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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)

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This article develops a differential extension of [[propositional calculus]] and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.

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This article develops a differential extension of propositional calculus and applies it to a context of problems arising in dynamic systems. The work pursued here is coordinated with a parallel application that focuses on neural network systems, but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.

==Review and Transition==

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Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.

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* A bracketed list of propositional expressions in the form <math>(e_1, e_2, \ldots, e_{k-1}, e_k)</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.

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* A bracketed list of propositional expressions in the form <math>\texttt{(} e_1, e_2, \ldots, e_{k-1}, e_k \texttt{)}\!</math> indicates that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false.

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* A concatenation of propositional expressions in the form <math>e_1~e_2~\ldots~e_{k-1}~e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.

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* A concatenation of propositional expressions in the form <math>e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k</math> indicates that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.

All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form, but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.

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This treatment of propositional logic is derived from the work of [[C.S. Peirce]] [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.

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This treatment of propositional logic is derived from the work of C.S. Peirce [P1, P2], who gave this approach an extensive development in his graphical systems of predicate, relational, and modal logic [Rob]. More recently, these ideas were revived and supplemented in an alternative interpretation by George Spencer-Brown [SpB]. Both of these authors used other forms of enclosure where I use parentheses, but the structural topologies of expression and the functional varieties of interpretation are fundamentally the same.

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While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.

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While working with expressions solely in propositional calculus, it is easiest to use plain parentheses for logical connectives. In contexts where parentheses are needed for other purposes “teletype” parentheses <math>\texttt{(} \ldots \texttt{)}\!</math> or barred parentheses <math>(\!| \ldots |\!)</math> may be used for logical operators.

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The briefest expression for logical truth is the empty word, usually denoted by <math>\varepsilon\!</math> or <math>\lambda\!</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression "<math>((~))\!</math>", or, especially if operating in an algebraic context, by a simple "<math>1\!</math>". Also when working in an algebraic mode, the plus sign "<math>+\!</math>" may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:

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The briefest expression for logical truth is the empty word, usually denoted by <math>{}^{\backprime\backprime} \boldsymbol\varepsilon {}^{\prime\prime}\!</math> or <math>{}^{\backprime\backprime} \boldsymbol\lambda {}^{\prime\prime}\!</math> in formal languages, where it forms the identity element for concatenation. To make it visible in this text, it may be denoted by the equivalent expression <math>{}^{\backprime\backprime} \texttt{((~))} {}^{\prime\prime},\!</math> or, especially if operating in an algebraic context, by a simple <math>{}^{\backprime\backprime} 1 {}^{\prime\prime}.\!</math> Also when working in an algebraic mode, the plus sign <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> may be used for [[exclusive disjunction]]. For example, we have the following paraphrases of algebraic expressions by bracket expressions:

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{| align="center" cellpadding="6" style="text-align:center"

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~~~~<math>\begin{matrix}

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|

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x + y & = & (x, y)

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<math>\begin{matrix}

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

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x + y ~=~ \texttt{(} x, y \texttt{)}

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\\[6pt]

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~~<math>~~\~~begin{matrix}~~

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x + y + z ~=~ \texttt{((} x, y \texttt{)}, z \texttt{)} ~=~ \texttt{(} x, \texttt{(} y, z \texttt{))}

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x + y + z & = & ((x, y), z) & = & (x, (y, z))

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\end{matrix}</math>

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

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|}

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~~</center>~~

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It is important to note that the last expressions are not equivalent to the triple bracket <math>(x, y, z).\!</math>

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It is important to note that the last expressions are not equivalent to the triple bracket <math>\texttt{(} x, y, z \texttt{)}.\!</math>

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| <math>1\!</math>

|-

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| <math>(~)\!</math>

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

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

| <math>0\!</math>

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| <math>x\!</math>

|-

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| <math>(x)\!</math>

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

| <math>\text{Not}~ x\!</math>

|

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| <math>x \land y \land z\!</math>

|-

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| <math>((x)(y)(z))\!</math>

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| <math>\texttt{((} x \texttt{)(} y \texttt{)(} z \texttt{))}\!</math>

| <math>x ~\text{or}~ y ~\text{or}~ z\!</math>

| <math>x \lor y \lor z\!</math>

|-

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| <math>(x~(y))\!</math>

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

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<math>\begin{matrix}

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| <math>x \Rightarrow y\!</math>

|-

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| <math>(x, y)\!</math>

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| <math>\texttt{(} x \texttt{,} y \texttt{)}\!</math>

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<math>\begin{matrix}

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\end{matrix}</math>

|-

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| <math>((x, y))\!</math>

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| <math>\texttt{((} x \texttt{,} y \texttt{))}\!</math>

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<math>\begin{matrix}

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\end{matrix}</math>

|-

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| <math>(x, y, z)\!</math>

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| <math>\texttt{(} x \texttt{,} y \texttt{,} z \texttt{)}\!</math>

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<math>\begin{matrix}

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\end{matrix}</math>

|-

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| <math>((x),(y),(z))~\!</math>

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| <math>\texttt{((} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math>

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<math>\begin{matrix}

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|

<math>\begin{matrix}

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((x, y), z)

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\texttt{((} x \texttt{,} y \texttt{),} z \texttt{)}

\\

&

\\

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(x, (y, z))

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\texttt{(} x \texttt{,(} y \texttt{,} z \texttt{))}

\end{matrix}\!</math>

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\end{matrix}\!</math>

|-

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| <math>(w, (x),(y),(z))\!</math>

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| <math>\texttt{(} w \texttt{,(} x \texttt{),(} y \texttt{),(} z \texttt{))}\!</math>

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<math>\begin{matrix}

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| <math>\mathbb{B}^n \to \mathbb{B}\!</math>

|-

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| <math>A^\~~circ~~\!</math>

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| <math>A^\bullet\!</math>

|

<math>\begin{matrix}

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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.

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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>

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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^\bullet = [\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.

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The ''linear propositions'', <math>\{ \ell : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{\ell} \mathbb{B}),\!</math> may be written as sums:

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{| align="center" ~~cellspacing~~="8" width="90%"

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{| align="center" cellpadding="8" width="90%"

|

<math>\sum_{i=1}^n e_i ~=~ e_1 + \ldots + e_n

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The ''positive propositions'', <math>\{ p : \mathbb{B}^n \to \mathbb{B} \} = (\mathbb{B}^n \xrightarrow{p} \mathbb{B}),\!</math> may be written as products:

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{| align="center" ~~cellspacing~~="8" width="90%"

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{| align="center" cellpadding="8" width="90%"

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<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n

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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:

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{| align="center" ~~cellspacing~~="8" width="90%"

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{| align="center" cellpadding="8" width="90%"

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<math>\prod_{i=1}^n e_i ~=~ e_1 \cdot \ldots \cdot e_n

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</ul>

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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>

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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>\texttt{(} a_1 \texttt{)(} a_2 \texttt{)(} a_3\texttt{)}.\!</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.

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Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.

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First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{(a_i)\}</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.

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First, all of these properties are relative to a particular basis. For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features. Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.

Second, the singular propositions <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B},</math> picking out as they do a single cell or a coordinate tuple <math>\mathbf{x}</math> of <math>\mathbb{B}^n,</math> become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B})</math> and infects the whole hierarchy of types built on them. In other words, the terms that signify the interpretations <math>\mathbf{x} : \mathbb{B}^n</math> and the singular propositions <math>\mathbf{x} : \mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}</math> are fully equivalent in information, and this means that every token of the type <math>\mathbb{B}^n</math> can be reinterpreted as an appearance of the subtype <math>\mathbb{B}^n \xrightarrow{\mathbf{x}} \mathbb{B}.</math> And vice versa, the two types can be exchanged with each other everywhere that they turn up. In practical terms, this allows the use of singular propositions as a way of denoting points, forming an alternative to coordinate tuples.

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For example, relative to the universe of discourse <math>[a_1, a_2, a_3]\!</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point ~~'''&lsaquo;~~1,~~ ~~1,~~ ~~1~~&rsaquo;''',~~ but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.

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For example, relative to the universe of discourse <math>[a_1, a_2, a_3]\!</math> the singular proposition <math>a_1 a_2 a_3 : \mathbb{B}^3 \xrightarrow{s} \mathbb{B}</math> could be explicitly retyped as <math>a_1 a_2 a_3 : \mathbb{B}^3</math> to indicate the point <math>(1, 1, 1)\!</math> but in most cases the proper interpretation could be gathered from context. Both notations remain dependent on a particular basis, but the code that is generated under the singular option has the advantage in its self-commenting features, in other words, it constantly reminds us of its basis in the process of denoting points. When the time comes to put a multiplicity of different bases into play, and to search for objects and properties that remain invariant under the transformations between them, this infinitesimal potential advantage may well evolve into an overwhelming practical necessity.

===The Analogy Between Real and Boolean Types===

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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.

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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>

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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>\textstyle \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.

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|

<math>\begin{matrix}

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X^\~~circ~~

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X^\bullet

\\

= & [\mathcal{X}]

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<math>\begin{matrix}

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\underline{X}^\~~circ~~

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\underline{X}^\bullet

\\

= & [\underline\mathcal{X}]

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<math>\begin{matrix}

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A^\~~circ~~

+

A^\bullet

\\

= & [\mathcal{A}]

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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:

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: <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}</math>

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{| align="center" cellpadding="8" width="90%"

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| <math>\underline{x}_i : \mathbb{R}^n \to \mathbb{B}\ \text{such that:}\!</math>

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: <math>\begin{matrix}

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|-

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\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i, \\

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|

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<math>\begin{matrix}

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\underline{x}_i(\mathbf{x}) = 1 & \text{if} & \mathbf{x} \in L_i,

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\\[4pt]

\underline{x}_i(\mathbf{x}) = 0 & \text{if} & \mathbf{x} \not\in L_i.

\end{matrix}</math>

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|}

In other notations that are sometimes used, the operator <math>\chi (\ldots)</math> or the corner brackets <math>\lceil\ldots\rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values in <math>\mathbb{B}.</math> Finally, it is not uncommon to use the name of the relation itself as a predicate that maps <math>n\!</math>-tuples into truth values. Thus we have the following notational variants of the above definition:

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: <math>\begin{matrix}

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{| align="center" cellpadding="8" width="90%"

+

|

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<math>\begin{matrix}

\underline{x}_i (\mathbf{x}) & = & \chi (\mathbf{x} \in L_i) & = & \lceil \mathbf{x} \in L_i \rceil & = & L_i (\mathbf{x}).

\end{matrix}</math>

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|}

−

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 ~~'''&lsaquo;''' ~~<math>0, \ldots, 0, r_i, 0, \ldots, 0</math>~~ '''&rsaquo;'''~~ 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^\text{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^\text{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^\~~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.

+

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^\text{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>\texttt{(} \ldots \texttt{)}\!</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>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!</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}^\bullet</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>

The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, graphically illustrating the links among the elementary cells <math>\underline\mathbf{x},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.

Line 855: Line 861:

To prepare for the next phase of discussion, Tables 6 and 7 collect and summarize all of the propositional forms on one and two variables. These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions. Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts. For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns. As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.

−

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^\bullet = [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^\bullet</math> are coordinated as a space of type <math>\mathbb{B}^1,</math> this in light of the universe <math>X^\bullet</math> being a functional domain where the coordinate projection <math>x\!</math> takes on its values in <math>\mathbb{B}.</math>

Line 879: Line 885:

| <math>f_{00}\!</math>

| <math>0~0\!</math>

−

| <math>(~)\!</math>

+

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

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

| <math>0\!</math>

Line 886: Line 892:

| <math>f_{01}\!</math>

| <math>0~1\!</math>

−

| <math>(x)\!</math>

+

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

| <math>\text{not}~ x\!</math>

| <math>\lnot x\!</math>

Line 900: Line 906:

| <math>f_{11}\!</math>

| <math>1~1\!</math>

−

| <math>((~))\!</math>

+

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

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

| <math>1\!</math>

Line 907: Line 913:

−

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^\bullet = [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^\bullet</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>

Line 991: Line 997:

| valign="bottom" |

<math>\begin{matrix}

−

(~)

+

\texttt{(~)}

\\[4pt]

−

(x)(y)

+

\texttt{(} x \texttt{)(} y \texttt{)}

\\[4pt]

−

(x)~y~

+

\texttt{(} x \texttt{)} ~ y ~

\\[4pt]

−

(x)

+

\texttt{(} x \texttt{)}

\\[4pt]

−

~x~(y)

+

~ x ~ \texttt{(} y \texttt{)}

\\[4pt]

−

(y)

+

\texttt{(} y \texttt{)}

\\[4pt]

−

(x,~y)

+

\texttt{(} x \texttt{,} ~ y \texttt{)}

\\[4pt]

−

(x~y)

+

\texttt{(} x ~ y \texttt{)}

\end{matrix}\!</math>

| valign="bottom" |

Line 1,100: Line 1,106:

| valign="bottom" |

<math>\begin{matrix}

−

x~y

+

x ~ y

\\[4pt]

−

((x,~y))

+

\texttt{((} x \texttt{,} ~ y \texttt{))}

\\[4pt]

y

\\[4pt]

−

(x~(y))

+

\texttt{(} x ~ \texttt{(} y \texttt{))}

\\[4pt]

x

\\[4pt]

−

((x)~y)

+

\texttt{((} x \texttt{)} ~ y \texttt{)}

\\[4pt]

−

((x)(y))

+

\texttt{((} x \texttt{)(} y \texttt{))}

\\[4pt]

−

((~))

+

\texttt{((~))}

\end{matrix}\!</math>

| valign="bottom" |

Line 1,180: Line 1,186:

|  

|-

−

| <math>~~f_0~~\!</math>

+

| <math>f_{0}\!</math>

| <math>f_{0000}\!</math>

| <math>0~0~0~0\!</math>

−

| <math>(~)\!</math>

+

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

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

| <math>0\!</math>

Line 1,189: Line 1,195:

| valign="bottom" |

<math>\begin{matrix}

−

~~f_1~~

+

f_{1}

\\[4pt]

−

~~f_2~~

+

f_{2}

\\[4pt]

−

~~f_4~~

+

f_{4}

\\[4pt]

−

~~f_8~~

+

f_{8}

\end{matrix}\!</math>

| valign="bottom" |

Line 1,219: Line 1,225:

| valign="bottom" |

<math>\begin{matrix}

−

(x)(y)

+

\texttt{(} x \texttt{)(} y \texttt{)}

\\[4pt]

−

(x)~y~

+

\texttt{(} x \texttt{)} ~ y ~

\\[4pt]

−

~x~(y)

+

~ x ~ \texttt{(} y \texttt{)}

\\[4pt]

−

~x~~y~

+

~ x ~~ y ~

\end{matrix}\!</math>

| valign="bottom" |

Line 1,250: Line 1,256:

| valign="bottom" |

<math>\begin{matrix}

−

~~f_3~~

+

f_{3}

\\[4pt]

f_{12}

Line 1,268: Line 1,274:

| valign="bottom" |

<math>\begin{matrix}

−

(x)

+

\texttt{(} x \texttt{)}

\\[4pt]

x

Line 1,287: Line 1,293:

| valign="bottom" |

<math>\begin{matrix}

−

~~f_6~~

+

f_{6}

\\[4pt]

−

~~f_9~~

+

f_{9}

\end{matrix}\!</math>

| valign="bottom" |

Line 1,305: Line 1,311:

| valign="bottom" |

<math>\begin{matrix}

−

(x, y)

+

\texttt{(} x \texttt{,} y \texttt{)}

\\[4pt]

−

((x, y))

+

\texttt{((} x \texttt{,} y \texttt{))}

\end{matrix}\!</math>

| valign="bottom" |

Line 1,324: Line 1,330:

| valign="bottom" |

<math>\begin{matrix}

−

~~f_5~~

+

f_{5}

\\[4pt]

f_{10}

Line 1,342: Line 1,348:

| valign="bottom" |

<math>\begin{matrix}

−

(y)

+

\texttt{(} y \texttt{)}

\\[4pt]

y

Line 1,361: Line 1,367:

| valign="bottom" |

<math>\begin{matrix}

−

~~f_7~~

+

f_{7}

\\[4pt]

f_{11}

Line 1,391: Line 1,397:

| valign="bottom" |

<math>\begin{matrix}

−

(~x~~y~)

+

\texttt{(} ~ x ~~ y ~ \texttt{)}

\\[4pt]

−

(~x~(y))

+

\texttt{(} ~ x ~ \texttt{(} y \texttt{))}

\\[4pt]

−

((x)~y~)

+

\texttt{((} x \texttt{)} ~ y ~ \texttt{)}

\\[4pt]

−

((x)(y))

+

\texttt{((} x \texttt{)(} y \texttt{))}

\end{matrix}\!</math>

| valign="bottom" |

Line 1,423: Line 1,429:

| <math>f_{1111}\!</math>

| <math>1~1~1~1\!</math>

−

| <math>((~))\!</math>

+

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

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

| <math>1\!</math>

Line 1,451: Line 1,457:

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>)

+

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.

Line 1,457: Line 1,463:

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>

+

{| align="center" cellpadding="8" width="90%"

+

| <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:

+

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>\texttt{(} \mathrm{d}a_i \texttt{)}\!</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>\{ \texttt{(} \mathrm{d}a_i \texttt{)}, \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>

+

{| align="center" cellpadding="8" width="90%"

+

| <math>\mathbb{D} = \{ \texttt{(} \mathrm{d}a_i \texttt{)}, \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.

+

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===

Line 1,476: Line 1,486:

|}

−

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>

+

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}

+

{| align="center" cellpadding="8" width="90%"

−

X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.

+

| <math>\begin{matrix}X^2 & = & \{ (u, v) : u = v \} & \cup & \{ (u, v) : u \ne v \}.\end{matrix}</math>

−

\end{matrix}</math>

+

|}

This partition may also be expressed in the following symbolic form:

−

: <math>\begin{matrix}

+

−

X^2 & \cong & \~~mathrm~~{diag}(X) & + & 2 \~~tbinom~~{X}{2}.

+

{| align="center" cellpadding="8" width="90%"

−

\end{matrix}</math>

+

| <math>\begin{matrix}X^2 & \cong & \operatorname{diag} (X) & + & 2 \binom{X}{2}.\end{matrix}</math>

+

|}

The separate terms of this formula are defined as follows:

−

: <math>\begin{matrix}

+

{| align="center" cellpadding="8" width="90%"

−

\~~mathrm~~{diag}(X) & = & \{ (x, x) : x \in X \}.

