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{{DISPLAYTITLE:Differential Logic and Dynamic Systems 2.0}}
{{DISPLAYTITLE:Differential Logic and Dynamic Systems 2.0}}
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
It is important to note that the last expressions are not equivalent to the triple bracket <math>(x, y, z).\!</math>
It is important to note that the last expressions are not equivalent to the triple bracket <math>(x, y, z).\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"
<br>
|+ '''Table 1. Syntax and Semantics of a Calculus for Propositional Logic'''
|- style="background:ghostwhite"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
! Expression
|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Syntax and Semantics of a Calculus for Propositional Logic}\!</math>
! Interpretation
|- style="height:40px; background:ghostwhite"
| <math>\text{Expression}~\!</math>
| <math>\text{Interpretation}\!</math>
| <math>\text{Other Notations}\!</math>
|-
|-
| <math>~</math>
|
| <math>\operatorname{True}</math>
| <math>\text{True}\!</math>
| <math>1\!</math>
| <math>1\!</math>
|-
|-
| <math>(~)</math>
| <math>(~)\!</math>
| <math>\operatorname{False}</math>
| <math>\text{False}\!</math>
| <math>0\!</math>
| <math>0\!</math>
|-
|-
|-
|-
| <math>(x)\!</math>
| <math>(x)\!</math>
| <math>\operatorname{Not}\ x</math>
| <math>\text{Not}~ x\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
x' \\
x'
\tilde{x} \\
\\
\lnot x \\
\tilde{x}
\end{matrix}</math>
\\
\lnot x
\end{matrix}\!</math>
|-
|-
| <math>x\ y\ z</math>
| <math>x~y~z\!</math>
| <math>x\ \operatorname{and}\ y\ \operatorname{and}\ z</math>
| <math>x ~\text{and}~ y ~\text{and}~ z\!</math>
| <math>x \land y \land z</math>
| <math>x \land y \land z\!</math>
|-
|-
| <math>((x)(y)(z))\!</math>
| <math>((x)(y)(z))\!</math>
| <math>x\ \operatorname{or}\ y\ \operatorname{or}\ z</math>
| <math>x ~\text{or}~ y ~\text{or}~ z\!</math>
| <math>x \lor y \lor z</math>
| <math>x \lor y \lor z\!</math>
|-
|-
| <math>(x\ (y))\!</math>
| <math>(x~(y))\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
x\ \operatorname{implies}\ y \\
x ~\text{implies}~ y
\operatorname{If}\ x\ \operatorname{then}\ y \\
\\
\mathrm{If}~ x ~\text{then}~ y
\end{matrix}</math>
\end{matrix}</math>
| <math>x \Rightarrow y\!</math>
| <math>x \Rightarrow y\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
x\ \operatorname{not~equal~to}\ y \\
x ~\text{not equal to}~ y
x\ \operatorname{exclusive~or}\ y \\
\\
x ~\text{exclusive or}~ y
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
x \neq y \\
x \ne y
x + y \\
\\
x + y
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
<math>\begin{matrix}
x\ \operatorname{is~equal~to}\ y \\
x ~\text{is equal to}~ y
x\ \operatorname{if~and~only~if}\ y \\
\\
x ~\text{if and only if}~ y
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
x = y \\
x = y
x \Leftrightarrow y \\
\\
x \Leftrightarrow y
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{Just~one~of} \\
\text{Just one of}
x, y, z \\
\\
\operatorname{is~false}. \\
x, y, z
\\
\text{is false}.
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
x'y~z~ & \lor \\
x'y~z~ & \lor
x~y'z~ & \lor \\
\\
x~y~z' & \\
x~y'z~ & \lor
\\
x~y~z' &
\end{matrix}</math>
\end{matrix}</math>
|-
|-
| <math>((x),(y),(z))\!</math>
| <math>((x),(y),(z))~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{Just~one~of} \\
\text{Just one of}
x, y, z \\
\\
\operatorname{is~true}. \\
x, y, z
& \\
\\
\operatorname{Partition~all} \\
\text{is true}.
\operatorname{into}\ x, y, z. \\
\\
&
\\
\text{Partition all}
\\
\text{into}~ x, y, z.
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
x~y'z' & \lor \\
x~y'z' & \lor
x'y~z' & \lor \\
\\
x'y'z~ & \\
x'y~z' & \lor
\\
x'y'z~ &
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((x, y), z) \\
((x, y), z)
& \\
\\
(x, (y, z)) \\
&
\end{matrix}</math>
\\
(x, (y, z))
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{Oddly~many~of} \\
\text{Oddly many of}
x, y, z \\
\\
\operatorname{are~true}. \\
x, y, z
\end{matrix}</math>
\\
\text{are true}.
\end{matrix}\!</math>
|
|
<p><math>x + y + z\!</math></p>
<p><math>x + y + z\!</math></p>
<br>
<br>
<p><math>\begin{matrix}
<p><math>\begin{matrix}
x~y~z~ & \lor \\
x~y~z~ & \lor
x~y'z' & \lor \\
\\
x'y~z' & \lor \\
x~y'z' & \lor
x'y'z~ & \\
\\
\end{matrix}</math></p>
x'y~z' & \lor
\\
x'y'z~ &
\end{matrix}\!</math></p>
|-
|-
| <math>(w, (x),(y),(z))\!</math>
| <math>(w, (x),(y),(z))\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{Partition}\ w \\
\text{Partition}~ w
\operatorname{into}\ x, y, z. \\
\\
& \\
\text{into}~ x, y, z.
\operatorname{Genus}\ w\ \operatorname{comprises} \\
\\
\operatorname{species}\ x, y, z. \\
&
\\
\text{Genus}~ w ~\text{comprises}
\\
\text{species}~ x, y, z.
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
w'x'y'z' & \lor \\
w'x'y'z' & \lor
w~x~y'z' & \lor \\
\\
w~x'y~z' & \lor \\
w~x~y'z' & \lor
w~x'y'z~ & \\
\\
w~x'y~z' & \lor
\\
w~x'y'z~ &
\end{matrix}</math>
\end{matrix}</math>
|}<br>
|}
<br>
'''Note.''' The usage that one often sees, of a plus sign "<math>+\!</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:
'''Note.''' The usage that one often sees, of a plus sign "<math>+\!</math>" to represent inclusive disjunction, and the reference to this operation as ''boolean addition'', is a misnomer on at least two counts. Boole used the plus sign to represent exclusive disjunction (at any rate, an operation of aggregation restricted in its logical interpretation to cases where the represented sets are disjoint (Boole, 32)), as any mathematician with a sensitivity to the ring and field properties of algebra would do:
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]\!</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n\!</math> elements.
Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams. Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]\!</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n\!</math> elements.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
<br>
|+ '''Table 2. Propositional Calculus : Basic Notation'''
|- style="background:ghostwhite"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
! Symbol
|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Propositional Calculus : Basic Notation}\!</math>
! Notation
|- style="height:40px; background:ghostwhite"
! Description
| <math>\text{Symbol}\!</math>
! Type
| <math>\text{Notation}\!</math>
| <math>\text{Description}\!</math>
| <math>\text{Type}\!</math>
|-
|-
| <math>\mathfrak{A}</math>
| <math>\mathfrak{A}\!</math>
| <math>\lbrace\!</math> “<math>a_1\!</math>” <math>, \ldots,\!</math> “<math>a_n\!</math>” <math>\rbrace\!</math>
| <math>\{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}\!</math>
| <math>\operatorname{Alphabet}</math>
| <math>\text{Alphabet}\!</math>
| <math>[n] = \mathbf{n}</math>
| <math>[n] = \mathbf{n}\!</math>
|-
|-
| <math>\mathcal{A}</math>
| <math>\mathcal{A}\!</math>
| <math>\{a_1, \ldots, a_n\}</math>
| <math>\{ a_1, \ldots, a_n \}\!</math>
| <math>\operatorname{Basis}</math>
| <math>\text{Basis}\!</math>
| <math>[n] = \mathbf{n}</math>
| <math>[n] = \mathbf{n}\!</math>
|-
|-
| <math>A_i\!</math>
| <math>A_i\!</math>
| <math>\{(a_i), a_i\}\!</math>
| <math>\{ (a_i), a_i \}\!</math>
| <math>\operatorname{Dimension}\ i</math>
| <math>\text{Dimension}~ i\!</math>
| <math>\mathbb{B}</math>
| <math>\mathbb{B}\!</math>
|-
|-
| <math>A\!</math>
| <math>A\!</math>
|
|
<math>\begin{matrix}
\langle \mathcal{A} \rangle
\\[2pt]
\langle a_1, \ldots, a_n \rangle
\\[2pt]
\{ (a_1, \ldots, a_n) \}
\\[2pt]
A_1 \times \ldots \times A_n
\\[2pt]
\textstyle \prod_{i=1}^n A_i
\end{matrix}</math>
|
|
<math>\begin{matrix}
\text{Set of cells},
\\[2pt]
\text{coordinate tuples},
\\[2pt]
| <math>\mathbb{B}^n</math>
\text{points, or vectors}
\\[2pt]
\text{in the universe}
\\[2pt]
\text{of discourse}
\end{matrix}</math>
| <math>\mathbb{B}^n\!</math>
|-
|-
| <math>A^*\!</math>
| <math>A^*\!</math>
| <math>(\operatorname{hom} : A \to \mathbb{B})</math>
| <math>(\mathrm{hom} : A \to \mathbb{B})\!</math>
| <math>\operatorname{Linear~functions}</math>
| <math>\text{Linear functions}\!</math>
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math>
| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n\!</math>
|-
|-
| <math>A^\uparrow</math>
| <math>A^\uparrow\!</math>
| <math>(A \to \mathbb{B})</math>
| <math>(A \to \mathbb{B})\!</math>
| <math>\operatorname{Boolean~functions}</math>
| <math>\text{Boolean functions}\!</math>
| <math>\mathbb{B}^n \to \mathbb{B}</math>
| <math>\mathbb{B}^n \to \mathbb{B}\!</math>
|-
|-
| <math>A^\circ</math>
| <math>A^\circ\!</math>
|
|
<math>\begin{matrix}
[\mathcal{A}]
\\[2pt]
(A, A^\uparrow)
\\[2pt]
(A ~+\!\to \mathbb{B})
\\[2pt]
(A, (A \to \mathbb{B}))
\\[2pt]
[a_1, \ldots, a_n]
\end{matrix}</math>
|
|
<math>\begin{matrix}
\text{Universe of discourse}
\\[2pt]
\text{based on the features}
\\[2pt]
\{ a_1, \ldots, a_n \}
\end{matrix}</math>
|
|
<math>\begin{matrix}
(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))
\\[2pt]
|}<br>
(\mathbb{B}^n ~+\!\to \mathbb{B})
\\[2pt]
[\mathbb{B}^n]
\end{matrix}</math>
|}
<br>
===Qualitative Logic and Quantitative Analogy===
===Qualitative Logic and Quantitative Analogy===
A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,\!</math> counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.
A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram. The dimension, <math>n,\!</math> counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions. Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters. Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram. The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets. Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition. To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,\!</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\circ = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math>
Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences. The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more. In general, abstract sets may be denoted by gothic, greek, or script capital variants of <math>A, B, C,\!</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math> Most of the time, a set such as <math>\mathcal{A} = \{a_i\}\!</math> will be employed as the ''alphabet'' of a [[formal language]]. These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse. When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations. If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\circ = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math>
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.
As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels. However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions. At any rate, these elaborations can be deferred until actually needed.
Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n\!</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n\!</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1\!</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.\!</math>
Among the <math>2^{2^n}</math> propositions in <math>[a_1, \ldots, a_n]</math> are several families of <math>2^n\!</math> propositions each that take on special forms with respect to the basis <math>\{ a_1, \ldots, a_n \}.</math> Three of these families are especially prominent in the present context, the ''linear'', the ''positive'', and the ''singular'' propositions. Each family is naturally parameterized by the coordinate <math>n\!</math>-tuples in <math>\mathbb{B}^n</math> and falls into <math>n + 1\!</math> ranks, with a binomial coefficient <math>\tbinom{n}{k}</math> giving the number of propositions that have rank or weight <math>k.\!</math>
<ul>
<li>
<p>The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),\!</math> may be written as sums:</p>
{| align="center" cellspacing="8" width="90%"
|
<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n
~\text{where}~
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 0 \end{matrix}\right\}
~\text{for}~ i = 1 ~\text{to}~ n.\!</math>
|}
</li>
<li>
\sum_{i=1}^n e_i & = & e_1 + \ldots + e_n &
<p>The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),\!</math> may be written as products:</p>
\operatorname{where}\ e_i = a_i\ \operatorname{or}\ e_i = 0 &
{| align="center" cellspacing="8" width="90%"
|
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n
~\text{where}~
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = 1 \end{matrix}\right\}
~\text{for}~ i = 1 ~\text{to}~ n.\!</math>
|}
</li>
<li>
\prod_{i=1}^n e_i & = & e_1 \cdot \ldots \cdot e_n &
<p>The ''singular propositions'', <math>\{ \mathbf{x} : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{s} \mathbb{B}),\!</math> may be written as products:</p>
\operatorname{where}\ e_i = a_i\ \operatorname{or}\ e_i = 1 &
{| align="center" cellspacing="8" width="90%"
|
<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n
~\text{where}~
\left\{\begin{matrix} e_i = a_i \\ \text{or} \\ e_i = \texttt{(} a_i \texttt{)} \end{matrix}\right\}
~\text{for}~ i = 1 ~\text{to}~ n.\!</math>
|}
</li>
</ul>
In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>n = 3,\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>(a_1)(a_2)(a_3).\!</math>
In each case the rank <math>k\!</math> ranges from <math>0\!</math> to <math>n\!</math> and counts the number of positive appearances of the coordinate propositions <math>a_1, \ldots, a_n\!</math> in the resulting expression. For example, for <math>{n = 3},\!</math> the linear proposition of rank <math>0\!</math> is <math>0,\!</math> the positive proposition of rank <math>0\!</math> is <math>1,\!</math> and the singular proposition of rank <math>0\!</math> is <math>(a_1)(a_2)(a_3).\!</math>
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.
The basic propositions <math>a_i : \mathbb{B}^n \to \mathbb{B}</math> are both linear and positive. So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry–Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42–46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table 3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.
Amazingly enough, these isomorphisms are themselves schematized by the axioms and theorems of propositional logic. This fact is known as the ''propositions as types'' analogy or the Curry–Howard isomorphism [How]. In another formulation it says that terms are to types as proofs are to propositions. See [LaS, 42–46] and [SeH] for a good discussion and further references. To anticipate the bearing of these issues on our immediate topic, Table 3 sketches a partial overview of the Real to Boolean analogy that may serve to illustrate the paradigm in question.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
<br>
|+ '''Table 3. Analogy Between Real and Boolean Types'''
|- style="background:ghostwhite"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Real Domain} ~ \mathbb{R}\!</math>
| <math>\longleftrightarrow\!</math>
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math>
|-
|-
| <math>\mathbb{R}^n</math>
| <math>\mathbb{R}^n\!</math>
| <math>\mbox{Basic Space}\!</math>
| <math>\text{Basic Space}\!</math>
| <math>\mathbb{B}^n</math>
| <math>\mathbb{B}^n\!</math>
|-
|-
| <math>\mathbb{R}^n \to \mathbb{R}</math>
| <math>\mathbb{R}^n \to \mathbb{R}\!</math>
| <math>\mbox{Function Space}\!</math>
| <math>\text{Function Space}\!</math>
| <math>\mathbb{B}^n \to \mathbb{B}</math>
| <math>\mathbb{B}^n \to \mathbb{B}\!</math>
|-
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}</math>
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math>
| <math>\mbox{Tangent Vector}\!</math>
| <math>\text{Tangent Vector}\!</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math>
|-
|-
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})</math>
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math>
| <math>\mbox{Vector Field}\!</math>
| <math>\text{Vector Field}\!</math>
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})</math>
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math>
|-
|-
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}</math>
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math>
| <font size="4">"</font>
| '''<font size="4">"</font>'''
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}</math>
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math>
|-
|-
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}</math>
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math>
| <font size="4">"</font>
| '''<font size="4">"</font>'''
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}</math>
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math>
|-
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})</math>
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math>
| <math>\mbox{Derivation}\!</math>
| <math>\text{Derivation}\!</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math>
|-
|-
| <math>\mathbb{R}^n \to \mathbb{R}^m</math>
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math>
|
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math>
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math>
| <math>\mathbb{B}^n \to \mathbb{B}^m</math>
|-
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})</math>
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math>
|
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math>
|}
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})</math>
|}<br>
<br>
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.\!</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.
The Table exhibits a sample of likely parallels between the real and boolean domains. The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table. These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.\!</math> Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.
|}
|}
In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,\!</math> and the elements of <math>X\!</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)\!</math> comprises the set of ''models'' of <math>f,\!</math> or examples of elements in <math>X\!</math> satisfying the proposition <math>f.\!</math> The fiber <math>f^{-1}(0)\!</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f\!</math> that exist in <math>X.\!</math> Of course, the space of functions <math>(X \to \mathbb{B})</math> is isomorphic to the set of all subsets of <math>X,\!</math> called the ''power set'' of <math>X,\!</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.\!</math>
In the boolean context a function <math>f : X \to \mathbb{B}</math> is tantamount to a ''proposition'' about elements of <math>X,\!</math> and the elements of <math>X\!</math> constitute the ''interpretations'' of that proposition. The fiber <math>f^{-1}(1)\!</math> comprises the set of ''models'' of <math>f,\!</math> or examples of elements in <math>X\!</math> satisfying the proposition <math>f.\!</math> The fiber <math>f^{-1}(0)\!</math> collects the complementary set of ''anti-models'', or the exceptions to the proposition <math>f\!</math> that exist in <math>X.\!</math> Of course, the space of functions <math>(X \to \mathbb{B})\!</math> is isomorphic to the set of all subsets of <math>X,\!</math> called the ''power set'' of <math>X,\!</math> and often denoted <math>\mathcal{P}(X)</math> or <math>2^X.\!</math>
The operation of replacing <math>X\!</math> by <math>(X \to \mathbb{B})</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,\!</math> in which one passes from a focus on the ostensibly individual elements of <math>X\!</math> to a concern with the states of information and uncertainty that one possesses about objects and situations in <math>X.\!</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.
The operation of replacing <math>X\!</math> by <math>(X \to \mathbb{B})\!</math> in a type schema corresponds to a certain shift of attitude towards the space <math>X,\!</math> in which one passes from a focus on the ostensibly individual elements of <math>X\!</math> to a concern with the states of information and uncertainty that one possesses about objects and situations in <math>X.\!</math> The conceptual obstacles in the path of this transition can be smoothed over by using singular functions <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> as stepping stones. First of all, it's an easy step from an element <math>\mathbf{x}</math> of type <math>\mathbb{B}^n</math> to the equivalent information of a singular proposition <math>\mathbf{x} : X \xrightarrow{s} \mathbb{B}, </math> and then only a small jump of generalization remains to reach the type of an arbitrary proposition <math>f : X \to \mathbb{B},</math> perhaps understood to indicate a relaxed constraint on the singularity of points or a neighborhood circumscribing the original <math>\mathbf{x}.</math> This is frequently a useful transformation, communicating between the ''objective'' and the ''intentional'' perspectives, in spite perhaps of the open objection that this distinction is transient in the mean time and ultimately superficial.
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.
It is hoped that this measure of flexibility, allowing us to stretch a point into a proposition, can be useful in the examination of inquiry driven systems, where the differences between empirical, intentional, and theoretical propositions constitute the discrepancies and the distributions that drive experimental activity. I can give this model of inquiry a cybernetic cast by realizing that theory change and theory evolution, as well as the choice and the evaluation of experiments, are actions that are taken by a system or its agent in response to the differences that are detected between observational contents and theoretical coverage.
The types collected in Table 3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.
The types collected in Table 3 (repeated below) serve to illustrate the themes of ''higher order propositional expressions'' and the ''propositions as types'' (PAT) analogy.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
<br>
|+ '''Table 3. Analogy Between Real and Boolean Types'''
|- style="background:ghostwhite"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Analogy Between Real and Boolean Types}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Real Domain} ~ \mathbb{R}\!</math>
| <math>\longleftrightarrow\!</math>
| <math>\text{Boolean Domain} ~ \mathbb{B}\!</math>
|-
|-
| <math>\mathbb{R}^n</math>
| <math>\mathbb{R}^n\!</math>
| <math>\mbox{Basic Space}\!</math>
| <math>\text{Basic Space}\!</math>
| <math>\mathbb{B}^n</math>
| <math>\mathbb{B}^n\!</math>
|-
|-
| <math>\mathbb{R}^n \to \mathbb{R}</math>
| <math>\mathbb{R}^n \to \mathbb{R}\!</math>
| <math>\mbox{Function Space}\!</math>
| <math>\text{Function Space}\!</math>
| <math>\mathbb{B}^n \to \mathbb{B}</math>
| <math>\mathbb{B}^n \to \mathbb{B}\!</math>
|-
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}</math>
| <math>(\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R}\!</math>
| <math>\mbox{Tangent Vector}\!</math>
| <math>\text{Tangent Vector}\!</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}\!</math>
|-
|-
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})</math>
| <math>\mathbb{R}^n \to ((\mathbb{R}^n \to \mathbb{R}) \to \mathbb{R})\!</math>
| <math>\mbox{Vector Field}\!</math>
| <math>\text{Vector Field}\!</math>
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})</math>
| <math>\mathbb{B}^n \to ((\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B})\!</math>
|-
|-
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}</math>
| <math>(\mathbb{R}^n \times (\mathbb{R}^n \to \mathbb{R})) \to \mathbb{R}\!</math>
| <font size="4">"</font>
| '''<font size="4">"</font>'''
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}</math>
| <math>(\mathbb{B}^n \times (\mathbb{B}^n \to \mathbb{B})) \to \mathbb{B}\!</math>
|-
|-
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}</math>
| <math>((\mathbb{R}^n \to \mathbb{R}) \times \mathbb{R}^n) \to \mathbb{R}\!</math>
| <font size="4">"</font>
| '''<font size="4">"</font>'''
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}</math>
| <math>((\mathbb{B}^n \to \mathbb{B}) \times \mathbb{B}^n) \to \mathbb{B}\!</math>
|-
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})</math>
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^n \to \mathbb{R})~\!</math>
| <math>\mbox{Derivation}\!</math>
| <math>\text{Derivation}\!</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^n \to \mathbb{B})\!</math>
|-
|-
| <math>\mathbb{R}^n \to \mathbb{R}^m</math>
| <math>\mathbb{R}^n \to \mathbb{R}^m\!</math>
|
| <math>\begin{matrix}\text{Basic}\\[2pt]\text{Transformation}\end{matrix}</math>
| <math>\mathbb{B}^n \to \mathbb{B}^m\!</math>
| <math>\mathbb{B}^n \to \mathbb{B}^m</math>
|-
|-
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})</math>
| <math>(\mathbb{R}^n \to \mathbb{R}) \to (\mathbb{R}^m \to \mathbb{R})\!</math>
|
| <math>\begin{matrix}\text{Function}\\[2pt]\text{Transformation}\end{matrix}</math>
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})\!</math>
|}
| <math>(\mathbb{B}^n \to \mathbb{B}) \to (\mathbb{B}^m \to \mathbb{B})</math>
|}<br>
<br>
First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta\!</math> takes a function on that space, an <math>f\!</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f\!</math> in the direction <math>\vartheta\!</math> [Che46, 76–77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.
First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math> At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta\!</math> takes a function on that space, an <math>f\!</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math> This value is known as the ''derivative'' of <math>f\!</math> in the direction <math>\vartheta\!</math> [Che46, 76–77]. In the boolean case <math>\vartheta : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.
Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows "<math>\to\!</math>" and products "<math>\times\!</math>" with the respective logical arrows "<math>\Rightarrow\!</math>" and products "<math>\land\!</math>". Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.
Next, by way of illustrating the propositions as types idea, consider a proposition of the form <math>X \Rightarrow (Y \Rightarrow Z).</math> One knows from propositional calculus that this is logically equivalent to a proposition of the form <math>(X \land Y) \Rightarrow Z.</math> But this equivalence should remind us of the functional isomorphism that exists between a construction of the type <math>X \to (Y \to Z)</math> and a construction of the type <math>(X \times Y) \to Z.</math> The propositions as types analogy permits us to take a functional type like this and, under the right conditions, replace the functional arrows “<math>\to~\!</math>” and products “<math>\times\!</math>” with the respective logical arrows “<math>\Rightarrow\!</math>” and products “<math>\land\!</math>”. Accordingly, viewing the result as a proposition, we can employ axioms and theorems of propositional calculus to suggest appropriate isomorphisms among the categorical and functional constructions.
