragmatic maxim.

We've seen a couple of groups, <math>V_4\!</math> and <math>S_3,\!</math> represented in various ways, and we've seen their representations presented in a variety of different manners. Let us look at one other stylistic variant for presenting a representation that is frequently seen, the so-called ''matrix representation'' of a group.

Line 3,345: Line 3,350:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"

−

|+ <math>\text{Permutation Substitutions in}~ \~~operatorname~~{Sym} \{ a, b, c \}</math>

+

|+ <math>\text{Permutation Substitutions in}~ \mathrm{Sym} \{ a, b, c \}\!</math>

|- 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>

|-

|

Line 3,361: Line 3,366:

\\[6pt]

a & b & c

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 3,369: Line 3,374:

\\[6pt]

c & a & b

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 3,377: Line 3,382:

\\[6pt]

b & c & a

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 3,385: Line 3,390:

\\[6pt]

a & c & b

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 3,393: Line 3,398:

\\[6pt]

c & b & a

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 3,401: Line 3,406:

\\[6pt]

b & a & c

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|}

Line 3,411: Line 3,416:

| align="center" |

<math>\begin{matrix}

−

\~~operatorname~~{e}

+

\mathrm{e}

& = & a\!:\!a

& + & b\!:\!b

& + & c\!:\!c

\\[4pt]

−

\~~operatorname~~{f}

+

\mathrm{f}

& = & a\!:\!c

& + & b\!:\!a

& + & c\!:\!b

\\[4pt]

−

\~~operatorname~~{g}

+

\mathrm{g}

& = & a\!:\!b

& + & b\!:\!c

& + & c\!:\!a

\\[4pt]

−

\~~operatorname~~{h}

+

\mathrm{h}

& = & a\!:\!a

& + & b\!:\!c

& + & c\!:\!b

\\[4pt]

−

\~~operatorname~~{i}

+

\mathrm{i}

& = & a\!:\!c

& + & b\!:\!b

& + & c\!:\!a

\\[4pt]

−

\~~operatorname~~{j}

+

\mathrm{j}

& = & a\!:\!b

& + & b\!:\!a

Line 3,443: Line 3,448:

|}

−

From the relational representation of <math>\~~operatorname~~{Sym} \{ a, b, c \} \cong S_3,</math> one easily derives a ''linear representation'' of the group by viewing each permutation as a linear transformation that maps the elements of a suitable vector space onto each other. Each of these linear transformations is in turn represented by a 2-dimensional array of coefficients in <math>\mathbb{B},</math> resulting in the following set of matrices for the group:

+

From the relational representation of <math>\mathrm{Sym} \{ a, b, c \} \cong S_3,</math> one easily derives a ''linear representation'' of the group by viewing each permutation as a linear transformation that maps the elements of a suitable vector space onto each other. Each of these linear transformations is in turn represented by a 2-dimensional array of coefficients in <math>\mathbb{B},</math> resulting in the following set of matrices for the group:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="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"

−

| 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>

|-

|

Line 3,464: Line 3,469:

\\

0 & 0 & 1

−

\end{matrix}</math>

+

\end{matrix}~\!</math>

|

<math>\begin{matrix}

Line 3,472: Line 3,477:

\\

0 & 1 & 0

−

\end{matrix}</math>

+

\end{matrix}~\!</math>

|

<math>\begin{matrix}

Line 3,480: Line 3,485:

\\

1 & 0 & 0

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|

<math>\begin{matrix}

Line 3,488: Line 3,493:

\\

0 & 1 & 0

−

\end{matrix}</math>

+

\end{matrix}~\!</math>

|

<math>\begin{matrix}

Line 3,496: Line 3,501:

\\

1 & 0 & 0

−

\end{matrix}</math>

+

\end{matrix}~\!</math>

|

<math>\begin{matrix}

Line 3,504: Line 3,509:

\\

0 & 0 & 1

−

\end{matrix}</math>

+

\end{matrix}~\!</math>

|}

Line 3,530: Line 3,535:

==Quick Review : Field Picture==

−

Let us summarize, in rough but intuitive terms, the outlook on differential logic that we have reached so far. We've been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse, <math>X^\circ,</math> to considering a larger universe of discourse, <math>\~~operatorname~~{E}X^\circ.</math> An operator <math>\~~operatorname~~{W}</math> of this general type, namely, <math>\~~operatorname~~{W} : X^\circ \to \~~operatorname~~{E}X^\circ,</math> acts on each proposition <math>f : X \to \mathbb{B}</math> of the source universe <math>X^\circ</math> to produce a proposition <math>\~~operatorname~~{W}f : \~~operatorname~~{E}X \to \mathbb{B}</math> of the target universe <math>\~~operatorname~~{E}X^\circ.</math>