+

| <math>\begin{matrix}\operatorname{diag} (X) & = & \{ (x, x) : x \in X \}.\end{matrix}\!</math>

−

\end{matrix}\!</math>

+

|}

−

: <math>\begin{matrix}

+

{| align="center" cellpadding="8" width="90%"

−

\~~tbinom~~{X}{k} & = & X\ ~~\mathrm~~{choose}\ k & = & \{ k\!\text{-sets from}\ X \}.

+

| <math>\begin{matrix}\binom{X}{k} & = & X ~\text{choose}~ k & = & \{ k\text{-sets from}~ X \}.\end{matrix}\!</math>

−

\end{matrix}\!</math>

+

|}

Thus we have:

−

: <math>\begin{matrix}

+

{| align="center" cellpadding="8" width="90%"

−

\~~tbinom~~{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.

+

| <math>\begin{matrix}\binom{X}{2} & = & \{ \{ u, v \} : u, v \in X \}.\end{matrix}</math>

−

\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}

+

{| align="center" cellpadding="8" width="90%"

+

|

+

<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>

+

\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>

+

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>{}^{\backprime\backprime} \mathbb{D} {}^{\prime\prime}</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:

−

~~: ~~<math>\begin{array}{lcrcl}

+

{| align="center" cellpadding="8" width="90%"

−

\mathrm{d}x_i ((u, v)) & = & (~~\!|\~~ x_i (u) & , & x_i (v)\ ~~|\!~~) \\

+

|

−

& = & x_i (u) & + & x_i (v) \\

+

<math>\begin{array}{lcrcl}

−

& = & x_i (v) & - & x_i (u). \\

+

\mathrm{d}x_i(u, v)

−

\end{array}</math>~~~~

+

& = & \texttt{(} ~ x_i(u) & \texttt{,} & x_i(v) ~ \texttt{)}

+

\\

+

& = & x_i(u) & + & x_i(v)

+

\\

+

& = & x_i(v) & - & x_i(u).

+

\end{array}</math>

+

|}

−

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.

+

In the above transcription, the operator bracket of the form <math>\texttt{(} \ldots \texttt{,} \ldots \texttt{)}\!</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.

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:

−

~~: ~~<math>\begin{array}{lcrcl}

+

{| align="center" cellpadding="8" width="90%"

−

\mathrm{d}x_i (q) & = & (~~\!|\~~ x_i (q_0) & , & x_i (q_1)\ ~~|\!~~) \\

+

|

−

& = & x_i (q_0) & + & x_i (q_1) \\

+

<math>\begin{array}{lcrcl}

−

& = & x_i (q_1) & - & x_i (q_0). \\

+

\mathrm{d}x_i (q)

−

\end{array}</math>~~~~

+

& = & \texttt{(} ~ x_i(q_0) & \texttt{,} & x_i(q_1) ~ \texttt{)}

−

+

\\

+

& = & x_i(q_0) & + & x_i(q_1)

+

\\

+

& = & x_i(q_1) & - & x_i(q_0).

+

\end{array}</math>

+

|}

+

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>

+

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>\texttt{(} \mathrm{d}x_1 \texttt{)} \cdots \texttt{(} \mathrm{d}x_n \texttt{)},\!</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.

Line 1,550: Line 1,576:

Next we define the ''extended alphabet'' or ''bundled alphabet'' <math>\mathrm{E}\mathcal{A}</math> as follows:

−

~~: ~~<math>\begin{array}{lclcl}

+

{| align="center" cellpadding="8" width="90%"

+

|

+

<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\}. \\

+

& = & \{ a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}.

−

\end{array}</math>~~~~

+

\end{array}</math>

+

|}

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:

−

~~: ~~<math>\begin{array}{lcl}

+

{| align="center" cellpadding="8" width="90%"

+

|

+

<math>\begin{array}{lcl}

\mathrm{E}A

−

& = & \langle \mathrm{E}\mathcal{A} \rangle \\

+

& = & \langle \mathrm{E}\mathcal{A} \rangle

−

& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle \\

+

\\[4pt]

−

& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle, \\

+

& = & \langle \mathcal{A} \cup \mathrm{d}\mathcal{A} \rangle

−

\end{array}</math>~~~~

+

\\[4pt]

+

& = & \langle a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n \rangle,

+

\end{array}</math>

+

|}

and also:

−

~~: ~~<math>\begin{array}{lcl}

+

{| align="center" cellpadding="8" width="90%"

+

|

+

<math>\begin{array}{lcl}

\mathrm{E}A

−

& = & A \times \mathrm{d}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. \\

+

\\[4pt]

−

\end{array}</math>~~~~

+

& = & A_1 \times \ldots \times A_n \times \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.

+

\end{array}</math>

+

|}

This gives <math>\mathrm{E}A</math> the type <math>\mathbb{B}^n \times \mathbb{D}^n.</math>

−

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:

+

Finally, the tangent universe <math>\mathrm{E}A^\bullet = [\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:

−

~~: ~~<math>\begin{array}{lclcl}

+

{| align="center" cellpadding="8" width="90%"

−

\mathrm{E}A^\~~circ~~

+

|

+

<math>\begin{array}{lclcl}

+

\mathrm{E}A^\bullet

& = & [\mathrm{E}\mathcal{A}]

−

& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n]. \\

+

& = & [a_1, \ldots, a_n, \mathrm{d}a_1, \ldots, \mathrm{d}a_n].

−

\end{array}</math>~~~~

+

\end{array}</math>

+

|}

−

This gives the tangent universe <math>\mathrm{E}A^\~~circ~~\!</math> the type:

+

This gives the tangent universe <math>\mathrm{E}A^\bullet\!</math> the type:

−

~~: ~~<math>\begin{array}{lcl}

+

{| align="center" cellpadding="8" width="90%"

+

|

+

<math>\begin{array}{lcl}

(\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})). \\

+

& = & (\mathbb{B}^n \times \mathbb{D}^n, (\mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B})).

−

\end{array}</math>~~~~

+

\end{array}</math>

+

|}

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.

Line 1,664: Line 1,708:

| <math>\mathbb{D}^n \to \mathbb{B}\!</math>

|-

−

| <math>\mathrm{d}A^\~~circ~~\!</math>

+

| <math>\mathrm{d}A^\bullet\!</math>

|

<math>\begin{matrix}

Line 1,679: Line 1,723:

|

<math>\begin{matrix}

−

\text{Tangent universe at a point of}~ A^\~~circ~~,

+

\text{Tangent universe at a point of}~ A^\bullet,

\\[2pt]

\text{based on the tangent features}

Line 1,784: Line 1,828:

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:

+

For the purposes of this discussion, an ''intentional proposition'' is defined as a proposition in the universe of discourse <math>\mathrm{Q}X^\bullet = [\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:

−

~~: ~~<math>\begin{array}{~~cclcc~~}

+

{| align="center" cellpadding="8" width="90%"

+

|

+

<math>\begin{array}{lllcl}

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

−

& \cong & (X & \times & X' \to \mathbb{B}) \\

+

& \cong & (X & \times & ~X' \to \mathbb{B})

−

& \cong & (X & \to & (X' \to \mathbb{B})) \\

+

\\[4pt]

−

& \cong & (X' & \to & (X \to \mathbb{B})). \\

+

& \cong & (X & \to & (X' \to \mathbb{B}))

−

\end{array}</math>~~~~

+

\\[4pt]

+

& \cong & (X' & \to & (X~ \to \mathbb{B})).

+

\end{array}</math>

+

|}

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.

+

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^\text{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.

Line 1,846: Line 1,895:

|}

−

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>

+

Let <math>\mathcal{X} = \{ x_1 \} = \{ A \}</math> be an alphabet that represents one boolean variable or a single logical feature. In this example the capital letter <math>{}^{\backprime\backprime} A {}^{\prime\prime}\!</math> is used usual informally, to name a feature and not a space, in departure from our 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 = \{ \texttt{(} A \texttt{)}, 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>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</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>{}^{\backprime\backprime} 1 {}^{\prime\prime}\!</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, \texttt{(} A \texttt{)}, 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''.

+

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:

+

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>\texttt{(} A \texttt{)}\!</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:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:50%"

+

{| align="center" border="1" cellpadding="12" 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%"~~

+

<math>\begin{matrix}

−

| &~~nbsp;~~

+

\text{From} & \texttt{(} A \texttt{)}

−

~~| From~~

+

& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}

−

~~| <math>~~(A)\~~!</math>~~

+

& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}

−

| and

+

\\[8pt]

−

~~| <math>~~(\mathrm{d}A)\~~!</math>~~

+

\text{From} & \texttt{(} A \texttt{)}

−

| infer

+

& \text{and} & \mathrm{d}A

−

~~| <math>~~(A)\~~!</math>~~

+

& \text{infer} & A & \text{next.}

−

| next.

+

\\[8pt]

−

~~|  ~~

+

\text{From} & A

−

|-

+

& \text{and} & \texttt{(} \mathrm{d}A \texttt{)}

−

| &~~nbsp;~~

+

& \text{infer} & A & \text{next.}

−

~~| From~~

+

\\[8pt]

−

~~| <math>~~(A)\~~!</math>~~

+

\text{From} & A

−

| and

+

& \text{and} & \mathrm{d}A

−

~~| <math>~~\mathrm{d}A\~~!</math>~~

+

& \text{infer} & \texttt{(} A \texttt{)} & \text{next.}

−

| infer

+

\end{matrix}</math>

−

~~| <math>~~A\~~!</math>~~

−

| next.

−

| &~~nbsp;~~

−

|-

−

| &~~nbsp;~~

−

~~| From~~

−

~~| <math>A~~\~~!</math>~~

−

| and

−

~~| <math>~~(\mathrm{d}A)\~~!</math>~~

−

| infer

−

~~| <math>~~A\~~!</math>~~

−

| next.

−

| &~~nbsp;~~

−

|-

−

| &~~nbsp;~~

−

~~| From~~

−

~~| <math>A~~\~~!</math>~~

−

| and

−

~~| <math>~~\mathrm{d}A\~~!</math>~~

−

| infer

−

~~| <math>~~(A)\!</math>

−

~~| next.~~

−

~~|  ~~

−

|}

Line 1,913: Line 1,938:

|}

−

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.

+

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>\{ \texttt{(} \mathrm{d}A \texttt{)}, \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===

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|}

−

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.

+

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 = \texttt{(} A \texttt{)}.\!</math> Since the equation <math>A = \mathrm{d}A\!</math> is logically equivalent to the disjunction <math>A ~ \mathrm{d}A ~\text{or}~ \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)},\!</math> we may infer two possible trajectories, as displayed in Table 11. In either case the state <math>A ~ \texttt{(} \mathrm{d}A \texttt{)(} \mathrm{d}^2 A \texttt{)}\!</math> is a stable attractor or a terminal condition for both starting points.

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|

<math>\begin{matrix}

−

A & \mathrm{d}A & (\mathrm{d}^2 A)

+

A & \mathrm{d}A & \texttt{(} \mathrm{d}^2 A \texttt{)}

\\[4pt]

−

(A) & \mathrm{d}A & \mathrm{d}^2 A

+

\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A

\\[4pt]

−

A & (\mathrm{d}A) & (\mathrm{d}^2 A)

+

A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}

\\[4pt]

−

A & (\mathrm{d}A) & (\mathrm{d}^2 A)

+

A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}

\\[4pt]

{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}

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|

<math>\begin{matrix}

−

(A) & (\mathrm{d}A) & \mathrm{d}^2 A

+

\texttt{(} A \texttt{)} & \texttt{(} \mathrm{d}A \texttt{)} & \mathrm{d}^2 A

\\[4pt]

−

(A) & \mathrm{d}A & \mathrm{d}^2 A

+

\texttt{(} A \texttt{)} & \mathrm{d}A & \mathrm{d}^2 A

\\[4pt]

−

A & (\mathrm{d}A) & (\mathrm{d}^2 A)

+

A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}

\\[4pt]

−

A & (\mathrm{d}A) & (\mathrm{d}^2 A)

+

A & \texttt{(} \mathrm{d}A \texttt{)} & \texttt{(} \mathrm{d}^2 A \texttt{)}

\\[4pt]

{}^{\shortparallel} & {}^{\shortparallel} & {}^{\shortparallel}

−

\end{matrix}</math>

+

\end{matrix}\!</math>

|}

−

Because the initial space ''X~~'' ~~=~~ 〈''~~A~~''〉~~ is one-dimensional, we can easily fit the second order extension E<~~sup~~>2~~''~~X~~'' ~~=~~ 〈''~~A'',~~ ~~d''A'',~~ ~~d~~~~2</~~sup~~>~~''A''〉~~ within the compass of a single venn diagram, charting the couple of converging trajectories as shown in Figure 12.

+

Because the initial space <math>X = \langle A \rangle\!</math> is one-dimensional, we can easily fit the second order extension <math>\mathrm{E}^2 X = \langle A, \mathrm{d}A, \mathrm{d}^2 A \rangle\!</math> 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%"

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|}

−

If we eliminate from view the regions of E<~~sup~~>2</~~sup~~>~~''X''~~ that are ruled out by the dynamic law d<~~sup~~>2~~''~~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~~~~2</~~sup~~>~~''A'').~~

+

If we eliminate from view the regions of <math>\mathrm{E}^2 X\!</math> that are ruled out by the dynamic law <math>\mathrm{d}^2 A = \texttt{(} A \texttt{)},\!</math> 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 <math>A\!</math> and <math>\mathrm{d}^2 A\!.</math> 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 <math>\texttt{(} A \texttt{,} \mathrm{d}^2 A \texttt{)}\!.</math>

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"

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|}

−

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.

+

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 <math>n\!</math>-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 <math>n\!</math>-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.

+

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.

+

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.

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|}

−

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~~~~X~~~~]~~ ~~=~~ ~~[''A'',~~ ~~d''A'']. Over the extended alphabet E<~~font face="lucida calligraphy"~~>X~~~~ = {~~''x''1~~,~~ ~~d~~''x''1~~} = {''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:

+

Let us assume that the sense intended for differential features is well enough established in the intuition, for now, that we may continue with outlining the structure of the differential extension <math>[\mathrm{E}\mathcal{X}] = [A, \mathrm{d}A].\!</math> Over the extended alphabet <math>\mathrm{E}\mathcal{X} = \{ x_1, \mathrm{d}x_1 \} = \{ A, \mathrm{d}A \}\!</math> of cardinality <math>2^n = 2\!</math> we generate the set of points <math>\mathrm{E}X\!</math> of cardinality <math>2^{2n} = 4\!</math> that bears the following chain of equivalent descriptions:

−

:{| cellpadding=2

+

{| align="center" cellpadding="8" width="90%"

−

| E''X''

+

|

−

| =

+

<math>\begin{array}{lll}

−

~~| 〈''~~A'', d''A~~''〉~~

+

\mathrm{E}X & = & \langle A, \mathrm{d}A \rangle

−

|-

+

\\[4pt]

−

| &~~nbsp;~~

+

& = & \{ \texttt{(} A \texttt{)}, A \} ~\times~ \{ \texttt{(} \mathrm{d}A \texttt{)}, \mathrm{d}A \}

−

| =

+

\\[4pt]

−

| {(''A''), ''A''} × {(d''A''), d''A''}

+

& = &

−

|-

+

\{

−

| &~~nbsp;~~

+

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

−

~~| =~~

+

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

−

| {(''A'')(d''A''), (''A'') d''A'', ''A'' (d''A''), ''A'' d''A''}.

+

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

+

A ~ \mathrm{d}A

+

\}.

+

\end{array}</math>

|}

−

The space E''X'' may be assigned the mnemonic type ~~'''~~B~~''' &~~times~~; '''~~D~~'''~~, which is really no different than ~~'''~~B~~''' &~~times~~; '''~~B~~''' ~~=~~ '''~~B~~'''~~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.

+

The space <math>\mathrm{E}X\!</math> may be assigned the mnemonic type <math>\mathbb{B} \times \mathbb{D},\!</math> which is really no different than <math>\mathbb{B} \times \mathbb{B} = \mathbb{B}^2.\!</math> An individual element of <math>\mathrm{E}X\!</math> 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~~'' • ~~=~~ ~~[~~''x''1~~,~~ ~~d~~''x''1~~]~~ ~~=~~ ~~[''A'',~~ ~~d''A''], one must add the set of differential propositions E''X''^~~ ~~=~~ ~~{''g~~'' ~~:~~ ~~E''X~~'' → '''~~B~~'''~~}~~ <math>~~\cong~~</math> ~~(~~'''~~B~~''' &~~times~~; '''~~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.

+

To complete the construction of the extended universe of discourse <math>\mathrm{E}X^\bullet = [x_1, \mathrm{d}x_1] = [A, \mathrm{d}A]\!</math> one must add the set of differential propositions <math>\mathrm{E}X^\uparrow = \{ g : \mathrm{E}X \to \mathbb{B} \} \cong (\mathbb{B} \times \mathbb{D} \to \mathbb{B})\!</math> to the set of dispositions in <math>\mathrm{E}X.\!</math> There are <math>2^{2^{2n}} = 16\!</math> propositions in <math>\mathrm{E}X^\uparrow,\!</math> as detailed in Table 14.