Finally, examine the middle four rows of Table 3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\xi : X \to \bigcup_{x \in X} \xi_x</math> that assigns to each point <math>x\!</math> of the space <math>X\!</math> a tangent vector to <math>X\!</math> at that point, namely, the tangent vector <math>\xi_x\!</math> [Che46, 82–83]. If <math>X\!</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi\!</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math>
Finally, examine the middle four rows of Table 3. These display a series of isomorphic types that stretch from the categories that are labeled ''Vector Field'' to those that are labeled ''Derivation''. A ''vector field'', also known as an ''infinitesimal transformation'', associates a tangent vector at a point with each point of a space. In symbols, a vector field is a function of the form <math>\xi : X \to \bigcup_{x \in X} \xi_x</math> that assigns to each point <math>x\!</math> of the space <math>X\!</math> a tangent vector to <math>X\!</math> at that point, namely, the tangent vector <math>\xi_x\!</math> [Che46, 82–83]. If <math>X\!</math> is of the type <math>\mathbb{K}^n,</math> then <math>\xi\!</math> is of the type <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K}).</math> This has the pattern <math>X \to (Y \to Z),</math> with <math>X = \mathbb{K}^n,</math> <math>Y = (\mathbb{K}^n \to \mathbb{K}),</math> and <math>Z = \mathbb{K}.</math>
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table 4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y\!</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,\!</math> initially viewed as attaching each tangent vector <math>\xi_x\!</math> to the site <math>x\!</math> where it acts in <math>X,\!</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.
Applying the propositions as types analogy, one can follow this pattern through a series of metamorphoses from the type of a vector field to the type of a derivation, as traced out in Table 4. Observe how the function <math>f : X \to \mathbb{K},</math> associated with the place of <math>Y\!</math> in the pattern, moves through its paces from the second to the first position. In this way, the vector field <math>\xi,\!</math> initially viewed as attaching each tangent vector <math>\xi_x\!</math> to the site <math>x\!</math> where it acts in <math>X,\!</math> now comes to be seen as acting on each scalar potential <math>f : X \to \mathbb{K}</math> like a generalized species of differentiation, producing another function <math>\xi f : X \to \mathbb{K}</math> of the same type.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"
<br>
|+ '''Table 4. An Equivalence Based on the Propositions as Types Analogy
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|- style="background:ghostwhite"
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{An Equivalence Based on the Propositions as Types Analogy}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Pattern}\!</math>
| <math>\text{Construct}\!</math>
| <math>\text{Instance}\!</math>
|-
|-
| <math>X \to (Y \to Z)</math>
| <math>X \to (Y \to Z)\!</math>
| <math>\mbox{Vector Field}\!</math>
| <math>\text{Vector Field}\!</math>
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})</math>
| <math>\mathbb{K}^n \to ((\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K})\!</math>
|-
|-
| <math>(X \times Y) \to Z</math>
| <math>(X \times Y) \to Z\!</math>
| <math>\Uparrow</math>
| <math>\Uparrow\!</math>
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}</math>
| <math>(\mathbb{K}^n \times (\mathbb{K}^n \to \mathbb{K})) \to \mathbb{K}\!</math>
|-
|-
| <math>(Y \times X) \to Z</math>
| <math>(Y \times X) \to Z\!</math>
| <math>\Downarrow</math>
| <math>\Downarrow\!</math>
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}</math>
| <math>((\mathbb{K}^n \to \mathbb{K}) \times \mathbb{K}^n) \to \mathbb{K}\!</math>
|-
|-
| <math>Y \to (X \to Z)</math>
| <math>Y \to (X \to Z)\!</math>
| <math>\mbox{Derivation}\!</math>
| <math>\text{Derivation}\!</math>
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})</math>
| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})\!</math>
|}<br>
|}
<br>
===Reality at the Threshold of Logic===
===Reality at the Threshold of Logic===
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.
Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
<br>
|+ '''Table 5. A Bridge Over Troubled Waters'''
|- style="background:ghostwhite"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"
| align="center" | <math>\mbox{Linear Space}\!</math>
|+ style="height:30px" | <math>\text{Table 5.} ~~ \text{A Bridge Over Troubled Waters}\!</math>
| align="center" | <math>\mbox{Liminal Space}\!</math>
|- style="height:40px; background:ghostwhite"
| align="center" | <math>\mbox{Logical Space}\!</math>
| align="center" | <math>\text{Linear Space}\!</math>
| align="center" | <math>\text{Liminal Space}\!</math>
| align="center" | <math>\text{Logical Space}\!</math>
|-
|-
|
| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math>
<math>\begin{matrix}
| <math>\begin{matrix}\underline\mathcal{X} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math>
\mathcal{X}
| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math>
& = & \{x_1, \ldots, x_n\} \\
\end{matrix}</math>
|
<math>\begin{matrix}
\underline\mathcal{X}
& = & \{\underline{x}_1, \ldots, \underline{x}_n\} \\
\end{matrix}</math>
|
<math>\begin{matrix}
\mathcal{A}
& = & \{a_1, \ldots, a_n\} \\
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
<math>\begin{matrix}
X_i
X_i & = & \langle x_i \rangle
& = & \langle x_i \rangle \\
\\
& \cong & \mathbb{K} \\
& \cong & \mathbb{K}
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\underline{X}_i
\underline{X}_i & = & \{ (\underline{x}_i), \underline{x}_i \}
& = & \{(\underline{x}_i), \underline{x}_i \} \\
\\
& \cong & \mathbb{B} \\
& \cong & \mathbb{B}
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
A_i
A_i & = & \{ (a_i), a_i \}
& = & \{(a_i), a_i \} \\
\\
& \cong & \mathbb{B} \\
& \cong & \mathbb{B}
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
<math>\begin{matrix}
X \\
X
= & \langle \mathcal{X} \rangle \\
\\
= & \langle x_1, \ldots, x_n \rangle \\
= & \langle \mathcal{X} \rangle
= & X_1 \times \ldots \times X_n \\
\\
= & \prod_{i=1}^n X_i \\
= & \langle x_1, \ldots, x_n \rangle
\cong & \mathbb{K}^n \\
\\
= & X_1 \times \ldots \times X_n
\\
= & \prod_{i=1}^n X_i
\\
\cong & \mathbb{K}^n
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\underline{X} \\
\underline{X}
= & \langle \underline\mathcal{X} \rangle \\
\\
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle \\
= & \langle \underline\mathcal{X} \rangle
= & \underline{X}_1 \times \ldots \times \underline{X}_n \\
\\
= & \prod_{i=1}^n \underline{X}_i \\
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle
\cong & \mathbb{B}^n \\
\\
= & \underline{X}_1 \times \ldots \times \underline{X}_n
\\
= & \prod_{i=1}^n \underline{X}_i
\\
\cong & \mathbb{B}^n
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
A \\
A
= & \langle \mathcal{A} \rangle \\
\\
= & \langle a_1, \ldots, a_n \rangle \\
= & \langle \mathcal{A} \rangle
= & A_1 \times \ldots \times A_n \\
\\
= & \prod_{i=1}^n A_i \\
= & \langle a_1, \ldots, a_n \rangle
\cong & \mathbb{B}^n \\
\\
= & A_1 \times \ldots \times A_n
\\
= & \prod_{i=1}^n A_i
\\
\cong & \mathbb{B}^n
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
<math>\begin{matrix}
X^*
X^* & = & (\ell : X \to \mathbb{K})
& = & (\ell : X \to \mathbb{K}) \\
\\
& \cong & \mathbb{K}^n \\
& \cong & \mathbb{K}^n
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\underline{X}^*
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B})
& = & (\ell : \underline{X} \to \mathbb{B}) \\
\\
& \cong & \mathbb{B}^n \\
& \cong & \mathbb{B}^n
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
A^*
A^* & = & (\ell : A \to \mathbb{B})
& = & (\ell : A \to \mathbb{B}) \\
\\
& \cong & \mathbb{B}^n \\
& \cong & \mathbb{B}^n
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
<math>\begin{matrix}
X^\uparrow
X^\uparrow & = & (X \to \mathbb{K})
& = & (X \to \mathbb{K}) \\
\\
& \cong & (\mathbb{K}^n \to \mathbb{K}) \\
& \cong & (\mathbb{K}^n \to \mathbb{K})
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\underline{X}^\uparrow
\underline{X}^\uparrow & = & (\underline{X} \to \mathbb{B})
& = & (\underline{X} \to \mathbb{B}) \\
\\
& \cong & (\mathbb{B}^n \to \mathbb{B}) \\
& \cong & (\mathbb{B}^n \to \mathbb{B})
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
A^\uparrow
A^\uparrow & = & (A \to \mathbb{B})
& = & (A \to \mathbb{B}) \\
\\
& \cong & (\mathbb{B}^n \to \mathbb{B}) \\
& \cong & (\mathbb{B}^n \to \mathbb{B})
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
<math>\begin{matrix}
X^\circ \\
X^\circ
= & [\mathcal{X}] \\
\\
= & [x_1, \ldots, x_n] \\
= & [\mathcal{X}]
= & (X, X^\uparrow) \\
\\
= & (X\ +\!\to \mathbb{K}) \\
= & [x_1, \ldots, x_n]
= & (X, (X \to \mathbb{K})) \\
\\
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K})) \\
= & (X, X^\uparrow)
= & (\mathbb{K}^n\ +\!\to \mathbb{K}) \\
\\
= & [\mathbb{K}^n] \\
= & (X ~+\!\to \mathbb{K})
\\
= & (X, (X \to \mathbb{K}))
\\
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K}))
\\
= & (\mathbb{K}^n ~+\!\to \mathbb{K})
\\
= & [\mathbb{K}^n]
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\underline{X}^\circ \\
\underline{X}^\circ
= & [\underline\mathcal{X}] \\
\\
= & [\underline{x}_1, \ldots, \underline{x}_n] \\
= & [\underline\mathcal{X}]
= & (\underline{X}, \underline{X}^\uparrow) \\
\\
= & (\underline{X}\ +\!\to \mathbb{B}) \\
= & [\underline{x}_1, \ldots, \underline{x}_n]
= & (\underline{X}, (\underline{X} \to \mathbb{B})) \\
\\
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\
= & (\underline{X}, \underline{X}^\uparrow)
= & (\mathbb{B}^n\ +\!\to \mathbb{B}) \\
\\
= & [\mathbb{B}^n] \\
= & (\underline{X} ~+\!\to \mathbb{B})
\\
= & (\underline{X}, (\underline{X} \to \mathbb{B}))
\\
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))
\\
= & (\mathbb{B}^n ~+\!\to \mathbb{B})
\\
= & [\mathbb{B}^n]
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
A^\circ \\
A^\circ
= & [\mathcal{A}] \\
\\
= & [a_1, \ldots, a_n] \\
= & [\mathcal{A}]
= & (A, A^\uparrow) \\
\\
= & (A\ +\!\to \mathbb{B}) \\
= & [a_1, \ldots, a_n]
= & (A, (A \to \mathbb{B})) \\
\\
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\
= & (A, A^\uparrow)
= & (\mathbb{B}^n\ +\!\to \mathbb{B}) \\
\\
= & [\mathbb{B}^n] \\
= & (A ~+\!\to \mathbb{B})
\\
= & (A, (A \to \mathbb{B}))
\\
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))
\\
= & (\mathbb{B}^n ~+\!\to \mathbb{B})
\\
= & [\mathbb{B}^n]
\end{matrix}</math>
\end{matrix}</math>
|}<br>
|}
<br>
The left side of the Table collects mostly standard notation for an <math>n\!</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math>
The left side of the Table collects mostly standard notation for an <math>n\!</math>-dimensional vector space over a field <math>\mathbb{K}.</math> The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field <math>\mathbb{K},</math> with a special interest in the continuous line <math>\mathbb{R},</math> to the qualitative and discrete situations that are instanced and typified by <math>\mathbb{B}.</math>
* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.
* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.
For the sake of concreteness, let us suppose that we start with a continuous <math>n\!</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i\!</math> we choose an <math>n\!</math>-ary relation <math>L_i\!</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\operatorname{th}\!</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:
For the sake of concreteness, let us suppose that we start with a continuous <math>n\!</math>-dimensional vector space like <math>X = \langle x_1, \ldots, x_n \rangle \cong \mathbb{R}^n.</math> The coordinate system <math>\mathcal{X} = \{x_i\}</math> is a set of maps <math>x_i : \mathbb{R}^n \to \mathbb{R},</math> also known as the ''coordinate projections''. Given a "dataset" of points <math>\mathbf{x}</math> in <math>\mathbb{R}^n,</math> there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each <math>i\!</math> we choose an <math>n\!</math>-ary relation <math>L_i\!</math> on <math>\mathbb{R}^n,</math> that is, a subset of <math>\mathbb{R}^n,</math> and then we define the <math>i^\mathrm{th}\!</math> threshold map, or ''limen'' <math>\underline{x}_i</math> as follows:
: <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \mbox{such that:}</math>
: <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}</math>
: <math>\begin{matrix}
: <math>\begin{matrix}
\underline{x}_i(\mathbf{x}) = 1 & \mbox{if} & \mathbf{x} \in L_i, \\
\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i, \\
\underline{x}_i(\mathbf{x}) = 0 & \mbox{if} & \mathbf{x} \not\in L_i.
\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.
\end{matrix}</math>
\end{matrix}</math>
\end{matrix}</math>
\end{matrix}</math>
Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\operatorname{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i\!</math> has points on the <math>i^\operatorname{th}\!</math> axis, that is, points of the form '''‹''' <math>0, \ldots, 0, r_i, 0, \ldots, 0</math> '''›''' where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.
Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\mathrm{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.\!</math> Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set <math>L_i\!</math> has points on the <math>i^\mathrm{th}\!</math> axis, that is, points of the form '''‹''' <math>0, \ldots, 0, r_i, 0, \ldots, 0</math> '''›''' where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline\mathcal{X} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\operatorname{th}\!</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a "hurdle" for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\operatorname{th}\!</math> threshold.
States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline\mathcal{X} = \{\underline{x}_i\}</math> as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\mathrm{th}\!</math> threshold map. This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a "hurdle" for it. In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\mathrm{th}\!</math> threshold.
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>(\ldots)</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>(a_1, \ldots, a_k)</math>, as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\circ</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math>
Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems. Parentheses <math>(\ldots)</math> may be used to indicate logical negation. Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>(a_1, \ldots, a_k)</math>, as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math> Together these form a new universe of discourse <math>\underline{X}^\circ</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math>
Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table 6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i\!</math> is indexed by the string of values that it takes on the points of the universe <math>X^\circ = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,\!</math> as shown in the first column of the Table. In their own right the <math>2^1\!</math> points of the universe <math>X^\circ</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\circ</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math>
Propositional forms on one variable correspond to boolean functions <math>f : \mathbb{B}^1 \to \mathbb{B}.</math> In Table 6 these functions are listed in a variant form of [[truth table]], one in which the axes of the usual arrangement are rotated through a right angle. Each function <math>f_i\!</math> is indexed by the string of values that it takes on the points of the universe <math>X^\circ = [x] \cong \mathbb{B}^1.</math> The binary index generated in this way is converted to its decimal equivalent and these are used as conventional names for the <math>f_i,\!</math> as shown in the first column of the Table. In their own right the <math>2^1\!</math> points of the universe <math>X^\circ</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\circ</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"
<br>
|+ '''Table 6. Propositional Forms on One Variable'''
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 6.} ~~ \text{Propositional Forms on One Variable}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math>
| style="width:16%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math>
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| style="width:16%" |
|
| align="right" | <math>x\colon\!</math>
| align="right" | <math>x\colon\!</math>
| <math>1~0</math>
| <math>1~0\!</math>
| <math>~</math>
|
| <math>~</math>
|
| <math>~</math>
|
|-
|-
| <math>f_0\!</math>
| <math>f_0\!</math>
| <math>f_{00}\!</math>
| <math>f_{00}\!</math>
| <math>0~0</math>
| <math>0~0\!</math>
| <math>(~)\!</math>
| <math>(~)\!</math>
| <math>\mbox{false}\!</math>
| <math>\text{false}\!</math>
| <math>0\!</math>
| <math>0\!</math>
|-
|-
| <math>f_1\!</math>
| <math>f_1\!</math>
| <math>f_{01}\!</math>
| <math>f_{01}\!</math>
| <math>0~1</math>
| <math>0~1\!</math>
| <math>(x)\!</math>
| <math>(x)\!</math>
| <math>\mbox{not}\ x</math>
| <math>\text{not}~ x\!</math>
| <math>\lnot x</math>
| <math>\lnot x\!</math>
|-
|-
| <math>f_2\!</math>
| <math>f_2\!</math>
| <math>f_{10}\!</math>
| <math>f_{10}\!</math>
| <math>1~0</math>
| <math>1~0\!</math>
| <math>x\!</math>
| <math>x\!</math>
| <math>x\!</math>
| <math>x\!</math>
| <math>f_3\!</math>
| <math>f_3\!</math>
| <math>f_{11}\!</math>
| <math>f_{11}\!</math>
| <math>1~1</math>
| <math>1~1\!</math>
| <math>((~))\!</math>
| <math>((~))\!</math>
| <math>\mbox{true}\!</math>
| <math>\text{true}\!</math>
| <math>1\!</math>
| <math>1\!</math>
|}<br>
|}
<br>
Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table 7 each function <math>f_i\!</math> is indexed by the values that it takes on the points of the universe <math>X^\circ = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2\!</math> points of the universe <math>X^\circ</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x\!</math> and <math>y\!</math> run through the various combinations of their values in <math>\mathbb{B}.</math>
Propositional forms on two variables correspond to boolean functions <math>f : \mathbb{B}^2 \to \mathbb{B}.</math> In Table 7 each function <math>f_i\!</math> is indexed by the values that it takes on the points of the universe <math>X^\circ = [x, y] \cong \mathbb{B}^2.</math> Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column. The <math>2^2\!</math> points of the universe <math>X^\circ</math> are coordinated as a space of type <math>\mathbb{B}^2,</math> as indicated under the heading of the Table, where the coordinate projections <math>x\!</math> and <math>y\!</math> run through the various combinations of their values in <math>\mathbb{B}.</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"
<br>
|+ '''Table 7. Propositional Forms on Two Variables'''
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 7-a.} ~~ \text{Propositional Forms on Two Variables}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}\!</math>
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}\!</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}\!</math>
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| style="width:16%" |
|
| align="right" | <math>x\colon\!</math>
| align="right" | <math>x\colon\!</math>
| <math>1~1~0~0\!</math>
| <math>1~1~0~0\!</math>
| <math>~\!</math>
|
| <math>~\!</math>
|
| <math>~\!</math>
|
|- style="background:ghostwhite"
|- style="background:ghostwhite"
| <math>~\!</math>
|
| align="right" | <math>y\colon\!</math>
| align="right" | <math>y\colon\!</math>
| <math>1~0~1~0\!</math>
| <math>1~0~1~0\!</math>
| <math>~\!</math>
|
| <math>~\!</math>
|
| <math>~\!</math>
|
|-
|-
| <math>f_{0}\!</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0}
\\[4pt]
f_{1}
\\[4pt]
f_{2}
\\[4pt]
f_{3}
\\[4pt]
f_{4}
\\[4pt]
f_{5}
\\[4pt]
f_{6}
\\[4pt]
f_{7}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0000}
\\[4pt]
f_{0001}
\\[4pt]
f_{0010}
\\[4pt]
f_{0011}
\\[4pt]
f_{0100}
\\[4pt]
| <math>0~1~0~0\!</math>
f_{0101}
| <math>x\ (y)\!</math>
\\[4pt]
f_{0110}
\\[4pt]
f_{0111}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~0~0
\\[4pt]
0~0~0~1
\\[4pt]
0~0~1~0
\\[4pt]
0~0~1~1
\\[4pt]
0~1~0~0
\\[4pt]
0~1~0~1
\\[4pt]
0~1~1~0
\\[4pt]
0~1~1~1
\end{matrix}\!</math>
| <math>\lnot x \lor \lnot y\!</math>
| valign="bottom" |
|-
<math>\begin{matrix}
(~)
\\[4pt]
(x)(y)
\\[4pt]
(x)~y~
\\[4pt]
(x)
\\[4pt]
~x~(y)
\\[4pt]
(y)
\\[4pt]
(x,~y)
\\[4pt]
(x~y)
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
\text{false}
\\[4pt]
\text{neither}~ x ~\text{nor}~ y
\\[4pt]
y ~\text{without}~ x
\\[4pt]
\text{not}~ x
\\[4pt]
x ~\text{without}~ y
\\[4pt]
\text{not}~ y
\\[4pt]
x ~\text{not equal to}~ y
\\[4pt]
\text{not both}~ x ~\text{and}~ y
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
| <math>f_{1101}\!</math>
0
\\[4pt]
\lnot x \land \lnot y
\\[4pt]
\lnot x \land y
\\[4pt]
\lnot x
\\[4pt]
x \land \lnot y
\\[4pt]
\lnot y
\\[4pt]
x \ne y
\\[4pt]
\lnot x \lor \lnot y
\end{matrix}\!</math>
| <math>1\!</math>
|-
|-
| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
| valign="bottom" |
<math>\begin{matrix}
f_{8}
\\[4pt]
f_{9}
\\[4pt]
f_{10}
\\[4pt]
f_{11}
\end{matrix}</math>
\\[4pt]
f_{12}
\\[4pt]
f_{13}
\\[4pt]
\end{matrix}</math>
f_{14}
\\[4pt]
f_{15}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
f_{1000}
\\[4pt]
f_{1001}
\tbinom{X}{k} & = & X\ \operatorname{choose}\ k & = & \{ k\!\mbox{-sets from}\ X \}.
\\[4pt]
\end{matrix}</math>
f_{1010}
\\[4pt]
f_{1011}
\\[4pt]
f_{1100}
\tbinom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.