+

Let us summarize, in rough but intuitive terms, the outlook on differential logic that we have reached so far. We've been considering a class of operators on universes of discourse, each of which takes us from considering one universe of discourse, <math>X^\circ,</math> to considering a larger universe of discourse, <math>\mathrm{E}X^\circ.</math> An operator <math>\mathrm{W}</math> of this general type, namely, <math>\mathrm{W} : X^\circ \to \mathrm{E}X^\circ,</math> acts on each proposition <math>f : X \to \mathbb{B}</math> of the source universe <math>X^\circ</math> to produce a proposition <math>\mathrm{W}f : \mathrm{E}X \to \mathbb{B}</math> of the target universe <math>\mathrm{E}X^\circ.</math>

−

The two main operators that we've examined so far are the enlargement or shift operator <math>\~~operatorname~~{E} : X^\circ \to \~~operatorname~~{E}X^\circ</math> and the difference operator <math>\~~operatorname~~{D} : X^\circ \to \~~operatorname~~{E}X^\circ.</math> The operators <math>\~~operatorname~~{E}</math> and <math>\~~operatorname~~{D}</math> act on propositions in <math>X^\circ,</math> that is, propositions of the form <math>f : X \to \mathbb{B}</math> that are said to be ''about'' the subject matter of <math>X,\!</math> and they produce extended propositions of the forms <math>\~~operatorname~~{E}f, \~~operatorname~~{D}f : \~~operatorname~~{E}X \to \mathbb{B},</math> propositions whose extended sets of variables allow them to be read as being about specified collections of changes that conceivably occur in <math>X.\!</math>

+

The two main operators that we've examined so far are the enlargement or shift operator <math>\mathrm{E} : X^\circ \to \mathrm{E}X^\circ</math> and the difference operator <math>\mathrm{D} : X^\circ \to \mathrm{E}X^\circ.</math> The operators <math>\mathrm{E}</math> and <math>\mathrm{D}</math> act on propositions in <math>X^\circ,</math> that is, propositions of the form <math>f : X \to \mathbb{B}</math> that are said to be ''about'' the subject matter of <math>X,\!</math> and they produce extended propositions of the forms <math>\mathrm{E}f, \mathrm{D}f : \mathrm{E}X \to \mathbb{B},</math> propositions whose extended sets of variables allow them to be read as being about specified collections of changes that conceivably occur in <math>X.\!</math>

At this point we find ourselves in need of visual representations, suitable arrays of concrete pictures to anchor our more earthy intuitions and to help us keep our wits about us as we venture higher into the ever more rarefied air of abstractions.

Line 3,548: Line 3,553:

|}

−

Each of the operators <math>\~~operatorname~~{E}, \~~operatorname~~{D} : X^\circ \to \~~operatorname~~{E}X^\circ</math> takes us from considering propositions <math>f : X \to \mathbb{B},</math> here viewed as ''scalar fields'' over <math>X,\!</math> to considering the corresponding ''differential fields'' over <math>X,\!</math> analogous to what are usually called ''vector fields'' over <math>X.\!</math>

+

Each of the operators <math>\mathrm{E}, \mathrm{D} : X^\circ \to \mathrm{E}X^\circ</math> takes us from considering propositions <math>f : X \to \mathbb{B},</math> here viewed as ''scalar fields'' over <math>X,\!</math> to considering the corresponding ''differential fields'' over <math>X,\!</math> analogous to what are usually called ''vector fields'' over <math>X.\!</math>

−

The structure of these differential fields can be described this way. With each point of <math>X\!</math> there is associated an object of the following type: a proposition about changes in <math>X,\!</math> that is, a proposition <math>g : \~~operatorname~~{d}X \to \mathbb{B}.</math> In this frame of reference, if <math>X^\circ</math> is the universe that is generated by the set of coordinate propositions <math>\{ p, q \},\!</math> then <math>\~~operatorname~~{d}X^\circ</math> is the differential universe that is generated by the set of differential propositions <math>\{ \~~operatorname~~{d}p, \~~operatorname~~{d}q \}.</math> These differential propositions may be interpreted as indicating <math>{}^{\backprime\backprime} \text{change in}\, p \, {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \text{change in}\, q \, {}^{\prime\prime},</math> respectively.

+

The structure of these differential fields can be described this way. With each point of <math>X\!</math> there is associated an object of the following type: a proposition about changes in <math>X,\!</math> that is, a proposition <math>g : \mathrm{d}X \to \mathbb{B}.</math> In this frame of reference, if <math>X^\circ</math> is the universe that is generated by the set of coordinate propositions <math>\{ p, q \},\!</math> then <math>\mathrm{d}X^\circ</math> is the differential universe that is generated by the set of differential propositions <math>\{ \mathrm{d}p, \mathrm{d}q \}.</math> These differential propositions may be interpreted as indicating <math>{}^{\backprime\backprime} \text{change in}\, p \, {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \text{change in}\, q \, {}^{\prime\prime},</math> respectively.