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|  

|-

−

| <math>~~f_0~~\!</math>

+

| <math>f_{0}\!</math>

−

| <math>~~g_0~~\!</math>

+

| <math>g_{0}\!</math>

| <math>0~0~0~0\!</math>

−

| <math>(~)\!</math>

+

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

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

| <math>0\!</math>

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|

<math>\begin{matrix}

−

~~g_1~~

+

g_{1}

\\[4pt]

−

~~g_2~~

+

g_{2}

\\[4pt]

−

~~g_4~~

+

g_{4}

\\[4pt]

−

~~g_8~~

+

g_{8}

\end{matrix}\!</math>

|

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|

<math>\begin{matrix}

−

(A)(\mathrm{d}A)

+

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

\\[4pt]

−

(A)~\mathrm{d}A~

+

\texttt{(} A \texttt{)} ~ \mathrm{d}A ~

\\[4pt]

−

~A~(\mathrm{d}A)

+

~ A ~ \texttt{(} \mathrm{d}A \texttt{)}

\\[4pt]

−

~A~~\mathrm{d}A~

+

~ A ~~ \mathrm{d}A ~

\end{matrix}\!</math>

|

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\text{neither}~ A ~\text{nor}~ \mathrm{d}A

\\[4pt]

−

\~~text~~{~~not~~}~ A ~\text{~~but~~}~ ~~\mathrm{d}~~A

+

\mathrm{d}A ~\text{and not}~ A

\\[4pt]

−

A ~\text{~~but~~ not}~ \mathrm{d}A

+

A ~\text{and not}~ \mathrm{d}A

\\[4pt]

A ~\text{and}~ \mathrm{d}A

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|

<math>\begin{matrix}

−

~~f_1~~

+

f_{1}

\\[4pt]

−

~~f_2~~

+

f_{2}

\end{matrix}\!</math>

|

<math>\begin{matrix}

−

~~g_3~~

+

g_{3}

\\[4pt]

g_{12}

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|

<math>\begin{matrix}

−

(A)

+

\texttt{(} A \texttt{)}

\\[4pt]

A

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|

<math>\begin{matrix}

−

~~g_6~~

+

g_{6}

\\[4pt]

−

~~g_9~~

+

g_{9}

\end{matrix}\!</math>

|

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|

<math>\begin{matrix}

−

(A, \mathrm{d}A)

+

\texttt{(} A \texttt{,} \mathrm{d}A \texttt{)}

\\[4pt]

−

((A, \mathrm{d}A))

+

\texttt{((} A \texttt{,} \mathrm{d}A \texttt{))}

\end{matrix}\!</math>

|

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|

<math>\begin{matrix}

−

~~g_5~~

+

g_{5}

\\[4pt]

g_{10}

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|

<math>\begin{matrix}

−

(\mathrm{d}A)

+

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

\\[4pt]

\mathrm{d}A

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|

<math>\begin{matrix}

−

~~g_7~~

+

g_{7}

\\[4pt]

g_{11}

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|

<math>\begin{matrix}

−

(~A~~\mathrm{d}A~)

+

\texttt{(} ~ A ~~ \mathrm{d}A ~ \texttt{)}

\\[4pt]

−

(~A~(\mathrm{d}A))

+

\texttt{(} ~ A ~ \texttt{(} \mathrm{d}A \texttt{))}

\\[4pt]

−

((A)~\mathrm{d}A~)

+

\texttt{((} A \texttt{)} ~ \mathrm{d}A ~ \texttt{)}

\\[4pt]

−

((A)(\mathrm{d}A))

+

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

\end{matrix}\!</math>

|

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\text{not}~ A ~\text{without}~ \mathrm{d}A

\\[4pt]

−

\text{~~without~~}~ A ~\text{~~not~~}~ ~~\mathrm{d}~~A

+

\text{not}~ \mathrm{d}A ~\text{without}~ A

\\[4pt]

A ~\text{or}~ \mathrm{d}A

Line 2,267: Line 2,295:

\end{matrix}\!</math>

|-

−

| <math>~~f_3~~\!</math>

+

| <math>f_{3}\!</math>

| <math>g_{15}\!</math>

| <math>1~1~1~1\!</math>

−

| <math>((~))\!</math>

+

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

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

| <math>1\!</math>

Line 2,277: Line 2,305:

−

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~~.

+

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 <math>X^\uparrow.\!</math> Thus the first set of propositions <math>\{ f_i \}\!</math> is automatically embedded in the present set <math>\{ g_j \}\!</math> and the corresponding inclusions are indicated at the far left margin of the Table.

===Tacit Extensions===

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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.

+

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>\boldsymbol\varepsilon 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>\boldsymbol\varepsilon 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>

+

{| align="center" cellpadding="8" width="90%"

+

| <math>\boldsymbol\varepsilon f(x_1, \ldots, x_n, \ldots, x_{n+k}) ~=~ f(x_1, \ldots, x_n).\!</math>

+

|}

−

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.

+

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>\boldsymbol\varepsilon : (\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>\boldsymbol\varepsilon\!</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.

+

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, \texttt{(} A \texttt{)}, A, 1 \},\!</math> the tacit extension <math>\boldsymbol\varepsilon 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>\boldsymbol\varepsilon f\!</math> of <math>f\!</math> to <math>\mathrm{E}X</math> may be explicated as shown in Table 15.

+

+

{| align="center" border="1" cellpadding="15" cellspacing="0" style="text-align:center; width:60%"

+

|+ style="height:30px" | <math>\text{Table 15.} ~~ \text{Tacit Extension of}~ [A] ~\text{to}~ [A, \mathrm{d}A]\!</math>

+

| valign="bottom" |

+

<math>\begin{matrix}

+

0

+

& = & 0 & \cdot & \texttt{(} \mathrm{d}A, ~ \texttt{(} \mathrm{d}A \texttt{))} & = & & 0

+

\\[8pt]

+

\texttt{(} A \texttt{)}

+

& = & \texttt{(} A \texttt{)} & \cdot & \texttt{(} \mathrm{d}A, ~ \texttt{(} \mathrm{d}A \texttt{))}

+

& = & \texttt{(} A \texttt{)} ~\mathrm{d}A~ & + & \texttt{(} A \texttt{)(} \mathrm{d}A \texttt{)}

+

\\[8pt]

+

A

+

& = & ~A~ & \cdot & \texttt{(} \mathrm{d}A, ~ \texttt{(} \mathrm{d}A \texttt{))}

+

& = & ~A~ ~\mathrm{d}A~ & + & ~A~ \texttt{(} \mathrm{d}A \texttt{)}

+

\\[8pt]

+

1

+

& = & 1 & \cdot & \texttt{(} \mathrm{d}A, ~ \texttt{(} \mathrm{d}A \texttt{))} & = & & 1

+

\end{matrix}</math>

+

|}

−

~~{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:75%"~~

+

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>\texttt{(} A \texttt{)},\!</math> to considering the collection of outcomes, the outgoing changes or the singular dispositions, that spring from that state.

−

~~|+ style="height:30px" |~~ <math>\~~text{Table 15~~.~~}~~~~\~~text~~{~~Tacit Extension of~~}~~~ [~~A~~] ~~~\~~text~~{to}~~~ [A~~, ~~\mathrm{d}A]~~\!</math>

+

−

|

+

===Example 2. Drives and Their Vicissitudes===

−

{| ~~align~~="~~center~~" ~~border~~="0~~" cellpadding="8~~" cellspacing="0" ~~style~~="~~text-align:center; width:100~~%"

+

−

|  

+

{| width="100%" cellpadding="0" cellspacing="0"

−

| ~~<math>0\!</math>~~

+

| width="40%" |  

−

~~| <math>~~=~~\!</math>~~

+

| width="60%" |

−

| ~~<math>0\!</math>~~

+

I open my scuttle at night and see the far-sprinkled systems,

−

| <~~math>\cdot\!</math~~>

+

And all I see, multiplied as high as I can cipher, edge but

−

~~| <math>((\mathrm{d}A) ~~~,~~~ \mathrm{d}A)\!~~<~~/math~~>

+

     the rim of the farther systems.

−

~~| <math>=\!</math>~~

−

~~| <math>0\!</math>~~

−

|  

|-

|  

−

~~| <math>(A)\!</math>~~

+

| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 81]

−

~~| <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>~~

−

|}

−

|}

−

~~ ~~

−

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, ~~

−

~~And all I see, multiplied as high as I can cipher, edge but ~~

−

~~     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~~ ~~=~~ ~~{~~''x''1~~}~~ ~~=~~ ~~{''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".

+

Again, let <math>\mathcal{X} = \{ x_1 \} = \{ A \}.\!</math> In the discussion that follows we will consider a class of trajectories having the property that <math>\mathrm{d}^k A = 0\!</math> for all <math>k\!</math> greater than some fixed <math>m\!</math> and we may 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 <math>\mathrm{d}^m A\!</math> 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 <math>\mathrm{d}^m A\!</math> the ''drive'' at that point. Curves of constant drive <math>\mathrm{d}^m A\!</math> are then referred to as ''<math>m^\text{th}\!</math>-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].

+

* '''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 1840s prefigured the Turing machines of the 1940s [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].

−

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~~~~2~~''~~A'',~~ ~~d~~~~3~~''~~A'',~~ ~~d~~~~4</~~sup~~>~~''A''〉.~~ These are the trajectories generated subject to the dynamic law d<~~sup~~>4~~''~~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~~''~~X'' = ~~〈''~~A'',~~ ~~d''A'',~~ ~~d~~~~2~~''~~A'',~~ ~~d~~~~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.

+

Given this language, the Example we take up here can be described as the family of <math>4^\text{th}\!</math>-gear curves through <math>\mathrm{E}^4 X\!</math> <math>=\!</math> <math>\langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A, ~\mathrm{d}^4\!A \rangle.</math> These are the trajectories generated subject to the dynamic law <math>\mathrm{d}^4 A = 1,\!</math> where it is understood in such a statement that all higher order differences are equal to <math>0.\!</math> Since <math>\mathrm{d}^4 A\!</math> and all higher <math>\mathrm{d}^k A\!</math> are fixed, the temporal or transitional conditions (initial, mediate, terminal — transient or stable states) vary only with respect to their projections as points of <math>\mathrm{E}^3 X = \langle A, ~\mathrm{d}A, ~\mathrm{d}^2\!A, ~\mathrm{d}^3\!A \rangle.</math> 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 <math>\mathrm{E}^3 X.\!</math> It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure 16.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"

Line 2,380: Line 2,385:

|}

−

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~~''~~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~~''~~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~~''''~~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'''~~.~~'''d1~~</~~sub~~>~~…d~~<~~sub~~>~~''m''~~</~~sub~~> ~~of the state ''q''~~ the coefficient d<~~sup~~>''k~~''''~~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:

+

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 <math>q \in \mathrm{E}^m X\!</math> with the dyadic rationals (or the binary fractions) in the half-open interval <math>[0, 2).\!</math> Formally and canonically, a state <math>q_r\!</math> is indexed by a fraction <math>r = \tfrac{s}{t}\!</math> whose denominator is the power of two <math>t = 2^m\!</math> and whose numerator is a binary numeral formed from the coefficients of state in a manner to be described next. The ''differential coefficients'' of the state <math>q\!</math> are just the values <math>\mathrm{d}^k\!A(q)</math> for <math>k = 0 ~\text{to}~ m,\!</math> where <math>\mathrm{d}^0\!A</math> is defined as being identical to <math>A.\!</math> To form the binary index <math>d_0.d_1 \ldots d_m\!</math> of the state <math>q\!</math> the coefficient <math>\mathrm{d}^k\!A(q)</math> is read off as the binary digit <math>d_k\!</math> associated with the place value <math>2^{-k}.\!</math> Expressed by way of algebraic formulas, the rational index <math>r\!</math> of the state <math>q\!</math> can be given by the following equivalent formulations:

Line 2,405: Line 2,410:

−

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''''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~~'' ~~=~~ ~~2~~~~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''''~~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''′ ~~=~~ ''d''''k'' ~~+~~ ''d''''~~k''+1</~~sub~~>.

+

Applied to the example of <math>4^\text{th}\!</math>-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 <math>(p_i, q_j),\!</math> where <math>p_i\!</math> may be read as a temporal parameter that indicates the present time of the state and where <math>j\!</math> is the decimal equivalent of the binary numeral <math>s.\!</math> Informally and more casually, the Tables exhibit the states <math>q_s\!</math> as subscripted with the numerators of their rational indices, taking for granted the constant denominators of <math>2^m\! = 2^4 = 16.\!</math> In this set-up the temporal successions of states can be reckoned as given by a kind of ''parallel round-up rule''. That is, if <math>(d_k, d_{k+1})\!</math> is any pair of adjacent digits in the state index <math>r,\!</math> then the value of <math>d_k\!</math> in the next state is <math>{d_k}' = d_k + d_{k+1}.\!</math>

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:52%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"

|+ style="height:30px" | <math>\text{Table 17-a.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 1}\!</math>

|- style="background:ghostwhite"

Line 2,415: Line 2,420:

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

| <math>A\!</math>

−

| <math>\mathrm{d}A</math>

+

| <math>\mathrm{d}A\!</math>

|  

Line 2,561: Line 2,566:

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:52%"

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:55%"

|+ style="height:30px" | <math>\text{Table 17-b.} ~~ \text{A Couple of Orbits in Fourth Gear : Orbit 2}\!</math>

|- style="background:ghostwhite"

Line 2,567: Line 2,572:

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

| <math>A\!</math>

−

| <math>\mathrm{d}A</math>

+

| <math>\mathrm{d}A\!</math>

|  

Line 2,741: Line 2,746:

|}

−

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~~~~]~~ → ~~[~~~~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.

+

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 <math>[\mathcal{X}] \to [\mathcal{Y}]\!</math> is implied any time that we make use of one alphabet <math>[\mathcal{X}]\!</math> that happens to be included in another alphabet <math>[\mathcal{Y}].\!</math> When we are discussing differential issues we usually have in mind that the extended alphabet <math>[\mathcal{Y}]\!</math> has a special construction or a specific lexical relation with respect to the initial alphabet <math>[\mathcal{X}],\!</math> 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~~ → '''~~B~~'''~~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.

+

Figure 18-a lays out the ''angular form'' of venn diagram for universes of 1 and 2 dimensions, indicating the embedding map of type <math>\mathbb{B}^1 \to \mathbb{B}^2\!</math> 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%"

Line 2,753: Line 2,758:

|}

−

Figure 18-b shows the differential extension from ~~''X''~~<~~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.

+

Figure 18-b shows the differential extension from <math>X^\bullet = [x]\!</math> to <math>\mathrm{E}X^\bullet = [x, \mathrm{d}x]\!</math> 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%"

Line 2,779: Line 2,784:

====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~~ → '''~~B~~'''~~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.

+

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 <math>\mathbb{B}^2 \to \mathbb{B}^4.\!</math> 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%"

Line 2,787: Line 2,792:

|}

−

Figure 19-b shows the differential extension from ~~''U''~~<~~sup~~>~~ • ~~=~~ ~~[''u'',~~ ''~~v''] ~~to E''U''~~<~~sup~~>~~ •~~<~~/sup~~>~~ ~~=~~ ~~[''u'',~~ ''~~v'',~~ ~~d''u'',~~ ~~d''v''] in the ''bundle of boxes'' form of venn diagram.

+

Figure 19-b shows the differential extension from <math>U^\bullet = [u, v]\!</math> to <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v]\!</math> in the ''bundle of boxes'' form of venn diagram.

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"

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|}

−

Figure 19-d gives the ''digraph'' form of representation for the differential extension ~~''U''~~<~~sup~~>~~ • → ~~E''U~~'' •~~</~~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.

+

Figure 19-d gives the ''digraph'' form of representation for the differential extension <math>U^\bullet \to \mathrm{E}U^\bullet,\!</math> where the 4 nodes marked with a circle <math>{}^{\bigcirc}\!</math> are the cells <math>uv,\, u \texttt{(} v \texttt{)},\, \texttt{(} u \texttt{)} v,\, \texttt{(} u \texttt{)(} v \texttt{)},\!</math> respectively, and where a 2-headed arc counts as 2 arcs of the differential digraph.

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−

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''~~·~~''v''~~ in [''u'',~~ ''~~v''].

+

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 considering the proposition <math>u\!\cdot\!v</math> in the universe <math>[u, v].\!</math> 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 <math>u\!\cdot\!v</math> a distinctive functional name <math>{}^{\backprime\backprime} J {}^{\prime\prime}.\!</math> Second, one has come to think explicitly about the target domain that contains the functional values of <math>J,\!</math> as when writing <math>J : \langle u, v \rangle \to \mathbb{B}.\!</math>

−

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''~~·~~''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~~'''~~.

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|}

−

In Figure 20-ii the proposition ''J'' is viewed explicitly as a transformation from one universe of discourse to another.

+

In Figure 20-ii the proposition <math>J\!</math> is viewed explicitly as a transformation from one universe of discourse to another.

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−

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.

+

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 <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> to serve as the name of its dependent variable <math>J : \mathbb{B}\!</math> 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".

+

The second venn diagram circumvents these difficulties by introducing a new variable name for each basic feature of the target universe, as when writing <math>J : \langle u, v \rangle \to \langle x \rangle,\!</math> and thereby assigns a concrete type <math>\langle x \rangle</math> to the abstract codomain <math>\mathbb{B}.\!</math> To make this induction of variables more formal one can append subscripts, as in <math>x_J,\!</math> to indicate the origin or derivation of the new characters. Or we may use a lexical modifier to convert function names into variable names, for example, associating the function name <math>J\!</math> with the variable name <math>\check{J}.\!</math> Thus we may think of <math>x = x_J = \check{J}\!</math> as the ''cache variable'' corresponding to the function <math>J\!</math> or the symbol <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> considered as a contingent variable.

−

In Figure 20-iii we arrive at a stage where the functional equations~~, ''~~J~~'' ~~=~~ ''~~u''·~~''v'' and ''~~x~~'' ~~=~~ ''~~u~~''·~~~~''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~~''·~~~~''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''.

+

In Figure 20-iii we arrive at a stage where the functional equations <math>J = u\!\cdot\!v</math> and <math>x = u\!\cdot\!v</math> are regarded as propositions in their own right, reigning in and ruling over the 3-feature universes of discourse <math>[u, v, J]~\!</math> and <math>[u, v, x],\!</math> respectively. Subject to the cautions already noted, the function name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> can be reinterpreted as the name of a feature <math>\check{J}</math> and the equation <math>J = u\!\cdot\!v</math> can be read as the logical equivalence <math>\texttt{((} J, u ~ v \texttt{))}.\!</math> To give it a generic name let us call this newly expressed, collateral proposition the ''thematization'' or the ''thematic extension'' of the original proposition <math>J.\!</math>

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−

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.

+

The first venn diagram represents the thematization of the conjunction <math>J\!</math> with shading in the appropriate regions of the universe <math>[u, v, J].\!</math> 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''~~·~~''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 &iota~~; ~~:~~ 〈''~~u'',~~ ''~~v'',~~ ''~~x~~''〉 → '''~~B~~'''~~.