\\[4pt]
\end{matrix}</math>
f_{1101}
\\[4pt]
f_{1110}
\\[4pt]
f_{1111}
\end{matrix}\!</math>
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
1~0~0~0
\\[4pt]
1~0~0~1
\\[4pt]
1~0~1~0
\\[4pt]
1~0~1~1
\\[4pt]
\end{array}</math></p>
1~1~0~0
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\\[4pt]
1~1~1~1
\operatorname{d}x_i (q) & = & (\!|\ x_i (q_0) & , & x_i (q_1)\ |\!) \\
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
\end{array}</math></p>
x~y
\\[4pt]
((x,~y))
\\[4pt]
y
\\[4pt]
(x~(y))
\\[4pt]
x
\\[4pt]
((x)~y)
\\[4pt]
((x)(y))
\\[4pt]
((~))
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
x ~\text{and}~ y
\\[4pt]
x ~\text{equal to}~ y
\\[4pt]
y
\\[4pt]
\text{not}~ x ~\text{without}~ y
\\[4pt]
x
\\[4pt]
\text{not}~ y ~\text{without}~ x
\\[4pt]
x ~\text{or}~ y
\\[4pt]
\text{true}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
x \land y
\\[4pt]
x = y
\\[4pt]
y
\\[4pt]
x \Rightarrow y
\\[4pt]
x
\\[4pt]
x \Leftarrow y
\\[4pt]
x \lor y
\\[4pt]
1
\end{matrix}\!</math>
|}
|}
<br>
: <p><math>\begin{array}{lclcl}
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 7-b.} ~~ \text{Propositional Forms on Two Variables}\!</math>
|- style="height:40px; background:ghostwhite"
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_1 \\ \text{Decimal}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_2 \\ \text{Binary}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_3 \\ \text{Vector}\end{matrix}</math>
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_4 \\ \text{Cactus}\end{matrix}</math>
| style="width:25%" | <math>\begin{matrix}\mathcal{L}_5 \\ \text{English}\end{matrix}</math>
: <p><math>\begin{array}{lcl}
| style="width:15%" | <math>\begin{matrix}\mathcal{L}_6 \\ \text{Ordinary}\end{matrix}</math>
|- style="background:ghostwhite"
|
| align="right" | <math>x\colon\!</math>
| <math>1~1~0~0\!</math>
\end{array}</math></p>
|
|
|
|- style="background:ghostwhite"
|
\operatorname{E}A
| align="right" | <math>y\colon\!</math>
| <math>1~0~1~0\!</math>
|
\end{array}</math></p>
|
|
: <p><math>\begin{array}{lclcl}
\end{array}</math></p>
\end{array}</math></p>
|+ '''Table 8. Differential Extension : Basic Notation'''
|- style="background:ghostwhite"
! Symbol
! Notation
|-
|-
| <math>\operatorname{d}\mathfrak{A}</math>
| <math>f_0\!</math>
| <math>\lbrace\!</math> “<math>\operatorname{d}a_1</math>” <math>, \ldots,\!</math> “<math>\operatorname{d}a_n</math>” <math>\rbrace\!</math>
| <math>f_{0000}\!</math>
| <math>0~0~0~0\!</math>
| <math>(~)\!</math>
| <math>\text{false}\!</math>
| <math>[n] = \mathbf{n}</math>
| <math>0\!</math>
|-
|-
| <math>\operatorname{d}\mathcal{A}</math>
| valign="bottom" |
<math>\begin{matrix}
f_1
\\[4pt]
f_2
\\[4pt]
f_4
\\[4pt]
f_8
\end{matrix}\!</math>
| valign="bottom" |
| <math>\mathbb{D}</math>
<math>\begin{matrix}
|-
f_{0001}
\\[4pt]
f_{0010}
\\[4pt]
f_{0100}
\\[4pt]
f_{1000}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~0~1
\\[4pt]
0~0~1~0
\\[4pt]
|-
0~1~0~0
| <math>\operatorname{d}A^*</math>
\\[4pt]
1~0~0~0
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
(x)(y)
\\[4pt]
(x)~y~
\\[4pt]
~x~(y)
\\[4pt]
|-
~x~~y~
| <math>\operatorname{d}A^\circ</math>
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
\text{neither}~ x ~\text{nor}~ y
\\[4pt]
y ~\text{without}~ x
\\[4pt]
x ~\text{without}~ y
\\[4pt]
x ~\text{and}~ y
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
\lnot x \land \lnot y
\\[4pt]
\lnot x \land y
\\[4pt]
x \land \lnot y
\\[4pt]
x \land y
\end{matrix}\!</math>
|
\operatorname{d}^0 \mathcal{A}
\operatorname{d}^1 \mathcal{A}
\end{array}</math></p>
|
\operatorname{E}^0 \mathcal{A}
\operatorname{E}^1 \mathcal{A}
\operatorname{E}^k \mathcal{A}
\operatorname{E}^\infty \mathcal{A}
\end{array}</math></p>
|-
|-
|
| valign="bottom" |
<math>\begin{matrix}
f_3
\\[4pt]
f_{12}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0011}
\\[4pt]
|
f_{1100}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
0~0~1~1
\operatorname{p}^1 \mathcal{A}
\\[4pt]
1~1~0~0
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
(x)
\\[4pt]
\end{array}</math></p>
x
\end{matrix}\!</math>
\operatorname{p}^k \mathcal{A}
| valign="bottom" |
<math>\begin{matrix}
\end{array}</math></p>
\text{not}~ x
|
\\[4pt]
x
\operatorname{Q}^0 \mathcal{A}
\end{matrix}\!</math>
| valign="bottom" |
\operatorname{Q}^1 \mathcal{A}
<math>\begin{matrix}
\lnot x
\cup \mathcal{A}' \\
\\[4pt]
\operatorname{Q}^2 \mathcal{A}
x
\end{matrix}\!</math>
\cup \mathcal{A}'
|-
| valign="bottom" |
<math>\begin{matrix}
f_6
\\[4pt]
f_9
\cup \ldots
\end{matrix}\!</math>
| valign="bottom" |
\end{array}</math></p>
<math>\begin{matrix}
|}<br>
f_{0110}
\\[4pt]
f_{1001}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
0~1~1~0
(\operatorname{Q}X \to \mathbb{B})
\\[4pt]
1~0~0~1
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
(x, y)
\\[4pt]
((x, y))
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
x ~\text{not equal to}~ y
\\[4pt]
x ~\text{equal to}~ y
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
x \ne y
\\[4pt]
x = y
\end{matrix}\!</math>
|-
|-
|
| valign="bottom" |
<math>\begin{matrix}
f_5
\\[4pt]
f_{10}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
f_{0101}
\\[4pt]
f_{1010}
\end{matrix}\!</math>
| width="4%" |
| valign="bottom" |
<math>\begin{matrix}
0~1~0~1
\\[4pt]
1~0~1~0
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
(y)
\\[4pt]
y
\end{matrix}\!</math>
{| width="100%" cellpadding="0" cellspacing="0"
| valign="bottom" |
<math>\begin{matrix}
| width="60%" |
\text{not}~ y
\\[4pt]
y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\lnot y
\\[4pt]
y
\end{matrix}\!</math>
|-
|-
|
| valign="bottom" |
<math>\begin{matrix}
f_7
\\[4pt]
f_{11}
\\[4pt]
f_{13}
\\[4pt]
f_{14}
\end{matrix}\!</math>
| valign="bottom" |
|
<math>\begin{matrix}
f_{0111}
\\[4pt]
| From
f_{1011}
\\[4pt]
f_{1101}
\\[4pt]
f_{1110}
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
0~1~1~1
\\[4pt]
1~0~1~1
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
(~x~~y~)
\\[4pt]
(~x~(y))
\\[4pt]
((x)~y~)
\\[4pt]
((x)(y))
\end{matrix}\!</math>
| valign="bottom" |
<math>\begin{matrix}
\text{not both}~ x ~\text{and}~ y
\\[4pt]
\text{not}~ x ~\text{without}~ y
\\[4pt]
\text{not}~ y ~\text{without}~ x
\\[4pt]
x ~\text{or}~ y
\end{matrix}</math>
| valign="bottom" |
<math>\begin{matrix}
\lnot x \lor \lnot y
\\[4pt]
x \Rightarrow y
\\[4pt]
x \Leftarrow y
\\[4pt]
x \lor y
\end{matrix}\!</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{1111}\!</math>
| <math>A\!</math>
| <math>1~1~1~1\!</math>
| <math>((~))\!</math>
| <math>(\operatorname{d}A)\!</math>
| <math>\text{true}\!</math>
| <math>1\!</math>
| <math>A\!</math>
|}
<br>
==A Differential Extension of Propositional Calculus==
| <math>A\!</math>
| <math>\operatorname{d}A\!</math>
| <math>(A)\!</math>
| next.
{| width="100%" cellpadding="0" cellspacing="0"
{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |
| width="40%" |
| width="60%" |
| width="60%" |
The clock indicates the moment . . . . but what does<br>
Fire over water:<br>
The image of the condition before transition.<br>
Thus the superior man is careful<br>
In the differentiation of things,<br>
So that each finds its place.
|-
|-
|
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 79]
| align="right" | — ''I Ching'', Hexagram 64, [Wil, 249]
|}
|}
This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage. I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems. The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors. The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.
===Example 1. A Square Rigging===
===Differential Propositions : The Qualitative Analogues of Differential Equations===
In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math> Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.
Therefore, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as <math>\mathrm{d}\mathcal{A}</math> <math>=\!</math> <math>\{\mathrm{d}a_1, \ldots, \mathrm{d}a_n\},</math> in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <math>\mathcal{A}</math> <math>=\!</math> <math>\{a_1, \ldots, a_n\},</math> that is intended to be interpreted in the way just indicated. It only remains to be understood that the precise interpretation of the symbols in <math>\mathrm{d}\mathcal{A}</math> is often conceived to be changeable from point to point of the underlying space <math>A.\!</math> (Indeed, for all we know, the state space <math>A\!</math> might well be the state space of a language interpreter, one that is concerned, among other things, with the idiomatic meanings of the dialect generated by <math>\mathcal{A}</math> and <math>\mathrm{d}\mathcal{A}.</math>)
The ''tangent space'' to <math>A\!</math> at one of its points <math>x,\!</math> sometimes written <math>\mathrm{T}_x(A),</math> takes the form <math>\mathrm{d}A</math> <math>=\!</math> <math>\langle \mathrm{d}\mathcal{A} \rangle</math> <math>=\!</math> <math>\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle.\!</math> Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.
Proceeding as we did with the base space <math>A,\!</math> the tangent space <math>\mathrm{d}A</math> at a point of <math>A\!</math> can be analyzed as a product of distinct and independent factors:
: <math>\mathrm{d}A\ =\ \prod_{i=1}^n \mathrm{d}A_i\ =\ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.</math>
Here, <math>\mathrm{d}A_i</math> is a set of two differential propositions, <math>\mathrm{d}A_i = \{(\mathrm{d}a_i), \mathrm{d}a_i\},</math> where <math>(\mathrm{d}a_i)</math> is a proposition with the logical value of "<math>\text{not}\ \mathrm{d}a_i</math>". Each component <math>\mathrm{d}A_i</math> has the type <math>\mathbb{B},</math> operating under the ordered correspondence <math>\{(\mathrm{d}a_i), \mathrm{d}a_i\} \cong \{0, 1\}.</math> However, clarity is often served by acknowledging this differential usage with a superficially distinct type <math>\mathbb{D},</math> whose intension may be indicated as follows:
: <math>\mathbb{D} = \{(\mathrm{d}a_i), \mathrm{d}a_i\} = \{\text{same}, \text{different}\} = \{\text{stay}, \text{change}\} = \{\text{stop}, \text{step}\}.</math>
Viewed within a coordinate representation, spaces of type <math>\mathbb{B}^n</math> and <math>\mathbb{D}^n</math> may appear to be identical sets of binary vectors, but taking a view at this level of abstraction would be like ignoring the qualitative units and the diverse dimensions that distinguish position and momentum, or the different roles of quantity and impulse.
===An Interlude on the Path===
{| width="100%" cellpadding="0" cellspacing="0"
{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |
| width="4%" |
| width="60%" |
| width="92%" |
There would have been no beginnings: instead, speech would proceed from me, while I stood in its path – a slender gap – the point of its possible disappearance.
| width="4%" |
|-
|-
| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 28]
|}
|}
A sense of the relation between <math>\mathbb{B}</math> and <math>\mathbb{D}</math> may be obtained by considering the ''path classifier'' (or the ''equivalence class of curves'') approach to tangent vectors. Consider a universe <math>[\mathcal{X}].\!</math> Given the boolean value system, a path in the space <math>X = \langle \mathcal{X} \rangle</math> is a map <math>q : \mathbb{B} \to X.</math> In this context the set of paths <math>(\mathbb{B} \to X)</math> is isomorphic to the cartesian square <math>X^2 = X \times X,</math> or the set of ordered pairs chosen from <math>X.\!</math>
We may analyze <math>X^2 = \{ (u, v) : u, v \in X \}</math> into two parts, specifically, the ordered pairs <math>(u, v)\!</math> that lie on and off the diagonal:
: <math>\begin{matrix}
X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.
\end{matrix}</math>
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center"
This partition may also be expressed in the following symbolic form:
: <math>\begin{matrix}
X^2 & \cong & \mathrm{diag}(X) & + & 2 \tbinom{X}{2}.
\end{matrix}</math>
The separate terms of this formula are defined as follows:
: <math>\begin{matrix}
\mathrm{diag}(X) & = & \{ (x, x) : x \in X \}.
\end{matrix}\!</math>
<br>
: <math>\begin{matrix}
\tbinom{X}{k} & = & X\ \mathrm{choose}\ k & = & \{ k\!\text{-sets from}\ X \}.
\end{matrix}\!</math>
Thus we have:
: <math>\begin{matrix}
\tbinom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.
\end{matrix}</math>
We may now use the features in <math>\mathrm{d}\mathcal{X} = \{ \mathrm{d}x_i \} = \{ \mathrm{d}x_1, \ldots, \mathrm{d}x_n \}</math> to classify the paths of <math>(\mathbb{B} \to X)</math> by way of the pairs in <math>X^2.\!</math> If <math>X \cong \mathbb{B}^n,</math> then a path <math>q\!</math> in <math>X\!</math> has the following form:
: <math>\begin{matrix}
q : (\mathbb{B} \to \mathbb{B}^n) & \cong & \mathbb{B}^n \times \mathbb{B}^n & \cong & \mathbb{B}^{2n} & \cong & (\mathbb{B}^2)^n.
\end{matrix}</math>
Intuitively, we want to map this <math>(\mathbb{B}^2)^n</math> onto <math>\mathbb{D}^n</math> by mapping each component <math>\mathbb{B}^2</math> onto a copy of <math>\mathbb{D}.</math> But in the presenting context "<math>\mathbb{D}</math>" is just a name associated with, or an incidental quality attributed to, coefficient values in <math>\mathbb{B}</math> when they are attached to features in <math>\mathrm{d}\mathcal{X}.</math>
Taking these intentions into account, define <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> in the following manner:
{| width="100%" cellpadding="0" cellspacing="0"
: <p><math>\begin{array}{lcrcl}
\mathrm{d}x_i ((u, v)) & = & (\!|\ x_i (u) & , & x_i (v)\ |\!) \\
& = & x_i (u) & + & x_i (v) \\
& = & x_i (v) & - & x_i (u). \\
green stuff woven.
\end{array}</math></p>
In the above transcription, the operator bracket of the form <math>(\!| \ldots\ ,\ \ldots |\!)</math> is a ''cactus lobe'', in general signifying that just one of the arguments listed is false. In the case of two arguments this is the same thing as saying that the arguments are not equal. The plus sign signifies boolean addition, in the sense of addition in <math>\mathrm{GF}(2),</math> and thus means the same thing in this context as the minus sign, in the sense of adding the additive inverse.
:{| cellpadding=2
The above definition of <math>\mathrm{d}x_i : X^2 \to \mathbb{B}</math> is equivalent to defining <math>\mathrm{d}x_i : (\mathbb{B} \to X) \to \mathbb{B}\!</math> in the following way:
| E''X''
| =
: <p><math>\begin{array}{lcrcl}
| 〈''A'', d''A''〉
\mathrm{d}x_i (q) & = & (\!|\ x_i (q_0) & , & x_i (q_1)\ |\!) \\
& = & x_i (q_0) & + & x_i (q_1) \\
& = & x_i (q_1) & - & x_i (q_0). \\
\end{array}</math></p>
In this definition <math>q_b = q(b),\!</math> for each <math>b\!</math> in <math>\mathbb{B}.</math> Thus, the proposition <math>\mathrm{d}x_i</math> is true of the path <math>q = (u, v)\!</math> exactly if the terms of <math>q,\!</math> the endpoints <math>u\!</math> and <math>v,\!</math> lie on different sides of the question <math>x_i.\!</math>
The language of features in <math>\langle \mathrm{d}\mathcal{X} \rangle,</math> indeed the whole calculus of propositions in <math>[\mathrm{d}\mathcal{X}],</math> may now be used to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions <math>g : \mathrm{d}X \to \mathbb{B}.</math> For example, the paths corresponding to <math>\mathrm{diag}(X)</math> fall under the description <math>(\!| \mathrm{d}x_1 |\!) \cdots (\!| \mathrm{d}x_n |\!),</math> which says that nothing changes against the backdrop of the coordinate frame <math>\{ x_1, \ldots, x_n \}.</math>
Finally, a few words of explanation may be in order. If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X\!</math> that contains its range. In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.
===The Extended Universe of Discourse===
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.
| width="4%" |
|-
|-
|
| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
| {(''A''), ''A''} × {(d''A''), d''A''}
|}
|}
Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\mathrm{E}\mathcal{A}</math> as follows:
: <p><math>\begin{array}{lclcl}
\mathrm{E}\mathcal{A}
& = & \mathcal{A} \cup \mathrm{d}\mathcal{A}
& = & \{a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n\}. \\
\end{array}</math></p>
This supplies enough material to construct the ''differential extension'' <math>\mathrm{E}A,</math> or the ''tangent bundle'' over the initial space <math>A,\!</math> in the following fashion:
: <p><math>\begin{array}{lcl}
\mathrm{E}A
& = & \langle \mathrm{E}\mathcal{A} \rangle \\
& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle \\
& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle, \\
\end{array}</math></p>
and also:
: <p><math>\begin{array}{lcl}
\mathrm{E}A
& = & A \times \mathrm{d}A \\
& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n. \\
\end{array}</math></p>
This gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math>
{| align="center" border="1" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
Finally, the tangent universe <math>\mathrm{E}A^\circ = [\mathrm{E}\mathcal{A}]\!</math> is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features <math>\mathrm{E}\mathcal{A},</math> and this fact is summed up in the following notation:
|+ '''Table 14. Differential Propositions'''
: <p><math>\begin{array}{lclcl}
|
\mathrm{E}A^\circ
& = & [\mathrm{E}\mathcal{A}]
& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n]. \\
\end{array}</math></p>
This gives the tangent universe <math>\mathrm{E}A^\circ\!</math> the type:
|- style="background:ghostwhite"
|
: <p><math>\begin{array}{lcl}
| align="right" | dA :
(\mathbb{B}^n \times \mathbb{D}^n\ +\!\to \mathbb{B})
& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})). \\
|
\end{array}</math></p>
A proposition in the tangent universe <math>[\mathrm{E}\mathcal{A}]</math> is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.
With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),\!</math> we have arrived, in main outline, at one of the major subgoals of this study. Table 8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.
| g<sub>0</sub>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Symbol}\!</math>
| <math>\text{Notation}\!</math>
| <math>\text{Description}\!</math>
| <math>\text{Type}\!</math>
|-
|-
| <math>\mathrm{d}\mathfrak{A}\!</math>
| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}\!</math>
|
|
{|
<math>\begin{matrix}
\text{Alphabet of}
\\[2pt]
\text{differential symbols}
\end{matrix}</math>
| <math>[n] = \mathbf{n}\!</math>
|-
| <math>\mathrm{d}\mathcal{A}\!</math>
| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}\!</math>
|
|
<math>\begin{matrix}
\text{Basis of}
\\[2pt]
\text{differential features}
|}
\end{matrix}</math>
| <math>[n] = \mathbf{n}\!</math>
|-
| <math>\mathrm{d}A_i\!</math>
| <math>\{ (\mathrm{d}a_i), \mathrm{d}a_i \}~\!</math>
| <math>\text{Differential dimension}~ i\!</math>
| <math>\mathbb{D}\!</math>
|-
| <math>\mathrm{d}A\!</math>
|
|
{|
<math>\begin{matrix}
\langle \mathrm{d}\mathcal{A} \rangle
\\[2pt]
\langle \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle
\\[2pt]
\{ (\mathrm{d}a_1, \ldots, \mathrm{d}a_n) \}
\\[2pt]
\mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n
\\[2pt]
\textstyle \prod_i \mathrm{d}A_i
\end{matrix}</math>
|
|
<math>\begin{matrix}
\text{Tangent space at a point:}
\\[2pt]
\text{Set of changes, motions,}
|}
\\[2pt]
\text{steps, tangent vectors}
\\[2pt]
\text{at a point}
\end{matrix}</math>
| <math>\mathbb{D}^n\!</math>
|-
| <math>\mathrm{d}A^*\!</math>
| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})\!</math>
| <math>\text{Linear functions on}~ \mathrm{d}A\!</math>
| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n\!</math>
|-
| <math>\mathrm{d}A^\uparrow\!</math>
| <math>(\mathrm{d}A \to \mathbb{B})\!</math>
| <math>\text{Boolean functions on}~ \mathrm{d}A\!</math>
| <math>\mathbb{D}^n \to \mathbb{B}\!</math>
|-
| <math>\mathrm{d}A^\circ\!</math>
|
|
{|
<math>\begin{matrix}
[\mathrm{d}\mathcal{A}]
\\[2pt]
(\mathrm{d}A, \mathrm{d}A^\uparrow)
\\[2pt]
(\mathrm{d}A ~+\!\to \mathbb{B})
\\[2pt]
(\mathrm{d}A, (\mathrm{d}A \to \mathbb{B}))
\\[2pt]
[\mathrm{d}a_1, \ldots, \mathrm{d}a_n]
\end{matrix}</math>
|
|
<math>\begin{matrix}
\text{Tangent universe at a point of}~ A^\circ,
\\[2pt]
\text{based on the tangent features}
\\[2pt]
\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}
\end{matrix}</math>
|
|
{|
<math>\begin{matrix}
(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))
(A)(dA)<br>
\\[2pt]
(A) dA <br>
(\mathbb{D}^n ~+\!\to \mathbb{B})
\\[2pt]
[\mathbb{D}^n]
\end{matrix}</math>
|}
|}
<br>
The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself. Strictly speaking, we probably ought to call <math>\mathrm{d}\mathcal{A}</math> the set of ''cotangent'' features derived from <math>\mathcal{A},</math> but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> from cotangent vectors as elements of type <math>\mathbb{D}^n.</math> In like fashion, having defined <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A},</math> we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of <math>{2n}\!</math> features.
It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table 9 provides a suggestion of how these further extensions can be carried out.
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"
|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}\!</math>
|
|
{|
<p><math>\begin{array}{lllll}
\mathrm{d}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}
\\
\mathrm{d}^1 \mathcal{A} & = & \{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \} & = & \mathrm{d}\mathcal{A}
\end{array}</math></p>
<p><math>\begin{array}{lll}
\mathrm{d}^k \mathcal{A} & = & \{ \mathrm{d}^k a_1, \ldots, \mathrm{d}^k a_n \}
\\
\mathrm{d}^* \mathcal{A} & = & \{ \mathrm{d}^0 \mathcal{A}, \ldots, \mathrm{d}^k \mathcal{A}, \ldots \}
\end{array}</math></p>
|
|
<math>\begin{array}{lll}
\mathrm{E}^0 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A}
A but not dA<br>
\\
A and dA
\mathrm{E}^1 \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \mathrm{d}^1 \mathcal{A}
\\
\mathrm{E}^k \mathcal{A} & = & \mathrm{d}^0 \mathcal{A} ~\cup~ \ldots ~\cup~ \mathrm{d}^k \mathcal{A}
{|
\\
\mathrm{E}^\infty \mathcal{A} & = & \bigcup~ \mathrm{d}^* \mathcal{A}
\end{array}</math>
A ∧ ¬dA<br>
|}
|}
<br>
===Intentional Propositions===
{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |
| width="60%" |
Do you guess I have some intricate purpose?<br>
Well I have . . . . for the April rain has, and the mica on<br>
the side of a rock has.
|-
|-
|
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 45]
|
|}
|}
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.
As a standard way of dealing with these situations, the following scheme of notation suggests a way of extending any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators <math>\mathrm{p}^k</math> and <math>\mathrm{Q}^k</math> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"
|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}\!</math>
|
|
{|
<p><math>\begin{array}{lllll}
\mathrm{p}^0 \mathcal{A} & = & \{ a_1, \ldots, a_n \} & = & \mathcal{A}
\\
\mathrm{p}^1 \mathcal{A} & = & \{ a_1^\prime, \ldots, a_n^\prime \} & = & \mathcal{A}^\prime
\\
\mathrm{p}^2 \mathcal{A} & = & \{ a_1^{\prime\prime}, \ldots, a_n^{\prime\prime} \} & = & \mathcal{A}^{\prime\prime}
\\
\cdots & & \cdots &
\end{array}</math></p>
<p><math>\begin{array}{lll}
\mathrm{p}^k \mathcal{A} & = & \{ \mathrm{p}^k a_1, \ldots, \mathrm{p}^k a_n \}
\end{array}</math></p>
|
|
<math>\begin{array}{lll}
\mathrm{Q}^0 \mathcal{A} & = & \mathcal{A}
\\
\mathrm{Q}^1 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}'
\\
\mathrm{Q}^2 \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \mathcal{A}''
\\
\cdots & & \cdots
\\
\mathrm{Q}^k \mathcal{A} & = & \mathcal{A} \cup \mathcal{A}' \cup \ldots \cup \mathrm{p}^k \mathcal{A}
\end{array}</math>
|}
|}
|
<br>
|
The resulting augmentations of our logical basis determine a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. This extension follows a pattern analogous to the differential extension, which was developed in terms of the operators <math>\mathrm{d}^k</math> and <math>\mathrm{E}^k,</math> and there is a natural relation between these two extensions that bears further examination. In contexts displaying this pattern, where a sequence of domains stretches from an anchoring domain <math>X\!</math> through an indefinite number of higher reaches, a particular collection of domains based on <math>X\!</math> will be referred to as a ''realm'' of <math>X,\!</math> and when the succession exhibits a temporal aspect, as a ''reign'' of <math>X.\!</math>
For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\circ = [\mathrm{Q}\mathcal{X}],</math> in other words, a map <math>q : \mathrm{Q}X \to \mathbb{B}.</math> The sense of this definition may be seen if we consider the following facts. First, the equivalence <math>\mathrm{Q}X = X \times X'</math> motivates the following chain of isomorphisms between spaces:
: <p><math>\begin{array}{cclcc}
(\mathrm{Q}X \to \mathbb{B})
& \cong & (X & \times & X' \to \mathbb{B}) \\
& \cong & (X & \to & (X' \to \mathbb{B})) \\
& \cong & (X' & \to & (X \to \mathbb{B})). \\
\end{array}</math></p>
Viewed in this light, an intentional proposition <math>q\!</math> may be rephrased as a map <math>q : X \times X' \to \mathbb{B},</math> which judges the juxtaposition of states in <math>X\!</math> from one moment to the next. Alternatively, <math>q\!</math> may be parsed in two stages in two different ways, as <math>q : X \to (X' \to \mathbb{B})</math> and as <math>q : X' \to (X \to \mathbb{B}),</math> which associate to each point of <math>X\!</math> or <math>X'\!</math> a proposition about states in <math>X'\!</math> or <math>X,\!</math> respectively. In this way, an intentional proposition embodies a type of value system, in effect, a proposal that places a value on a collection of ends-in-view, or a project that evaluates a set of goals as regarded from each point of view in the state space of a system.
In sum, the intentional proposition <math>q\!</math> indicates a method for the systematic selection of local goals. As a general form of description, a map of the type <math>q : \mathrm{Q}^i X \to \mathbb{B}\!</math> may be referred to as an "<math>i^\mathrm{th}</math> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.
As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts — aims, ends, goals, objectives, purposes, and so on — metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.
===Life on Easy Street===
{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |
| width="60%" |
Failing to fetch me at first keep encouraged,<br>
Missing me one place search another,<br>
I stop some where waiting for you
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 88]
|}
|}
{|
The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least,
in principle. The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\mathrm{E}A.</math> Finding all the models of <math>q,\!</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.
(A)<br>
In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.
==Back to the Beginning : Exemplary Universes==
{|
|
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
|}
I would have preferred to be enveloped in words, borne way beyond all possible beginnings.
|
| width="4%" |
|-
|
| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
¬A<br>
|}
|}
To anchor our understanding of differential logic, let us look at how the various concepts apply in the simplest possible concrete cases, where the initial dimension is only 1 or 2. In spite of the obvious simplicity of these cases, it is possible to observe how central difficulties of the subject begin to arise already at this stage.