−

A differential operator <math>\~~operatorname~~{W},</math> of the first order class that we have been considering, takes a proposition <math>f : X \to \mathbb{B}</math> and gives back a differential proposition <math>\~~operatorname~~{W}f : \~~operatorname~~{E}X \to \mathbb{B}.</math> In the field view, we see the proposition <math>f : X \to \mathbb{B}</math> as a scalar field and we see the differential proposition <math>\~~operatorname~~{W}f : \~~operatorname~~{E}X \to \mathbb{B}</math> as a vector field, specifically, a field of propositions about contemplated changes in <math>X.\!</math>

+

A differential operator <math>\mathrm{W},</math> of the first order class that we have been considering, takes a proposition <math>f : X \to \mathbb{B}</math> and gives back a differential proposition <math>\mathrm{W}f : \mathrm{E}X \to \mathbb{B}.</math> In the field view, we see the proposition <math>f : X \to \mathbb{B}</math> as a scalar field and we see the differential proposition <math>\mathrm{W}f : \mathrm{E}X \to \mathbb{B}</math> as a vector field, specifically, a field of propositions about contemplated changes in <math>X.\!</math>

−

The field of changes produced by <math>\~~operatorname~~{E}</math> on <math>pq\!</math> is shown in the next venn diagram:

+

The field of changes produced by <math>\mathrm{E}</math> on <math>pq\!</math> is shown in the next venn diagram:

{| align="center" cellpadding="10" style="text-align:center"

| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]

|-

−

| <math>\text{Enlargement}~ \~~operatorname~~{E}(pq) : \~~operatorname~~{E}X \to \mathbb{B}</math>

+

| <math>\text{Enlargement}~ \mathrm{E}(pq) : \mathrm{E}X \to \mathbb{B}</math>

|-

|

<math>\begin{array}{rcccccc}

−

\~~operatorname~~{E}(pq)

+

\mathrm{E}(pq)

& = &

p

Line 3,569: Line 3,574:

q

& \cdot &

−

\texttt{(} \~~operatorname~~{d}p \texttt{)}

+

\texttt{(} \mathrm{d}p \texttt{)}

−

\texttt{(} \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} \mathrm{d}q \texttt{)}

\\[4pt]

& + &

Line 3,577: Line 3,582:

\texttt{(} q \texttt{)}

& \cdot &

−

\texttt{(} \~~operatorname~~{d}p \texttt{)}

+

\texttt{(} \mathrm{d}p \texttt{)}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

\\[4pt]

& + &

Line 3,585: Line 3,590:

q

& \cdot &

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{(} \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} \mathrm{d}q \texttt{)}

\\[4pt]

& + &

Line 3,593: Line 3,598:

\texttt{(} q \texttt{)}

& \cdot &

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

−

\end{array}</math>

+

\end{array}\!</math>

|}

−

The differential field <math>\~~operatorname~~{E}(pq)</math> specifies the changes that need to be made from each point of <math>X\!</math> in order to reach one of the models of the proposition <math>pq,\!</math> that is, in order to satisfy the proposition <math>pq.\!</math>

+

The differential field <math>\mathrm{E}(pq)</math> specifies the changes that need to be made from each point of <math>X\!</math> in order to reach one of the models of the proposition <math>pq,\!</math> that is, in order to satisfy the proposition <math>pq.\!</math>

−

The field of changes produced by <math>\~~operatorname~~{D}\!</math> on <math>pq\!</math> is shown in the following venn diagram:

+

The field of changes produced by <math>\mathrm{D}\!</math> on <math>pq\!</math> is shown in the following venn diagram:

{| align="center" cellpadding="10" style="text-align:center"

| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]

|-

−

| <math>\text{Difference}~ \~~operatorname~~{D}(pq) : \~~operatorname~~{E}X \to \mathbb{B}</math>

+

| <math>\text{Difference}~ \mathrm{D}(pq) : \mathrm{E}X \to \mathbb{B}</math>

|-

|

<math>\begin{array}{rcccccc}

−

\~~operatorname~~{D}(pq)

+

\mathrm{D}(pq)

& = &

p

Line 3,616: Line 3,621:

& \cdot &

\texttt{(}

−

\texttt{(} \~~operatorname~~{d}p \texttt{)}

+

\texttt{(} \mathrm{d}p \texttt{)}

−

\texttt{(} \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} \mathrm{d}q \texttt{)}

\texttt{)}

\\[4pt]

Line 3,626: Line 3,631:

& \cdot &

\texttt{~}

−

\texttt{(} \~~operatorname~~{d}p \texttt{)}

+

\texttt{(} \mathrm{d}p \texttt{)}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

\texttt{~}

\\[4pt]

Line 3,636: Line 3,641:

& \cdot &

\texttt{~}

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{(} \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} \mathrm{d}q \texttt{)}

\texttt{~}

\\[4pt]

Line 3,646: Line 3,651:

& \cdot &

\texttt{~}

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

\texttt{~}

\end{array}</math>

|}

−

The differential field <math>\~~operatorname~~{D}(pq)</math> specifies the changes that need to be made from each point of <math>X\!</math> in order to feel a change in the felt value of the field <math>pq.\!</math>

+

The differential field <math>\mathrm{D}(pq)</math> specifies the changes that need to be made from each point of <math>X\!</math> in order to feel a change in the felt value of the field <math>pq.\!</math>

===Proposition and Tacit Extension===

−

Now that we've introduced the field picture as an aid to thinking about propositions and their analytic series, a very pleasing way of picturing the relationships among a proposition <math>f : X \to \mathbb{B},</math> its enlargement or shift map <math>\~~operatorname~~{E}f : \~~operatorname~~{E}X \to \mathbb{B},</math> and its difference map <math>\~~operatorname~~{D}f : \~~operatorname~~{E}X \to \mathbb{B}</math> can now be drawn.