+

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 <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> are resolved by introducing a new variable name <math>{}^{\backprime\backprime} x {}^{\prime\prime}</math> to take the place of <math>\check{J},\!</math> and the region that represents this fresh featured <math>x\!</math> is circumscribed in a more conventional symmetry of form and placement. Just as we once gave the name <math>{}^{\backprime\backprime} J {}^{\prime\prime}</math> to the proposition <math>u\!\cdot\!v,</math> we now give the name <math>{}^{\backprime\backprime} \iota {}^{\prime\prime}</math> to its thematization <math>\texttt{((} x, u ~ v \texttt{))}.\!</math> Already, again, at this culminating stage of reflection, we begin to think of the newly named proposition as a distinctive individual, a particular function <math>\iota : \langle u, v, x \rangle \to \mathbb{B}.\!</math>

−

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 = &theta~~;''~~J'' in the present instance. The operator θ, in the present situation bearing the type &theta~~; ~~:~~ ~~[''u'',~~ ''~~v'']~~ →> ~~[''u'',~~ ''~~v'',~~ ''~~x''], provides us with a convenient way of recapitulating and summarizing the complete cycle of thematic developments.

+

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 <math>J\!</math> to <math>\iota,\!</math> we introduce a class of operators symbolized by the Greek letter <math>\theta,\!</math> writing <math>\iota = \theta J\!</math> in the present instance. The operator <math>\theta,\!</math> in the present situation bearing the type <math>\theta : [u, v] \to [u, v, x],\!</math> 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~~ = &theta~~;''~~f'' and γ = &theta~~;''~~g''.

+

Figure 21 shows how the thematic extension operator <math>\theta\!</math> acts on two further examples, the disjunction <math>\texttt{((} u \texttt{)(} v \texttt{))}\!</math> and the equality <math>\texttt{((} u, v \texttt{))}.\!</math> Referring to the disjunction as <math>f(u, v)\!</math> and the equality as <math>f(u, v),\!</math> we may express the thematic extensions as <math>\varphi = \theta f\!</math> and <math>\gamma = \theta g.\!</math>

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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'')).

+

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 <math>f(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math> and <math>g(u, v) = \texttt{((} u, v \texttt{))}.\!</math>

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−

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 &theta~~;''~~f'' and &theta~~;''~~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 &theta~~;''~~f'' and &theta~~;''~~g'' are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions ''f'' and ''g''.

+

Next, each propositional form is individually represented in the fashion shown in Tables 23-i and 23-ii, using <math>{}^{\backprime\backprime} f {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} g {}^{\prime\prime}\!</math> as function names and creating new variables <math>x\!</math> and <math>y\!</math> to hold the associated functional values. This pair of Tables outlines the first stage in the transition from the 2-dimensional universes of <math>f\!</math> and <math>g\!</math> to the 3-dimensional universes of <math>\theta f\!</math> and <math>\theta g.\!</math> The top halves of the Tables replicate the truth table patterns for <math>f\!</math> and <math>g\!</math> in the form <math>f : [u, v] \to [x]\!</math> and <math>g : [u, v] \to [y].\!</math> The bottom halves of the tables print the negatives of these pictures, as it were, and paste the truth tables for <math>\texttt{(} f \texttt{)}\!</math> and <math>\texttt{(} g \texttt{)}\!</math> under the copies for <math>f\!</math> and <math>g.\!</math> At this stage, the columns for <math>\theta f\!</math> and <math>\theta g\!</math> are appended almost as afterthoughts, amounting to indicator functions for the sets of ordered triples that make up the functions <math>f\!</math> and <math>g.\!</math>

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−

All the data are now in place to give the truth tables for &theta~~;''~~f'' and &theta~~;''~~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 &theta~~;''~~f~~'' ~~=~~ φ ~~:~~ ~~[''u'',~~ ''~~v'',~~ ''~~x'']~~ → '''~~B~~'''~~ and &theta~~;''~~g~~'' ~~=~~ &~~gamma~~; ~~:~~ ~~[''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~~> are now ~~to be regarded~~ as independent variables.

+

All the data are now in place to give the truth tables for <math>\theta f\!</math> and <math>\theta g.\!</math> All that remains to be done is to permute the rows and change the roles of <math>x\!</math> and <math>y\!</math> 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 <math>(u, v, x)\!</math> and <math>(u, v, y)\!</math> in binary numerical order, suitable for viewing as the arguments of the maps <math>\theta f = \varphi : [u, v, x] \to \mathbb{B}\!</math> and <math>\theta g = \gamma : [u, v, y] \to \mathbb{B}.\!</math> Moreover, the structure of the tables is altered slightly, allowing the now vestigial functions <math>\theta f\!</math> and <math>\theta g\!</math> 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 <math>x := \check{f}\!</math> and <math>y := \check{g}\!</math> are now treated as independent variables.

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−

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 &gamma~~; ~~:~~ ~~[''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 &theta~~;''~~F'' exhibits the pattern of the original ''F'', and when ~~''F''~~<~~sup~~>~~ ¢~~</~~sup~~> is false then &theta~~;''~~F'' exhibits the pattern of its negation (''F'').

+

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 <math>x\!</math> and <math>y\!</math> as the primary variables in their respective 3-tuples. Regarding the thematic extensions in the form <math>\varphi : [x, u, v] \to \mathbb{B}\!</math> and <math>\gamma : [y, u, v] \to \mathbb{B}\!</math> 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 <math>\check{F}\!</math> is true then <math>\theta F\!</math> exhibits the pattern of the original <math>F,\!</math> and when <math>\check{F}\!</math> is false then <math>\theta F\!</math> exhibits the pattern of its negation <math>\texttt{(} F \texttt{)}.\!</math>

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−

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.

+

Finally, Tables 26-i and 26-ii compare the tacit extensions <math>\boldsymbol\varepsilon : [u, v] \to [u, v, x]\!</math> and <math>\boldsymbol\varepsilon : [u, v] \to [u, v, y]\!</math> with the thematic extensions of the same types, as applied to the propositions <math>f\!</math> and <math>g,\!</math> respectively.

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| <math>v\!</math>

| <math>x\!</math>

−

| style="border-left:1px solid black" | <math>\varepsilon f\!</math>

+

| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon f\!</math>

−

| <math>\~~vartheta~~ f\!</math>

+

| <math>\theta f\!</math>

|-

| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>

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| <math>v\!</math>

| <math>y\!</math>

−

| style="border-left:1px solid black" | <math>\varepsilon g\!</math>

+

| style="border-left:1px solid black" | <math>\boldsymbol\varepsilon g\!</math>

−

| <math>\~~vartheta~~ g\!</math>

+

| <math>\theta g\!</math>

|-

| style="border-top:1px solid black" | <math>\begin{matrix}0\\0\\0\\0\end{matrix}\!</math>

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−

Table 27 summarizes the thematic extensions of all propositions on two variables. Column 4 lists the equations of form ((~~ ''f'' ¢ ~~,~~ ''f'' ¢‹''~~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.)~~

+

Table 27 summarizes the thematic extensions of all propositions on two variables. Column 4 lists the equations of form <math>\texttt{((} \check{f_i}, f_i (u, v) \texttt{))}\!</math> and Column 5 simplifies these equations into the form of algebraic expressions. As always, <math>{}^{\backprime\backprime} + {}^{\prime\prime}\!</math> refers to exclusive disjunction and each <math>{}^{\backprime\backprime} \check{f} {}^{\prime\prime}\!</math> appearing in the last two Columns refers to the corresponding variable name <math>{}^{\backprime\backprime} \check{f_i} {}^{\prime\prime}.~\!</math>

+

−

~~ ~~

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:75%"

−

{| align="center" border="1" cellpadding="4" cellspacing="0" style="~~font-weight:bold;~~ text-align:center; width:96%"

+

|+ style="height:30px" | <math>\text{Table 27.} ~~ \text{Thematization of Bivariate Propositions}\!</math>

−

|+ Table 27. Thematization of Bivariate Propositions

+

|- style="height:30px; background:ghostwhite"

+

|  

+

|  

+

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

+

| <math>\theta f\!</math>

+

| <math>\theta f\!</math>

|- style="background:ghostwhite"

−

|

+

| align="right" | <math>u\colon\!</math>

−

{| align="right" ~~style="background:ghostwhite; text-align:right"~~

+

| <math>1~1~0~0\!</math>

−

| u :

+

|  

−

|-

+

|  

−

| ~~v :~~

+

|  

−

|}

+

|- style="background:ghostwhite"

−

|

+

| align="right" | <math>v\colon\!</math>

−

{| style="background:ghostwhite"

+

| <math>1~0~1~0\!</math>

−

| ~~1100~~

+

|  

−

|-

−

~~| 1010~~

−

|}

−

|

−

~~{| style~~="~~background:ghostwhite~~"

−

| f

−

|-

|  

−

|}

−

|

−

~~{| style="background:ghostwhite"~~

−

~~| θf~~

−

|-

|  

−

|}

−

|

−

~~{| style="background:ghostwhite"~~

−

~~| θf~~

|-

−

| ~~ ~~

+

| <math>f_{0}\!</math>

−

|}

+

| <math>0~0~0~0\!</math>

+

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

+

| align="left" | <math>\texttt{((} \check{f} \texttt{,~(~)~))}\!</math>

+

| align="left" | <math>\check{f} + 1\!</math>

|-

|

−

~~{| cellpadding="2"~~

+

<math>\begin{matrix}

−

~~| f~~<~~sub>0</sub~~>

+

f_{1}

−

|-

+

\\[4pt]

−

~~| f~~1~~~~

+

f_{2}

−

|-

+

\\[4pt]

−

~~| f~~2~~~~

+

f_{4}

−

|-

+

\\[4pt]

−

~~| f3~~

+

f_{8}

−

|-

+

\end{matrix}\!</math>

−

~~| f~~4~~~~

−

|-

−

~~| f5~~

−

|-

−

~~| f6~~

−

|-

−

~~| f7~~</~~sub~~>

−

|}

|

−

{~~| cellpadding="2"~~

+

<math>\begin{matrix}

−

~~| 0000~~

+

0~0~0~1

−

|-

+

\\[4pt]

−

~~| 0001~~

+

0~0~1~0

−

|-

+

\\[4pt]

−

~~| 0010~~

+

0~1~0~0

−

|-

+

\\[4pt]

−

~~| 0011~~

+

1~0~0~0

−

|-

+

\end{matrix}\!</math>

−

~~| 0100~~

−

|-

−

~~| 0101~~

−

|-

−

~~| 0110~~

−

|-

−

~~| 0111~~

−

|}

|

−

{| ~~cellpadding~~="2"

+

<math>\begin{matrix}

−

| ()

+

\texttt{(} u \texttt{)(} v \texttt{)}

−

|-

+

\\[4pt]

−

~~|  ~~(u)(v)~~ ~~

+

\texttt{(} u \texttt{)~} v \texttt{~}

−

|-

+

\\[4pt]

−

~~|  ~~(u)~~ v  ~~

+

\texttt{~} u \texttt{~(} v \texttt{)}

−

|-

+

\\[4pt]

−

~~|  ~~(u)~~    ~~

+

\texttt{~} u \texttt{~~} v \texttt{~}

−

|-

+

\end{matrix}\!</math>

−

~~|   ~~u~~ (~~v~~) ~~

+

| align="left" |

+

<math>\begin{array}{l}

+

\texttt{((} \check{f} \texttt{,~(u)(v)~))}

+

\\[4pt]

+

\texttt{((} \check{f} \texttt{,~(u)~v~~))}

+

\\[4pt]

+

\texttt{((} \check{f} \texttt{,~~u~(v)~))}

+

\\[4pt]

+

\texttt{((} \check{f} \texttt{,~~u~~v~~))}

+

\end{array}</math>

+

| align="left" |

+

<math>\begin{array}{l}

+

\check{f} + u + v + uv

+

\\[4pt]

+

\check{f} + v + uv + 1

+

\\[4pt]

+

\check{f} + u + uv + 1

+

\\[4pt]

+

\check{f} + uv + 1

+

\end{array}\!</math>

|-

−

~~|     (v) ~~

−

|-

−

~~|  (u, v) ~~

−

|-

−

~~|  (u  v) ~~

−

|}

|

−

{~~| cellpadding="2"~~

+

<math>\begin{matrix}

−

~~| (( f ,    ()    ))~~

+

f_{3}

−

|-

+

\\[4pt]

−

~~| (( f ,  (u)(v)  ))~~

+

f_{12}

−

|-

+

\end{matrix}\!</math>

−

| ~~(( f ,  (u) v   ))~~

+

|

−

|-

+

<math>\begin{matrix}

−

~~| (( f ,  (u)     ))~~

+

0~0~1~1

−

|-

+

\\[4pt]

−

~~| (( f ,   u (v)  ))~~

+

1~1~0~0

−

|-

+

\end{matrix}\!</math>

−

~~| (( f ,     (v)  ))~~

−

|-

−

~~| (( f ,  (u, v)  ))~~

−

|-

−

~~| (( f ,  (u  v)  ))~~

−

|}

|

−

{| align="left" ~~cellpadding="2" style="text-align:left"~~

+

<math>\begin{matrix}

−

| ~~ f + 1~~

+

\texttt{(} u \texttt{)}

−

|-

+

\\[4pt]

−

~~|  ~~f + u ~~+ v + uv~~

+

\texttt{~} u \texttt{~}

−

|-

+

\end{matrix}\!</math>

−

~~|  ~~f ~~+ v + uv + 1~~

+

| align="left" |

−

|-

+

<math>\begin{array}{l}

−

| ~~ f + u~~

+

\texttt{((} \check{f} \texttt{,~(u)~))}

−

|-

+

\\[4pt]

−

~~|  ~~f + u ~~+ uv + 1~~

+

\texttt{((} \check{f} \texttt{,~~u~~))}

−

|-

+

\end{array}\!</math>

−

~~|  f + v~~

+

| align="left" |

−

|-

+

<math>\begin{array}{l}

−

~~|  ~~f + u ~~+ v~~ + 1

+

\check{f} + u

−

|-

+

\\[4pt]

−

~~|  f + uv~~

+

\check{f} + u + 1

−

|}

+

\end{array}\!</math>

|-

|

−

{~~| cellpadding="2"~~

+

<math>\begin{matrix}

−

~~| f8~~

+

f_{6}

−

|-

+

\\[4pt]

−

~~| f~~9~~~~

+

f_{9}

−

|-

+

\end{matrix}\!</math>

−

~~| f10~~</~~sub~~>

−

|-

−

~~| f11~~

−

|-

−

~~| f12~~

−

|-

−

~~| f13~~

−

|-

−

~~| f14~~

−

|-

−

~~| f15~~

−

|}

|

−

{~~| cellpadding="2"~~

+

<math>\begin{matrix}

−

~~| 1000~~

+

0~1~1~0

−

|-

+

\\[4pt]

−

~~| 1001~~

+

1~0~0~1

−

|-

+

\end{matrix}\!</math>

−

~~| 1010~~

−

|-

−

~~| 1011~~

−

|-

−

~~| 1100~~

−

|-

−

~~| 1101~~

−

|-

−

~~| 1110~~

−

|-

−

~~| 1111~~

−

|}

|

−

{~~| cellpadding="2"~~

+

<math>\begin{matrix}

−

~~|   ~~u~~  ~~v~~  ~~

+

\texttt{(} u \texttt{,} v \texttt{)}

−

|-

+

\\[4pt]

−

| ((u,~~ ~~v))

+

\texttt{((} u \texttt{,} v \texttt{))}

−

|-

+

\end{matrix}\!</math>

−

~~|      ~~v~~  ~~

+

| align="left" |

−

|-

+

<math>\begin{array}{l}

−

~~|  ~~(u~~ (~~v))

+

\texttt{((} \check{f} \texttt{,~~(} u \texttt{,} v \texttt{)~~))}

−

|-

+

\\[4pt]

−

| ~~  u     ~~

+

\texttt{((} \check{f} \texttt{,~((} u \texttt{,} v \texttt{))~))}

−

|-

+

\end{array}\!</math>

−

~~| ((~~u~~) ~~v~~) ~~

+

| align="left" |

−

|-

+

<math>\begin{array}{l}

−

~~| ((~~u)(v))

+

\check{f} + u + v + 1

+

\\[4pt]

+

\check{f} + u + v

+

\end{array}\!</math>

|-

−

~~| (())~~

−

|}

|

−

{~~| cellpadding="2"~~

+

<math>\begin{matrix}

−

~~| (( f ,   u  v   ))~~

+

f_{5}

−

|-

+

\\[4pt]

−

~~| (( f , ((u, v)) ))~~

+

f_{10}

−

|-

+

\end{matrix}\!</math>

−

~~| (( f ,      v   ))~~

−

|-

−

~~| (( f ,  (u (v)) ))~~

−

|-

−

~~| (( f ,   u      ))~~

−

|-

−

~~| (( f , ((u) v)  ))~~

−

|-

−

~~| (( f , ((u)(v)) ))~~

−

|-

−

~~| (( f ,   (())   ))~~

−

|}

|

−

{| align="left" ~~cellpadding~~="~~2" style="text-align:~~left"

+

<math>\begin{matrix}

−

~~|  ~~f + uv + 1

+

0~1~0~1

+

\\[4pt]

+

1~0~1~0

+

\end{matrix}\!</math>

+

|

+

<math>\begin{matrix}

+

\texttt{(} v \texttt{)}

+

\\[4pt]

+

\texttt{~} v \texttt{~}

+

\end{matrix}</math>

+

| align="left" |

+

<math>\begin{array}{l}

+

\texttt{((} \check{f} \texttt{,~(} v \texttt{)~))}

+

\\[4pt]

+

\texttt{((} \check{f} \texttt{,~~} v \texttt{~~))}

+

\end{array}\!</math>

+

| align="left" |

+

<math>\begin{array}{l}

+

\check{f} + v

+

\\[4pt]

+

\check{f} + v + 1

+

\end{array}\!</math>

|-

−

| ~~ ~~f + u + v

+

|

+

<math>\begin{matrix}

+

f_{7}

+

\\[4pt]

+

f_{11}

+

\\[4pt]

+

f_{13}

+

\\[4pt]

+

f_{14}

+

\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}

+

\texttt{(~} u \texttt{~~} v \texttt{~)}

+

\\[4pt]

+

\texttt{(~} u \texttt{~(} v \texttt{))}

+

\\[4pt]

+

\texttt{((} u \texttt{)~} v \texttt{~)}

+

\\[4pt]

+

\texttt{((} u \texttt{)(} v \texttt{))}

+

\end{matrix}\!</math>

+

| align="left" |

+

<math>\begin{array}{l}

+

\texttt{((} \check{f} \texttt{,~(~} u \texttt{~~} v \texttt{~)~))}

+

\\[4pt]

+

\texttt{((} \check{f} \texttt{,~(~} u \texttt{~(} v \texttt{))~))}

+

\\[4pt]