===A One-Dimensional Universe===
{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |
| width="60%" |
There was never any more inception than there is now,<br>
Nor any more youth or age than there is now;<br>
And will never be any more perfection than there is now,<br>
Nor any more heaven or hell than there is now.
|-
|-
|
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 28]
<br>
|}
|}
Let <math>\mathcal{X} = \{ x_1 \} = \{ A \}</math> be an alphabet that represents one boolean variable or a single logical feature. In this example I am using the capital letter "<math>A\!</math>" in a more usual informal way, to name a feature and not a space, in departure from my formerly stated formal conventions. At any rate, the basis element <math>A = x_1\!</math> may be interpreted as a simple proposition or a coordinate projection <math>A = x_1 : \mathbb{B} \xrightarrow{i} \mathbb{B}.</math> The space <math>X = \langle A \rangle = \{ (A), A \}</math> of points (cells, vectors, interpretations) has cardinality <math>2^n = 2^1 = 2\!</math> and is isomorphic to <math>\mathbb{B} = \{ 0, 1 \}.</math> Moreover, <math>X\!</math> may be identified with the set of singular propositions <math>\{ x : \mathbb{B} \xrightarrow{s} \mathbb{B} \}.</math> The space of linear propositions <math>X^* = \{ \mathrm{hom} : \mathbb{B} \xrightarrow{\ell} \mathbb{B} \} = \{ 0, A \}</math> is algebraically dual to <math>X\!</math> and also has cardinality <math>2.\!</math> Here, "<math>0\!</math>" is interpreted as denoting the constant function <math>0 : \mathbb{B} \to \mathbb{B},</math> amounting to the linear proposition of rank <math>0,\!</math> while <math>A\!</math> is the linear proposition of rank <math>1.\!</math> Last but not least we have the positive propositions <math>\{ \mathrm{pos} : \mathbb{B} \xrightarrow{p} \mathbb{B} \} = \{ A, 1 \},\!</math> of rank <math>1\!</math> and <math>0,\!</math> respectively, where "<math>1\!</math>" is understood as denoting the constant function <math>1 : \mathbb{B} \to \mathbb{B}.</math> In sum, there are <math>2^{2^n} = 2^{2^1} = 4</math> propositions altogether in the universe of discourse, comprising the set <math>X^\uparrow = \{ f : X \to \mathbb{B} \} = \{ 0, (A), A, 1 \} \cong (\mathbb{B} \to \mathbb{B}).</math>
The first order differential extension of <math>\mathcal{X}</math> is <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}.</math> If the feature <math>A\!</math> is understood as applying to some object or state, then the feature <math>\mathrm{d}A</math> may be interpreted as an attribute of the same object or state that says that it is changing ''significantly'' with respect to the property <math>A,\!</math> or that it has an ''escape velocity'' with respect to the state <math>A.\!</math>. In practice, differential features acquire their logical meaning through a class of ''temporal inference rules''.
For example, relative to a frame of observation that is left implicit for now, one is permitted to make the following sorts of inference: From the fact that <math>A\!</math> and <math>\mathrm{d}A</math> are true at a given moment one may infer that <math>(A)\!</math> will be true in the next moment of observation. Altogether in the present instance, there is the fourfold scheme of inference that is shown below:
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"
|
|
{|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
|
|
| From
| <math>(A)\!</math>
|}
| and
|
| <math>(\mathrm{d}A)\!</math>
| infer
|
| <math>(A)\!</math>
| next.
|
|}
|-
|
|
| From
|
| <math>(A)\!</math>
(A, dA)<br>
| and
| <math>\mathrm{d}A\!</math>
| infer
| <math>A\!</math>
| next.
|
|-
|
| From
| <math>A\!</math>
| and
| <math>(\mathrm{d}A)\!</math>
| infer
| <math>A\!</math>
| next.
|
|-
|
| From
| <math>A\!</math>
| and
| <math>\mathrm{d}A\!</math>
| infer
| <math>(A)\!</math>
| next.
|
|}
|}
|}
|}
<br>
It might be thought that an independent time variable needs to be brought in at this point, but it is an insight of fundamental importance that the idea of process is logically prior to the notion of time. A time variable is a reference to a ''clock'' — a canonical, conventional process that is accepted or established as a standard of measurement, but in essence no different than any other process. This raises the question of how different subsystems in a more global process can be brought into comparison, and what it means for one process to serve the function of a local standard for others. But these inquiries only wrap up puzzles in further riddles, and are obviously too involved to be handled at our current level of approximation.
{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |
| width="60%" |
The clock indicates the moment . . . . but what does<br>
eternity indicate?
|-
|-
|
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 79]
<br>
|}
|}
{|
Observe that the secular inference rules, used by themselves, involve a loss of information, since nothing in them can tell us whether the momenta <math>\{ (\mathrm{d}A), \mathrm{d}A \}</math> are changed or unchanged in the next instance. In order to know this, one would have to determine <math>\mathrm{d}^2 A,</math> and so on, pursuing an infinite regress. Ultimately, in order to rest with a finitely determinate system, it is necessary to make an infinite assumption, for example, that <math>\mathrm{d}^k A = 0</math> for all <math>k\!</math> greater than some fixed value <math>M.\!</math> Another way to escape the regress is through the provision of a dynamic law, in typical form making higher order differentials dependent on lower degrees and estates.
===Example 1. A Square Rigging===
|}
{| width="100%" cellpadding="0" cellspacing="0"
|
| width="40%" |
| width="60%" |
|
Urge and urge and urge,<br>
Always the procreant urge of the world.
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 28]
|}
|}
By way of example, suppose that we are given the initial condition <math>A = \mathrm{d}A\!</math> and the law <math>\mathrm{d}^2 A = (A).</math> Since the equation <math>A = \mathrm{d}A\!</math> is logically equivalent to the disjunction <math>A ~\mathrm{d}A ~\mathrm{or}~ (A)(\mathrm{d}A),\!</math> we may infer two possible trajectories, as displayed in Table 11. In either case the state <math>A\ (\mathrm{d}A)(\mathrm{d}^2 A)</math> is a stable attractor or a terminal condition for both starting points.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 11.} ~~ \text{A Pair of Commodious Trajectories}\!</math>
|- style="height:40px; background:ghostwhite"
| <math>\text{Time}\!</math>
| <math>\text{Trajectory 1}\!</math>
| <math>\text{Trajectory 2}\!</math>
|-
|
|
{|
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
2
\\[4pt]
3
\\[4pt]
4
\end{matrix}</math>
|
|
(dA)<br>
<math>\begin{matrix}
A & \mathrm{d}A & (\mathrm{d}^2 A)
\\[4pt]
(A) & \mathrm{d}A & \mathrm{d}^2 A
\\[4pt]
A & (\mathrm{d}A) & (\mathrm{d}^2 A)
\\[4pt]
A & (\mathrm{d}A) & (\mathrm{d}^2 A)
\\[4pt]
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}
\end{matrix}</math>
|
|
{|
<math>\begin{matrix}
(A) & (\mathrm{d}A) & \mathrm{d}^2 A
\\[4pt]
(A) & \mathrm{d}A & \mathrm{d}^2 A
\\[4pt]
A & (\mathrm{d}A) & (\mathrm{d}^2 A)
\\[4pt]
A & (\mathrm{d}A) & (\mathrm{d}^2 A)
\\[4pt]
{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}
\end{matrix}</math>
|}
|}
|
<br>
|
Because the initial space ''X'' = 〈''A''〉 is one-dimensional, we can easily fit the second order extension E<sup>2</sup>''X'' = 〈''A'', d''A'', d<sup>2</sup>''A''〉 within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure 12.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 12 -- The Anchor.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 12.} ~~ \text{The Anchor}\!</math>
|}
|}
If we eliminate from view the regions of E<sup>2</sup>''X'' that are ruled out by the dynamic law d<sup>2</sup>''A'' = (''A''), then what remains is the quotient structure that is shown in Figure 13. This picture makes it easy to see that the dynamically allowable portion of the universe is partitioned between the properties ''A'' and d<sup>2</sup>''A''. As it happens, this fact might have been expressed "right off the bat" by an equivalent formulation of the differential law, one that uses the exclusive disjunction to state the law as (''A'', d<sup>2</sup>''A'').
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 13 -- The Tiller.gif|center]]
|-
|-
|
| height="20px" valign="top" | <math>\text{Figure 13.} ~~ \text{The Tiller}\!</math>
|}
|}
What we have achieved in this example is to give a differential description of a simple dynamic process. In effect, we did this by embedding a directed graph, which can be taken to represent the state transitions of a finite automaton, in a dynamically allotted quotient structure that is created from a boolean lattice or an ''n''-cube by nullifying all of the regions that the dynamics outlaws. With growth in the dimensions of our contemplated universes, it becomes essential, both for human comprehension and for computer implementation, that the dynamic structures of interest to us be represented not actually, by acquaintance, but virtually, by description. In our present study, we are using the language of propositional calculus to express the relevant descriptions, and to comprehend the structure that is implicit in the subsets of a ''n''-cube without necessarily being forced to actualize all of its points.
One of the reasons for engaging in this kind of extremely reduced, but explicitly controlled case study is to throw light on the general study of languages, formal and natural, in their full array of syntactic, semantic, and pragmatic aspects. Propositional calculus is one of the last points of departure where we can view these three aspects interacting in a non-trivial way without being immediately and totally overwhelmed by the complexity they generate. Often this complexity causes investigators of formal and natural languages to adopt the strategy of focusing on a single aspect and to abandon all hope of understanding the whole, whether it's the still living natural language or the dynamics of inquiry that lies crystallized in formal logic.
From the perspective that I find most useful here, a language is a syntactic system that is designed or evolved in part to express a set of descriptions. When the explicit symbols of a language have extensions in its object world that are actually infinite, or when the implicit categories and generative devices of a linguistic theory have extensions in its subject matter that are potentially infinite, then the finite characters of terms, statements, arguments, grammars, logics, and rhetorics force an excess of intension to reside in all these symbols and functions, across the spectrum from the object language to the metalinguistic uses. In the aphorism from W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30] and [Cho93, 49], language requires "the infinite use of finite means". This is necessarily true when the extensions are infinite, when the referential symbols and grammatical categories of a language possess infinite sets of models and instances. But it also voices a practical truth when the extensions, though finite at every stage, tend to grow at exponential rates.
This consequence of dealing with extensions that are "practically infinite" becomes crucial when one tries to build neural network systems that learn, since the learning competence of any intelligent system is limited to the objects and domains that it is able to represent. If we want to design systems that operate intelligently with the full deck of propositions dealt by intact universes of discourse, then we must supply them with succinct representations and efficient transformations in this domain. Furthermore, in the project of constructing inquiry driven systems, we find ourselves forced to contemplate the level of generality that is embodied in propositions, because the dynamic evolution of these systems is driven by the measurable discrepancies that occur among their expectations, intentions, and observations, and because each of these subsystems or components of knowledge constitutes a propositional modality that can take on the fully generic character of an empirical summary or an axiomatic theory.
A compression scheme by any other name is a symbolic representation, and this is what the differential extension of propositional calculus, through all of its many universes of discourse, is intended to supply. Why is this particular program of mental calisthenics worth carrying out in general? By providing a uniform logical medium for describing dynamic systems we can make the task of understanding complex systems much easier, both in looking for invariant representations of individual cases and in finding points of comparison among diverse structures that would otherwise appear as isolated systems. All of this goes to facilitate the search for compact knowledge and to adapt what is learned from individual cases to the general realm.
===Tacit Extensions===
===Back to the Feature===
{| width="100%" cellpadding="0" cellspacing="0"
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="40%" |
| width="92%" |
| width="60%" |
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.
I guess it must be the flag of my disposition, out of hopeful<br>
green stuff woven.
|-
|-
| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 31]
|}
|}
Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that I may continue with outlining the structure of the differential extension [E<font face="lucida calligraphy">X</font>] = [''A'', d''A'']. Over the extended alphabet E<font face="lucida calligraphy">X</font> = {''x''<sub>1</sub>, d''x''<sub>1</sub>} = {''A'', d''A''}, of cardinality 2<sup>''n''</sup> = 2, we generate the set of points, E''X'', of cardinality 2<sup>2''n''</sup> = 4, that bears the following chain of equivalent descriptions:
:{| cellpadding=2
| E''X''
| =
| 〈''A'', d''A''〉
|-
|
| =
| {(''A''), ''A''} × {(d''A''), d''A''}
|-
|
| =
| {(''A'')(d''A''), (''A'') d''A'', ''A'' (d''A''), ''A'' d''A''}.
|}
The space E''X'' may be assigned the mnemonic type '''B''' × '''D''', which is really no different than '''B''' × '''B''' = '''B'''<sup>2</sup>. An individual element of E''X'' may be regarded as a ''disposition at a point'' or a ''situated direction'', in effect, a singular mode of change occurring at a single point in the universe of discourse. In applications, the modality of this change can be interpreted in various ways, for example, as an expectation, an intention, or an observation with respect to the behavior of a system.
To complete the construction of the extended universe of discourse E''X''<sup> •</sup> = [''x''<sub>1</sub>, d''x''<sub>1</sub>] = [''A'', d''A''], one must add the set of differential propositions E''X''^ = {''g'' : E''X'' → '''B'''} <math>\cong</math> ('''B''' × '''D''' → '''B''') to the set of dispositions in E''X''. There are <math>2^{2^{2n}}</math> = 16 propositions in E''X''^, as detailed in Table 14.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|+ style="height:30px" | <math>\text{Table 14.} ~~ \text{Differential Propositions}\!</math>
|- style="background:ghostwhite"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"
|
|+ '''Table 15. Tacit Extension of <math>[A]\!</math> to <math>[A, \operatorname{d}A]</math>'''
| align="right" | <math>A\colon\!</math>
|
| <math>1~1~0~0\!</math>
|
|
|
|
|
|-
|- style="background:ghostwhite"
|
|
| <math>(A)\!</math>
| align="right" | <math>\mathrm{d}A\colon\!</math>
| <math>1~0~1~0\!</math>
| <math>((\operatorname{d}A),\ \operatorname{d}A)\!</math>
| <math>=\!</math>
|
|
|
|
|
|
|-
|-
| <math>f_0\!</math>
| <math>1\!</math>
| <math>g_0\!</math>
| <math>=\!</math>
| <math>0~0~0~0\!</math>
| <math>1\!</math>
| <math>(~)\!</math>
| <math>\cdot\!</math>
| <math>\text{false}\!</math>
| <math>((\operatorname{d}A),\ \operatorname{d}A)\!</math>
| <math>0\!</math>
| <math>=\!</math>
|-
|-
|
|
|
|
<math>\begin{matrix}
g_1
\\[4pt]
g_2
| <math>=</math>
\\[4pt]
g_4
\\[4pt]
g_8
\end{matrix}\!</math>
|
<math>\begin{matrix}
0~0~0~1
| <math>=</math>
\\[4pt]
0~0~1~0
\\[4pt]
0~1~0~0
\\[4pt]
1~0~0~0
\end{matrix}\!</math>
|
<math>\begin{matrix}
(A)(\mathrm{d}A)
\\[4pt]
(A)~\mathrm{d}A~
\\[4pt]
~A~(\mathrm{d}A)
\\[4pt]
~A~~\mathrm{d}A~
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
\text{neither}~ A ~\text{nor}~ \mathrm{d}A
\\[4pt]
\text{not}~ A ~\text{but}~ \mathrm{d}A
\\[4pt]
A ~\text{but not}~ \mathrm{d}A
\\[4pt]
A ~\text{and}~ \mathrm{d}A
\end{matrix}\!</math>
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center"
<math>\begin{matrix}
\lnot A \land \lnot \mathrm{d}A
\\[4pt]
\lnot A \land \mathrm{d}A
\\[4pt]
A \land \lnot \mathrm{d}A
\\[4pt]
A \land \mathrm{d}A
\end{matrix}\!</math>
|-
|-
| ''q''<sub>03</sub>
|
<math>\begin{matrix}
f_1
\\[4pt]
f_2
\end{matrix}\!</math>
|
<math>\begin{matrix}
g_3
\\[4pt]
g_{12}
\end{matrix}\!</math>
|
<math>\begin{matrix}
0~0~1~1
\\[4pt]
1~1~0~0
\end{matrix}\!</math>
|
<math>\begin{matrix}
(A)
\\[4pt]
A
\end{matrix}\!</math>
|
<math>\begin{matrix}
\text{not}~ A
\\[4pt]
A
\end{matrix}\!</math>
|
<math>\begin{matrix}
\lnot A
\\[4pt]
A
\end{matrix}\!</math>
|-
|-
|
|
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center"
<math>\begin{matrix}
| ''p''<sub>0</sub>
g_6
|-
\\[4pt]
g_9
|-
\end{matrix}\!</math>
|
<math>\begin{matrix}
0~1~1~0
\\[4pt]
1~0~0~1
\end{matrix}\!</math>
|
<math>\begin{matrix}
(A, \mathrm{d}A)
\\[4pt]
((A, \mathrm{d}A))
\end{matrix}\!</math>
|
<math>\begin{matrix}
A ~\text{not equal to}~ \mathrm{d}A
\\[4pt]
A ~\text{equal to}~ \mathrm{d}A
\end{matrix}\!</math>
|
<math>\begin{matrix}
A \ne \mathrm{d}A
\\[4pt]
A = \mathrm{d}A
\end{matrix}\!</math>
|-
|-
| ''p''<sub>3</sub>
|
|
<math>\begin{matrix}
g_5
\\[4pt]
g_{10}
\end{matrix}\!</math>
|
<math>\begin{matrix}
0~1~0~1
\\[4pt]
1~0~1~0
\end{matrix}\!</math>
|
<math>\begin{matrix}
(\mathrm{d}A)
\\[4pt]
\mathrm{d}A
\end{matrix}\!</math>
|
<math>\begin{matrix}
\text{not}~ \mathrm{d}A
\\[4pt]
\mathrm{d}A
\end{matrix}\!</math>
|
<math>\begin{matrix}
\lnot \mathrm{d}A
\\[4pt]
\mathrm{d}A
\end{matrix}\!</math>
|-
|-
| ''p''<sub>4</sub>
|
|-
|
<math>\begin{matrix}
g_7
\\[4pt]
g_{11}
\\[4pt]
g_{13}
\\[4pt]
g_{14}
\end{matrix}\!</math>
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center"
<math>\begin{matrix}
0~1~1~1
|-
\\[4pt]
1~0~1~1
|-
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\end{matrix}\!</math>
|
<math>\begin{matrix}
(~A~~\mathrm{d}A~)
|-
\\[4pt]
(~A~(\mathrm{d}A))
\\[4pt]
((A)~\mathrm{d}A~)
\\[4pt]
((A)(\mathrm{d}A))
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
\end{matrix}\!</math>
|
<math>\begin{matrix}
\text{not both}~ A ~\text{and}~ \mathrm{d}A
\\[4pt]
\text{not}~ A ~\text{without}~ \mathrm{d}A
\\[4pt]
\text{without}~ A ~\text{not}~ \mathrm{d}A
\\[4pt]
A ~\text{or}~ \mathrm{d}A
\end{matrix}\!</math>
|
<math>\begin{matrix}
\lnot A \lor \lnot \mathrm{d}A
\\[4pt]
A \Rightarrow \mathrm{d}A
\\[4pt]
A \Leftarrow \mathrm{d}A
\\[4pt]
A \lor \mathrm{d}A
\end{matrix}\!</math>
|-
|-
| 1. || 1 || 1 || 0 || 1
| <math>f_3\!</math>
|-
| <math>g_{15}\!</math>
| 0. || 0 || 1 || 1 || 1
| <math>1~1~1~1\!</math>
| <math>((~))\!</math>
| <math>\text{true}\!</math>
|-
| <math>1\!</math>
| 1. || 1 || 0 || 1 || 1
| 0. || 1 || 1 || 0 || 1
|}
|}
<br>
<br>
==Transformations of Discourse==
Aside from changing the names of variables and shuffling the order of rows, this Table follows the format that was used previously for boolean functions of two variables. The rows are grouped to reflect natural similarity classes among the propositions. In a future discussion, these classes will be given additional explanation and motivation as the orbits of a certain transformation group acting on the set of 16 propositions. Notice that four of the propositions, in their logical expressions, resemble those given in the table for ''X''^. Thus the first set of propositions {''f''<sub>''i''</sub>} is automatically embedded in the present set {''g''<sub>''j''</sub>}, and the corresponding inclusions are indicated at the far left margin of the table.
===Tacit Extensions===
{| width="100%" cellpadding="0" cellspacing="0"
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="4%" |
| width="92%" |
| width="92%" |
I would really like to have slipped imperceptibly into this lecture, as into all the others I shall be delivering, perhaps over the years ahead.
| width="4%" |
| width="4%" |
|-
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 39]
| align="right" colspan="3" | — Michel Foucault, ''The Discourse on Language'', [Fou, 215]
|}
|}
Strictly speaking, however, there is a subtle distinction in type between the function <math>f_i : X \to \mathbb{B}</math> and the corresponding function <math>g_j : \mathrm{E}X \to \mathbb{B},</math> even though they share the same logical expression. Naturally, we want to maintain the logical equivalence of expressions that represent the same proposition while appreciating the full diversity of that proposition's functional and typical representatives. Both perspectives, and all the levels of abstraction extending through them, have their reasons, as will develop in time.
Because this special circumstance points up an important general theme, it is a good idea to discuss it more carefully. Whenever there arises a situation like this, where one alphabet <math>\mathcal{X}</math> is a subset of another alphabet <math>\mathcal{Y},</math> then we say that any proposition <math>f : \langle \mathcal{X} \rangle \to \mathbb{B}</math> has a ''tacit extension'' to a proposition <math>\epsilon f : \langle \mathcal{Y} \rangle \to \mathbb{B},</math> and that the space <math>(\langle \mathcal{X} \rangle \to \mathbb{B})</math> has an ''automatic embedding'' within the space <math>(\langle \mathcal{Y} \rangle \to \mathbb{B}).</math> The extension is defined in such a way that <math>\epsilon f\!</math> puts the same constraint on the variables of <math>\mathcal{X}</math> that are contained in <math>\mathcal{Y}</math> as the proposition <math>f\!</math> initially did, while it puts no constraint on the variables of <math>\mathcal{Y}</math> outside of <math>\mathcal{X},</math> in effect, conjoining the two constraints.
If the variables in question are indexed as <math>\mathcal{X} = \{ x_1, \ldots, x_n \}</math> and <math>\mathcal{Y} = \{ x_1, \ldots, x_n, \ldots, x_{n+k} \},</math> then the definition of the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> may be expressed in the form of an equation:
: <math>\epsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) = f(x_1, \ldots, x_n).</math>
{| width="100%" cellpadding="0" cellspacing="0"
On formal occasions, such as the present context of definition, the tacit extension from <math>\mathcal{X}</math> to <math>\mathcal{Y}</math> is explicitly symbolized by the operator <math>\epsilon : (\langle \mathcal{X} \rangle \to \mathbb{B}) \to (\langle \mathcal{Y} \rangle \to \mathbb{B}),</math> where the appropriate alphabets <math>\mathcal{X}</math> and <math>\mathcal{Y}</math> are understood from context, but normally one may leave the "<math>\epsilon\!</math>" silent.
Let's explore what this means for the present Example. Here, <math>\mathcal{X} = \{ A \}</math> and <math>\mathcal{Y} = \mathrm{E}\mathcal{X} = \{ A, \mathrm{d}A \}.</math> For each of the propositions <math>f_i\!</math> over <math>X\!,</math> specifically, those whose expression <math>e_i\!</math> lies in the collection <math>\{ 0, (A), A, 1 \},\!</math> the tacit extension <math>\epsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> can be phrased as a logical conjunction of two factors, <math>f_i = e_i \cdot \tau\ ,</math> where <math>\tau\!</math> is a logical tautology that uses all the variables of <math>\mathcal{Y} - \mathcal{X}.</math> Working in these terms, the tacit extensions <math>\epsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> may be explicated as shown in Table 15.
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"
|+ style="height:30px" | <math>\text{Table 15.}~~\text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math>
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
|
| <math>0\!</math>
| <math>=\!</math>
| <math>0\!</math>
| <math>\cdot\!</math>
| <math>((\mathrm{d}A) ~,~ \mathrm{d}A)\!</math>
| <math>=\!</math>
| <math>0\!</math>
|
|-
|
| <math>(A)\!</math>
| <math>=\!</math>
| <math>(A)\!</math>
| <math>\cdot\!</math>
| <math>((\mathrm{d}A) ~,~ \mathrm{d}A)\!</math>
| <math>=\!</math>
| <math>(A)(\mathrm{d}A) ~+~ (A)~\mathrm{d}A\!</math>
|
|-
|
| <math>A\!</math>
| <math>=\!</math>
| <math>A\!</math>
| <math>\cdot\!</math>
| <math>((\mathrm{d}A) ~,~ \mathrm{d}A)\!</math>
| <math>=\!</math>
| <math>A~(\mathrm{d}A) ~+~ A~\mathrm{d}A\!</math>
|
|-
|
| <math>1\!</math>
| <math>=\!</math>
| <math>1\!</math>
| <math>\cdot\!</math>
| <math>((\mathrm{d}A) ~,~ \mathrm{d}A)\!</math>
| <math>=\!</math>
| <math>1\!</math>
|}
|}
<br>
<br>
In its effect on the singular propositions over <math>X,\!</math> this analysis has an interesting interpretation. The tacit extension takes us from thinking about a particular state, like <math>A\!</math> or <math>(A),\!</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.