+

Now that we've introduced the field picture as an aid to thinking about propositions and their analytic series, a very pleasing way of picturing the relationships among a proposition <math>f : X \to \mathbb{B},</math> its enlargement or shift map <math>\mathrm{E}f : \mathrm{E}X \to \mathbb{B},</math> and its difference map <math>\mathrm{D}f : \mathrm{E}X \to \mathbb{B}</math> can now be drawn.

To illustrate this possibility, let's return to the differential analysis of the conjunctive proposition <math>f(p, q) = pq,\!</math> giving the development a slightly different twist at the appropriate point.

Line 3,668: Line 3,673:

|}

−

Given a proposition <math>f : X \to \mathbb{B},</math> the ''tacit extension'' of <math>f\!</math> to <math>\~~operatorname~~{E}X</math> is denoted <math>\varepsilon f : \~~operatorname~~{E}X \to \mathbb{B}</math> and defined by the equation <math>\varepsilon f = f,</math> so it's really just the same proposition residing in a bigger universe. Tacit extensions formalize the intuitive idea that a function on a particular set of variables can be extended to a function on a superset of those variables in such a way that the new function obeys the same constraints on the old variables, with a "don't care" condition on the new variables.

+

Given a proposition <math>f : X \to \mathbb{B},</math> the ''tacit extension'' of <math>f\!</math> to <math>\mathrm{E}X</math> is denoted <math>\varepsilon f : \mathrm{E}X \to \mathbb{B}</math> and defined by the equation <math>\varepsilon f = f,</math> so it's really just the same proposition residing in a bigger universe. Tacit extensions formalize the intuitive idea that a function on a particular set of variables can be extended to a function on a superset of those variables in such a way that the new function obeys the same constraints on the old variables, with a "don't care" condition on the new variables.

−

The tacit extension of the scalar field <math>pq : X \to \mathbb{B}</math> to the differential field <math>\varepsilon (pq) : \~~operatorname~~{E}X \to \mathbb{B}</math> is shown in the following venn diagram:

+

The tacit extension of the scalar field <math>pq : X \to \mathbb{B}</math> to the differential field <math>\varepsilon (pq) : \mathrm{E}X \to \mathbb{B}</math> is shown in the following venn diagram:

{| align="center" cellpadding="10" style="text-align:center"

| [[Image:Field Picture PQ Tacit Extension Conjunction.jpg|500px]]

|-

−

| <math>\text{Tacit Extension}~ \varepsilon (pq) : \~~operatorname~~{E}X \to \mathbb{B}</math>

+

| <math>\text{Tacit Extension}~ \varepsilon (pq) : \mathrm{E}X \to \mathbb{B}</math>

|-

|

Line 3,682: Line 3,687:

& = &

p & \cdot & q & \cdot &

−

\texttt{(} \~~operatorname~~{d}p \texttt{)}

+

\texttt{(} \mathrm{d}p \texttt{)}

−

\texttt{(} \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} \mathrm{d}q \texttt{)}

\\[4pt]

& + &

p & \cdot & q & \cdot &

−

\texttt{(} \~~operatorname~~{d}p \texttt{)}

+

\texttt{(} \mathrm{d}p \texttt{)}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

\\[4pt]

& + &

p & \cdot & q & \cdot &

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{(} \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} \mathrm{d}q \texttt{)}

\\[4pt]

& + &

p & \cdot & q & \cdot &

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

\end{array}</math>

|}

Line 3,704: Line 3,709:

===Enlargement and Difference Maps===

−

Continuing with the example <math>pq : X \to \mathbb{B},</math> the next venn diagram shows the enlargement or shift map <math>\~~operatorname~~{E}(pq) : \~~operatorname~~{E}X \to \mathbb{B}</math> in the same style of differential field picture that we drew for the tacit extension <math>\varepsilon (pq) : \~~operatorname~~{E}X \to \mathbb{B}.</math>

+

Continuing with the example <math>pq : X \to \mathbb{B},</math> the next venn diagram shows the enlargement or shift map <math>\mathrm{E}(pq) : \mathrm{E}X \to \mathbb{B}</math> in the same style of differential field picture that we drew for the tacit extension <math>\varepsilon (pq) : \mathrm{E}X \to \mathbb{B}.</math>

{| align="center" cellpadding="10" style="text-align:center"

| [[Image:Field Picture PQ Enlargement Conjunction.jpg|500px]]

|-

−

| <math>\text{Enlargement Map}~ \~~operatorname~~{E}(pq) : \~~operatorname~~{E}X \to \mathbb{B}</math>

+

| <math>\text{Enlargement Map}~ \mathrm{E}(pq) : \mathrm{E}X \to \mathbb{B}</math>

|-

|

<math>\begin{array}{rcccccc}

−

\~~operatorname~~{E}(pq)