+

\texttt{((} \check{f} \texttt{,~((} u \texttt{)~} v \texttt{~)~))}

+

\\[4pt]

+

\texttt{((} \check{f} \texttt{,~((} u \texttt{)(} v \texttt{))~))}

+

\end{array}\!</math>

+

| align="left" |

+

<math>\begin{array}{l}

+

\check{f} + uv

+

\\[4pt]

+

\check{f} + u + uv

+

\\[4pt]

+

\check{f} + v + uv

+

\\[4pt]

+

\check{f} + u + v + uv + 1

+

\end{array}\!</math>

|-

−

| ~~ f + v + 1~~

+

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

−

|-

+

| <math>1~1~1~1\!</math>

−

~~|  f + u + uv~~

+

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

−

|-

+

| align="left" | <math>\texttt{((} \check{f} \texttt{,~((~))~))}\!</math>

−

~~|  f + u +~~ 1

+

| align="left" | <math>\check{f}\!</math>

−

|-

−

| ~~ f + v + uv~~

−

|-

−

~~|  ~~f ~~+ u + v + uv + 1~~

−

|-

−

| ~~ ~~f

|}

−

|}

−

~~ ~~

−

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'' ~~:~~ '''~~B~~'''~~2</~~sup~~>~~ → '''B'''~~ and for the corresponding thematizations ~~θ''f''~~<~~sub~~>~~''i'' ~~=~~ φ''i'' ~~:~~ '''~~B~~'''~~3</~~sup~~>~~ → '''B'''.~~

+

+

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 <math>f_i : \mathbb{B}^2 \to \mathbb{B}\!</math> and for the corresponding thematizations <math>\theta f_i = \varphi_i : \mathbb{B}^3 \to \mathbb{B}.\!</math>

−

{| align="center~~" border="1~~" cellpadding="2" cellspacing="0" style="~~font~~-~~weight~~:~~bold~~; text-align:center; width:96%"

+

−

|+ Table 28. Propositions on Two Variables

+

{| 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:75%"

−

|

+

|+ style="height:30px" | <math>\text{Table 28.} ~~ \text{Propositions on Two Variables}\!</math>

−

~~{| align~~="~~center" border="0" cellpadding="4" cellspacing="0~~" style="~~font-weight~~:~~bold~~; ~~text~~-~~align~~:~~center; width:100%~~"

+

|- style="height:35px; background:ghostwhite"

−

|- style="~~background~~:~~ghostwhite~~"

+

| style="width:5%; border-bottom:1px solid black" | <math>u\!</math>

−

~~| u |~~| v |~~| &nbsp~~;

+

| style="width:5%; border-bottom:1px solid black" | <math>v\!</math>

−

|f<~~sub~~>00</~~sub~~>||f<~~sub~~>01</~~sub~~>||f<~~sub~~>02</~~sub~~>||f<~~sub~~>03</~~sub~~>

+

| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>f_{0}\!</math>

−

|f<~~sub~~>04</~~sub~~>||f<~~sub~~>05</~~sub~~>||f<~~sub~~>06</~~sub~~>||f<~~sub~~>07</~~sub~~>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{1}\!</math>

−

|f<~~sub~~>08</~~sub~~>||f<~~sub~~>09</~~sub~~>||f<~~sub~~>10</~~sub~~>||f<~~sub~~>11</~~sub~~>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{2}\!</math>

−

|f<~~sub~~>12</~~sub~~>||f<~~sub~~>13</~~sub~~>||f<~~sub~~>14</~~sub~~>||f<~~sub~~>15</~~sub~~>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{3}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{4}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{5}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{6}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{7}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{8}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{9}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{10}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{11}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{12}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{13}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{14}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>f_{15}\!</math>

|-

−

| 0 || 0 ||  

+

| <math>0\!</math>

−

|~~0||~~1||~~0||~~1||~~0||~~1||~~0||~~1||~~0||~~1||~~0||~~1||~~0||~~1||~~0||~~1

+

| <math>0\!</math>

+

| style="border-left:1px solid black" |  

+

| <math>1\!</math>

+

|  

+

| <math>1\!</math>

+

|  

+

| <math>1\!</math>

+

|  

+

| <math>1\!</math>

+

|  

+

| <math>1\!</math>

+

|  

+

| <math>1\!</math>

+

|  

+

| <math>1\!</math>

+

|  

+

| <math>1\!</math>

|-

−

| 0 || 1 ||  

+

| <math>0\!</math>

−

|~~0||0|~~|1||1||~~0||0|~~|1||1||~~0||0|~~|1||1||~~0||0|~~|1||1

+

| <math>1\!</math>

+

| style="border-left:1px solid black" |  

+

|  

+

| <math>1\!</math>

+

| <math>1\!</math>

+

|  

+

|  

+

| <math>1\!</math>

+

| <math>1\!</math>

+

|  

+

|  

+

| <math>1\!</math>

+

| <math>1\!</math>

+

|  

+

|  

+

| <math>1\!</math>

+

| <math>1\!</math>

|-

−

| 1 || 0 ||  

+

| <math>1\!</math>

−

|0||~~0||0||0|~~|1||1||1||1||0||~~0||0||0|~~|1||1||1||1

+

| <math>0\!</math>

+

| style="border-left:1px solid black" |  

+

|  

+

|  

+

|  

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

|  

+

|  

+

|  

+

|  

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

|-

−

| 1 || 1 ||  

+

| <math>1\!</math>

−

|0||0||0||~~0||0||0||0||0|~~|1||1||1||1||1||1||1||1

+

| <math>1\!</math>

−

|}

+

| style="border-left:1px solid black" |  

+

|  

+

|  

+

|  

+

|  

+

|  

+

|  

+

|  

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

|}

−

~~ ~~

−

~~{| align="center" border="1" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:96%"~~

−

~~|+ Table 29. Thematic Extensions of Bivariate Propositions~~

−

|

−

~~{| align="center" border="0" cellpadding="2" cellspacing="0" style="font-weight:bold; text-align:center; width:100%"~~

−

~~|- style="background:ghostwhite"~~

−

~~| u || v || f¢~~

−

~~| φ00 || φ01~~

−

~~| φ02 || φ03~~

−

~~| φ04 || φ05~~

−

~~| φ06 || φ07~~

−

~~| φ08 || φ09~~

−

~~| φ10 || φ11~~

−

~~| φ12 || φ13~~

−

~~| φ14 || φ15~~

−

|-

−

~~| 0 || 0 || 0 ||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0~~

−

|-

−

~~| 0 || 0 || 1 ||0||1||0||1||0||1||0||1||0||1||0||1||0||1||0||1~~

−

|-

−

~~| 0 || 1 || 0 ||1||1||0||0||1||1||0||0||1||1||0||0||1||1||0||0~~

−

|-

−

~~| 0 || 1 || 1 ||0||0||1||1||0||0||1||1||0||0||1||1||0||0||1||1~~

−

|-

−

~~| 1 || 0 || 0 ||1||1||1||1||0||0||0||0||1||1||1||1||0||0||0||0~~

−

|-

−

~~| 1 || 0 || 1 ||0||0||0||0||1||1||1||1||0||0||0||0||1||1||1||1~~

−

|-

−

~~| 1 || 1 || 0 ||1||1||1||1||1||1||1||1||0||0||0||0||0||0||0||0~~

−

|-

−

~~| 1 || 1 || 1 ||0||0||0||0||0||0||0||0||1||1||1||1||1||1||1||1~~

−

|}

−

|}

−

===~~Propositional Transformations~~===

+

{| 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:75%"

−

+

|+ style="height:30px" | <math>\text{Table 29.} ~~ \text{Thematic Extensions of Bivariate Propositions}\!</math>

−

{| width="~~100~~%" ~~cellpadding~~="0" ~~cellspacing~~="0"

+

|- style="height:35px; background:ghostwhite"

−

| width="4%" | ~~&nbsp~~;

+

| style="width:5%; border-bottom:1px solid black" | <math>u\!</math>

−

| width="92%" |

+

| style="width:5%; border-bottom:1px solid black" | <math>v\!</math>

−

~~If only the word &lsquo~~;~~artificial&rsquo~~; 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.

+

| style="width:5%; border-bottom:1px solid black" | <math>\check{f}\!</math>

−

| width="4%" | ~~&nbsp~~;

+

| style="width:5%; border-bottom:1px solid black; border-left:1px solid black" | <math>\varphi_{0}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{1}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{2}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{3}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{4}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{5}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{6}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{7}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{8}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{9}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{10}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{11}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{12}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{13}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{14}\!</math>

+

| style="width:5%; border-bottom:1px solid black" | <math>\varphi_{15}\!</math>

|-

−

| ~~align~~="~~right" colspan="3~~" | &~~mdash~~; ~~John Dewey, ''How We Think'', [Dew, 56~~&~~ndash~~;~~57]~~

+

| <math>0\!</math>

−

|}

+

| <math>0\!</math>

−

+

| <math>0\!</math>

−

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.

+

| style="border-left:1px solid black" | <math>1\!</math>

−

+

|  

−

~~====Alias and Alibi Transformations====~~

+

| <math>1\!</math>

−

+

|  

−

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:

+

| <math>1\!</math>

−

+

|  

−

~~# 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.~~

+

| <math>1\!</math>

−

~~# 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.~~

+

|  

−

+

| <math>1\!</math>

−

~~(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].)~~

+

|  

−

+

| <math>1\!</math>

−

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.

+

|  

−

+

| <math>1\!</math>

−

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.

+

|  

−

+

| <math>1\!</math>

−

~~====Transformations of General Type====~~

+

|  

−

{| ~~width="100%" cellpadding="0" cellspacing="0"~~

−

~~| width="4%"~~ |  

−

| ~~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%"~~ |  

|-

−

| ~~align~~="~~right" colspan="3~~" | &~~mdash~~; ~~Robert Musil, ''The Man Without Qualities'', [Mus, 34]~~

+

| <math>0\!</math>

−

|}

+

| <math>0\!</math>

−

+

| <math>1\!</math>

−

~~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''].~~

+

| style="border-left:1px solid black" |  

−

+

| <math>1\!</math>

−

{| ~~align="center" border="0" cellpadding="10"~~

+

|  

−

|

+

| <math>1\!</math>

−

<~~pre~~>

+

|  

−

~~o-------------------------------------------------------o~~

+

| <math>1\!</math>

−

~~| U~~ |

+

|  

−

| |

+

| <math>1\!</math>

−

~~| o-----------o o-----------o |~~

+

|  

−

~~| /~~ \ / ~~\ |~~

+

| <math>1\!</math>

−

~~| / o \~~ |

+

|  

−

| ~~/ /~~ \ ~~\ |~~

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| <math>1\!</math>

−

| / ~~/ \ \ |~~

+

|  

−

~~| o o o o~~ |

+

| <math>1\!</math>

−

| ~~| | | | |~~

+

|  

−

~~| | u | | v | |~~

+

| <math>1\!</math>

−

~~| | | | | |~~

+

|-

−

~~| o o o o |~~

+

| <math>0\!</math>

−

| \ \ / / |

+

| <math>1\!</math>

−

| \ \ / ~~/ |~~

+

| <math>0\!</math>

−

| \ o / |

+

| style="border-left:1px solid black" | <math>1\!</math>

−

| \ / ~~\ / |~~

+

| <math>1\!</math>

−

| o-~~----------o o-----------o~~ |

+

|  

−

~~| |~~

+

|  

−

~~| |~~

+

| <math>1\!</math>

−

~~o---------------------------o---------------------------o~~

+

| <math>1\!</math>

−

/ \ / ~~\ / \~~

+

|  

−

/ \ / ~~\ / \~~

+

|  

−

~~/ \ / \ / \~~

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| <math>1\!</math>

−

~~/ \ / \ / \~~

+

| <math>1\!</math>

−

/ \ / ~~\ / \~~

+

|  

−

/ \ / ~~\ / \~~

+

|  

−

~~/ \ / \ / \~~

+

| <math>1\!</math>

−

~~/ \ / \ / \~~

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| <math>1\!</math>

−

~~/ \ / \ / \~~

+

|  

−

~~/ \ / \ / \~~

+

|  

−

~~/ \ / \ / \~~

+

|-

−

~~/ \ / \ / \~~

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| <math>0\!</math>

−

~~o-------------------------o o-------------------------o o-------------------------o~~

+

| <math>1\!</math>

−

| ~~U | | U | | U |~~

+

| <math>1\!</math>

−

+

| style="border-left:1px solid black" |  

−

| / \ / \ | ~~| /~~ \ / \ | | ~~/ \ / \ |~~

+

|  

−

| ~~/ o~~ \ ~~| |~~ / ~~o \ | | / o \ |~~

+

| <math>1\!</math>

−

| ~~/ /~~ \ ~~\ | |~~ / ~~/ \ \ | | / / \ \ |~~

+

| <math>1\!</math>

−

+

|  

−

| ~~| u | | v | | | | u | | v | | | | u | | v | |~~

+

|  

−

+

| <math>1\!</math>

−

| \ \ / ~~/ | | \ \ / / | | \ \ / / |~~

+

| <math>1\!</math>

−

| \ o / ~~| | \ o / | | \ o / |~~

+

|  

−

| \ / ~~\ / | | \ / \ / | | \ / \ / |~~

+

|  

−

+

| <math>1\!</math>

−

| ~~| | | | |~~

+

| <math>1\!</math>

−

~~o-------------------------o o-------------------------o o-------------------------o~~

+

|  

−

\ | \ ~~/ |~~ /

+

|  

−

\ | \ / | /

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| <math>1\!</math>

−

\ | ~~\ / | /~~

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| <math>1\!</math>

−

\ | \ ~~/ |~~ /

+

|-

−

~~\ g~~ | \ f / | ~~h /~~

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| <math>1\!</math>

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\ | ~~\ / | /~~

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| <math>0\!</math>

−

\ | \ ~~/ |~~ /

+

| <math>0\!</math>

−

\ | \ ~~/ |~~ /

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| style="border-left:1px solid black" | <math>1\!</math>

−

\ | ~~\ / | /~~

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| <math>1\!</math>

−

~~\ o----------~~|~~-----------\-----/-----------|----------o /~~

+

| <math>1\!</math>

−

\ | ~~X |~~ \ ~~/ | |~~ /

+

| <math>1\!</math>

−

\ | | \ / ~~| | /~~

+

|  

−

\ | ~~| o~~-~~----o-----o | | /~~

+

|  

−

\| ~~| /~~ \ ~~| |~~/

+

|  

−

\ | / \ | /

+

|  

−

|\ | / ~~\ | /|~~

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| <math>1\!</math>

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| \ | / \ | / |

+

| <math>1\!</math>

−

| \ | / ~~\ | / |~~

+

| <math>1\!</math>

−

| \ ~~| o x o |~~ / |

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| <math>1\!</math>

−

| \ ~~| | | |~~ / |

+

|  

−

| \ | ~~| | | /~~ |

+

|  

−

| \ ~~| | | |~~ / |

+

|  

−

| \ ~~| | | |~~ / |

+

|  

−

| \ ~~| | | |~~ / |

+

|-

−

| \~~| | | |~~/ |

+

| <math>1\!</math>

−

| ~~o--o--------o o--------o--o~~ |

+

| <math>0\!</math>

−

~~| / \ \ / / \~~ |

+

| <math>1\!</math>

−

~~| / \ \ / / \~~ |

+

| style="border-left:1px solid black" |  

−

| / \ o / ~~\ |~~

+

|  

−

| / \ / ~~\ / \ |~~

+

|  

−

| / \ / ~~\ / \ |~~

+

|  

−

| ~~o o~~-~~-o-----o--o o~~ |

+

| <math>1\!</math>

−

~~| | | | |~~ |

+

| <math>1\!</math>

−

~~| | | | |~~ |

+

| <math>1\!</math>

−

~~| | | | |~~ |

+

| <math>1\!</math>

−

| ~~| y | | z | |~~

+

|  

−

~~| | | | | |~~

+

|  

−

~~| | | | | |~~

+

|  

−

~~| o o o o |~~

+

|  

−

| \ \ / ~~/ |~~

+

| <math>1\!</math>

−

| \ \ / ~~/ |~~

+

| <math>1\!</math>

−

| \ o / |

+

| <math>1\!</math>

−

| \ / ~~\ /~~ |

+

| <math>1\!</math>

−

~~| \ / \ /~~ |

+

|-

−

~~| o-----------o o-----------o~~ |

+

| <math>1\!</math>

−

| |

+

| <math>1\!</math>

−

| |

+

| <math>0\!</math>

−

~~o---------------------------------------------------o~~

+

| style="border-left:1px solid black" | <math>1\!</math>

−

\ /

+

| <math>1\!</math>

−

\ /

+

| <math>1\!</math>

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\ /

+

| <math>1\!</math>

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\ /

+

| <math>1\!</math>

−

~~\ /~~

+

| <math>1\!</math>

−

\ ~~p , q~~ /

+

| <math>1\!</math>

−

\ /

+

| <math>1\!</math>

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\ /

+

|  

−

~~\ /~~

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|  

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~~\ /~~

+

|  

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~~\ /~~

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|  

−

~~\ /~~

+

|  

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\ /

+

|  

−

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.

−

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~~>.

−

<~~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''›~~ ||  

+

| <math>1\!</math>

−

|-

+

| <math>1\!</math>

−

|   || ~~''z''~~ || = || ~~''h''‹''u'', ''v''›~~ || ~~ ~~

+

| <math>1\!</math>

−

|}

+

| style="border-left:1px solid black" |  

−

|}

+

|  

−

<~~/font~~>

+

|  

+

|  

+

|  

+

|  

+

|  

+

|  

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

| <math>1\!</math>

+

|}

−

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~~===

+

===Propositional Transformations===

{| width="100%" cellpadding="0" cellspacing="0"

| width="4%" |  

| width="92%" |

−

~~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~~.

+

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%" |  

|-

−

| align="right" colspan="3" | — ~~C.S. Peirce~~, ~~"The Maxim of Pragmatism"~~, ~~CP 5.438~~

+

| align="right" colspan="3" | — John Dewey, ''How We Think'', [Dew, 56–57]

|}

−

~~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~~.

+

In this section we develop a comprehensive set of concepts for dealing with transformations between universes of discourse. In this most general setting the source and target universes of a transformation are allowed to be different, but may be the same. When we apply these concepts to dynamic systems we focus on the important special case of transformations that map a universe into itself, regarding them as the state transitions of a discrete dynamical process and placing them among the myriad ways that a universe of discourse might change, and by that change turn into itself.