===Example 2. Drives and Their Vicissitudes===
{| width="100%" cellpadding="0" cellspacing="0"
| width="40%" |
| width="60%" |
I open my scuttle at night and see the far-sprinkled systems,<br>
And all I see, multiplied as high as I can cipher, edge but<br>
the rim of the farther systems.
|-
|
| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 81]
|}
Before we leave the one-feature case let's look at a more substantial example, one that illustrates a general class of curves that can be charted through the extended feature spaces and that provides an opportunity to discuss a number of important themes concerning their structure and dynamics.
Again, let <font face="lucida calligraphy">X</font> = {''x''<sub>1</sub>} = {''A''}. In the discussion that follows I will consider a class of trajectories having the property that d<sup>''k''</sup>''A'' = 0 for all ''k'' greater than some fixed ''m'', and I indulge in the use of some picturesque terms that describe salient classes of such curves. Given the finite order condition, there is a highest order non-zero difference d<sup>''m''</sup>''A'' exhibited at each point in the course of any determinate trajectory that one may wish to consider. With respect to any point of the corresponding orbit or curve let us call this highest order differential feature d<sup>''m''</sup>''A'' the ''drive'' at that point. Curves of constant drive d<sup>''m''</sup>''A'' are then referred to as "''m''<sup>th</sup> gear curves".
* '''Scholium.''' The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, § 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic. And of course there is the strange but true story of how the Turin machines of the 1840's prefigured the Turing machines of the 1940's [Men, 225-297]. At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].
====Extension from 2 to 4 Dimensions====
Given this language, the particular Example that I take up here can be described as the family of 4<sup>th</sup> gear curves through E<sup>4</sup>''X'' = 〈''A'', d''A'', d<sup>2</sup>''A'', d<sup>3</sup>''A'', d<sup>4</sup>''A''〉. These are the trajectories generated subject to the dynamic law d<sup>4</sup>''A'' = 1, where it is understood in such a statement that all higher order differences are equal to 0. Since d<sup>4</sup>''A'' and all higher d<sup>''k''</sup>''A'' are fixed, the temporal or transitional conditions (initial, mediate, terminal — transient or stable states) vary only with respect to their projections as points of E<sup>3</sup>''X'' = 〈''A'', d''A'', d<sup>2</sup>''A'', d<sup>3</sup>''A''〉. Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of E<sup>3</sup>''X''. It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure 16.
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type '''B'''<sup>2</sup> → '''B'''<sup>4</sup>. In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]
|-
| height="20px" valign="top" | <math>\text{Figure 16.} ~~ \text{A Couple of Fourth Gear Orbits}\!</math>
|}
With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation. With this end in sight, let us index the states ''q'' in E<sup>''m''</sup>''X'' with the dyadic rationals (or the binary fractions) in the half-open interval [0, 2). Formally and canonically, a state ''q''<sub>''r''</sub> is indexed by a fraction ''r'' = ''s''/''t'' whose denominator is the power of two ''t'' = 2<sup>''m''</sup> and whose numerator is a binary numeral that is formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state ''q'' are just the values d<sup>''k''</sup>''A''(''q''), for ''k'' = 0 to ''m'', where d<sup>0</sup>''A'' is defined as being identical to ''A''. To form the binary index d<sub>0</sub>'''.'''d<sub>1</sub>…d<sub>''m''</sub> of the state ''q'' the coefficient d<sup>''k''</sup>''A''(''q'') is read off as the binary digit ''d''<sub>''k''</sub> associated with the place value 2<sup>–''k''</sup>. Expressed by way of algebraic formulas, the rational index ''r'' of the state ''q'' can be given by the following equivalent formulations:
<p><center><font size="+1">'''Figure 19-a. Extension from 2 to 4 Dimensions: Areal'''</font></center></p>
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:75%"
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="text-align:center; width:100%"
| <math>r(q)\!</math>
<br>
| <math>=\!</math>
| <math>\sum_k d_k \cdot 2^{-k}\!</math>
<p><center><font size="+1">'''Figure 19-c. Extension from 2 to 4 Dimensions: Compact'''</font></center></p>
| <math>=\!</math>
| <math>\sum_k \text{d}^k A(q) \cdot 2^{-k}\!</math>
|-
| <math>=\!</math>
<br>
|-
<p>[[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]</p>
| <math>\frac{s(q)}{t}\!</math>
<p><center><font size="+1">'''Figure 19-d. Extension from 2 to 4 Dimensions: Digraph'''</font></center></p>
| <math>=\!</math>
| <math>\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}\!</math>
| <math>=\!</math>
| <math>\frac{\sum_k \text{d}^k A(q) \cdot 2^{(m-k)}}{2^m}\!</math>
|}
|}
|}
|}
<br>
<br>
Applied to the example of fourth gear curves, this scheme results in the data of Tables 17-a and 17-b, which exhibit one period for each orbit. The states in each orbit are listed as ordered pairs ‹''p''<sub>''i''</sub>, ''q''<sub>''j''</sub>›, where ''p''<sub>''i''</sub> may be read as a temporal parameter that indicates the present time of the state, and where ''j'' is the decimal equivalent of the binary numeral ''s''. Informally and more casually, the Tables exhibit the states ''q''<sub>''s''</sub> as subscripted with the numerators of their rational indices, taking for granted the constant denominators of 2<sup>''m''</sup> = 2<sup>4</sup> = 16. Within this set-up, the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if ‹''d''<sub>''k''</sub>, ''d''<sub>''k''+1</sub>› is any pair of adjacent digits in the state index ''r'', then the value of ''d''<sub>''k''</sub> in the next state is ''d''<sub>''k''</sub>′ = ''d''<sub>''k''</sub> + ''d''<sub>''k''+1</sub>.
<br>
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:52%"
|+ style="height:30px" | <math>\text{Table 17-a.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 1}\!</math>
| | | |
|- style="background:ghostwhite"
| <math>\text{Time}\!</math>
| / \ / \ | | / \ / \ |
| <math>\text{State}\!</math>
| <math>A\!</math>
| <math>\mathrm{d}A</math>
|
|
|
|- style="background:ghostwhite"
| <math>p_i\!</math>
| <math>q_j\!</math>
| <math>\mathrm{d}^0\!A</math>
| <math>\mathrm{d}^1\!A</math>
| <math>\mathrm{d}^2\!A</math>
| <math>\mathrm{d}^3\!A</math>
| <math>\mathrm{d}^4\!A</math>
|-
|
<math>\begin{matrix}
p_0
| \ / | | \ / |
\\[4pt]
p_1
\\[4pt]
| /`````````````\ | | /`````````````\ |
p_2
\\[4pt]
| /`````````````````\ | | /`````````````````\ |
p_3
\\[4pt]
p_4
\\[4pt]
p_5
\\[4pt]
p_6
| \```````````````/ | | \```````````````/ |
\\[4pt]
p_7
\end{matrix}\!</math>
|
| | | |
<math>\begin{matrix}
q_{01}
\\[4pt]
q_{03}
\\[4pt]
</pre>
q_{05}
\\[4pt]
q_{15}
\\[4pt]
q_{17}
\\[4pt]
q_{19}
\\[4pt]
<br>
q_{21}
\\[4pt]
<p><center><font size="+1">'''Figure 20-iii. Thematization of Conjunction (Stage 3)'''</font></center></p>
q_{31}
\end{matrix}\!</math>
| colspan="5" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"
|
<math>\begin{matrix}
0.
\\[4pt]
0.
\\[4pt]
0.
\\[4pt]
<p><center><font size="+1">'''Figure 21. Thematization of Disjunction and Equality'''</font></center></p>
0.
\\[4pt]
1.
\\[4pt]
{| width="100%" cellpadding="0" cellspacing="0"
1.
\\[4pt]
1.
\\[4pt]
1.
\end{matrix}\!</math>
<br><font face="courier new">
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Table 22. Disjunction ''f'' and Equality ''g'' '''
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
0
\\[4pt]
1
\\[4pt]
0
\\[4pt]
0
\\[4pt]
0
\\[4pt]
1
\end{matrix}\!</math>
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
1
\\[4pt]
1
\\[4pt]
0
\\[4pt]
0
\\[4pt]
1
\\[4pt]
1
\end{matrix}\!</math>
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
0
\\[4pt]
1
\\[4pt]
|
0
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
\\[4pt]
1
\\[4pt]
0
\\[4pt]
1
\end{matrix}\!</math>
|
<math>\begin{matrix}
1
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\end{matrix}\!</math>
|}
|}
|}
|}
<br>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:52%"
{| align="center" style="width:96%"
|+ style="height:30px" | <math>\text{Table 17-b.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 2}\!</math>
|+ '''Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)'''
|- style="background:ghostwhite"
| <math>\text{Time}\!</math>
| <math>\text{State}\!</math>
| <math>A\!</math>
| <math>\mathrm{d}A</math>
|
|
|
|- style="background:ghostwhite"
| <math>p_i\!</math>
| <math>q_j\!</math>
| <math>\mathrm{d}^0\!A</math>
| <math>\mathrm{d}^1\!A</math>
| <math>\mathrm{d}^2\!A</math>
| <math>\mathrm{d}^3\!A</math>
| <math>\mathrm{d}^4\!A</math>
|-
|
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
<math>\begin{matrix}
p_0
\\[4pt]
p_1
\\[4pt]
p_2
\\[4pt]
p_3
\\[4pt]
p_4
\\[4pt]
p_5
\\[4pt]
p_6
\\[4pt]
p_7
\end{matrix}\!</math>
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
<math>\begin{matrix}
q_{25}
\\[4pt]
|-
q_{11}
|
\\[4pt]
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
q_{29}
\\[4pt]
q_{07}
\\[4pt]
q_{09}
\\[4pt]
q_{27}
\\[4pt]
q_{13}
\\[4pt]
q_{23}
\end{matrix}\!</math>
| colspan="5" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="text-align:center; width:100%"
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
<math>\begin{matrix}
1.
\\[4pt]
0.
\\[4pt]
1.
\\[4pt]
0.
\\[4pt]
0.
\\[4pt]
1.
\\[4pt]
0.
\\[4pt]
1.
\end{matrix}\!</math>
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
<math>\begin{matrix}
1
\\[4pt]
1
\\[4pt]
1
\\[4pt]
0
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\\[4pt]
0
\end{matrix}\!</math>
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
1
\\[4pt]
1
\\[4pt]
0
\\[4pt]
0
\\[4pt]
1
\\[4pt]
1
\end{matrix}\!</math>
|
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
0
\\[4pt]
1
\\[4pt]
0
\\[4pt]
1
\\[4pt]
0
\\[4pt]
1
\end{matrix}\!</math>
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
<math>\begin{matrix}
1
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\end{matrix}\!</math>
|}
|}
|}
|}
<br>
==Transformations of Discourse==
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
It is understandable that an engineer should be completely absorbed in his speciality, instead of pouring himself out into the freedom and vastness of the world of thought, even though his machines are being sent off to the ends of the earth; for he no more needs to be capable of applying to his own personal soul what is daring and new in the soul of his subject than a machine is in fact capable of applying to itself the differential calculus on which it is based. The same thing cannot, however, be said about mathematics; for here we have the new method of thought, pure intellect, the very well-spring of the times, the ''fons et origo'' of an unfathomable transformation.
| width="4%" |
|-
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 39]
|}
| 0 || 0 || →
In this section we take up the general study of logical transformations, or maps that relate one universe of discourse to another. In many ways, and especially as applied to the subject of intelligent dynamic systems, my argument develops the antithesis of the statement just quoted. Along the way, if incidental to my ends, I hope this essay can pose a fittingly irenic epitaph to the frankly ironic epigraph inscribed at its head.
My goal in this section is to answer a single question: What is a propositional tangent functor? In other words, my aim is to develop a clear conception of what manner of thing would pass in the logical realm for a genuine analogue of the tangent functor, an object conceived to generalize as far as possible in the abstract terms of category theory the ordinary notions of functional differentiation and the all too familiar operations of taking derivatives.
As a first step I discuss the kinds of transformations that we already know as ''extensions'' and ''projections'', and I use these special cases to illustrate several different styles of logical and visual representation that will figure heavily in the sequel.
===Foreshadowing Transformations : Extensions and Projections of Discourse===
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
And, despite the care which she took to look behind her at every moment, she failed to see a shadow which followed her like her own shadow, which stopped when she stopped, which started again when she did, and which made no more noise than a well-conducted shadow should.
| width="4%" |
|-
|-
| 0 || 1 || →
| align="right" colspan="3" | — Gaston Leroux, ''The Phantom of the Opera'', [Ler, 126]
|}
Many times in our discussion we have occasion to place one universe of discourse in the context of a larger universe of discourse. An embedding of the general type [<font face="lucida calligraphy">X</font>] → [<font face="lucida calligraphy">Y</font>] is implied any time that we make use of one alphabet <font face="lucida calligraphy">X</font> that happens to be included in another alphabet <font face="lucida calligraphy">Y</font>. When we are discussing differential issues we usually have in mind that the extended alphabet <font face="lucida calligraphy">Y</font> has a special construction or a specific lexical relation with respect to the initial alphabet <font face="lucida calligraphy">X</font>, one that is marked by characteristic types of accents, indices, or inflected forms.
====Extension from 1 to 2 Dimensions====
Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type '''B'''<sup>1</sup> → '''B'''<sup>2</sup> and detailing the coordinates that are associated with individual cells. Because all points, cells, or logical interpretations are represented as connected geometric areas, we can say that these pictures provide us with an ''areal view'' of each universe of discourse.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 18-a -- Extension from 1 to 2 Dimensions.gif|center]]
|-
|-
| 1 || 0 || →
| height="20px" valign="top" | <math>\text{Figure 18-a.} ~~ \text{Extension from 1 to 2 Dimensions : Areal}\!</math>
|}
Figure 18-b shows the differential extension from ''X''<sup> •</sup> = [''x''] to E''X''<sup> •</sup> = [''x'', d''x''] in a ''bundle of boxes'' form of venn diagram. As awkward as it may seem at first, this type of picture is often the most natural and the most easily available representation when we want to conceptualize the localized information or momentary knowledge of an intelligent dynamic system. It gives a ready picture of a ''proposition at a point'', in the present instance, of a proposition about changing states which is itself associated with a particular dynamic state of a system. It is easy to see how this application might be extended to conceive of more general types of instantaneous knowledge that are possessed by a system.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 18-b -- Extension from 1 to 2 Dimensions.gif|center]]
|-
|-
| 1 || 1 || →
| height="20px" valign="top" | <math>\text{Figure 18-b.} ~~ \text{Extension from 1 to 2 Dimensions : Bundle}\!</math>
|}
|}
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
Figure 18-c shows the same extension in a ''compact'' style of venn diagram, where the differential features at each position are represented by arrows extending from that position that cross or do not cross, as the case may be, the corresponding feature boundaries.
| 1 || 1
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 18-c -- Extension from 1 to 2 Dimensions.gif|center]]
|-
|-
| 0 || 1
| height="20px" valign="top" | <math>\text{Figure 18-c.} ~~ \text{Extension from 1 to 2 Dimensions : Compact}\!</math>
|-
|}
| 0 || 1
Figure 18-d compresses the picture of the differential extension even further, yielding a directed graph or ''digraph'' form of representation. (Notice that my definition of a digraph allows for loops or ''slings'' at individual points, in addition to arcs or ''arrows'' between the points.)
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 18-d -- Extension from 1 to 2 Dimensions.gif|center]]
|-
|-
| 1 || 1
| height="20px" valign="top" | <math>\text{Figure 18-d.} ~~ \text{Extension from 1 to 2 Dimensions : Digraph}\!</math>
|}
|}
====Extension from 2 to 4 Dimensions====
Figure 19-a lays out the ''areal view'' or the ''angular form'' of venn diagram for universes of 2 and 4 dimensions, indicating the embedding map of type '''B'''<sup>2</sup> → '''B'''<sup>4</sup>. In many ways these pictures are the best kind there is, giving full canvass to an ideal vista. Their style allows the clearest, the fairest, and the plainest view that we can form of a universe of discourse, affording equal representation to all dispositions and maintaining a balance with respect to ordinary and differential features. If only we could extend this view! Unluckily, an obvious difficulty beclouds this prospect, and that is how precipitately we run into the limits of our plane and visual intuitions. Even within the scope of the spare few dimensions that we have scanned up to this point subtle discrepancies have crept in already. The circumstances that bind us and the frameworks that block us, the flat distortion of the planar projection and the inevitable ineffability that precludes us from wrapping its rhomb figure into rings around a torus, all of these factors disguise the underlying but true connectivity of the universe of discourse.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 19-a -- Extension from 2 to 4 Dimensions.gif|center]]
|-
|-
|
| height="20px" valign="top" | <math>\text{Figure 19-a.} ~~ \text{Extension from 2 to 4 Dimensions : Areal}\!</math>
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
|}
| 0 || 0 ||
Figure 19-b shows the differential extension from ''U''<sup> •</sup> = [''u'', ''v''] to E''U''<sup> •</sup> = [''u'', ''v'', d''u'', d''v''] in the ''bundle of boxes'' form of venn diagram.
| 0 || 1 ||
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 19-b -- Extension from 2 to 4 Dimensions.gif|center]]
|-
|-
| 1 || 0 ||
| height="20px" valign="top" | <math>\text{Figure 19-b.} ~~ \text{Extension from 2 to 4 Dimensions : Bundle}\!</math>
|}
As dimensions increase, this factorization of the extended universe along the lines that are marked out by the bundle picture begins to look more and more like a practical necessity. But whenever we use a propositional model to address a real situation in the context of nature we need to remain aware that this articulation into factors, affecting our description, may be wholly artificial in nature and cleave to nothing, no joint in nature, nor any juncture in time to be in or out of joint.
Figure 19-c illustrates the extension from 2 to 4 dimensions in the ''compact'' style of venn diagram. Here, just the changes with respect to the center cell are shown.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 19-c -- Extension from 2 to 4 Dimensions.gif|center]]
|-
|-
| 1 || 1 ||
| height="20px" valign="top" | <math>\text{Figure 19-c.} ~~ \text{Extension from 2 to 4 Dimensions : Compact}\!</math>
|}
|}
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
Figure 19-d gives the ''digraph'' form of representation for the differential extension ''U''<sup> •</sup> → E''U''<sup> •</sup>, where the 4 nodes marked "@" are the cells ''uv'', ''u''(''v''), (''u'')''v'', (''u'')(''v''), respectively, and where a 2-headed arc counts as two arcs of the differential digraph.
| 0 || 0
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 19-d -- Extension from 2 to 4 Dimensions.gif|center]]
| 1 || 0
|-
|-
| 0 || 0
| height="20px" valign="top" | <math>\text{Figure 19-d.} ~~ \text{Extension from 2 to 4 Dimensions : Digraph}\!</math>
|}
|}
===Thematization of Functions : And a Declaration of Independence for Variables===
{| width="100%"
| align="left" |
''And as imagination bodies forth''<br>
''The forms of things unknown, the poet's pen''<br>
''Turns them to shapes, and gives to airy nothing''<br>
''A local habitation and a name.''
| align="right" valign="bottom" | A Midsummer Night's Dream, 5.1.18
|}
|}
In the representation of propositions as functions it is possible to notice different degrees of explicitness in the way their functional character is symbolized. To indicate what I mean by this, the next series of Figures illustrates a set of graphic conventions that will be put to frequent use in the remainder of this discussion, both to mark the relevant distinctions and to help us convert between related expressions at different levels of explicitness in their functionality.
====Thematization : Venn Diagrams====
{| width="100%" cellpadding="0" cellspacing="0"
{| align="center" style="width:96%"
| width="4%" |
| width="92%" |
The known universe has one complete lover and that is the greatest poet. He consumes an eternal passion and is indifferent which chance happens and which possible contingency of fortune or misfortune and persuades daily and hourly his delicious pay.
| width="4%" |
|
|
| φ
|-
|-
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 11–12]
|}
|}
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
Figure 20-i traces the first couple of steps in this order of ''thematic'' progression, that will gradually run the gamut through a complete series of degrees of functional explicitness in the expression of logical propositions. The first venn diagram represents a situation where the function is indicated by a shaded figure and a logical expression. At this stage one may be thinking of the proposition only as expressed by a formula in a particular language and its content only as a subset of the universe of discourse, as when one considers the proposition ''u''<b>·</b>''v'' in [''u'', ''v''].
| 1
The second venn diagram depicts a situation in which two significant steps have been taken. First, one has taken the trouble to give the proposition ''u''<b>·</b>''v'' a distinctive functional name "''J'' ". Second, one has come to think explicitly about the target domain that contains the functional values of ''J'', as when one writes ''J'' : 〈''u'', ''v''〉 → '''B'''.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 20-i -- Thematization of Conjunction (Stage 1).gif|center]]
|-
|-
| 0
| height="20px" valign="top" | <math>\text{Figure 20-i.} ~~ \text{Thematization of Conjunction (Stage 1)}\!</math>
|-
|}
|}
In Figure 20-ii the proposition ''J'' is viewed explicitly as a transformation from one universe of discourse to another.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 20-ii -- Thematization of Conjunction (Stage 2).gif|center]]
|-
|-
|
| height="20px" valign="top" | <math>\text{Figure 20-ii.} ~~ \text{Thematization of Conjunction (Stage 2)}\!</math>
|}
|}
{| align="center" border="0" cellpadding="10"
|
|
<pre>
| 0
o-------------------------------o o-------------------------------o
|-
| | | |
| 1
| o-----o o-----o | | o-----o o-----o |
|-
| / \ / \ | | / \ / \ |
| 0
| / o \ | | / o \ |
|-
| / /`\ \ | | / /`\ \ |
| 1
| o o```o o | | o o```o o |
|}
| | u |```| v | | | | u |```| v | |
|}
| o o```o o | | o o```o o |
|
| \ \`/ / | | \ \`/ / |
| \ o / | | \ o / |
|+ '''Table 24-ii. Equality ''g'' '''
| \ / \ / | | \ / \ / |
|
| o-----o o-----o | | o-----o o-----o |
| | | |
| ''u'' || ''v'' || ''g'' || ''y''
o-------------------------------o o-------------------------------o
|}
\ / \ /
|
\ / \ /
\ / \ J /
\ / \ /
\ / \ /
o----------\---------/----------o o----------\---------/----------o
| \ / | | \ / |
| \ / | | \ / |
| o-----@-----o | | o-----@-----o |
| /`````````````\ | | /`````````````\ |
| /```````````````\ | | /```````````````\ |
| /`````````````````\ | | /`````````````````\ |
| o```````````````````o | | o```````````````````o |
| |```````````````````| | | |```````````````````| |
| |```````` J ````````| | | |```````` x ````````| |
| |```````````````````| | | |```````````````````| |
| o```````````````````o | | o```````````````````o |
| \`````````````````/ | | \`````````````````/ |
| \```````````````/ | | \```````````````/ |
| \`````````````/ | | \`````````````/ |
| o-----------o | | o-----------o |
| | | |
| | | |
o-------------------------------o o-------------------------------o
J = u v x = J<u, v>
Figure 20-ii. Thematization of Conjunction (Stage 2)
</pre>
|}
|}
In the first venn diagram the name that is assigned to a composite proposition, function, or region in the source universe is delegated to a simple feature in the target universe. This can result in a single character or term exceeding the responsibilities it can carry off well. Allowing the name of a function ''J'' : 〈''u'', ''v''〉 → '''B''' to serve as the name of its dependent variable ''J'' : '''B''' does not mean that one has to confuse a function with any of its values, but it does put one at risk for a number of obvious problems, and we should not be surprised, on numerous and limiting occasions, when quibbling arises from the attempts of a too original syntax to serve these two masters.
The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when one writes ''J'' : 〈''u'', ''v''〉 → 〈''x''〉 and thereby assigns a concrete type 〈''x''〉 to the abstract codomain '''B'''. To make this induction of variables more formal one can append subscripts, as in ''x''<sub>''J''</sub>, to indicate the origin or the derivation of these parvenu characters. However, it is not always convenient to keep inventing new variable names in this way. For use at these times, I introduce a lexical operator "¢", read ''cents'' or ''obelus'', that converts a function name into a variable name. For example, one may think of ''x'' = ''x''<sub>''J''</sub> = ¢(''J'') = ''J'' ¢ = ''J''<sup> ¢</sup> as "the cache variable of ''J'' ", "''J'' circumscript", "''J'' made circumstantial", or "''J'' considered as a contingent variable".