+

\mathrm{E}(pq)

& = &

p

Line 3,719: Line 3,724:

q

& \cdot &

−

\texttt{(} \~~operatorname~~{d}p \texttt{)}

+

\texttt{(} \mathrm{d}p \texttt{)}

−

\texttt{(} \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} \mathrm{d}q \texttt{)}

\\[4pt]

& + &

Line 3,727: Line 3,732:

\texttt{(} q \texttt{)}

& \cdot &

−

\texttt{(} \~~operatorname~~{d}p \texttt{)}

+

\texttt{(} \mathrm{d}p \texttt{)}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

\\[4pt]

& + &

Line 3,735: Line 3,740:

q

& \cdot &

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{(} \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} \mathrm{d}q \texttt{)}

\\[4pt]

& + &

Line 3,743: Line 3,748:

\texttt{(} q \texttt{)}

& \cdot &

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

−

\end{array}</math>

+

\end{array}\!</math>

|}

−

A very important conceptual transition has just occurred here, almost tacitly, as it were. Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields <math>\varepsilon f</math> and <math>\~~operatorname~~{E}f,</math> both of the type <math>\~~operatorname~~{E}X \to \mathbb{B},</math> is very useful, because it allows us to consider these fields as integral mathematical objects that can be operated on and combined in the ways that we usually associate with algebras.

+

A very important conceptual transition has just occurred here, almost tacitly, as it were. Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields <math>\varepsilon f</math> and <math>\mathrm{E}f,</math> both of the type <math>\mathrm{E}X \to \mathbb{B},</math> is very useful, because it allows us to consider these fields as integral mathematical objects that can be operated on and combined in the ways that we usually associate with algebras.

−

In this case one notices that the tacit extension <math>\varepsilon f</math> and the enlargement <math>\~~operatorname~~{E}f</math> are in a certain sense dual to each other. The tacit extension <math>\varepsilon f</math> indicates all the arrows out of the region where <math>f\!</math> is true and the enlargement <math>\~~operatorname~~{E}f</math> indicates all the arrows into the region where <math>f\!</math> is true. The only arc they have in common is the no-change loop <math>\texttt{(} \~~operatorname~~{d}p \texttt{)(} \~~operatorname~~{d}q \texttt{)}</math> at <math>pq.\!</math> If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of <math>\~~operatorname~~{D}(pq) = \varepsilon(pq) + \~~operatorname~~{E}(pq)</math> that are illustrated below:

+

In this case one notices that the tacit extension <math>\varepsilon f</math> and the enlargement <math>\mathrm{E}f</math> are in a certain sense dual to each other. The tacit extension <math>\varepsilon f</math> indicates all the arrows out of the region where <math>f\!</math> is true and the enlargement <math>\mathrm{E}f</math> indicates all the arrows into the region where <math>f\!</math> is true. The only arc they have in common is the no-change loop <math>\texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)}</math> at <math>pq.\!</math> If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of <math>\mathrm{D}(pq) = \varepsilon(pq) + \mathrm{E}(pq)</math> that are illustrated below:

{| align="center" cellpadding="10" style="text-align:center"

| [[Image:Field Picture PQ Difference Conjunction.jpg|500px]]

|-

−

| <math>\text{Difference Map}~ \~~operatorname~~{D}(pq) : \~~operatorname~~{E}X \to \mathbb{B}</math>

+

| <math>\text{Difference Map}~ \mathrm{D}(pq) : \mathrm{E}X \to \mathbb{B}</math>

|-

|

<math>\begin{array}{rcccccc}

−

\~~operatorname~~{D}(pq)

+

\mathrm{D}(pq)

& = &

p

Line 3,766: Line 3,771:

& \cdot &

\texttt{(}

−

\texttt{(} \~~operatorname~~{d}p \texttt{)}

+

\texttt{(} \mathrm{d}p \texttt{)}

−

\texttt{(} \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} \mathrm{d}q \texttt{)}

\texttt{)}

\\[4pt]

Line 3,776: Line 3,781:

& \cdot &

\texttt{~}

−

\texttt{(} \~~operatorname~~{d}p \texttt{)}

+

\texttt{(} \mathrm{d}p \texttt{)}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

\texttt{~}

\\[4pt]

Line 3,786: Line 3,791:

& \cdot &

\texttt{~}

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{(} \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} \mathrm{d}q \texttt{)}

\texttt{~}

\\[4pt]

Line 3,796: Line 3,801:

& \cdot &

\texttt{~}

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

\texttt{~}

\end{array}</math>

Line 3,806: Line 3,811:

If we follow the classical line that singles out linear functions as ideals of simplicity, then we may complete the analytic series of the proposition <math>f = pq : X \to \mathbb{B}</math> in the following way.