+

====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.

+

# 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].)

−

~~====Operators on Propositions~~ and ~~Transformations====~~

+

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.

−

~~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~~.

+

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.

−

~~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.

+

====Transformations of General Type====

−

{| align="center" border="0" cellpadding="20"

+

{| width="100%" cellpadding="0" cellspacing="0"

+

| width="4%" |  

+

| 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%" |  

+

|-

+

| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 34]

+

|}

+

Consider the situation illustrated in Figure 30, where the alphabets <math>\mathcal{U} = \{ u, v \}\!</math> and <math>\mathcal{X} = \{ x, y, z \}\!</math> are used to label basic features in two different logical universes, <math>U^\bullet = [u, v]\!</math> and <math>X^\bullet = [x, y, z].\!</math>

+

{| align="center" border="0" cellpadding="10"

|

<pre>

−

o-------------------------------------~~--o~~

+

o-------------------------------------------------------o

−

~~| |~~

+

| U |

−

~~| |~~

+

| |

−

~~| U% F X% |~~

+

| o-----------o o-----------o |

−

~~| o~~------------------>o |

+

| / \ / \ |

−

~~| | |~~ |

+

| / o \ |

−

| ~~| |~~ |

+

| / / \ \ |

−

| ~~| | |~~

+

| / / \ \ |

−

| ~~| |~~ |

+

| o o o o |

−

| ~~!W!~~ | | ~~!W!~~ |

+

| | | | | |

−

| | | |

+

| | u | | v | |

−

| | | |

+

| | | | | |

−

| | | |

+

| o o o o |

−

| ~~v v~~ |

+

| \ \ / / |

−

| ~~o------------------>o~~ |

+

| \ \ / / |

−

| ~~!W!U% !W!F !W!X%~~ |

+

| \ o / |

−

| |

+

| \ / \ / |

−

| |

+

| o-----------o o-----------o |

−

~~o--~~-------------------------------------o

+

| |

−

~~Figure 31. Operator Diagram (1)~~

+

| |

−

~~</pre>~~

+

o---------------------------o---------------------------o

−

|}

+

/ \ / \ / \

−

+

/ \ / \ / \

−

~~In this Figure "'''W'''<~~/~~font>" serves as a generic name for an operator, in this case one that takes a logical transformation ''F'' of type (''U'' •<~~/~~sup> → ''X'' •<~~/~~sup>) into a logical transformation '''W'''<~~/~~font>''F'' of type ('''W'''<~~/~~font>''U'' •<~~/~~sup> → '''W'''<~~/~~font>''X'' •<~~/~~sup>). Thus, the operator '''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> and ''X'' •<~~/~~sup> and for logical transformations like ''F''.~~

+

/ \ / \ / \

−

+

/ \ / \ / \

−

~~NB. Strictly speaking, an operator like '''W'''<~~/~~font> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of '''W'''<~~/~~font>. Given this setting, '''W'''<~~/~~font> specifies for each universe ''U'' •<~~/~~sup> in its source category a definite universe '''W'''<~~/~~font>''U'' •<~~/~~sup> in its target category, and to each transformation ''F'' in its source category it assigns a unique transformation '''W'''<~~/~~font>''F'' in its target category. Naturally, this only works if '''W'''<~~/~~font> takes the source ''U'' •<~~/~~sup> and the target ''X'' •<~~/~~sup> of the map F over to the source '''W'''<~~/~~font>''U'' •<~~/~~sup> and the target '''W'''<~~/~~font>''X'' •<~~/~~sup> of the map '''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 '''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 "'''W'''<~~/~~font>''U'' •<~~/~~sup>" and "'''W'''<~~/font>''X'' •") that '''W''' 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 '''W''' what universes they are.

+

/ \ / \ / \

−

+

/ \ / \ / \

−

In Figure 31 the maps ''F'' and '''W'''''F'' are displayed horizontally, the way that one normally orients functional arrows in a written text, and '''W''' rolls the map ''F'' downward into the images that are associated with '''W'''''F''. In Figure 32 the same information is redrawn so that the maps ''F'' and '''W'''''F'' flow down the page, and '''W''' unfurls the map ''F'' rightward into domains that are the eminent purview of '''W'''''F''.

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/ \ / \ / \

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/ \ / \ / \

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~~{| align="center" border="0" cellpadding="20"~~

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/ \ / \ / \

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|

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/ \ / \ / \

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~~<pre>~~

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/ \ / \ / \

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~~o----~~-----------------------------------o

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/ \ / \ / \

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| |

+

o-------------------------o o-------------------------o o-------------------------o

−

~~| |~~

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| U | | U | | U |

−

~~| U% !W! !W!U% |~~

+

−

| o------~~------------>o~~ |

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| / \ / \ | | / \ / \ | | / \ / \ |

−

| | | |

+

| / o \ | | / o \ | | / o \ |

−

| | | |

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| / / \ \ | | / / \ \ | | / / \ \ |

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| | | |

+

−

| | | |

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| | u | | v | | | | u | | v | | | | u | | v | |

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| F | ~~| !W!F~~ |

+

−

| | | |

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| \ \ / / | | \ \ / / | | \ \ / / |

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| ~~| |~~ |

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| \ o / | | \ o / | | \ o / |

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| | | |

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| \ / \ / | | \ / \ / | | \ / \ / |

−

| v v |

+

−

| ~~o------------------>o~~ |

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| | | | | |

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| ~~X% !W! !W!X%~~ |

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o-------------------------o o-------------------------o o-------------------------o

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| |

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\ | \ / | /

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| |

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\ | \ / | /

−

o~~---------------------------------------o~~

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\ | \ / | /

−

~~Figure 32. Operator Diagram (2)~~

+

\ | \ / | /

−

</~~pre>~~

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\ g | \ f / | h /

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|}

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\ | \ / | /

−

+

\ | \ / | /

−

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====~~

+

\ o----------|-----------\-----/-----------|----------o /

−

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\ | X | \ / | | /

−

{| ~~width="100%" cellpadding="0" cellspacing="0"~~

+

\ | | \ / | | /

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| ~~width="4%" |  ~~

+

\ | | o-----o-----o | | /

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| ~~width="92%"~~ | ~~The resultant metaphysical problem now is this: ''Does the man go round the squirrel or not?''~~

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\| | / \ | |/

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| ~~width="4%"~~ | ~~ ~~

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\ | / \ | /

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|-

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|\ | / \ | /|

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| ~~align="right" colspan="3"~~ | ~~— William James, ''Pragmatism'', [Jam, 43]~~

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| \ | / \ | / |

−

|}

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| \ | / \ | / |

−

+

| \ | o x o | / |

−

~~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 '''W'''<~~/~~font> that act on propositions ''F'' or on transformations ''F'' to yield the corresponding operator maps '''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 '''W''' : (''U'' • → ''X'' •) → (E''U'' • → E''X'' •). If we assume that the source universe ''U'' • and the target universe ''X'' • have finite dimensions ''n'' and ''k'', respectively, then each operator '''W''' is encompassed by the same

+

| \ | | | | / |

−

~~abstract type:~~

+

| \| | | |/ |

−

+

| o--o--------o o--------o--o |

−

:{| ~~cellpadding=1 style="height:40px"~~

+

| / \ \ / / \ |

−

| ~~'''W'''<~~/~~font>~~

+

| / \ \ / / \ |

−

| :

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| / \ o / \ |

−

| (

+

| / \ / \ / \ |

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| [

+

| / \ / \ / \ |

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~~| '''B'''''n''<~~/~~sup>~~

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| o o--o-----o--o o |

−

| ]

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| | | | | |

−

| ~~→~~

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| | | | | |

−

| [

+

| | | | | |

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| ~~'''B'''''k''<~~/~~sup>~~

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| | y | | z | |

−

| ]

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| | | | | |

−

| )

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| | | | | |

−

| ~~ ~~

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| o o o o |

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| ~~→~~

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| \ \ / / |

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| ~~ ~~

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| \ \ / / |

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| (

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| \ o / |

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| [

+

| \ / \ / |

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| ~~'''B'''''n''<~~/~~sup>~~

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| \ / \ / |

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| ~~×~~

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| o-----------o o-----------o |

−

~~| '''D'''''n''<~~/~~sup>~~

+

| |

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~~| ]~~

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| |

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~~| →~~

+

o---------------------------------------------------o

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| [

+

\ /

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~~| '''B'''''k''~~

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\ /

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~~| ×~~

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\ /

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~~| '''D'''''k''<~~/~~sup>~~

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\ /

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| ]

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\ /

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| )

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\ p , q /

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| .

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\ /

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|}

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\ /

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\ /

−

~~Since the range features of the operator result '''W'''<~~/~~font>''F'' : ['''B'''''n''<~~/~~sup> × '''D'''''n''<~~/~~sup>] → ['''B'''''k''<~~/~~sup> × '''D'''''k''<~~/~~sup>] can be sorted out by their ordinary versus their differential qualities and the component maps can be examined independently, the complete operator '''W'''<~~/~~font> can be separated accordingly into two components, in the form '''W''' = ‹<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%"~~

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\ /

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| ~~align=left width=20%| Concrete type~~

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\ /

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| ~~width=8% | <math>~~\~~epsilon<~~/~~math>~~

+

o

−

| :

+

−

| (

+

Figure 30. Generic Frame of a Logical Transformation

−

| ~~''U'' •~~

+

</pre>

−

~~| →~~

−

| ~~''X'' •<~~/~~sup>~~

−

| )

−

| ~~width=16%~~ | ~~→~~

−

| (

−

| ~~E''U'' •~~

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| ~~→~~

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| ~~''X'' •<~~/~~sup>~~

−

| )

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|-

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| ~~align=left width=20%~~| ~~Abstract type~~

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| ~~width=8%~~ | ~~<math>\epsilon<~~/~~math>~~

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| :

−

| (

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| ~~['''B'''''n'']~~

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| ~~→~~

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| ~~['''B'''''k''<~~/~~sup>]~~

−

| )

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~~| width=16% | →~~

−

~~| (~~

−

~~| ['''B'''''n'' × '''D'''''n''</sup~~>]

−

~~| →~~

−

~~| ['''B'''''k'']~~

−

~~| )~~

|}

−

On the ~~other hand~~, the ~~operator W is specific to each~~ <~~font face=georgia~~>~~'''W'''~~</~~font~~>. In this ~~context W always has~~ the ~~form~~:

+

Enter the picture, as we usually do, in the middle of things, with features like <math>x, y , z\!</math> that present themselves to be simple enough in their own right and that form a satisfactory, if 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 <math>p, q : X \to \mathbb{B}.\!</math> Then we discover that the simple features <math>\{ x, y, z \}\!</math> are really more complex than we thought at first, and it becomes useful to regard them as functions <math>\{ f, g, h \}\!</math> of other features <math>\{ u, v \}\!</math> that we place in a preface to our original discourse, or suppose as topics of a preliminary universe of discourse <math>U^\bullet = [u, v].\!</math> 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.

−

~~:{| style="height:80px; text-align:center; width:90%"~~

+

A particular transformation <math>F : [u, v] \to [x, y, z]\!</math> may be expressed by a system of equations, as shown below. Here, <math>F\!</math> is defined by its component maps <math>F = (F_1, F_2, F_3) = (f, g, h),\!</math> where each component map in <math>\{ f, g, h \}\!</math> is a proposition of type <math>\mathbb{B}^n \to \mathbb{B}^1.\!</math>

−

~~| align=left width=20%| Concrete type~~

−

~~| width=8% | W~~

−

~~| :~~

−

~~| (~~

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~~| ''U''~~<~~sup~~>~~ •~~

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~~| →~~

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~~| ''X'' •~~</~~sup~~>

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~~| )~~

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~~| width=16% | →~~

−

~~| (~~

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~~| E''U''~~<~~sup~~>~~ •~~</~~sup>~~

−

~~| →~~

−

~~| d''X''<sup~~>~~ •~~<~~/sup~~>

−

~~| )~~

−

|-

−

~~| align~~=~~left width=20%| Abstract type~~

−

~~| width=8% | W~~

−

~~| :~~

−

| (

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~~| ['''B'''''n'']~~

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~~| →~~

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~~| ['''B'''''k'']~~

−

| )

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~~| width~~=~~16% | →~~

−

| (

−

~~| ['''B'''''n''~~</~~sup~~> ~~× '''D'''~~<~~sup~~>~~''n''~~</~~sup~~>]

−

~~| →~~

−

~~| ['''D'''~~<~~sup~~>~~''k''~~</~~sup~~>]

−

~~| )~~

−

|}

−

~~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%"

+

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:50%"

−

| ~~width=6% |~~ <math>\~~epsilon</math>''F''~~

+

|

−

~~| width=2% | :~~

+

<math>\begin{matrix}

−

~~| width=2% | (~~

+

x & = & f(u, v)

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~~| width=8% | E''U'' •~~

+

\\[10pt]

−

~~| width=4% |~~ &~~rarr;~~

+

y & = & g(u, v)

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~~| width~~=~~8% | ''X''~~&~~nbsp;•~~

+

\\[10pt]

−

~~| width=4% | &sube;~~

+

z & = & h(u, v)

−

~~| width=8% | E''X'' •~~

+

\end{matrix}</math>

−

~~| width=2% |~~ )

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~~| width=4% | <math>~~\~~cong</math>~~

−

~~| width=2% | (~~

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~~| width=16% |~~ [~~'''B'''''n'' × '''D'''''n''~~]

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~~| width~~=~~4% |~~ &~~rarr;~~

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~~| width=8% |~~ [~~'''B'''''k''~~]

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~~| width=4% |~~ &~~sube;~~

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~~| width~~=~~16% | ['''B'''''k''~~ &~~times; '''D'''''k'']~~

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~~| width=2% | )~~

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|-

−

~~| width=6% | W''F''~~

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~~| width=2% | :~~

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~~| width=2% |~~ (

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~~| width=8% | E''U'' •~~

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~~| width=4% | →~~

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~~| width=8% | d''X'' •~~

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~~| width=4% | &sube;~~

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~~| width=8% | E''X'' •~~

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~~| width=2% |~~ )

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~~| width=4% | <math>~~\~~cong~~</math>

−

~~| width=2% | (~~

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~~| width=16% | ['''B'''''n'' × '''D'''''n'']~~

−

~~| width=4% | →~~

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~~| width=8% | ['''D'''''k'']~~

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~~| width=4% | &sube;~~

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~~| width=16% | ['''B'''''k'' × '''D'''''k'']~~

−

~~| width=2% | )~~

|}

−

~~Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.~~

+

−

~~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 '''W''' = ‹<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 '''W'''''F''~~ and W''F'' can be fixed in this frame of reference by tethering the operative adjective for '''W''' or W to the anchoring epithet ''map'', in conformity with an already standard practice.

+

Regarded as a logical statement, this system of equations expresses a relation between a collection of freely chosen propositions <math>\{ f, g, h \}\!</math> in one universe of discourse and the special collection of simple propositions <math>\{ x, y, z \}\!</math> on which is 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.

−

===~~==The Secant Operator~~ : ~~'''E'''==~~===

+

===Analytic Expansions : Operators and Functors===

{| width="100%" cellpadding="0" cellspacing="0"

| width="4%" |  

| 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~~.

+

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%" |  

|-

−

| align="right" colspan="3" | — ~~William James~~, ''Pragmatism'', ~~[Jam, 46]~~

+

| align="right" colspan="3" | — C.S. Peirce, “The Maxim of Pragmatism”, CP 5.438

|}

−

~~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 "'''E'''", which receives the principal investment of analytic attention, and on the constituent parts of '''E''', ~~which derive their shares of significance~~ as ~~developed~~ by the analysis. In the sequel, I refer to '''E''' 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 '''E''' ~~= ‹<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~~.)

+

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 explicitly considered in our discussion will be of this kind. Figure 31 illustrates the typical situation.

+

{| align="center" border="0" cellpadding="20"

+

|

<pre>

−

~~U% $E$~~ ~~$E$~~U% ~~$E$U% $E$U~~%

+

o---------------------------------------o

−

o------------------>o~~============o============o~~

+

| |

−

| | | |

+

| |

−

| | | |

+

| U% F X% |

−

| | | |

+

| o------------------>o |

−

| | | |

+

| | | |

−

F | | ~~$E$F =~~ | ~~$d$^0.F +~~ | ~~$r$^0.F~~

+

| | | |

−

| | | |

+

| | | |

−

| | | |

+

| | | |

−

| | | |

+

| !W! | | !W! |

−

v v ~~v v~~

+

| | | |

−

o------------------>o~~============~~o~~============~~o

+

| | | |

−

X~~% $E$ $E$~~X~~% $E$~~X~~% $E$~~X%

+

| | | |

+

| v v |

+

| o------------------>o |

+

| !W!U% !W!F !W!X% |

+

| |

+

| |

+

o---------------------------------------o

+

Figure 31. Operator Diagram (1)

+

</pre>

+

|}

+

In this Figure <math>{}^{\backprime\backprime} \mathsf{W} {}^{\prime\prime}\!</math> stands for a generic operator <math>\mathsf{W},\!</math> in this case one that takes a logical transformation <math>F\!</math> of type <math>(U^\bullet \to X^\bullet)\!</math> into a logical transformation <math>\mathsf{W}F\!</math> of type <math>(\mathsf{W}U^\bullet \to \mathsf{W}X^\bullet).\!</math> Thus, the operator <math>\mathsf{W}\!</math> must be viewed as making assignments for both families of objects we have previously considered, that is, for universes of discourse like <math>{U^\bullet}\!</math> and <math>{X^\bullet}\!</math> and for logical transformations like <math>F.\!</math>

+

'''Note.''' Strictly speaking, an operator like <math>\mathsf{W}\!</math> works between two whole categories of universes and transformations, which we call the ''source'' and the ''target'' categories of <math>\mathsf{W}.\!</math> Given this setting, <math>\mathsf{W}\!</math> specifies for each universe <math>U^\bullet\!</math> in its source category a definite universe <math>\mathsf{W}U^\bullet\!</math> in its target category, and to each transformation <math>F\!</math> in its source category it assigns a unique transformation <math>\mathsf{W}F\!</math> in its target category. Naturally, this only works if <math>\mathsf{W}\!</math> takes the source <math>U^\bullet</math> and the target <math>X^\bullet</math> of the map <math>F\!</math> over to the source <math>\mathsf{W}U^\bullet\!</math> and the target <math>\mathsf{W}X^\bullet\!</math> of the map <math>\mathsf{W}F.\!</math> 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 <math>F,\!</math> and thus we can take it for granted that the assignment of universes under <math>\mathsf{W}\!</math> is defined appropriately at the source and target ends of <math>F.\!</math> It is not always the case, though, that we need to use the particular names (like <math>{}^{\backprime\backprime} \mathsf{W}U^\bullet {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathsf{W}X^\bullet {}^{\prime\prime}\!</math>) that <math>\mathsf{W}\!</math> 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 necessary only that we can tell from the information associated with an operator <math>\mathsf{W}\!</math> what universes they are.