In Figure 20-iii we arrive at a stage where the functional equations, ''J'' = ''u''<b>·</b>''v'' and ''x'' = ''u''<b>·</b>''v'', are regarded as propositions in their own right, reigning in and ruling over 3-feature universes of discourse, [''u'', ''v'', ''J''] and [''u'', ''v'', ''x''], respectively. Subject to the cautions already noted, the function name "''J'' " can be reinterpreted as the name of a feature ''J''<sup> ¢</sup>, and the equation ''J'' = ''u''<b>·</b>''v'' can be read as the logical equivalence ((''J'', ''u'' ''v'')). To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition ''J''.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 20-iii -- Thematization of Conjunction (Stage 3).gif|center]]
|-
|-
|
| height="20px" valign="top" | <math>\text{Figure 20-iii.} ~~ \text{Thematization of Conjunction (Stage 3)}\!</math>
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
|}
| 0 || 0 || || 0
The first venn diagram represents the thematization of the conjunction ''J'' with shading in the appropriate regions of the universe [''u'', ''v'', ''J'']. Also, it illustrates a quick way of constructing a thematic extension. First, draw a line, in practice or the imagination, that bisects every cell of the original universe, placing half of each cell under the aegis of the thematized proposition and the other half under its antithesis. Next, on the scene where the theme applies leave the shade wherever it lies, and off the stage, where it plays otherwise, stagger the pattern in a harlequin guise.
In the final venn diagram of this sequence the thematic progression comes full circle and completes one round of its development. The ambiguities that were occasioned by the changing role of the name "''J'' " are resolved by introducing a new variable name "''x'' " to take the place of ''J''<sup> ¢</sup>, and the region that represents this fresh featured ''x'' is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name "''J'' " to the proposition ''u''<b>·</b>''v'', we now give the name "ι" to its thematization ((''x'', ''u'' ''v'')). Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function ι : 〈''u'', ''v'', ''x''〉 → '''B'''.
| 0 || 1 || → || 0
From now on, the terms ''thematic extension'' and ''thematization'' will be used to describe both the process and degree of explication that progresses through this series of pictures, both the operation of increasingly explicit symbolization and the dimension of variation that is swept out by it. To speak of this change in general, that takes us in our current example from ''J'' to ι, I introduce a class of operators symbolized by the Greek letter θ, writing ι = θ''J'' in the present instance. The operator θ, in the present situation bearing the type θ : [''u'', ''v''] →> [''u'', ''v'', ''x''], provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.
Figure 21 shows how the thematic extension operator θ acts on two further examples, the disjunction ((''u'')(''v'')) and the equality ((''u'', ''v'')). Referring to the disjunction as ''f''‹''u'', ''v''› and the equality as ''g''‹''u'', ''v''›, I write the thematic extensions as φ = θ''f'' and γ = θ''g''.
{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"
| [[Image:Diff Log Dyn Sys -- Figure 21 -- Thematization of Disjunction and Equality.gif|center]]
|-
|-
| 0 || 1 || || 1
| height="20px" valign="top" | <math>\text{Figure 21.} ~~ \text{Thematization of Disjunction and Equality}\!</math>
|}
|}
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
====Thematization : Truth Tables====
| 0
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
That which distorts honest shapes or which creates unearthly beings or places or contingencies is a nuisance and a revolt.
| width="4%" |
|-
|-
| 1
| align="right" colspan="3" | — Walt Whitman, ''Leaves of Grass'', [Whi, 19]
|-
|}
|}
Tables 22 through 25 outline a method for computing the thematic extensions of propositions in terms of their coordinate values.
A preliminary step, as illustrated in Table 22, is to write out the truth table representations of the propositional forms whose thematic extensions one wants to compute, in the present instance, the functions ''f''‹''u'', ''v''› = ((''u'')(''v'')) and ''g''‹''u'', ''v''› = ((''u'', ''v'')).
<br>
<br>
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:50%"
|+ style="height:30px" | <math>\text{Table 22.} ~~ \text{Disjunction}~ f ~\text{and Equality}~ g\!</math>
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
|- style="height:35px; background:ghostwhite; width:100%"
|+ '''Table 25-i. Disjunction ''f'' '''
| <math>u\!</math>
|
| <math>v\!</math>
| style="border-left:1px solid black" | <math>f\!</math>
| ''u'' || ''v'' || ''f'' || ''x''
| <math>g\!</math>
|}
| φ
|}
|-
|-
|
| style="border-top:1px solid black" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
<math>\begin{matrix}
0
\\[4pt]
0
\\[4pt]
1
\\[4pt]
1
|}
\end{matrix}\!</math>
|
| style="border-top:1px solid black" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
<math>\begin{matrix}
| 1
0
|-
\\[4pt]
| 0
1
\\[4pt]
0
\\[4pt]
1
\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" |
<math>\begin{matrix}
0
\\[4pt]
1
\\[4pt]
1
\\[4pt]
1
\end{matrix}\!</math>
| style="border-top:1px solid black" |
<math>\begin{matrix}
1
\\[4pt]
0
\\[4pt]
0
\\[4pt]
1
\end{matrix}\!</math>
|}
<br>
Next, each propositional form is individually represented in the fashion shown in Tables 23-i and 23-ii, using "''f'' " and "''g'' " as function names and creating new variables ''x'' and ''y'' to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the 2-dimensional universes of ''f'' and ''g'' to the 3-dimensional universes of θ''f'' and θ''g''. The top halves of the Tables replicate the truth table patterns for ''f'' and ''g'' in the form ''f'' : [''u'', ''v''] → [''x''] and ''g'' : [''u'', ''v''] → [''y'']. The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for (''f'') and (''g'') under the copies for ''f'' and ''g''. At this stage, the columns for θ''f'' and θ''g'' are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions ''f'' and ''g''.
<br>
{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 23-i and 23-ii.} ~~ \text{Thematics of Disjunction and Equality (1)}\!</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 23-i.} ~~ \text{Disjunction}~ f\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| style="border-left:1px solid black; border-right:1px solid black" | <math>f\!</math>
| <math>x\!</math>
| <math>\varphi\!</math>
|-
|-
| 0
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
|-
|-
| 0
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
|}
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 23-ii.} ~~ \text{Equality}~ g\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| style="border-left:1px solid black; border-right:1px solid black" | <math>g\!</math>
| <math>y\!</math>
| <math>\gamma\!</math>
|-
|-
|
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| 0 || 0 || || 1
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |
<math>\begin{matrix}\to\\\to\\\to\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
|-
|-
| 0 || 1 || → || 1
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-right:1px solid black; border-top:1px solid black" |
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
|}
|}
<br>
All the data are now in place to give the truth tables for θ''f'' and θ''g''. In the remaining steps all we do is to permute the rows and change the roles of ''x'' and ''y'' from dependent to independent variables. In Tables 24-i and 24-ii the rows are arranged in such a way as to put the 3-tuples ‹''u'', ''v'', ''x''› and ‹''u'', ''v'', ''y''› in binary numerical order, suitable for viewing as the arguments of the maps θ''f'' = φ : [''u'', ''v'', ''x''] → '''B''' and θ''g'' = γ : [''u'', ''v'', ''y''] → '''B'''. Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions ''f'' and ''g'' to be passed over without further attention and shifting the heavy vertical bars a notch to the right. In effect, this clinches the fact that the thematic variables ''x'' := ''f''<sup> ¢</sup> and ''y'' := ''g''<sup> ¢</sup> are now to be regarded as independent variables.
<br>
{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 24-i and 24-ii.} ~~ \text{Thematics of Disjunction and Equality (2)}\!</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 24-i.} ~~ \text{Disjunction}~ f\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| <math>f\!</math>
| <math>x\!</math>
| style="border-left:1px solid black" | <math>\varphi\!</math>
|-
|-
| 1 || 0 || → || 1
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
|-
|-
| 1 || 1 || → || 1
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <font size="+2"> <br></font><math>\begin{matrix}\to\\~\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
|}
|}
|
| width="50%" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
| 0
|+ style="height:25px" | <math>\text{Table 24-ii.} ~~ \text{Equality}~ g\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| <math>g\!</math>
| <math>y\!</math>
| style="border-left:1px solid black" | <math>\gamma\!</math>
|-
|-
| 1
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
|-
|-
| 1
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| 1
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
|}
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
|}
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
|}
|
| γ
|}
|}
<br>
An optional reshuffling of the rows brings additional features of the thematic extensions to light. Leaving the columns in place for the sake of comparison, Tables 25-i and 25-ii sort the rows in a different order, in effect treating ''x'' and ''y'' as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form φ : [''x'', ''u'', ''v''] → '''B''' and γ : [''y'', ''u'', ''v''] → '''B''' makes it easier to see in this tabular setting a property that was graphically obvious in the venn diagrams above. Specifically, when the thematic variable ''F''<sup> ¢</sup> is true then θ''F'' exhibits the pattern of the original ''F'', and when ''F''<sup> ¢</sup> is false then θ''F'' exhibits the pattern of its negation (''F'').
<br>
{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 25-i and 25-ii.} ~~ \text{Thematics of Disjunction and Equality (3)}\!</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 25-i.} ~~ \text{Disjunction}~ f\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| <math>f\!</math>
| <math>x\!</math>
| style="border-left:1px solid black" | <math>\varphi\!</math>
|-
|-
|
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| 0 || 0 || || 0
| style="border-top:1px solid black" | <math>{\to}\!</math><br><font size="+2"> <br> <br> <br></font>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\0\end{matrix}\!</math>
|-
|-
| 0 || 1 || → || 0
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <font size="+2"> <br></font><math>\begin{matrix}\to\\\to\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\1\end{matrix}\!</math>
|}
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 25-ii.} ~~ \text{Equality}~ g\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| <math>g\!</math>
| <math>y\!</math>
| style="border-left:1px solid black" | <math>\gamma\!</math>
|-
|-
| 1 || 0 || → || 0
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <br><math>\begin{matrix}\to\\\to\end{matrix}\!</math><br>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
|-
|-
| 1 || 1 || || 0
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}\to\\~\\~\\\to\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
| style="border-left:1px solid black; border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
|}
|}
|}
|}
<br>
Finally, Tables 26-i and 26-ii compare the tacit extensions <math>\epsilon</math> : [''u'', ''v''] → [''u'', ''v'', ''x''] and <math>\epsilon</math> : [''u'', ''v''] → [''u'', ''v'', ''y''] with the thematic extensions of the same types, as applied to the propositions ''f'' and ''g'', respectively.
<br>
{| align="center" border="0" style="width:90%"
|+ style="height:25px" | <math>\text{Tables 26-i and 26-ii.} ~~ \text{Tacit Extension and Thematization}\!</math>
| width="50%" |
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
|+ style="height:25px" | <math>\text{Table 26-i.} ~~ \text{Disjunction}~ f\!</math>
|- style="height:25px; background:ghostwhite; width:100%"
| <math>u\!</math>
| <math>v\!</math>
| <math>x\!</math>
| style="border-left:1px solid black" | <math>\varepsilon f\!</math>
| <math>\vartheta f\!</math>
|-
|-
|
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| 0 || 0 || → || 1
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
|-
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| 0 || 1 || || 1
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
|-
| 1 || 0 || || 1
|-
|-
| 1 || 1 || → || 1
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
|}
|}
|
| width="50%" |
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; text-align:center; width:90%"
| 1
|+ style="height:25px" | <math>\text{Table 26-ii.} ~~ \text{Equality}~ g\!</math>
|-
|- style="height:25px; background:ghostwhite; width:100%"
| 0
| <math>u\!</math>
| <math>v\!</math>
| <math>y\!</math>
| style="border-left:1px solid black" | <math>\varepsilon g\!</math>
| <math>\vartheta g\!</math>
|-
|-
| 0
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}1\\1\\0\\0\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\1\\0\end{matrix}\!</math>
|-
|-
| 1
| style="border-top:1px solid black" | <math>\begin{matrix}1\\1\\1\\1\end{matrix}\!</math>
|}
| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}0\\1\\0\\1\end{matrix}\!</math>
| style="border-top:1px solid black; border-left:1px solid black" | <math>\begin{matrix}0\\0\\1\\1\end{matrix}\!</math>
| style="border-top:1px solid black" | <math>\begin{matrix}1\\0\\0\\1\end{matrix}\!</math>
|}
|}
|}
|}
<br>
<br>
Table 27 summarizes the thematic extensions of all propositions on two variables. Column 4 lists the equations of form (( ''f''<sup> ¢</sup> , ''f''<sup> ¢</sup>‹''u'', ''v''› )) and Column 5 simplifies these equations into the form of algebraic expressions. (As always, "+" refers to exclusive disjunction, and "''f'' " should be read as "''f''<sub>''i''</sub><sup>¢</sup>" in the body of the Table.)
<br>
<br><font face="courier new">
{| align="center" style="width:96%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ '''Tables 26-i and 26-ii. Tacit Extension and Thematization'''
|+ Table 27. Thematization of Bivariate Propositions
|- style="background:ghostwhite"
|
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
{| align="right" style="background:ghostwhite; text-align:right"
|+ '''Table 26-i. Disjunction ''f'' '''
| u :
|-
| v :
|}
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
{| style="background:ghostwhite"
| ''u'' || ''v'' || ''x''
| 1100
|-
| 1010
|}
|}
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="background:ghostwhite; font-weight:bold; text-align:center; width:100%"
{| style="background:ghostwhite"
| ε''f'' || θ''f''
| f
|-
|
|}
|}
|
{| style="background:ghostwhite"
| θf
|-
|-
|
|}
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
{| style="background:ghostwhite"
| 0 || 0 || 0
| θf
|-
|-
| 0 || 0 || 1
|
|}
|-
|-
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
{| cellpadding="2"
| 0 || 1
| f<sub>0</sub>
|-
|-
| 0 || 0
| f<sub>1</sub>
|-
|-
| 1 || 0
| f<sub>2</sub>
|-
|-
| 1 || 1
| f<sub>3</sub>
|-
|-
|
| f<sub>4</sub>
|-
|-
| 1 || 0 || 1
| f<sub>5</sub>
|-
|-
| 1 || 1 || 0
| f<sub>6</sub>
|-
|-
| 1 || 1 || 1
| f<sub>7</sub>
|}
|}
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
{| cellpadding="2"
| 1 || 0
| 0000
|-
|-
| 1 || 1
| 0001
|-
|-
| 1 || 0
| 0010
|-
|-
| 1 || 1
| 0011
|}
|-
| 0100
|-
| 0101
|-
| 0110
|-
| 0111
|}
|}
|
|
{| align="center" border="1" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:center; width:90%"
{| cellpadding="2"
|+ '''Table 26-ii. Equality ''g'' '''
| ()
|
|-
| (u)(v)
| ''u'' || ''v'' || ''y''
|-
|}
| (u) v
|
|-
| (u)
|-
|-
|
| u (v)
|-
|-
| 0 || 0 || 1
| (v)
|-
|-
| 0 || 1 || 0
| (u, v)
|-
|-
| 0 || 1 || 1
| (u v)
|}
|}
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
{| cellpadding="2"
| (( f , () ))
|-
|-
| 1 || 1
| (( f , (u)(v) ))
|-
|-
| 0 || 1
| (( f , (u) v ))
|-
|-
| 0 || 0
| (( f , (u) ))
|-
|-
|
| (( f , u (v) ))
|-
|-
| 1 || 0 || 1
| (( f , (v) ))
|-
|-
| 1 || 1 || 0
| (( f , (u, v) ))
|-
|-
| 1 || 1 || 1
| (( f , (u v) ))
|}
|}
|
|
{| align="center" border="0" cellpadding="6" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
{| align="left" cellpadding="2" style="text-align:left"
| 0 || 1
| f + 1
|-
|-
| 0 || 0
| f + u + v + uv
|-
|-
| 1 || 0
| f + v + uv + 1
|-
|-
| 1 || 1
| f + u
|-
|-
| v :
| f + u + uv + 1
|-
|-
| 1010
| f + v
|-
|-
|
| f + u + v + 1
|-
|-
|
| f + uv
|}
|}
|-
|-
|
|
{| cellpadding="2"
{| cellpadding="2"
| f<sub>0</sub>
| f<sub>8</sub>
|-
|-
| f<sub>1</sub>
| f<sub>9</sub>
|-
|-
| f<sub>2</sub>
| f<sub>10</sub>
|-
|-
| f<sub>3</sub>
| f<sub>11</sub>
|-
|-
| f<sub>4</sub>
| f<sub>12</sub>
|-
|-
| f<sub>5</sub>
| f<sub>13</sub>
|-
|-
| f<sub>6</sub>
| f<sub>14</sub>
|-
|-
| f<sub>7</sub>
| f<sub>15</sub>
|}
|}
|
|
{| cellpadding="2"
{| cellpadding="2"
| 0000
| 1000
|-
|-
| 0001
| 1001
|-
|-
| 0010
| 1010
|-
|-
| 0011
| 1011
|-
|-
| 0100
| 1100
|-
|-
| 0101
| 1101
|-
|-
| 0110
| 1110
|-
|-
| 0111
| 1111
|}
|}
|
|
{| cellpadding="2"
{| cellpadding="2"
| ()
| u v
|-
|-
| (u)(v)
| ((u, v))
|-
|-
| (u) v
| v
|-
|-
| (u)
| (u (v))
|-
|-
| u (v)
| u
|-
|-
| (v)
| ((u) v)
|-
|-
| (u, v)
| ((u)(v))
|-
|-
| (u v)
| (())
|}
|}
|
|
{| cellpadding="2"
{| cellpadding="2"
| (( f , () ))
| (( f , u v ))
|-
|-
| (( f , (u)(v) ))
| (( f , ((u, v)) ))
|-
|-
| (( f , (u) v ))
| (( f , v ))
|-
|-
| (( f , (u) ))
| (( f , (u (v)) ))
|-
|-
| (( f , u (v) ))
| (( f , u ))
|-
|-
| (( f , (v) ))
| (( f , ((u) v) ))
|-
|-
| (( f , (u, v) ))
| (( f , ((u)(v)) ))
|-
|-
| (( f , (u v) ))
| (( f , (()) ))
|}
|}
|
|
{| align="left" cellpadding="2" style="text-align:left"
{| align="left" cellpadding="2" style="text-align:left"
| f + 1
| f + uv + 1
|-
|-
| f + u + v + uv
| f + u + v
|-
|-
| f + v + uv + 1
| f + v + 1
|-
|-
| f + u
| f + u + uv
|-
|-
| f + u + uv + 1
| f + u + 1
|-
|-
| f + v
| f + v + uv
|-
|-
| f + u + v + 1
| f + u + v + uv + 1
|-
|-
| f + uv
| f
|}
|}
|}
|-
</font><br>
In order to show what all of the thematic extensions from two dimensions to three dimensions look like in terms of coordinates, Tables 28 and 29 present ordinary truth tables for the functions ''f''<sub>''i''</sub> : '''B'''<sup>2</sup> → '''B''' and for the corresponding thematizations θ''f''<sub>''i''</sub> = φ<sub>''i''</sub> : '''B'''<sup>3</sup> → '''B'''.
<br>
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ Table 28. Propositions on Two Variables
|
|
{| cellpadding="2"
{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
| f<sub>8</sub>
|- style="background:ghostwhite"
| u || v ||
|f<sub>00</sub>||f<sub>01</sub>||f<sub>02</sub>||f<sub>03</sub>
|f<sub>04</sub>||f<sub>05</sub>||f<sub>06</sub>||f<sub>07</sub>
|f<sub>08</sub>||f<sub>09</sub>||f<sub>10</sub>||f<sub>11</sub>
|f<sub>12</sub>||f<sub>13</sub>||f<sub>14</sub>||f<sub>15</sub>
|-
|-
| f<sub>9</sub>
| 0 || 0 ||
|0||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1
|-
|-
| f<sub>10</sub>
| 0 || 1 ||
|0||0||1||1||0||0||1||1||0||0||1||1||0||0||1||1
|-
|-
| f<sub>11</sub>
| 1 || 0 ||
|0||0||0||0||1||1||1||1||0||0||0||0||1||1||1||1
|-
|-
| f<sub>12</sub>
| 1 || 1 ||
|-
|0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1
| f<sub>13</sub>
|}
|-
| f<sub>14</sub>
|-
| f<sub>15</sub>
|}
|}
<br>
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|+ Table 29. Thematic Extensions of Bivariate Propositions
|
|
{| cellpadding="2"
{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"
| 1000
|- style="background:ghostwhite"
|-
| u || v || f<sup>¢</sup>
| 1001
| φ<sub>00</sub> || φ<sub>01</sub>
| φ<sub>02</sub> || φ<sub>03</sub>
| φ<sub>04</sub> || φ<sub>05</sub>
| φ<sub>06</sub> || φ<sub>07</sub>
| φ<sub>08</sub> || φ<sub>09</sub>
| φ<sub>10</sub> || φ<sub>11</sub>
| φ<sub>12</sub> || φ<sub>13</sub>
| φ<sub>14</sub> || φ<sub>15</sub>
|-
|-
| 1010
| 0 || 0 || 0 ||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0
|-
|-
| 1011
| 0 || 0 || 1 ||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1
|-
|-
| 1100
| 0 || 1 || 0 ||1||1||0||0||1||1||0||0||1||1||0||0||1||1||0||0
|-
|-
| 1101
| 0 || 1 || 1 ||0||0||1||1||0||0||1||1||0||0||1||1||0||0||1||1
|-
|-
| 1110
| 1 || 0 || 0 ||1||1||1||1||0||0||0||0||1||1||1||1||0||0||0||0
|-
|-
| 1111
| 1 || 0 || 1 ||0||0||0||0||1||1||1||1||0||0||0||0||1||1||1||1
|}
|
| u v
|-
|-
| ((u, v))
| 1 || 1 || 0 ||1||1||1||1||1||1||1||1||0||0||0||0||0||0||0||0
|-
|-
| v
| 1 || 1 || 1 ||0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1
|}
|}
<br>
===Propositional Transformations===
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
If only the word ‘artificial’ were associated with the idea of ''art'', or expert skill gained through voluntary apprenticeship (instead of suggesting the factitious and unreal), we might say that ''logical'' refers to artificial thought.
| width="4%" |
|-
|-
| (u (v))
| align="right" colspan="3" | — John Dewey, ''How We Think'', [Dew, 56–57]
| u
|}
|}
{| cellpadding="2"
In this Subdivision I develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general context the source and the target universes of a transformation are allowed to be distinct, but may also be one and the same. When these concepts are applied to dynamic systems one focuses on the important special cases of transformations that map a universe into itself, and transformations of this shape may be interpreted as the state transitions of a discrete dynamical process, as these take place among the myriad ways that a universe of discourse might change, and by that change turn into itself.
| (( f , u v ))
|-
====Alias and Alibi Transformations====
There are customarily two modes of understanding a transformation, at least, when we try to interpret its relevance to something in reality. A transformation always refers to a changing prospect, to say it in a unified but equivocal way, but this can be taken to mean either a subjective change in the interpreting observer's point of view or an objective change in the systematic subject of discussion. In practice these variant uses of the transformation concept are distinguished in the following terms:
|-
# A ''perspectival'' or ''alias'' transformation refers to a shift in perspective or a change in language that takes place in the observer's frame of reference.
| (( f , (u (v)) ))
# A ''transitional'' or ''alibi'' transformation refers to a change of position or an alteration of state that occurs in the object system as it falls under study.
(For a recent discussion of the alias vs. alibi issue, as it relates to linear transformations in vector spaces and to other issues of an algebraic nature, see [MaB, 256, 582-4].)
Naturally, when we are concerned with the dynamical properties of a system, the transitional aspect of transformation is the factor that comes to the fore, and this involves us in contemplating all of the ways of changing a universe into itself while remaining under the rule of established dynamical laws. In the prospective application to dynamic systems, and to neural networks viewed in this light, our interest lies chiefly with the transformations of a state space into itself that constitute the state transitions of a discrete dynamic process. Nevertheless, many important properties of these transformations, and some constructions that we need to see most clearly, are independent of the transitional interpretation and are likely to be confounded with irrelevant features if presented first and only in that association.
In addition, and in partial contrast, intelligent systems are exactly that species of dynamic agents that have the capacity to have a point of view, and we cannot do justice to their peculiar properties without examining their ability to form and transform their own frames of reference in exposure to the elements of their own experience. In this setting, the perspectival aspect of transformation is the facet that shines most brightly, perhaps too often leaving us fascinated with mere glimmerings of its actual potential. It needs to be emphasized that nothing of the ordinary sort needs be moved in carrying out a transformation under the alias interpretation, that it may only involve a change in the forms of address, an amendment of the terms which are customed to approach and fashioned to describe the very same things in the very same world. But again, working within a discipline of realistic computation, we know how formidably complex and resource-consuming such transformations of perspective can be to implement in practice, much less to endow in the self-governed form of a nascently intelligent dynamical system.
====Transformations of General Type====
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| width="92%" |
''Es ist passiert'', "it just sort of happened", people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.
| width="4%" |
|-
|-
| (( f , u ))
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 34]
|-
|}
|}
Consider the situation illustrated in Figure 30, where the alphabets <font face="lucida calligraphy">U</font> = {''u'', ''v''} and <font face="lucida calligraphy">X</font> = {''x'', ''y'', ''z''} are used to label basic features in two different logical universes, ''U''<sup> •</sup> = [''u'', ''v''] and ''X''<sup> •</sup> = [''x'', ''y'', ''z''].