−

The next venn diagram shows the differential proposition <math>\~~operatorname~~{d}f = \~~operatorname~~{d}(pq) : \~~operatorname~~{E}X \to \mathbb{B}</math> that we get by extracting the cell-wise linear approximation to the difference map <math>\~~operatorname~~{D}f = \~~operatorname~~{D}(pq) : \~~operatorname~~{E}X \to \mathbb{B}.</math> This is the logical analogue of what would ordinarily be called ''the'' differential of <math>pq,\!</math> but since I've been attaching the adjective ''differential'' to just about everything in sight, the distinction tends to be lost. For the time being, I'll resort to using the alternative name ''tangent map'' for <math>\~~operatorname~~{d}f.\!</math>

+

The next venn diagram shows the differential proposition <math>\mathrm{d}f = \mathrm{d}(pq) : \mathrm{E}X \to \mathbb{B}</math> that we get by extracting the cell-wise linear approximation to the difference map <math>\mathrm{D}f = \mathrm{D}(pq) : \mathrm{E}X \to \mathbb{B}.</math> This is the logical analogue of what would ordinarily be called ''the'' differential of <math>pq,\!</math> but since I've been attaching the adjective ''differential'' to just about everything in sight, the distinction tends to be lost. For the time being, I'll resort to using the alternative name ''tangent map'' for <math>\mathrm{d}f.\!</math>

{| align="center" cellpadding="10" style="text-align:center"

| [[Image:Field Picture PQ Differential Conjunction.jpg|500px]]

|-

−

| <math>\text{Tangent Map}~ \~~operatorname~~{d}(pq) : \~~operatorname~~{E}X \to \mathbb{B}</math>

+

| <math>\text{Tangent Map}~ \mathrm{d}(pq) : \mathrm{E}X \to \mathbb{B}</math>

|}

Line 3,819: Line 3,824:

|

<math>\begin{array}{rcccccc}

−

\~~operatorname~~{d}(pq)

+

\mathrm{d}(pq)

& = &

p & \cdot & q & \cdot &

−

\texttt{(} \~~operatorname~~{d}p \texttt{,} \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} \mathrm{d}p \texttt{,} \mathrm{d}q \texttt{)}

\\[4pt]

& + &

p & \cdot & \texttt{(} q \texttt{)} & \cdot &

−

\~~operatorname~~{d}q

+

\mathrm{d}q

\\[4pt]

& + &

\texttt{(} p \texttt{)} & \cdot & q & \cdot &

−

\~~operatorname~~{d}p

+

\mathrm{d}p

\\[4pt]

& + &

Line 3,843: Line 3,848:

<math>\begin{matrix}

\texttt{(}

−

\~~operatorname~~{d}p

+

\mathrm{d}p

\texttt{,}

−

\~~operatorname~~{d}q

+

\mathrm{d}q

\texttt{)}

& = &

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{(} \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} \mathrm{d}q \texttt{)}

& + &

−

\texttt{(} \~~operatorname~~{d}p \texttt{)}

+

\texttt{(} \mathrm{d}p \texttt{)}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

\\[4pt]

dp

& = &

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

& + &

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{(} \~~operatorname~~{d}q \texttt{)}

+

\texttt{(} \mathrm{d}q \texttt{)}

\\[4pt]

−

\~~operatorname~~{d}q

+

\mathrm{d}q

& = &

−

\texttt{~} \~~operatorname~~{d}p \texttt{~}

+

\texttt{~} \mathrm{d}p \texttt{~}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

& + &

−

\texttt{(} \~~operatorname~~{d}p \texttt{)}

+

\texttt{(} \mathrm{d}p \texttt{)}

−

\texttt{~} \~~operatorname~~{d}q \texttt{~}

+

\texttt{~} \mathrm{d}q \texttt{~}

\end{matrix}</math>

|}

−

Capping the series that analyzes the proposition <math>pq\!</math> in terms of succeeding orders of linear propositions, the final venn diagram in this series shows the remainder map <math>\~~operatorname~~{r}(pq) : \~~operatorname~~{E}X \to \mathbb{B},</math> that happens to be linear in pairs of variables.

+

Capping the series that analyzes the proposition <math>pq\!</math> in terms of succeeding orders of linear propositions, the final venn diagram in this series shows the remainder map <math>\mathrm{r}(pq) : \mathrm{E}X \to \mathbb{B},</math> that happens to be linear in pairs of variables.

{| align="center" cellpadding="10" style="text-align:center"

| [[Image:Field Picture PQ Remainder Conjunction.jpg|500px]]

|-

−

| <math>\text{Remainder Map}~ \~~operatorname~~{r}(pq) : \~~operatorname~~{E}X \to \mathbb{B}</math>

+

| <math>\text{Remainder Map}~ \mathrm{r}(pq) : \mathrm{E}X \to \mathbb{B}</math>

|}

Line 3,885: Line 3,890:

|

<math>\begin{array}{rcccccc}

−

\~~operatorname~~{r}(pq)

+

\mathrm{r}(pq)

& = &

p & \cdot & q & \cdot &

−

\~~operatorname~~{d}p ~ \~~operatorname~~{d}q

+

\mathrm{d}p ~ \mathrm{d}q

\\[4pt]

& + &

p & \cdot & \texttt{(} q \texttt{)} & \cdot &

−

\~~operatorname~~{d}p ~ \~~operatorname~~{d}q

+

\mathrm{d}p ~ \mathrm{d}q

\\[4pt]