−

Figure ~~33-i~~. ~~Analytic Diagram (1)~~

+

In Figure 31 the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> are displayed horizontally, the way one normally orients functional arrows in a written text, and <math>\mathsf{W}\!</math> rolls the map <math>F\!</math> downward into the images that are associated with <math>\mathsf{W}F.\!</math> In Figure 32 the same information is redrawn so that the maps <math>F\!</math> and <math>\mathsf{W}F\!</math> flow down the page, and <math>\mathsf{W}\!</math> unfurls the map <math>F\!</math> rightward into domains that are the eminent purview of <math>\mathsf{W}F.\!</math>

−

</~~pre~~>

+

{| align="center" border="0" cellpadding="20"

+

|

<pre>

−

U% ~~$E$ $E$U% $E$U% $E$U% $E$~~U%

+

o---------------------------------------o

−

o------------------>o~~============o============o============o~~

+

| |

−

| | | | |

+

| |

−

| | | | |

+

| U% !W! !W!U% |

−

| | | | |

+

| o------------------>o |

−

| | | | |

+

| | | |

−

F | | ~~$E$~~F = | ~~$d$^0.F +~~ | ~~$d$^1.F + | $r$^1.F~~

+

| | | |

−

| | | | |

+

| | | |

−

| | | | |

+

| | | |

−

| | | | |

+

| F | | !W!F |

−

v v ~~v v v~~

+

| | | |

−

o------------------>o~~============o============o============o~~

+

| | | |

−

X% ~~$E$ $E$X% $E$X% $E$X% $E$~~X%

+

| | | |

−

+

| v v |

−

Figure ~~33-ii~~. ~~Analytic~~ Diagram (2)

+

| o------------------>o |

+

| X% !W! !W!X% |

+

| |

+

| |

+

o---------------------------------------o

+

Figure 32. Operator Diagram (2)

</pre>

+

|}

−

~~In its action on universes '''E''' yields the same result~~ as ~~E, a fact that can be expressed~~ in ~~equational form by writing '''E'''''U''~~ ~~• = E''U'' • for any universe ''U'' •. 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 '''E'''''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.

+

The latter arrangement, as exhibited in Figure 32, is more congruent with the thinking about operators that we shall do in the rest of this discussion, since all logical transformations from here on out will be pictured vertically, after the fashion of Figure 30.

−

~~Acting on a transformation ''F'' from universe ''U'' • to universe ''X'' •, the operator '''E''' determines a transformation '''E'''''F'' from '''E'''''U'' • to '''E'''''X'' •. The map '''E'''''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 '''E'''''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 '''E'''''F'' along the lines of the generic frame in Figure 30.~~

+

====Differential Analysis of Propositions and Transformations====

−

~~If one~~ is ~~working within the confines of propositional calculus, it is possible to give an elementary definition of ~~'''E'''~~~~''~~F'' by means of a system of propositional equations~~, ~~as will now be described.~~

+

{| 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]

+

|}

−

~~Given~~ a ~~transformation:~~

+

The approach to the differential analysis of logical propositions and transformations of discourse to be pursued here is carried out in terms of particular operators <math>\mathsf{W}\!</math> that act on propositions <math>F\!</math> or on transformations <math>F\!</math> to yield the corresponding operator maps <math>\mathsf{W}F.\!</math> 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.

−

: ''F'' ~~= ‹~~''~~F''1, …~~, ''F''~~~~''k''~~› : '''B'''''n'' → '''B'''''k''~~

+

* '''Remark on Strategy.''' At this point we run into a set of conceptual difficulties that force us to make a strategic choice in how we proceed. Part of the problem can be remedied by extending our discussion of tacit extensions to the transformational context. But the troubles that remain are much more obstinate and lead us to try two different types of solution. The approach that we 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 we currently use, but are defined subject to certain internal constraints. The extra work needed to set up this method forces us to put it off to a later stage. However, as a compromise, and to prepare the ground for the next pass, we call attention to the various conceptual difficulties as they arise along the way and try to give an honest estimate of how well our first approach deals with them.

−

of ~~concrete type~~:

+

We 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:

−

~~: ''F'' : [''u''~~<~~sub~~>~~1, …, ''u''''n''] → [''x''1, …, ''x''''k'']~~

+

−

~~the transformation~~:

+

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"

+

|

+

<math>\begin{matrix}

+

\mathsf{W}

+

& : &

+

( U^\bullet \to X^\bullet )

+

& \to &

+

( \mathrm{E}U^\bullet \to \mathrm{E}X^\bullet )

+

\end{matrix}\!</math>

+

|}

−

: '''E'''''F'' = ‹''F''1, …, ''F''''k'', E''F''1, …, E''F''''k''› : '''B'''''n'' × '''D'''''n'' → '''B'''''k'' × '''D'''''k''

+

−

~~of concrete~~ type:

+

If we assume that the source universe <math>U^\bullet</math> and the target universe <math>X^\bullet</math> have finite dimensions <math>n\!</math> and <math>k,\!</math> respectively, then each operator <math>\mathsf{W}\!</math> is encompassed by the same abstract type:

−

: <~~font face=georgia~~>'''E'''''F'' : [''u''1, …, ''u''''n'', d''u''1, …, d''u''''n''] → [''x''1, …, ''x''''k'', d''x''1, …, d''x''''k'']

+

−

~~is defined by means of the following system of logical equations:~~

+

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"

−

~~ ~~

−

{| 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%"~~

+

<math>\begin{matrix}

−

~~| width="8%" | ''x''~~<~~sub>1</sub~~>

+

\mathsf{W}

−

~~| width="4%" | =~~

+

& : &

−

~~| width="44%" | <math>\epsilon</math>''F''1‹''u''1,~~ &~~hellip;, ''u''''n'', d''u''1,~~ &~~hellip;, d''u''''n''›~~

+

( [\mathbb{B}^n] \to [\mathbb{B}^k] )

−

~~| width="4%" | =~~

+

& \to &

−

~~| width="40%" | ''F''1‹''u''1, …, ''u''''~~n~~''›~~

+

( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k] )

−

|-

+

\end{matrix}\!</math>

−

~~| ...~~

−

|-

−

~~| width="8%" | ''x''''~~k~~''~~

−

~~| width="4%" | =~~

−

~~| width="44%" | <math>~~\~~epsilon</math>''F''''k''‹''u''1, …, ''u''''~~n~~'', d''u''1, …, d''u''''~~n~~''›~~

−

~~| width="4%" | =~~

−

~~| width="40%" | ''F''''k''~~</~~sub>‹''u''1, …, ''u''''n''</sub~~>›

|}

−

|-

+

+

Since the range features of the operator result <math>\mathsf{W}F : [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k \times \mathbb{D}^k]</math> can be sorted by their ordinary versus differential qualities and the component maps can be examined independently, the complete operator <math>\mathsf{W}\!</math> can be separated accordingly into two components, in the form <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W}).\!</math> Given a fixed context of source and target universes, <math>\boldsymbol\varepsilon\!</math> is always the same type of operator, a multiple component version of the tacit extension operators that were described earlier. In this context <math>\boldsymbol\varepsilon\!</math> has the form:

+

+

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"

|

−

{~~| align="center" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold;~~ text~~-align:left; width:100%"~~

+

<math>\begin{array}{lccccc}

−

~~| width="8%" | d''x''1~~

+

\text{Concrete type}

−

~~| width="4%" | =~~

+

& \boldsymbol\varepsilon

−

~~| width="44%" | E''F''1‹''u''1,~~ &~~hellip;, ''u''''n'', d''u''1,~~ &~~hellip;, d''u''''n''›~~

+

& : &

−

~~| width="4%" | =~~

+

( U^\bullet \to X^\bullet )

−

~~| width="40%" | ''F''1‹''u''1 + d''u''1,~~ &~~hellip;, ''u''''n'' + d''u''''n''›~~

+

& \to &

−

|-

+

( \mathrm{E}U^\bullet \to X^\bullet )

−

~~| ...~~

+

\\[10pt]

−

|-

+

\text{Abstract type}

−

~~| width="8%" | d''x''''k''~~

+

& \boldsymbol\varepsilon

−

~~| width="4%" | =~~

+

& : &

−

~~| width="44%" | E''F''''~~k~~''‹''u''1,~~ &~~hellip;, ''u''''n'', d''u''1,~~ &~~hellip;, d''u''''n''›~~

+

( [\mathbb{B}^n] \to [\mathbb{B}^k] )

−

~~| width="4%" | =~~

+

& \to &

−

~~| width="40%" | ''F''''k''‹''u''1 + d''u''1, …, ''u''''~~n~~'' + d''u''''~~n''</~~sub~~>›

+

( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] )

+

\end{array}</math>

|}

+

+

On the other hand, the operator <math>\mathrm{W}\!</math> is specific to each <math>\mathsf{W}.\!</math> In this context <math>\mathrm{W}\!</math> always has the form:

+

+

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"

+

|

+

<math>\begin{array}{lccccc}

+

\text{Concrete type}

+

& W

+

& : &

+

( U^\bullet \to X^\bullet )

+

& \to &

+

( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet )

+

\\[10pt]

+

\text{Abstract type}

+

& W

+

& : &

+

( [\mathbb{B}^n] \to [\mathbb{B}^k] )

+

& \to &

+

( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] )

+

\end{array}</math>

|}

−

~~ ~~

−

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''1~~</~~sub~~>, ~~…, ''x''''k'', d''u''1, …, d''u''''n'', d''x''1, …, d''x''''k''].~~

+

In the types just assigned to <math>\boldsymbol\varepsilon\!</math> and <math>\mathrm{W}\!</math> and by implication to their results <math>\boldsymbol\varepsilon F\!</math> and <math>\mathrm{W}F,\!</math> we 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:

−

~~In this light, it should be clear that the system of equations defining~~ <~~font face=georgia~~>'''E'''''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'''''n'' → '''B'''.

+

−

~~The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing '''E'''''F''~~ = ‹<math>\~~epsilon</math>''~~F~~'',~~&~~nbsp;~~E~~''F''›, for any map ''F''. This is tantamount~~ to ~~regarding '''~~E~~''' as a complex operator, '''~~E~~''' = ‹<math>~~\~~epsilon~~</math>~~, E›, with a form of application that distributes each component of the operator to work on each component of the operand:~~

+

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"

+

|

+

<math>\begin{matrix}

+

\boldsymbol\varepsilon F

+

& : &

+

( \mathrm{E}U^\bullet \to X^\bullet \subseteq \mathrm{E}X^\bullet )

+

& \cong &

+

( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{B}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )

+

\\[10pt]

+

WF

+

& : &

+

( \mathrm{E}U^\bullet \to \mathrm{d}X^\bullet \subseteq \mathrm{E}X^\bullet )

+

& \cong &

+

( [\mathbb{B}^n \times \mathbb{D}^n] \to [\mathbb{D}^k] \subseteq [\mathbb{B}^k \times \mathbb{D}^k] )

+

\end{matrix}</math>

+

|}

−

: <~~font face=georgia~~>'''E'''''F'' = ‹<math>\epsilon</math>, E›''F'' = ‹<math>\epsilon</math>''F'', E''F''› = ‹<math>\epsilon</math>''F''1, …, <math>\epsilon</math>''F''''k'', E''F''1, …, E''F''''k''›.

+

−

~~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 '''E'''F.

+

Hopefully, though, a general appreciation of these subsumptions will prevent us from having to make such declarations more often than absolutely necessary.

−

The ~~generic notations~~ <~~font face=georgia~~>~~'''d'''~~</~~font~~>~~0''F'', '''d'''1''F'',~~ &~~hellip~~;, <~~font face=georgia~~>~~'''d'''~~</~~font><sup~~>~~''m''''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''~~, ~~'''r'''1''F''~~, ~~…~~, <~~font face=georgia~~>~~'''r'''~~</~~font~~><~~sup~~>~~''m''''~~F~~'' may be used to summarize the contributions to '''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''~~.

+

In giving names to these operators we try 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 “sans serif” operators <math>\mathsf{W}\!</math> and their “serified” components <math>\mathrm{W},\!</math> which forces us to find two distinct but parallel sets of terminology. Here is a plan to that purpose. First, the component operators <math>\mathrm{W}\!</math> are named by analogy with the corresponding operators in the classical difference calculus. Next, the complete operators <math>\mathsf{W} = (\boldsymbol\varepsilon, \mathrm{W})</math> are assigned 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 we are still working toward, comes out fit with its customary name. Finally, the operator results <math>\mathsf{W}F\!</math> and <math>\mathrm{W}F\!</math> can be fixed in our frame of reference by tethering the operative adjective for <math>\mathsf{W}\!</math> or <math>\mathrm{W}\!</math> to the anchoring epithet “map”, in conformity with an already standard practice.

−

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 Secant Operator : '''E'''=====

−

=====The ~~Radius~~ Operator : ~~~~'''e'''~~~~=====

{| width="100%" cellpadding="0" cellspacing="0"

| width="4%" |  

| 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~~.

+

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%" |  

|-

Line 4,057: Line 4,138:

|}

−

~~The operator identified as '''d'''0 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, '''d'''0~~ is ~~equivalent to the doubly tacit extension~~ operator ‹<math>\~~epsilon~~</math>,~~ ~~<math>\~~epsilon~~</math>~~›, in recognition~~ of ~~which let us redub it~~ as ~~"'''e'''"~~. ~~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:~~

+

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 <math>{}^{\backprime\backprime} \mathsf{E} {}^{\prime\prime},\!</math> which receives the principal investment of analytic attention, and on the constituent parts of <math>\mathsf{E},\!</math> which derive their shares of significance as developed by the analysis. In the sequel, we refer to <math>\mathsf{E}\!</math> 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 <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> and its active ingredient <math>\mathrm{E}\!</math> is known as the ''enlargement operator''. (Here, we name <math>\mathrm{E}\!</math> after the literal ancestor of the shift operator in the calculus of finite differences, defined so that <math>\mathrm{E}f(x) = f(x+1)\!</math> for any suitable function <math>f,\!</math> though of course the logical analogue that we take up here must have a rather different definition.)

−

~~:{| cellpadding=2~~

+

{| align="center" border="0" cellpadding="10"

−

~~| '''e'''''F''~~

−

~~| =~~

−

~~| ‹<math>\epsilon</math>, <math>\epsilon</math>›''F''~~

−

|-

−

~~|  ~~

−

~~| =~~

−

~~| ‹<math>\epsilon</math>''F'', <math>\epsilon</math>''F''›~~

−

|-

−

~~|  ~~

−

~~| =~~

−

~~| ‹<math>\epsilon</math>''F''1, …, <math>\epsilon</math>F''k'', <math>\epsilon</math>F1, …, <math>\epsilon</math>F''k''› ,~~

−

|}

−

~~which is tantamount to the system of equations given below.~~

−

~~ ~~

−

{| 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%"~~

+

<pre>

−

| ~~width="8%"~~ | ~~''x''1~~

+

U% $E$ $E$U% $E$U% $E$U%

−

| ~~width="4%"~~ | =

+

o------------------>o============o============o

−

| ~~width="44%"~~ | ~~<math>\epsilon</math>''F''1‹''u''1, …, ''u''''n'', d''u''1, …, d''u''''n''›~~

+

| | | |

−

| ~~width="4%"~~ | =

+

| | | |

−

| ~~width="40%"~~ | ~~''F''1‹''u''1, …, ''u''''n''›~~

+

| | | |

−

|-

+

| | | |

−

| ~~...~~

+

F | | $E$F = | $d$^0.F + | $r$^0.F

−

|-

+

| | | |

−

| ~~width="8%"~~ | ~~''x''''k''~~

+

| | | |

−

| ~~width="4%"~~ | =

+

| | | |

−

| ~~width="44%"~~ | ~~<math>\epsilon</math>''~~F~~''''k''‹''u''1, …, ''u''''n'', d''u''1, …, d''u''''n''›~~

+

v v v v

−

| ~~width="4%"~~ | =

+

o------------------>o============o============o

−

| ~~width="40%"~~ | ~~''F''''k''‹''u''1, …, ''u''''n''›~~

+

X% $E$ $E$X% $E$X% $E$X%

−

|}

+

−

|-

+

Figure 33-i. Analytic Diagram (1)

−

|

+

</pre>

−

~~{| align~~=~~"center" border~~=~~"0" cellpadding~~=~~"4" cellspacing~~=~~"0" style~~=~~"font-weight:bold; text-align:left; width:100%"~~

−

~~| width~~=~~"8%" | d''x''1~~

−

~~| width~~=~~"4%" |~~ =

−

~~| width~~=~~"44%" | <math>\epsilon</math>''F''1‹''u''1, …, ''u''''n'', d''u''1, …, d''u''''n''›~~

−

~~| width~~=~~"4%" |~~ =

−

~~| width~~=~~"40%" | ''F''1‹''u''1, …, ''u''''n''›~~

−

|-

−

~~| ...~~

−

|-

−

~~| width~~=~~"8%" | d''x''''k''~~

−

~~| width~~=~~"4%" |~~ =

−

~~| width~~=~~"44%" | <math>\epsilon</math>''F''''k''‹''u''1, …, ''u''''n'', d''u''1, …, d''u''''n''›~~

−

~~| width~~=~~"4%" |~~ =

−

~~| width="40~~%~~" | ''F''''k''‹''u''~~1</~~sub>, …, ''u''''n''</sub~~>›

|}

+

{| align="center" border="0" cellpadding="10"

+

|

+

<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>

|}

−

~~ ~~

−

=~~====The Phantom~~ of the ~~Operators : '''~~&~~eta~~;~~'''=====~~

+

In its action on universes <math>\mathsf{E}\!</math> yields the same result as <math>\mathrm{E},\!</math> a fact that can be expressed in equational form by writing <math>\mathsf{E}U^\bullet = \mathrm{E}U^\bullet\!</math> for any universe <math>U^\bullet.\!</math> 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 <math>\mathsf{E}F\!</math> 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.