{| align="center" border="0" cellpadding="10"
{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"
|
|
<pre>
o-------------------------------------------------------o
| u || v ||
| U |
|f<sub>00</sub>||f<sub>01</sub>||f<sub>02</sub>||f<sub>03</sub>
| |
|f<sub>04</sub>||f<sub>05</sub>||f<sub>06</sub>||f<sub>07</sub>
| o-----------o o-----------o |
|f<sub>08</sub>||f<sub>09</sub>||f<sub>10</sub>||f<sub>11</sub>
| / \ / \ |
|f<sub>12</sub>||f<sub>13</sub>||f<sub>14</sub>||f<sub>15</sub>
| / o \ |
|-
| / / \ \ |
| 0 || 0 ||
| / / \ \ |
|0||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1
| o o o o |
| | | | | |
| | u | | v | |
|0||0||1||1||0||0||1||1||0||0||1||1||0||0||1||1
| | | | | |
|-
| o o o o |
| 1 || 0 ||
| \ \ / / |
|0||0||0||0||1||1||1||1||0||0||0||0||1||1||1||1
| \ \ / / |
|-
| \ o / |
| 1 || 1 ||
| \ / \ / |
|0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1
| o-----------o o-----------o |
| |
| |
o---------------------------o---------------------------o
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
| u || v || f<sup>¢</sup>
/ \ / \ / \
| φ<sub>00</sub> || φ<sub>01</sub>
/ \ / \ / \
| φ<sub>02</sub> || φ<sub>03</sub>
/ \ / \ / \
| φ<sub>04</sub> || φ<sub>05</sub>
/ \ / \ / \
| φ<sub>06</sub> || φ<sub>07</sub>
/ \ / \ / \
| φ<sub>08</sub> || φ<sub>09</sub>
/ \ / \ / \
| φ<sub>10</sub> || φ<sub>11</sub>
o-------------------------o o-------------------------o o-------------------------o
| φ<sub>12</sub> || φ<sub>13</sub>
| U | | U | | U |
| φ<sub>14</sub> || φ<sub>15</sub>
| o---o o---o | | o---o o---o | | o---o o---o |
|-
| / \ / \ | | / \ / \ | | / \ / \ |
| 0 || 0 || 0 ||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0
| / o \ | | / o \ | | / o \ |
|-
| / / \ \ | | / / \ \ | | / / \ \ |
| 0 || 0 || 1 ||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1
| o o o o | | o o o o | | o o o o |
|-
| | u | | v | | | | u | | v | | | | u | | v | |
| 0 || 1 || 0 ||1||1||0||0||1||1||0||0||1||1||0||0||1||1||0||0
| o o o o | | o o o o | | o o o o |
|-
| \ \ / / | | \ \ / / | | \ \ / / |
| 0 || 1 || 1 ||0||0||1||1||0||0||1||1||0||0||1||1||0||0||1||1
| \ o / | | \ o / | | \ o / |
| \ / \ / | | \ / \ / | | \ / \ / |
| o---o o---o | | o---o o---o | | o---o o---o |
| | | | | |
o-------------------------o o-------------------------o o-------------------------o
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ g | \ f / | h /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ o----------|-----------\-----/-----------|----------o /
\ | X | \ / | | /
\ | | \ / | | /
\ | | o-----o-----o | | /
\| | / \ | |/
\ | / \ | /
|\ | / \ | /|
| \ | / \ | / |
| \ | / \ | / |
| \ | o x o | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \| | | |/ |
| o--o--------o o--------o--o |
| / \ \ / / \ |
| / \ \ / / \ |
| / \ o / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o--o-----o--o o |
| | | | | |
| | | | | |
| | | | | |
| | y | | z | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------------------------------o
\ /
\ /
\ /
\ /
\ /
\ p , q /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
o
Figure 30. Generic Frame of a Logical Transformation
</pre>
|}
|}
Enter the picture, as we usually do, in the middle of things, with features like ''x'', ''y'', ''z'' that present themselves to be simple enough in their own right and that form a satisfactory, if a temporary, foundation to provide a basis for discussion. In this universe and on these terms we find expression for various propositions and questions of principal interest to ourselves, as indicated by the maps ''p'', ''q'' : ''X'' → '''B'''. Then we discover that the simple features {''x'', ''y'', ''z''} are really more complex than we thought at first, and it becomes useful to regard them as functions {''f'', ''g'', ''h''} of other features {''u'', ''v''}, that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse ''U''<sup> •</sup> = [''u'', ''v'']. It may happen that these late-blooming but pre-ambling features are found to lie closer, in a sense that may be our job to determine, to the central nature of the situation of interest, in which case they earn our regard as being more fundamental, but these functions and features are only required to supply a critical stance on the universe of discourse or an alternate perspective on the nature of things in order to be preserved as useful.
====Alias and Alibi Transformations====
A particular transformation ''F'' : [''u'', ''v''] → [''x'', ''y'', ''z''] may be expressed by a system of equations, as shown below. Here, ''F'' is defined by its component maps ''F'' = ‹F<sub>1</sub>, F<sub>2</sub>, F<sub>3</sub>› = ‹''f'', ''g'', ''h''›, where each component map in {''f'', ''g'', ''h''} is a proposition of type '''B'''<sup>''n''</sup> → '''B'''<sup>1</sup>.
<br><font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
|
{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"
| width="20%" |
| width="20%" | ''x''
| width="20%" | =
| width="20%" | ''f''‹''u'', ''v''›
| width="20%" |
|-
| || ''y'' || = || ''g''‹''u'', ''v''› ||
|-
| || ''z'' || = || ''h''‹''u'', ''v''› ||
|}
|}
</font><br>
Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions {''f'', ''g'', ''h''} in one universe of discourse and the special collection of simple propositions {''x'', ''y'', ''z''} on which are founded another universe of discourse. Growing familiarity with a particular transformation of discourse, and the desire to achieve a ready understanding of its implications, requires that we be able to convert this information about generals and simples into information about all the main subtypes of propositions, including the linear and singular propositions.
===Analytic Expansions : Operators and Functors===
====Transformations of General Type====
{| width="100%" cellpadding="0" cellspacing="0"
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="4%" |
| width="92%" |
| width="92%" |
''Es ist passiert'', "it just sort of happened", people said there when other people in other places thought heaven knows what had occurred. It was a peculiar phrase, not known in this sense to the Germans and with no equivalent in other languages, the very breath of it transforming facts and the bludgeonings of fate into something light as eiderdown, as thought itself.
Consider what effects that might ''conceivably'' have practical bearings you ''conceive'' the objects of your ''conception'' to have. Then, your ''conception'' of those effects is the whole of your ''conception'' of the object.
| width="4%" |
| width="4%" |
|-
|-
| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 34]
| align="right" colspan="3" | — C.S. Peirce, "The Maxim of Pragmatism", CP 5.438
|}
|}
Given the barest idea of a logical transformation, as suggested by the sketch in Figure 30, and having conceptualized the universe of discourse, with all of its points and propositions, as a beginning object of discussion, we are ready to enter the next phase of our investigation.
====Operators on Propositions and Transformations====
The next step is naturally inclined toward objects of the next higher order, namely, with operators that take in argument lists of logical transformations and that give back specified types of logical transformations as their results. For our present aims, we do not need to consider the most general class of such operators, nor any one of them for its own sake. Rather, we are interested in the special sorts of operators that arise in the study and analysis of logical transformations. Figuratively speaking, these operators serve as instruments for the live tomography (and hopefully not the vivisection) of the forms of change under view. Beyond that, they open up ways to implement the changes of view that we need to grasp all the variations on a transformational theme, or to appreciate enough of its significant features to "get the drift" of the change occurring, to form a passing acquaintance or a synthetic comprehension of its general character and disposition.
The simplest type of operator is one that takes a single transformation as an argument and returns a single transformation as a result, and most of the operators that I will explicitly consider here are of this kind. Figure 31 illustrates the typical situation.
{| align="center" border="0" cellpadding="20"
|
<pre>
<pre>
o---------------------------------------o
| |
| |
| U% F X% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| !W! | | !W! |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
| !W!U% !W!F !W!X% |
| |
| |
o---------------------------------------o
Figure 31. Operator Diagram (1)
</pre>
|}
In this Figure "<font face=georgia>'''W'''</font>" serves as a generic name for an operator, in this case one that takes a logical transformation ''F'' of type (''U''<sup> •</sup> → ''X''<sup> •</sup>) into a logical transformation <font face=georgia>'''W'''</font>''F'' of type (<font face=georgia>'''W'''</font>''U''<sup> •</sup> → <font face=georgia>'''W'''</font>''X''<sup> •</sup>). Thus, the operator <font face=georgia>'''W'''</font> must be viewed as making assignments for both families of objects that we have previously considered, both for universes of discourse like ''U''<sup> •</sup> and ''X''<sup> •</sup> and for logical transformations like ''F''.
NB. Strictly speaking, an operator like <font face=georgia>'''W'''</font> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <font face=georgia>'''W'''</font>. Given this setting, <font face=georgia>'''W'''</font> specifies for each universe ''U''<sup> •</sup> in its source category a definite universe <font face=georgia>'''W'''</font>''U''<sup> •</sup> in its target category, and to each transformation ''F'' in its source category it assigns a unique transformation <font face=georgia>'''W'''</font>''F'' in its target category. Naturally, this only works if <font face=georgia>'''W'''</font> takes the source ''U''<sup> •</sup> and the target ''X''<sup> •</sup> of the map F over to the source <font face=georgia>'''W'''</font>''U''<sup> •</sup> and the target <font face=georgia>'''W'''</font>''X''<sup> •</sup> of the map <font face=georgia>'''W'''</font>''F''. With luck or care enough, we can avoid ever having to put anything like that in words again, letting diagrams do the work. In the situations of present concern we are usually focused on a single transformation ''F'', and thus we can take it for granted that the assignment of universes under <font face=georgia>'''W'''</font> is defined appropriately at the source and the target ends of ''F''. It is not always the case, though, that we need to use the particular names (like "<font face=georgia>'''W'''</font>''U''<sup> •</sup>" and "<font face=georgia>'''W'''</font>''X''<sup> •</sup>") that <font face=georgia>'''W'''</font> assigns by default to its operative image universes. In most contexts we will usually have a prior acquaintance with these universes under other names, and it is only necessary that we can tell from the information associated with an operator <font face=georgia>'''W'''</font> what universes they are.
In Figure 31 the maps ''F'' and <font face=georgia>'''W'''</font>''F'' are displayed horizontally, the way that one normally orients functional arrows in a written text, and <font face=georgia>'''W'''</font> rolls the map ''F'' downward into the images that are associated with <font face=georgia>'''W'''</font>''F''. In Figure 32 the same information is redrawn so that the maps ''F'' and <font face=georgia>'''W'''</font>''F'' flow down the page, and <font face=georgia>'''W'''</font> unfurls the map ''F'' rightward into domains that are the eminent purview of <font face=georgia>'''W'''</font>''F''.
{| align="center" border="0" cellpadding="20"
|
<pre>
/ \ / \ / \
o---------------------------------------o
| |
| |
| U% !W! !W!U% |
| o------------------>o |
| | | |
| | | |
| | | |
| | | |
| F | | !W!F |
| | | |
| | | |
| | | |
| v v |
| o------------------>o |
o-------------------------o o-------------------------o o-------------------------o
| X% !W! !W!X% |
| |
| |
o---------------------------------------o
Figure 32. Operator Diagram (2)
</pre>
|}
The latter arrangement, as it appears in Figure 32, is more congruent with the thinking about operators that we shall be doing in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure 30.
====Differential Analysis of Propositions and Transformations====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''
| width="4%" |
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 43]
|}
The approach to the differential analysis of logical propositions and transformations of discourse that will be pursued here is carried out in terms of particular operators <font face=georgia>'''W'''</font> that act on propositions ''F'' or on transformations ''F'' to yield the corresponding operator maps <font face=georgia>'''W'''</font>''F''. The operator results then become the subject of a series of further stages of analysis, which take them apart into their propositional components, rendering them as a set of purely logical constituents. After this is done, all the parts are then re-integrated to reconstruct the original object in the light of a more complete understanding, at least in ways that enable one to appreciate certain aspects of it with fresh insight.
* '''Remark on Strategy.''' At this point I run into a set of conceptual difficulties that force me to make a strategic choice in how I proceed. Part of the problem can be remedied by extending my discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead me to try two different types of solution. The approach that I develop first makes use of a variant type of extension operator, the ''trope extension'', to be defined below. This method is more conservative and requires less preparation, but has features which make it seem unsatisfactory in the long run. A more radical approach, but one with a better hope of long term success, makes use of the notion of ''contingency spaces''. These are an even more generous type of extended universe than the kind I currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces me to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, I call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well my first approach deals with them.
I now describe in general terms the particular operators that are instrumental to this form of analysis. The main series of operators all have the form <font face=georgia>'''W'''</font> : (''U''<sup> •</sup> → ''X''<sup> •</sup>) → (E''U''<sup> •</sup> → E''X''<sup> •</sup>). If we assume that the source universe ''U''<sup> •</sup> and the target universe ''X''<sup> •</sup> have finite dimensions ''n'' and ''k'', respectively, then each operator <font face=georgia>'''W'''</font> is encompassed by the same
abstract type:
:{| cellpadding=1 style="height:40px"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
| <font face=georgia>'''W'''</font>
|
| :
| (
| width="20%" |
| [
| width="20%" | ''x''
| '''B'''<sup>''n''</sup>
| width="20%" | =
| ]
| width="20%" | ''f''‹''u'', ''v''›
| →
| width="20%" |
| [
|-
| '''B'''<sup>''k''</sup>
| || ''y'' || = || ''g''‹''u'', ''v''› ||
| ]
|-
| )
| || ''z'' || = || ''h''‹''u'', ''v''› ||
|
| →
|
| (
| [
| '''B'''<sup>''n''</sup>
| ×
| '''D'''<sup>''n''</sup>
| ]
| →
| [
| '''B'''<sup>''k''</sup>
| ×
| '''D'''<sup>''k''</sup>
| ]
| )
| .
|}
|}
Since the range features of the operator result <font face=georgia>'''W'''</font>''F'' : ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] can be sorted out by their ordinary versus their differential qualities and the component maps can be examined independently, the complete operator <font face=georgia>'''W'''</font> can be separated accordingly into two components, in the form <font face=georgia>'''W'''</font> = ‹<math>\epsilon</math>, W›. Given a fixed context of source and target universes of discourse, <math>\epsilon</math> is always the same type of operator, a multiple component elaboration of the tacit extension operators that were articulated earlier. In this context <math>\epsilon</math> has the shape:
:{| style="height:80px; text-align:center; width:90%"
| align=left width=20%| Concrete type
{| width="100%" cellpadding="0" cellspacing="0"
| width=8% | <math>\epsilon</math>
| width="4%" |
| :
| width="92%" |
| (
| ''U''<sup> •</sup>
| →
| ''X''<sup> •</sup>
| )
| width=16% | →
| (
| E''U''<sup> •</sup>
| →
| ''X''<sup> •</sup>
| )
|-
|-
| align="right" colspan="3" | — C.S. Peirce, "The Maxim of Pragmatism", CP 5.438
| align=left width=20%| Abstract type
| width=8% | <math>\epsilon</math>
| :
| (
| ['''B'''<sup>''n''</sup>]
| →
| ['''B'''<sup>''k''</sup>]
| )
| width=16% | →
| (
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>]
| →
| ['''B'''<sup>''k''</sup>]
| )
|}
|}
On the other hand, the operator W is specific to each <font face=georgia>'''W'''</font>. In this context W always has the form:
====Operators on Propositions and Transformations====
:{| style="height:80px; text-align:center; width:90%"
| align=left width=20%| Concrete type
| width=8% | W
| :
| (
| ''U''<sup> •</sup>
<pre>
| →
| ''X''<sup> •</sup>
| |
| )
| |
| width=16% | →
| U% F X% |
| (
| o------------------>o |
| E''U''<sup> •</sup>
| | | |
| →
| | | |
| d''X''<sup> •</sup>
| | | |
| )
| | | |
|-
| !W! | | !W! |
| align=left width=20%| Abstract type
| | | |
| width=8% | W
| | | |
| :
| | | |
| (
| v v |
| ['''B'''<sup>''n''</sup>]
| o------------------>o |
| →
| !W!U% !W!F !W!X% |
| ['''B'''<sup>''k''</sup>]
| |
| )
| |
| width=16% | →
| (
| ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>]
| →
| ['''D'''<sup>''k''</sup>]
| )
|}
In this Figure "<font face=georgia>'''W'''</font>" serves as a generic name for an operator, in this case one that takes a logical transformation ''F'' of type (''U''<sup> •</sup> → ''X''<sup> •</sup>) into a logical transformation <font face=georgia>'''W'''</font>''F'' of type (<font face=georgia>'''W'''</font>''U''<sup> •</sup> → <font face=georgia>'''W'''</font>''X''<sup> •</sup>). Thus, the operator <font face=georgia>'''W'''</font> must be viewed as making assignments for both families of objects that we have previously considered, both for universes of discourse like ''U''<sup> •</sup> and ''X''<sup> •</sup> and for logical transformations like ''F''.
In the types just assigned to <math>\epsilon</math> and W, and implicitly to their results <math>\epsilon</math>''F'' and W''F'', I have listed the most restrictive ranges defined for them, rather than the more expansive target spaces that subsume these ranges. When there is need to recognize both, we may use type indications like the following:
:{| style="height:80px; text-align:center; width:90%"
| width=6% | <math>\epsilon</math>''F''
| width=2% | :
| width=2% | (
<pre>
| width=8% | E''U''<sup> •</sup>
| width=4% | →
| |
| width=8% | ''X''<sup> •</sup>
| width=4% | ⊆
| width=8% | E''X''<sup> •</sup>
| o------------------>o |
| width=2% | )
| width=4% | <math>\cong</math>
| width=2% | (
| width=16% | ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>]
| width=4% | →
| width=8% | ['''B'''<sup>''k''</sup>]
| width=4% | ⊆
| width=16% | ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
| width=2% | )
|-
| width=6% | W''F''
| width=2% | :
| width=2% | (
| width=8% | E''U''<sup> •</sup>
| width=4% | →
| width=8% | d''X''<sup> •</sup>
| width=4% | ⊆
| width=8% | E''X''<sup> •</sup>
| width=2% | )
| width=4% | <math>\cong</math>
| width=2% | (
| width=16% | ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>]
| width=4% | →
| width=8% | ['''D'''<sup>''k''</sup>]
| width=4% | ⊆
| width=16% | ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
| width=2% | )
|}
Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.
The latter arrangement, as it appears in Figure 32, is more congruent with the thinking about operators that we shall be doing in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure 30.
In giving names to these operators I am attempting to preserve as much of the traditional nomenclature and as many of the classical associations as possible. The chief difficulty in doing this is occasioned by the distinction between the operators <font face=georgia>'''W'''</font> and their components W, which forces me to find two distinct but parallel sets of terminology. Here is the plan that I have settled on. First, the component operators W are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <font face=georgia>'''W'''</font> = ‹<math>\epsilon</math>, W› are assigned their titles according to their roles in a geometric or trigonometric allegory, if only to ensure that the tangent functor, that belongs to this family and whose exposition I am still working toward, comes out fit with its customary name. Finally, the operator results <font face=georgia>'''W'''</font>''F'' and W''F'' can be fixed in this frame of reference by tethering the operative adjective for <font face=georgia>'''W'''</font> or W to the anchoring epithet ''map'', in conformity with an already standard practice.
====Differential Analysis of Propositions and Transformations====
=====The Secant Operator : <font face=georgia>'''E'''</font>=====
{| width="100%" cellpadding="0" cellspacing="0"
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="4%" |
| width="92%" | The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''
| width="92%" |
Mr. Peirce, after pointing out that our beliefs are really rules for action, said that, to develop a thought's meaning, we need only determine what conduct it is fitted to produce: that conduct is for us its sole significance.
| width="4%" |
| width="4%" |
|-
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 43]
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 46]
|}
|}
Figures 33-i and 33-ii depict two stages in the form of analysis that will be applied to transformations throughout the remainder of this study. From now on our interest is staked on an operator denoted "<font face=georgia>'''E'''</font>", which receives the principal investment of analytic attention, and on the constituent parts of <font face=georgia>'''E'''</font>, which derive their shares of significance as developed by the analysis. In the sequel, I refer to <font face=georgia>'''E'''</font> as the ''secant operator'', taking it for granted that a context has been chosen that defines its type. The secant operator has the component description <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E›, and its active ingredient E is known as the ''enlargement operator''. (Here, I have named E after the literal ancestor of the shift operator in the calculus of finite differences, defined so that E''f''(''x'') = ''f''(''x''+1) for any suitable function ''f'', though of course the logical analogue that we take up here must have a rather different definition.)
<pre>
U% $E$ $E$U% $E$U% $E$U%
o------------------>o============o============o
| | | |
| | | |
| | | |
| | | |
F | | $E$F = | $d$^0.F + | $r$^0.F
| | | |
| | | |
| | | |
v v v v
o------------------>o============o============o
X% $E$ $E$X% $E$X% $E$X%
Figure 33-i. Analytic Diagram (1)
</pre>
<pre>
U% $E$ $E$U% $E$U% $E$U% $E$U%
o------------------>o============o============o============o
| | | | |
| | | | |
| | | | |
| | | | |
F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F
| | | | |
| | | | |
| | | | |
v v v v v
o------------------>o============o============o============o
X% $E$ $E$X% $E$X% $E$X% $E$X%
Figure 33-ii. Analytic Diagram (2)
</pre>
In its action on universes <font face=georgia>'''E'''</font> yields the same result as E, a fact that can be expressed in equational form by writing <font face=georgia>'''E'''</font>''U''<sup> •</sup> = E''U''<sup> •</sup> for any universe ''U''<sup> •</sup>. Notice that the extended universes across the top and bottom of the diagram are indicated to be strictly identical, rather than requiring a corresponding decomposition for them. In a certain sense, the functional parts of <font face=georgia>'''E'''</font>''F'' are partitioned into separate contexts that have to be re-integrated again, but the best image to use is that of making transparent copies of each universe and then overlapping their functional contents once more at the conclusion of the analysis, as suggested by the graphic conventions that are used at the top of Figure 30.
Acting on a transformation ''F'' from universe ''U''<sup> •</sup> to universe ''X''<sup> •</sup>, the operator <font face=georgia>'''E'''</font> determines a transformation <font face=georgia>'''E'''</font>''F'' from <font face=georgia>'''E'''</font>''U''<sup> •</sup> to <font face=georgia>'''E'''</font>''X''<sup> •</sup>. The map <font face=georgia>'''E'''</font>''F'' forms the main body of evidence to be investigated in performing a differential analysis of ''F''. Because we shall frequently be focusing on small pieces of this map for considerable lengths of time, and consequently lose sight of the "big picture", it is critically important to emphasize that the map <font face=georgia>'''E'''</font>''F'' is a transformation that determines a relation from one extended universe into another. This means that we should not be satisfied with our understanding of a transformation ''F'' until we can lay out the full "parts diagram" of <font face=georgia>'''E'''</font>''F'' along the lines of the generic frame in Figure 30.
If one is working within the confines of propositional calculus, it is possible to give an elementary definition of <font face=georgia>'''E'''</font>''F'' by means of a system of propositional equations, as will now be described.
Given a transformation:
:{| style="height:80px; text-align:center; width:90%"
: ''F'' = ‹''F''<sub>1</sub>, …, ''F''<sub>''k''</sub>› : '''B'''<sup>''n''</sup> → '''B'''<sup>''k''</sup>
of concrete type:
:{| style="height:80px; text-align:center; width:90%"
: ''F'' : [''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>] → [''x''<sub>1</sub>, …, ''x''<sub>''k''</sub>]
the transformation:
: <font face=georgia>'''E'''</font>''F'' = ‹''F''<sub>1</sub>, …, ''F''<sub>''k''</sub>, E''F''<sub>1</sub>, …, E''F''<sub>''k''</sub>› : '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>
of concrete type:
: <font face=georgia>'''E'''</font>''F'' : [''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>] → [''x''<sub>1</sub>, …, ''x''<sub>''k''</sub>, d''x''<sub>1</sub>, …, d''x''<sub>''k''</sub>]
is defined by means of the following system of logical equations:
:{| style="height:80px; text-align:center; width:90%"
<br><font face="courier new">
| width=6% | <math>\epsilon</math>''F''
{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
| width=2% | :
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"
| width=8% | E''U''<sup> •</sup>
| width="8%" | ''x''<sub>1</sub>
| width=4% | →
| width="4%" | =
| width=8% | ''X''<sup> •</sup>
| width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| width=4% | ⊆
| width="4%" | =
| width=8% | E''X''<sup> •</sup>
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>›
| width=2% | )
|-
| width=4% | <math>\cong</math>
| ...
| width=2% | (
|-
| width=16% | ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>]
| width="8%" | ''x''<sub>''k''</sub>
| width=4% | →
| width="4%" | =
| width=8% | ['''B'''<sup>''k''</sup>]
| width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| width=4% | ⊆
| width="4%" | =
| width=16% | ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>›
| width=2% | )
|}
|-
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"
| width="8%" | d''x''<sub>1</sub>
| width="4%" | =
| width="44%" | E''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub> + d''u''<sub>1</sub>, …, ''u''<sub>''n''</sub> + d''u''<sub>''n''</sub>›
|-
| ...
|-
|-
| width=6% | W''F''
| width="8%" | d''x''<sub>''k''</sub>
| width=2% | :
| width="4%" | =
| width="44%" | E''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| width=8% | E''U''<sup> •</sup>
| width="4%" | =
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub> + d''u''<sub>1</sub>, …, ''u''<sub>''n''</sub> + d''u''<sub>''n''</sub>›
|}
| width=4% | →
| width=8% | ['''D'''<sup>''k''</sup>]
| width=2% | )
|}
|}
</font><br>
It is important to note that this system of equations can be read as a conjunction of equational propositions, in effect, as a single proposition in the universe of discourse that is generated by all of the named variables. Specifically, this is the universe of discourse over 2(''n''+''k'') variables that is denoted by:
: E[<font face="lucida calligraphy">U</font> ∪ <font face="lucida calligraphy">X</font>] = [''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, ''x''<sub>1</sub>, …, ''x''<sub>''k''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>, d''x''<sub>1</sub>, …, d''x''<sub>''k''</sub>].