& + &

\texttt{(} p \texttt{)} & \cdot & q & \cdot &

−

\~~operatorname~~{d}p ~ \~~operatorname~~{d}q

+

\mathrm{d}p ~ \mathrm{d}q

\\[4pt]

& + &

\texttt{(} p \texttt{)} & \cdot & \texttt{(} q \texttt{)} & \cdot &

−

\~~operatorname~~{d}p ~ \~~operatorname~~{d}q

+

\mathrm{d}p ~ \mathrm{d}q

\end{array}</math>

|}

−

In short, <math>\~~operatorname~~{r}(pq)</math> is a constant field, having the value <math>\~~operatorname~~{d}p~\~~operatorname~~{d}q</math> at each cell.

+

In short, <math>\mathrm{r}(pq)</math> is a constant field, having the value <math>\mathrm{d}p~\mathrm{d}q</math> at each cell.

==Least Action Operators==

−

We have been contemplating functions of the type <math>f : X \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 : X \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''.

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{| align="center" cellpadding="10"

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| [[Image:Minimal Negation Operator (p,q,r).jpg|500px]]

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In relation to the center cell indicated by the conjunction <math>pqr,\!</math> the region indicated by <math>\texttt{(} p, q, r \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, reached as if by way of Leibniz's ''minimal changes'' from the point of origin, in this case, <math>pqr.\!</math>

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In relation to the center cell indicated by the conjunction <math>pqr,\!</math> the region indicated by <math>\texttt{(} p, q, r \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, reached as if by way of Leibniz's ''minimal changes'' from the point of origin, in this case, <math>pqr.~\!</math>

More generally speaking, in a <math>k\!</math>-dimensional universe of discourse that is based on the ''alphabet'' of features <math>\mathcal{X} = \{ x_1, \ldots, x_k \},</math> the same form of boundary relationship is manifested 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>

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I want to continue developing the basic tools of differential logic, which arose from exploring the connections between dynamics and logic, but I also wanted to give some hint of the applications that have motivated this work all along. One of these applications is to cybernetic systems, whether we see these systems as agents or cultures, individuals or species, organisms or organizations.

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A cybernetic system has goals and actions for reaching them. It has a state space <math>X,\!</math> giving us all of the states that the system can be in, plus it has a goal space <math>G \subseteq X,</math> the set of states that the system "likes" to be in, in other words, the distinguished subset of possible states where the system is regarded as living, surviving, or thriving, depending on the type of goal that one has in mind for the system in question. As for actions, there is to begin with the full set <math>\mathcal{T}</math> of all possible actions, each of which is a transformation of the form <math>T : X \to X,</math> but a given cybernetic system will most likely have but a subset of these actions available to it at any given time. And even if we begin by thinking of actions in very general and very global terms, as arbitrarily complex transformations acting on the whole state space <math>X,\!</math> we quickly find a need to analyze and approximate them in terms of simple transformations acting locally. The preferred measure of "simplicity" will of course vary from one paradigm of research to another.

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A cybernetic system has goals and actions for reaching them. It has a state space <math>X,\!</math> giving us all of the states that the system can be in, plus it has a goal space <math>G \subseteq X,\!</math> the set of states that the system “likes” to be in, in other words, the distinguished subset of possible states where the system is regarded as living, surviving, or thriving, depending on the type of goal that one has in mind for the system in question. As for actions, there is to begin with the full set <math>\mathcal{T}\!</math> of all possible actions, each of which is a transformation of the form <math>T : X \to X,\!</math> but a given cybernetic system will most likely have but a subset of these actions available to it at any given time. And even if we begin by thinking of actions in very general and very global terms, as arbitrarily complex transformations acting on the whole state space <math>X,\!</math> we quickly find a need to analyze and approximate them in terms of simple transformations acting locally. The preferred measure of “simplicity” will of course vary from one paradigm of research to another.

A generic enough picture at this stage of the game, and one that will remind us of these fundamental features of the cybernetic system even as things get far more complex, is afforded by Figure 23.

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A more detailed presentation of Differential Logic can be found here:

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:* [~~[Directory~~:~~Jon_Awbrey~~/~~Papers~~/Differential_Logic_and_Dynamic_Systems_2.0|Differential Logic and Dynamic Systems]]

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:* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]

==Document History==

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===Ontology List ~~(Apr–Jul~~ 2002)===

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===Differential Logic • Ontology List 2002===

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* http://suo.ieee.org/ontology/thrd28.html#04040

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* http://web.archive.org/web/20110612002240/http://suo.ieee.org/ontology/thrd28.html#04040

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# http://suo.ieee.org/ontology/msg04040.html

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# http://web.archive.org/web/20140406040004/http://suo.ieee.org/ontology/msg04040.html

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# http://suo.ieee.org/ontology/msg04041.html

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# http://web.archive.org/web/20110612001949/http://suo.ieee.org/ontology/msg04041.html

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# http://suo.ieee.org/ontology/msg04045.html

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# http://web.archive.org/web/20110612010502/http://suo.ieee.org/ontology/msg04045.html

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# http://suo.ieee.org/ontology/msg04046.html

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# http://web.archive.org/web/20110612005212/http://suo.ieee.org/ontology/msg04046.html

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# http://suo.ieee.org/ontology/msg04047.html

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# http://web.archive.org/web/20110612001954/http://suo.ieee.org/ontology/msg04047.html

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# http://suo.ieee.org/ontology/msg04048.html

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# http://web.archive.org/web/20110612010620/http://suo.ieee.org/ontology/msg04048.html

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# http://suo.ieee.org/ontology/msg04052.html

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# http://web.archive.org/web/20110612010550/http://suo.ieee.org/ontology/msg04052.html

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# http://suo.ieee.org/ontology/msg04054.html

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# http://web.archive.org/web/20110612010724/http://suo.ieee.org/ontology/msg04054.html

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# http://suo.ieee.org/ontology/msg04055.html

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# http://web.archive.org/web/20110612000847/http://suo.ieee.org/ontology/msg04055.html

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# http://suo.ieee.org/ontology/msg04067.html

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# http://web.archive.org/web/20110612001959/http://suo.ieee.org/ontology/msg04067.html

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# http://suo.ieee.org/ontology/msg04068.html

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# http://web.archive.org/web/20110612010507/http://suo.ieee.org/ontology/msg04068.html

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# http://suo.ieee.org/ontology/msg04069.html

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# http://web.archive.org/web/20110612002014/http://suo.ieee.org/ontology/msg04069.html

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# http://suo.ieee.org/ontology/msg04070.html

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# http://web.archive.org/web/20110612010701/http://suo.ieee.org/ontology/msg04070.html

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# http://suo.ieee.org/ontology/msg04072.html

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# http://web.archive.org/web/20110612003540/http://suo.ieee.org/ontology/msg04072.html

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# http://suo.ieee.org/ontology/msg04073.html

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# http://web.archive.org/web/20110612005229/http://suo.ieee.org/ontology/msg04073.html

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# http://suo.ieee.org/ontology/msg04074.html

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# http://web.archive.org/web/20110610153117/http://suo.ieee.org/ontology/msg04074.html

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# http://suo.ieee.org/ontology/msg04077.html

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# http://web.archive.org/web/20110612010555/http://suo.ieee.org/ontology/msg04077.html

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# http://suo.ieee.org/ontology/msg04079.html

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# http://web.archive.org/web/20110612001918/http://suo.ieee.org/ontology/msg04079.html

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# http://suo.ieee.org/ontology/msg04080.html

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# http://web.archive.org/web/20110612005244/http://suo.ieee.org/ontology/msg04080.html

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# http://suo.ieee.org/ontology/msg04268.html

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# http://web.archive.org/web/20110612005249/http://suo.ieee.org/ontology/msg04268.html

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# http://suo.ieee.org/ontology/msg04269.html

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# http://web.archive.org/web/20110612010626/http://suo.ieee.org/ontology/msg04269.html

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# http://suo.ieee.org/ontology/msg04272.html

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# http://web.archive.org/web/20110612000853/http://suo.ieee.org/ontology/msg04272.html

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# http://suo.ieee.org/ontology/msg04273.html

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# http://web.archive.org/web/20110612010514/http://suo.ieee.org/ontology/msg04273.html

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# http://suo.ieee.org/ontology/msg04290.html

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# http://web.archive.org/web/20110612002235/http://suo.ieee.org/ontology/msg04290.html

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===Inquiry List ~~(May & Jul~~ 2004)===

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===Dynamics And Logic • Inquiry List 2004===

* http://stderr.org/pipermail/inquiry/2004-May/thread.html#1400

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# http://stderr.org/pipermail/inquiry/2004-July/001688.html

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===NKS Forum ~~(May & Jul~~ 2004)===

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===Dynamics And Logic • NKS Forum 2004===

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* http://forum.wolframscience.com/archive/topic/420-1.html

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* http://forum.wolframscience.com/archive/topic/420.html

* http://forum.wolframscience.com/printthread.php?threadid=420

* http://forum.wolframscience.com/showthread.php?threadid=420

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[[Category:Artificial Intelligence]]

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[[Category:Boolean Algebra]]

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[[Category:Boolean Functions]]

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[[Category:Charles Sanders Peirce]]

[[Category:Combinatorics]]

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[[Category:Computational Complexity]]

[[Category:Computer Science]]

[[Category:Cybernetics]]

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[[Category:Formal Systems]]

[[Category:Graph Theory]]

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[[Category:Inquiry]]

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[[Category:Inquiry Driven Systems]]

[[Category:Knowledge Representation]]

[[Category:Logic]]

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[[Category:Mathematics]]

[[Category:Philosophy]]

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[[Category:Propositional Calculus]]

[[Category:Semiotics]]

[[Category:Visualization]]

Jon Awbrey

12,080

edits

Changes

Directory:Jon Awbrey/Papers/Differential Logic : Introduction (view source)

Revision as of 22:10, 8 December 2014

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