In this light, it should be clear that the system of equations defining <font face=georgia>'''E'''</font>''F'' embodies, in a higher rank and in a differentially extended version, an analogy with the process of thematization that was treated earlier for propositions of the type ''F'' : '''B'''<sup>''n''</sup> → '''B'''.
{| width="100%" cellpadding="0" cellspacing="0"
The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <font face=georgia>'''E'''</font>''F'' = ‹<math>\epsilon</math>''F'', E''F''›, for any map ''F''. This is tantamount to regarding <font face=georgia>'''E'''</font> as a complex operator, <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E›, with a form of application that distributes each component of the operator to work on each component of the operand:
: <font face=georgia>'''E'''</font>''F'' = ‹<math>\epsilon</math>, E›''F'' = ‹<math>\epsilon</math>''F'', E''F''› = ‹<math>\epsilon</math>''F''<sub>1</sub>, …, <math>\epsilon</math>''F''<sub>''k''</sub>, E''F''<sub>1</sub>, …, E''F''<sub>''k''</sub>›.
Quite a lot of "thematic infrastructure" or interpretive information is being swept under the rug in the use of such abbreviations. When confusion arises about the meaning of such constructions, one always has recourse to the defining system of equations, in its totality a purely propositional expression. This means that the angle brackets, which were used in this context to build an image of multi-component transformations, should not be expected to determine a well-defined product in themselves, but only to serve as reminders of the prior thematic decisions (choices of variable names, etc.) that have to be made in order to determine one. Accordingly, the angle bracket notation ‹ , › can be regarded as a kind of ''thematic frame'', an interpretive storage device that preserves the proper associations of concrete logical features between the extended universes at the source and target of <font face=georgia>'''E'''</font>F.
The generic notations <font face=georgia>'''d'''</font><sup>0</sup>''F'', <font face=georgia>'''d'''</font><sup>1</sup>''F'', …, <font face=georgia>'''d'''</font><sup>''m''</sup>''F'' in Figure 33 refer to the increasing orders of differentials that are extracted in the course of analyzing ''F''. When the analysis is halted at a partial stage of development, notations like <font face=georgia>'''r'''</font><sup>0</sup>''F'', <font face=georgia>'''r'''</font><sup>1</sup>''F'', …, <font face=georgia>'''r'''</font><sup>''m''</sup>''F'' may be used to summarize the contributions to <font face=georgia>'''E'''</font>''F'' that remain to be analyzed. The Figure illustrates a convention that renders the remainder term <font face=georgia>'''r'''</font><sup>''m''</sup>''F'', in effect, the sum of all differentials of order strictly greater than ''m''.
I next discuss the set of operators that figure into this form of analysis, describing their effects on transformations. In simplified or specialized contexts these operators tend to take on a variety of different names and notations, some of whose number I will introduce along the way.
=====The Radius Operator : <font face=georgia>'''e'''</font>=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="4%" |
| width="92%" |
| width="92%" |
And the tangible fact at the root of all our thought-distinctions, however subtle, is that there is no one of them so fine as to consist in anything but a possible difference of practice.
| width="4%" |
| width="4%" |
|-
|-
|}
|}
The operator identified as <font face=georgia>'''d'''</font><sup>0</sup> in the analytic diagram (Figure 33) has the sole purpose of creating a proxy for ''F'' in the appropriately extended context. Construed in terms of its broadest components, <font face=georgia>'''d'''</font><sup>0</sup> is equivalent to the doubly tacit extension operator ‹<math>\epsilon</math>, <math>\epsilon</math>›, in recognition of which let us redub it as "<font face=georgia>'''e'''</font>". Pursuing a geometric analogy, we may refer to <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\epsilon</math>› = <font face=georgia>'''d'''</font><sup>0</sup> as the ''radius operator''. The operation that is intended by all of these forms is defined by the equation:
<pre>
:{| cellpadding=2
| <font face=georgia>'''e'''</font>''F''
| =
| ‹<math>\epsilon</math>, <math>\epsilon</math>›''F''
|-
|
| =
| ‹<math>\epsilon</math>''F'', <math>\epsilon</math>''F''›
|-
|
| =
| ‹<math>\epsilon</math>''F''<sub>1</sub>, …, <math>\epsilon</math>F<sub>''k''</sub>, <math>\epsilon</math>F<sub>1</sub>, …, <math>\epsilon</math>F<sub>''k''</sub>› ,
|}
which is tantamount to the system of equations given below.
<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"
| width="8%" | ''x''<sub>1</sub>
| width="4%" | =
<br><font face="courier new">
{| align="center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"
| width="8%" | ''x''<sub>1</sub>
| width="4%" | =
| width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="4%" | =
| width="8%" | d''x''<sub>1</sub>
| width="8%" | d''x''<sub>1</sub>
| width="4%" | =
| width="4%" | =
| width="44%" | E''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="4%" | =
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub> + d''u''<sub>1</sub>, …, ''u''<sub>''n''</sub> + d''u''<sub>''n''</sub>›
| width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>›
|-
|-
| ...
| ...
| width="8%" | d''x''<sub>''k''</sub>
| width="8%" | d''x''<sub>''k''</sub>
| width="4%" | =
| width="4%" | =
| width="44%" | E''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| width="4%" | =
| width="4%" | =
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub> + d''u''<sub>1</sub>, …, ''u''<sub>''n''</sub> + d''u''<sub>''n''</sub>›
| width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>›
|}
|}
|}
|}
</font><br>
</font><br>
=====The Phantom of the Operators : '''η'''=====
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
{| width="100%" cellpadding="0" cellspacing="0"
| width="4%" |
| width="92%" |
| width="92%" |
I was wondering what the reason could be, when I myself raised my head and everything within me seemed drawn towards the Unseen, ''which was playing the most perfect music''!
| width="4%" |
| width="4%" |
|-
|-
| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 46]
| align="right" colspan="3" | — Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]
|}
|}
I now describe an operator whose persistent but elusive action behind the scenes, whose slightly twisted and ambivalent character, and whose fugitive disposition, caught somewhere in flight between the arrantly negative and the positive but errant intent, has cost me some painstaking trouble to detect. In the end I shall place it among the other extensions and projections, as a shade among shadows, of muted tones and motley hue, that adumbrates its own thematic frame and paradoxically lights the way toward a whole new spectrum of values.
Given a transformation ''F'' : [''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>] → [''x''<sub>1</sub>, …, ''x''<sub>''k''</sub>], we often need to make a separate treatment of a related family of transformations of the form ''F''* : [''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>] → [d''x''<sub>1</sub>, …, d''x''<sub>''k''</sub>]. The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:
:{| cellpadding=2
: <math>\eta</math>''F'' : [''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>] → [d''x''<sub>1</sub>, …, d''x''<sub>''k''</sub>]
which is defined by the equations:
which is defined by the equations:
<br><font face="courier new">
<br><font face="courier new">
<br>
<br>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J.gif|center]]</p>
<p>[[Image:Diff Log Dyn Sys -- Figure 56-b4 -- Tangent Map of J ISW.gif|center]]</p>
<p><center><font size="+1">'''Figure 56-b4. Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
<p><center><font size="+1">'''Figure 56-b4. Tangent Map of the Conjunction ''J'' = ''uv'''''</font></center></p>
\\[4pt]
\\[4pt]
f_7
f_7
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
f_{0111}
f_{0111}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
0~1~1~1
0~1~1~1
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x~~y)
(x~~y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
\text{not both}~ x ~\text{and}~ y
\text{not both}~ x ~\text{and}~ y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
\lnot x \lor \lnot y
\lnot x \lor \lnot y
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{15}
f_{15}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
f_{1111}
f_{1111}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
1~1~1~1
1~1~1~1
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
\text{true}
\text{true}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
1
1
\end{matrix}</math>
\end{matrix}\!</math>
|}
|}
\\[4pt]
\\[4pt]
f_8
f_8
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
f_{1000}
f_{1000}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
1~0~0~0
1~0~0~0
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x ~\text{and}~ y
x ~\text{and}~ y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x \land y
x \land y
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{12}
f_{12}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
f_{1100}
f_{1100}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
1~1~0~0
1~1~0~0
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x
x
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x
x
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_9
f_9
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
f_{1001}
f_{1001}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
1~0~0~1
1~0~0~1
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x ~\text{equal to}~ y
x ~\text{equal to}~ y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
x = y
x = y
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{10}
f_{10}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
f_{1010}
f_{1010}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
1~0~1~0
1~0~1~0
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
y
y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
y
y
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{14}
f_{14}
\end{matrix}</math>
\end{matrix}\!</math>
|
<math>\begin{matrix}
f_{0111}
\\[4pt]
f_{1011}
\\[4pt]
f_{1101}
\\[4pt]
f_{1110}
\end{matrix}\!</math>
|
<math>\begin{matrix}
0~1~1~1
\\[4pt]
1~0~1~1
\\[4pt]
1~1~0~1
\\[4pt]
1~1~1~0
\end{matrix}\!</math>
|
<math>\begin{matrix}
~(x~~y)~
\\[4pt]
~(x~(y))
\\[4pt]
((x)~y)~
\\[4pt]
((x)(y))
\end{matrix}\!</math>
|
<math>\begin{matrix}
\text{not both}~ x ~\text{and}~ y
\\[4pt]
\text{not}~ x ~\text{without}~ y
\\[4pt]
\text{not}~ y ~\text{without}~ x
\\[4pt]
x ~\text{or}~ y
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\lnot x \lor \lnot y
\\[4pt]
x \Rightarrow y
\\[4pt]
x \Leftarrow y
\\[4pt]
x \lor y
\end{matrix}\!</math>
|-
| <math>f_{15}\!</math>
| <math>f_{1111}\!</math>
| <math>1~1~1~1\!</math>
\lnot x \lor \lnot y
\\[4pt]
x \Rightarrow y
\\[4pt]
x \Leftarrow y
\\[4pt]
x \lor y
\end{matrix}</math>
|-
| <math>f_{15}\!</math>
| <math>f_{1111}\!</math>
| <math>1~1~1~1</math>
| <math>((~))</math>
| <math>((~))</math>
| <math>\text{true}\!</math>
| <math>\text{true}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table A3.}~~\operatorname{E}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
|+ <math>\text{Table A3.}~~\mathrm{E}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="10%" |
| width="10%" |
| width="18%" | <math>f\!</math>
| width="18%" | <math>f\!</math>
| width="18%" |
| width="18%" |
<p><math>\operatorname{T}_{11} f</math></p>
<p><math>\mathrm{T}_{11} f</math></p>
<p><math>\operatorname{E}f|_{\operatorname{d}x~\operatorname{d}y}</math></p>
<p><math>\mathrm{E}f|_{\mathrm{d}x~\mathrm{d}y}</math></p>
| width="18%" |
| width="18%" |
<p><math>\operatorname{T}_{10} f</math></p>
<p><math>\mathrm{T}_{10} f</math></p>
<p><math>\operatorname{E}f|_{\operatorname{d}x(\operatorname{d}y)}</math></p>
<p><math>\mathrm{E}f|_{\mathrm{d}x(\mathrm{d}y)}</math></p>
| width="18%" |
| width="18%" |
<p><math>\operatorname{T}_{01} f</math></p>
<p><math>\mathrm{T}_{01} f</math></p>
<p><math>\operatorname{E}f|_{(\operatorname{d}x)\operatorname{d}y}</math></p>
<p><math>\mathrm{E}f|_{(\mathrm{d}x)\mathrm{d}y}</math></p>
| width="18%" |
| width="18%" |
<p><math>\operatorname{T}_{00} f</math></p>
<p><math>\mathrm{T}_{00} f</math></p>
<p><math>\operatorname{E}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math></p>
<p><math>\mathrm{E}f|_{(\mathrm{d}x)(\mathrm{d}y)}</math></p>
|-
|-
| <math>f_0\!</math>
| <math>f_0\!</math>
\\[4pt]
\\[4pt]
f_8
f_8
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)(y)
(x)(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)~y~
(x)~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~(y)
~x~(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{12}
f_{12}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)
(x)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)
(x)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_9
f_9
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~(x,~y)~
~(x,~y)~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~(x,~y)~
~(x,~y)~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{10}
f_{10}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(y)
(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(y)
(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{14}
f_{14}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)(y))
((x)(y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~x~~y~)
(~x~~y~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~x~(y))
(~x~(y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)~y~)
((x)~y~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)(y))
((x)(y))
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table A4.}~~\operatorname{D}f ~\text{Expanded Over Differential Features}~ \{ \operatorname{d}x, \operatorname{d}y \}</math>
|+ <math>\text{Table A4.}~~\mathrm{D}f ~\text{Expanded Over Differential Features}~ \{ \mathrm{d}x, \mathrm{d}y \}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="10%" |
| width="10%" |
| width="18%" | <math>f\!</math>
| width="18%" | <math>f\!</math>
| width="18%" |
| width="18%" |
<math>\operatorname{D}f|_{\operatorname{d}x~\operatorname{d}y}</math>
<math>\mathrm{D}f|_{\mathrm{d}x~\mathrm{d}y}</math>
| width="18%" |
| width="18%" |
<math>\operatorname{D}f|_{\operatorname{d}x(\operatorname{d}y)}</math>
<math>\mathrm{D}f|_{\mathrm{d}x(\mathrm{d}y)}</math>
| width="18%" |
| width="18%" |
<math>\operatorname{D}f|_{(\operatorname{d}x)\operatorname{d}y}</math>
<math>\mathrm{D}f|_{(\mathrm{d}x)\mathrm{d}y}</math>
| width="18%" |
| width="18%" |
<math>\operatorname{D}f|_{(\operatorname{d}x)(\operatorname{d}y)}</math>
<math>\mathrm{D}f|_{(\mathrm{d}x)(\mathrm{d}y)}</math>
|-
|-
| <math>f_0\!</math>
| <math>f_0\!</math>
\\[4pt]
\\[4pt]
f_8
f_8
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{12}
f_{12}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_9
f_9
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{10}
f_{10}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((~))
((~))
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{14}
f_{14}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)(y))
((x)(y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(y)
(y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(x)
(x)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
(~)
(~)
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table A5.}~~\operatorname{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|+ <math>\text{Table A5.}~~\mathrm{E}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="10%" |
| width="10%" |
| width="18%" | <math>f\!</math>
| width="18%" | <math>f\!</math>
| width="18%" | <math>\operatorname{E}f|_{xy}</math>
| width="18%" | <math>\mathrm{E}f|_{xy}</math>
| width="18%" | <math>\operatorname{E}f|_{x(y)}</math>
| width="18%" | <math>\mathrm{E}f|_{x(y)}</math>
| width="18%" | <math>\operatorname{E}f|_{(x)y}</math>
| width="18%" | <math>\mathrm{E}f|_{(x)y}</math>
| width="18%" | <math>\operatorname{E}f|_{(x)(y)}</math>
| width="18%" | <math>\mathrm{E}f|_{(x)(y)}</math>
|-
|-
| <math>f_0\!</math>
| <math>f_0\!</math>
\\[4pt]
\\[4pt]
f_8
f_8
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}x~~\operatorname{d}y~
~\mathrm{d}x~~\mathrm{d}y~
\\[4pt]
\\[4pt]
~\operatorname{d}x~(\operatorname{d}y)
~\mathrm{d}x~(\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x)~\operatorname{d}y~
(\mathrm{d}x)~\mathrm{d}y~
\\[4pt]
\\[4pt]
(\operatorname{d}x)(\operatorname{d}y)
(\mathrm{d}x)(\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}x~(\operatorname{d}y)
~\mathrm{d}x~(\mathrm{d}y)
\\[4pt]
\\[4pt]
~\operatorname{d}x~~\operatorname{d}y~
~\mathrm{d}x~~\mathrm{d}y~
\\[4pt]
\\[4pt]
(\operatorname{d}x)(\operatorname{d}y)
(\mathrm{d}x)(\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x)~\operatorname{d}y~
(\mathrm{d}x)~\mathrm{d}y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x)~\operatorname{d}y~
(\mathrm{d}x)~\mathrm{d}y~
\\[4pt]
\\[4pt]
(\operatorname{d}x)(\operatorname{d}y)
(\mathrm{d}x)(\mathrm{d}y)
\\[4pt]
\\[4pt]
~\operatorname{d}x~~\operatorname{d}y~
~\mathrm{d}x~~\mathrm{d}y~
\\[4pt]
\\[4pt]
~\operatorname{d}x~(\operatorname{d}y)
~\mathrm{d}x~(\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x)(\operatorname{d}y)
(\mathrm{d}x)(\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x)~\operatorname{d}y~
(\mathrm{d}x)~\mathrm{d}y~
\\[4pt]
\\[4pt]
~\operatorname{d}x~(\operatorname{d}y)
~\mathrm{d}x~(\mathrm{d}y)
\\[4pt]
\\[4pt]
~\operatorname{d}x~~\operatorname{d}y~
~\mathrm{d}x~~\mathrm{d}y~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{12}
f_{12}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}x~
~\mathrm{d}x~
\\[4pt]
\\[4pt]
(\operatorname{d}x)
(\mathrm{d}x)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}x~
~\mathrm{d}x~
\\[4pt]
\\[4pt]
(\operatorname{d}x)
(\mathrm{d}x)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x)
(\mathrm{d}x)
\\[4pt]
\\[4pt]
~\operatorname{d}x~
~\mathrm{d}x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x)
(\mathrm{d}x)
\\[4pt]
\\[4pt]
~\operatorname{d}x~
~\mathrm{d}x~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_9
f_9
\end{matrix}</math>
\end{matrix}~</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}~</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~(\operatorname{d}x,~\operatorname{d}y)~
~(\mathrm{d}x,~\mathrm{d}y)~
\\[4pt]
\\[4pt]
((\operatorname{d}x,~\operatorname{d}y))
((\mathrm{d}x,~\mathrm{d}y))
\end{matrix}</math>
\end{matrix}~</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((\operatorname{d}x,~\operatorname{d}y))
((\mathrm{d}x,~\mathrm{d}y))
\\[4pt]
\\[4pt]
~(\operatorname{d}x,~\operatorname{d}y)~
~(\mathrm{d}x,~\mathrm{d}y)~
\end{matrix}</math>
\end{matrix}~</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((\operatorname{d}x,~\operatorname{d}y))
((\mathrm{d}x,~\mathrm{d}y))
\\[4pt]
\\[4pt]
~(\operatorname{d}x,~\operatorname{d}y)~
~(\mathrm{d}x,~\mathrm{d}y)~
\end{matrix}</math>
\end{matrix}~</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~(\operatorname{d}x,~\operatorname{d}y)~
~(\mathrm{d}x,~\mathrm{d}y)~
\\[4pt]
\\[4pt]
((\operatorname{d}x,~\operatorname{d}y))
((\mathrm{d}x,~\mathrm{d}y))
\end{matrix}</math>
\end{matrix}~</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{10}
f_{10}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}y~
~\mathrm{d}y~
\\[4pt]
\\[4pt]
(\operatorname{d}y)
(\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}y)
(\mathrm{d}y)
\\[4pt]
\\[4pt]
~\operatorname{d}y~
~\mathrm{d}y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~\operatorname{d}y~
~\mathrm{d}y~
\\[4pt]
\\[4pt]
(\operatorname{d}y)
(\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}y)
(\mathrm{d}y)
\\[4pt]
\\[4pt]
~\operatorname{d}y~
~\mathrm{d}y~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{14}
f_{14}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)(y))
((x)(y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
((\operatorname{d}x)~\operatorname{d}y~)
((\mathrm{d}x)~\mathrm{d}y~)
\\[4pt]
\\[4pt]
(~\operatorname{d}x~(\operatorname{d}y))
(~\mathrm{d}x~(\mathrm{d}y))
\\[4pt]
\\[4pt]
(~\operatorname{d}x~~\operatorname{d}y~)
(~\mathrm{d}x~~\mathrm{d}y~)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((\operatorname{d}x)~\operatorname{d}y~)
((\mathrm{d}x)~\mathrm{d}y~)
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
(~\operatorname{d}x~~\operatorname{d}y~)
(~\mathrm{d}x~~\mathrm{d}y~)
\\[4pt]
\\[4pt]
(~\operatorname{d}x~(\operatorname{d}y))
(~\mathrm{d}x~(\mathrm{d}y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(~\operatorname{d}x~(\operatorname{d}y))
(~\mathrm{d}x~(\mathrm{d}y))
\\[4pt]
\\[4pt]
(~\operatorname{d}x~~\operatorname{d}y~)
(~\mathrm{d}x~~\mathrm{d}y~)
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
((\operatorname{d}x)~\operatorname{d}y~)
((\mathrm{d}x)~\mathrm{d}y~)
\end{matrix}</math>
\end{matrix}~\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(~\operatorname{d}x~~\operatorname{d}y~)
(~\mathrm{d}x~~\mathrm{d}y~)
\\[4pt]
\\[4pt]
(~\operatorname{d}x~(\operatorname{d}y))
(~\mathrm{d}x~(\mathrm{d}y))
\\[4pt]
\\[4pt]
((\operatorname{d}x)~\operatorname{d}y~)
((\mathrm{d}x)~\mathrm{d}y~)
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; text-align:center; width:90%"
|+ <math>\text{Table A6.}~~\operatorname{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|+ <math>\text{Table A6.}~~\mathrm{D}f ~\text{Expanded Over Ordinary Features}~ \{ x, y \}</math>
|- style="background:#f0f0ff"
|- style="background:#f0f0ff"
| width="10%" |
| width="10%" |
| width="18%" | <math>f\!</math>
| width="18%" | <math>f\!</math>
| width="18%" | <math>\operatorname{D}f|_{xy}</math>
| width="18%" | <math>\mathrm{D}f|_{xy}</math>
| width="18%" | <math>\operatorname{D}f|_{x(y)}</math>
| width="18%" | <math>\mathrm{D}f|_{x(y)}</math>
| width="18%" | <math>\operatorname{D}f|_{(x)y}</math>
| width="18%" | <math>\mathrm{D}f|_{(x)y}</math>
| width="18%" | <math>\operatorname{D}f|_{(x)(y)}</math>
| width="18%" | <math>\mathrm{D}f|_{(x)(y)}</math>
|-
|-
| <math>f_0\!</math>
| <math>f_0\!</math>
\\[4pt]
\\[4pt]
f_8
f_8
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~~y~
~x~~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\\[4pt]
\\[4pt]
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{12}
f_{12}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~x~
~x~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}x
\mathrm{d}x
\\[4pt]
\\[4pt]
\operatorname{d}x
\mathrm{d}x
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}x
\mathrm{d}x
\\[4pt]
\\[4pt]
\operatorname{d}x
\mathrm{d}x
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}x
\mathrm{d}x
\\[4pt]
\\[4pt]
\operatorname{d}x
\mathrm{d}x
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}x
\mathrm{d}x
\\[4pt]
\\[4pt]
\operatorname{d}x
\mathrm{d}x
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_9
f_9
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x,~y))
((x,~y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\\[4pt]
\\[4pt]
(\operatorname{d}x,~\operatorname{d}y)
(\mathrm{d}x,~\mathrm{d}y)
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{10}
f_{10}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
~y~
~y~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}y
\mathrm{d}y
\\[4pt]
\\[4pt]
\operatorname{d}y
\mathrm{d}y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}y
\mathrm{d}y
\\[4pt]
\\[4pt]
\operatorname{d}y
\mathrm{d}y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}y
\mathrm{d}y
\\[4pt]
\\[4pt]
\operatorname{d}y
\mathrm{d}y
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\operatorname{d}y
\mathrm{d}y
\\[4pt]
\\[4pt]
\operatorname{d}y
\mathrm{d}y
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
|
|
\\[4pt]
\\[4pt]
f_{14}
f_{14}
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\\[4pt]
\\[4pt]
((x)(y))
((x)(y))
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\\[4pt]
\\[4pt]
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\end{matrix}</math>
\end{matrix}\!</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
~~\operatorname{d}x~~\operatorname{d}y~~
~~\mathrm{d}x~~\mathrm{d}y~~
\\[4pt]
\\[4pt]
~~\operatorname{d}x~(\operatorname{d}y)~
~~\mathrm{d}x~(\mathrm{d}y)~
\\[4pt]
\\[4pt]
~(\operatorname{d}x)~\operatorname{d}y~~
~(\mathrm{d}x)~\mathrm{d}y~~
\\[4pt]
\\[4pt]
((\operatorname{d}x)(\operatorname{d}y))
((\mathrm{d}x)(\mathrm{d}y))
\end{matrix}</math>
\end{matrix}\!</math>
|-
|-
| <math>f_{15}\!</math>
| <math>f_{15}\!</math>