−

{~~| width="100%" cellpadding="0" cellspacing="0"~~

+

Acting on a transformation <math>F\!</math> from universe <math>U^\bullet\!</math> to universe <math>X^\bullet,\!</math> the operator <math>\mathsf{E}\!</math> determines a transformation <math>\mathsf{E}F\!</math> from <math>\mathsf{E}U^\bullet\!</math> to <math>\mathsf{E}X^\bullet.\!</math> The map <math>\mathsf{E}F\!</math> forms the main body of evidence to be investigated in performing a differential analysis of <math>F.\!</math> 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 <math>\mathsf{E}F\!</math> 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 <math>F\!</math> until we can lay out the full “parts diagram” of <math>\mathsf{E}F\!</math> along the lines of the generic frame in Figure 30.

−

~~| width="4%"~~ ~~|  ~~

−

~~| 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%"~~ | &~~nbsp~~;

−

|-

−

~~| align="right" colspan="3" |~~ &~~mdash~~; ~~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~~.

+

Working within the confines of propositional calculus, it is possible to give an elementary definition of <math>\mathsf{E}F\!</math> by means of a system of propositional equations, as we now describe.

−

Given a transformation ''F'' : [''u''1, …, ''u''''n''] → [''x''1, …, ''x''''k''], we often need to make a separate treatment of a related family of transformations of the form ''F''* : [''u''1, …, ''u''''n'', d''u''1, …, d''u''''n''] → [d''x''1, …, d''x''''k'']. The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:

+

Given a transformation

−

~~: <math>\eta~~</math>''F~~'' : [''u''1~~,~~ …~~,~~ ''u''''~~n~~'', d''u''1, …, d''u''''n''] → [d''x''1, …, d''x''''~~k''</~~sub~~>]

+

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

+

| <math>F = (F_1, \ldots, F_k) : \mathbb{B}^n \to \mathbb{B}^k\!</math>

+

|}

−

~~which is defined by the equations:~~

+

of concrete type

−

~~ ~~

+

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

−

{| align="center~~" border="1~~" cellpadding="~~12" cellspacing="0" style=~~"~~font-weight:bold; text-align:left;~~ width~~:96%"~~

+

| <math>F : [u_1, \ldots, u_n] \to [x_1, \ldots, x_k],\!</math>

−

|

−

~~{| align="center" border~~="~~0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100~~%"

−

~~| width="8%" | d''x''1~~

−

~~| width="4%" | =~~

−

~~| width="44%"~~ | <math>~~\epsilon</math>''~~F~~''1‹''u''1~~, ~~…~~, ~~''u''''n''~~, ~~d''u''1~~, ~~…~~, ~~d''u''''n''›~~

−

~~| width="4%" | =~~

−

~~| width="40%" | ''F''1‹''u''1, …, ''u''''n''›~~

−

|-

−

~~| ...~~

−

|-

−

~~| width="8%" | d''x''''k''~~

−

~~| width="4%" | =~~

−

~~| width="44%" | <math>~~\~~epsilon~~</math>~~''F''''k''‹''u''1, …, ''u''''n'', d''u''1, …, d''u''''n''›~~

−

~~| width="4%" | =~~

−

~~| width="40%" | ''F''''k''‹''u''1, …, ''u''''n''›~~

|}

+

the transformation

+

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

+

| <math>\mathsf{E}F = (F_1, \ldots, F_k, \mathrm{E}F_1, \ldots, \mathrm{E}F_k) : \mathbb{B}^n \times \mathbb{D}^n \to \mathbb{B}^k \times \mathbb{D}^k\!</math>

|}

−

~~ ~~

−

~~In effect, the operator <math>\eta</math> is nothing but the stand-alone version~~ of a procedure that is otherwise invoked subordinate to the work of the radius operator '''e'''. Operating independently, <math>\eta</math> achieves precisely the same results that the second <math>\epsilon</math> in ‹<math>\epsilon</math>, <math>\epsilon</math>› accomplishes by working within the context of its adjuvant thematic frame, "‹ , ›". From this point on, because the use of <math>\epsilon</math> and <math>\eta</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta</math> reflects in regard to <math>\epsilon</math> little more than the application of a differential twist, a mere turn of phrase, I refer to <math>\eta</math> as the ''trope extension'' operator.

+

of concrete type

−

===~~==The Chord Operator :~~ <~~font face=georgia~~>~~'''D'''~~</~~font~~>~~=====~~

+

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

+

| <math>\mathsf{E}F : [u_1, \dots, u_n, \mathrm{d}u_1, \dots, \mathrm{d}u_n] \to [x_1, \ldots, x_k, \mathrm{d}x_1, \ldots, \mathrm{d}x_k]\!</math>

+

|}

−

{| ~~width~~="~~100%~~" cellpadding="0" cellspacing="0"

+

is defined by means of the following system of logical equations:

−

~~| width~~="4%" | &~~nbsp;~~

+

−

~~| width~~=~~"92%" |~~

+

−

~~What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced~~, ~~then the alternatives mean practically the same thing~~, ~~and all dispute is idle.~~

+

−

~~| width~~=~~"4%" |~~ &~~nbsp;~~

+

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"

−

|-

+

|

−

~~| align~~=~~"right" colspan~~=~~"3" |~~ &~~mdash; William James~~, ~~''Pragmatism''~~, [~~Jam~~, ~~45]~~

+

<math>\begin{matrix}

+

x_1

+

& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)

+

& = & F_1 (u_1, \ldots, u_n)

+

\\[4pt]

+

\cdots && \cdots && \cdots

+

\\[4pt]

+

x_k

+

& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)

+

& = & F_k (u_1, \ldots, u_n)

+

\\[16pt]

+

\mathrm{d}x_1

+

& = & \mathrm{E}F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)

+

& = & F_1 (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)

+

\\[4pt]

+

\cdots && \cdots && \cdots

+

\\[4pt]

+

\mathrm{d}x_k

+

& = & \mathrm{E}F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)

+

& = & F_k (u_1 + \mathrm{d}u_1, \ldots, u_n + \mathrm{d}u_n)

+

\end{matrix}</math>

|}

−

Next I discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.

+

−

~~This~~ is ~~the operator~~ that ~~is referred to~~ as ~~'''r'''0~~ in ~~the initial stage of analysis (Figure 33-i)~~, ~~and that is expanded~~ as ~~'''d'''1 + '''r'''1~~ in the ~~subsequent step (Figure 33-ii). In congruence, but not quite harmony, with my allusions~~ of ~~analogy that are not quite geometry, I call this~~ the ~~''chord operator'' and denote it '''D'''~~. ~~In the more casual terms that are here introduced~~, ~~'''D'''~~ is ~~defined as~~ the ~~remainder~~ of <~~font face=georgia~~>~~'''E''' and '''e'''~~</~~font~~>, and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise '''E''' and the bar of exigency '''e'''.

+

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 generated by all the named variables. Specifically, this is the universe of discourse over <math>2(n+k)\!</math> variables denoted by:

−

The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we may write '''D'''~~<~~/font~~>&~~nbsp;~~=&~~nbsp;‹<math>~~\~~epsilon</math>~~,~~ D›~~, calling D the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, ~~we should note that the chord '''D''' is not one that need be lost at any stage of development~~. ~~At the ''m''th~~</~~sup~~> ~~stage of play it can always be reconstituted in the following form:

+

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

+

|

+

<math>\begin{matrix}

+

\mathrm{E}[\mathcal{U} \cup \mathcal{X}]

+

& = &

+

[u_1, \ldots, u_n, ~ x_1, \ldots, x_k, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n, ~ \mathrm{d}x_1, \ldots, \mathrm{d}x_k].

+

\end{matrix}</math>

+

|}

−

<~~font face~~=~~"courier new"~~>

+

In this light, it should be clear that the system of equations defining <math>\mathsf{E}F\!</math> embodies, in a higher rank and differentially extended version, an analogy with the process of thematization that we treated earlier for propositions of type <math>F : \mathbb{B}^n \to \mathbb{B}.\!</math>

−

{| align="center~~" border="1~~" cellpadding="12" ~~cellspacing="0" style~~="~~font-weight:bold; text-align:left; width:96~~%"

+

The entire collection of constraints that is represented in the above system of equations may be abbreviated by writing <math>\mathsf{E}F = (\boldsymbol\varepsilon F, \mathrm{E}F),\!</math> for any map <math>F.\!</math> This is tantamount to regarding <math>\mathsf{E}\!</math> as a complex operator, <math>\mathsf{E} = (\boldsymbol\varepsilon, \mathrm{E}),\!</math> with a form of application that distributes each component of the operator to work on each component of the operand, as follows:

+

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

|

−

~~{| align="center" border="0" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"~~

+

<math>\begin{matrix}

−

| <~~font face=georgia>'''D'''</font~~>

+

\mathsf{E}F

−

~~| =~~

+

& = &

−

~~| '''E'''~~ &~~ndash; '''e'''~~

+

(\boldsymbol\varepsilon, \mathrm{E})F

−

|-

+

& = &

−

| &~~nbsp;~~

+

(\boldsymbol\varepsilon F, \mathrm{E}F)

−

~~| =~~

+

& = &

−

~~| '''r'''0~~

+

(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \mathrm{E}F_1, \ldots, \mathrm{E}F_k).

−

|-

+

\end{matrix}</math>

−

| &~~nbsp;~~

−

~~| =~~

−

~~| '''d'''1 + '''r'''1~~

−

|-

−

| &~~nbsp;~~

−

~~| =~~

−

~~| '''d'''1 +~~ &~~hellip; + '''d'''''m'' + '''r'''''m''~~

−

|-

−

| &~~nbsp;~~

−

~~| =~~

−

~~| ∑~~(~~''i'' = 1 … ''m''~~)</~~sub> '''d'''''i'' + '''r'''''m''</sup~~>

|}

−

|}

−

~~ ~~

−

=====The ~~Tangent~~ Operator : ~~~~'''T'''~~~~=====

+

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 parenthesized argument lists, that 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 argument list 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 <math>\mathsf{E}F.\!</math>

+

The generic notations <math>\mathsf{d}^0\!F, \mathsf{d}^1\!F, \ldots, \mathsf{d}^m\!F\!</math> in Figure 33 refer to the increasing orders of differentials that are extracted in the course of analyzing <math>F.\!</math> When the analysis is halted at a partial stage of development, notations like <math>\mathsf{r}^0\!F, \mathsf{r}^1\!F, \ldots, \mathsf{r}^m\!F\!</math> may be used to summarize the contributions to <math>\mathsf{E}F\!</math> that remain to be analyzed. The Figure illustrates a convention that makes <math>\mathsf{r}^m\!F,\!</math> in effect, the sum of all differentials of order strictly greater than <math>m.\!</math>

+

We next discuss the 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 we introduce along the way.

+

=====The Radius Operator : '''e'''=====

{| width="100%" cellpadding="0" cellspacing="0"

| width="4%" |  

| width="92%" |

−

~~They take part in scenes~~ of ~~whose significance they have~~ no ~~inkling. They are merely tangent to curves of history the beginnings and ends and forms~~ of ~~which pass wholly beyond their ken. So we are tangent~~ to ~~the wider life~~ of ~~things~~.

+

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%" |  

|-

−

| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, ~~300~~]

+

| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 46]

|}

−

The operator ~~tagged~~ as <~~font face=georgia~~>~~'''~~d~~'''~~</~~font>1</sup~~> in the analytic diagram (Figure 33) ~~is called~~ the ~~''tangent operator'', and is usually denoted in this text as~~ <~~font face=georgia~~>~~'''d'''~~</~~font~~> or <~~font face=georgia~~>~~'''T'''~~</~~font~~>~~. Because it has the properties required~~ to ~~qualify as a functor, namely, preserving~~ the ~~identity element of the composition operation and the articulated form of every composure among transformations~~, ~~it also earns the title of a ''tangent functor''. According to the custom adopted here~~, ~~we dissect it as '''T'''~~</~~font~~>~~ = ~~<~~font face=georgia~~>~~'''d'''~~</~~font~~>~~ = ‹~~<math>\~~epsilon~~</math>~~, d›, where d is the operator that yields~~ the ~~first order differential d~~''F'' ~~when applied to a transformation ''F'', and whose name~~ is ~~legion.~~

+

The operator identified as <math>\mathrm{d}^0\!</math> in the analytic diagram (Figure 33) has the sole purpose of creating a proxy for <math>F\!</math> in the appropriately extended context. Construed in terms of its broadest components, <math>\mathrm{d}^0\!</math> is equivalent to the doubly tacit extension operator <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon),\!</math> in recognition of which let us redub it as <math>{}^{\backprime\backprime} \mathsf{e} {}^{\prime\prime}.\!</math> Pursuing a geometric analogy, we may refer to <math>\mathsf{e} =(\boldsymbol\varepsilon, \boldsymbol\varepsilon) = \mathrm{d}^0\!</math> as the ''radius operator''. The operation intended by all of these forms is defined by the following equation:

−

~~Figure 34 illustrates a stage of analysis where we ignore everything but the tangent functor '''T'''~~<~~/font~~>~~, and attend to it chiefly as it bears on the first order differential d''~~F~~'' in the analytic expansion of ''F''. In this situation, we often refer to the extended universes E''U'' ~~&~~bull; and E''X'' • under the equivalent designations '''T'''''U''~~&~~nbsp;~~&~~bull; and '''T'''''X''~~&~~nbsp;•, respectively. The purpose of the tangent functor '''T''' is to extract the tangent map '''T'''''~~F~~'' at each point of ''U'' •~~, ~~and the tangent map '''T'''''~~F''&~~nbsp;~~=&~~nbsp;‹<math>~~\~~epsilon</math>~~, d›''F'' tells us not only what the transformation ''F'' is doing at each point of the universe ''U'' • but also what ''F'' is doing to states in the neighborhood of that point, ~~approximately~~, ~~linearly~~, ~~and relatively speaking~~.

+

{| align="center" cellpadding="8" width="90%"

−

+

|

−

<~~pre~~>

+

<math>\begin{array}{lll}

−

~~U% $T$ $T$U% $T$U%~~

+

\mathsf{e}F

−

~~o------------------>o============o~~

+

& = &

−

~~| | |~~

+

(\boldsymbol\varepsilon, \boldsymbol\varepsilon)F

−

| ~~| |~~

+

\\[4pt]

−

~~| | |~~

+

& = &

−

~~| | |~~

+

(\boldsymbol\varepsilon F, ~ \boldsymbol\varepsilon F)

−

~~F | | $T$F = | <!e!, d> F~~

+

\\[4pt]

−

~~| | |~~

+

& = &

−

~~| | |~~

+

(\boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k, ~ \boldsymbol\varepsilon F_1, \ldots, \boldsymbol\varepsilon F_k).

−

~~| | |~~

+

\end{array}</math>

−

~~v v v~~

+

|}

−

~~o------------------>o============o~~

−

~~X% $T$ $T$X% $T$X%~~

−

~~Figure 34~~. ~~Tangent Functor Diagram~~

+

which is tantamount to the system of equations below.

−

~~</pre>~~

−

* NB. There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators W in {<~~math~~>\eta</math>, E, D, d, r} so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps WF have equally good reasons for placing their values in differential stocks. The only explanation I can devise at present is that, without doing this, I cannot justify the comparison and combination of their values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now, the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.

+

−

===~~Transformations of Type '''B'''~~<~~sup>2</sup~~> &~~rarr; '''B'''1~~</~~sup~~>~~===~~

+

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"

+

|

+

<math>\begin{matrix}

+

x_1

+

& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)

+

& = & F_1 (u_1, \ldots, u_n)

+

\\[4pt]

+

\cdots && \cdots && \cdots

+

\\[4pt]

+

x_k

+

& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)

+

& = & F_k (u_1, \ldots, u_n)

+

\\[16pt]

+

\mathrm{d}x_1

+

& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)

+

& = & F_1 (u_1, \ldots, u_n)

+

\\[4pt]

+

\cdots && \cdots && \cdots

+

\\[4pt]

+

\mathrm{d}x_k

+

& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n)

+

& = & F_k (u_1, \ldots, u_n)

+

\end{matrix}</math>

+

|}

−

To study the effects of these analytic operators in the simplest possible situation, let us revert to a still more primitive case. Consider the singular proposition ''J''‹''u'', ''v''› = ''uv'', regarded either as the functional product of the maps ''u'' and ''v'' or as the logical conjunction of the features ''u'' and ''v'', a map whose fiber of truth ''J''<~~sup~~>–1(1) picks out the single cell of that logical description in the universe of discourse ''U'' •. Thus ''J'', or ''uv'', may be treated as a pseudonym for the point whose coordinates are ‹1, 1› in ''U'' •.

+

−

====~~Analytic Expansion~~ of ~~Conjunction~~====

+

=====The Phantom of the Operators : '''η'''=====

{| width="100%" cellpadding="0" cellspacing="0"

| width="4%" |  

| width="92%" |

−

~~In her sufferings she read a great deal~~ and ~~discovered that she had lost something~~, the ~~possession of which she had previously not been much aware of: a soul.~~

+

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''!

−

~~What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.~~

| width="4%" |  

|-

−

| align="right" colspan="3" | — ~~Robert Musil~~, ''The ~~Man Without Qualities~~'', [~~Mus~~, ~~118~~]

+

| align="right" colspan="3" | — Gaston Leroux, ''The Phantom of the Opera'', [Ler, 81]

|}

−

~~Figure 35 pictures~~ the ~~form of conjunction ''J'' : '''B'''2 → '''B''' as a transformation from~~ the ~~2-dimensional universe [''u'', ''v''] to~~ the ~~1-dimensional universe [''x'']. This is a subtle~~ but ~~significant change of viewpoint on the proposition~~, ~~attaching an arbitrary but concrete quality~~ to ~~its functional value~~. ~~Using~~ the ~~language introduced earlier,~~ we ~~can express this change by saying that~~ the ~~proposition ''J'' : 〈''u''~~,~~ ''v''〉 → '''B''' is being recast into the thematized role of~~ a ~~transformation ''J'' : [''u''~~,~~ ''v''] → [''x'']~~, ~~where~~ the new ~~variable ''x'' takes the part~~ of ~~a thematic variable ¢(''J'')~~.

+

We 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 us some painstaking trouble to detect. In the end we 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.

−

<~~br>~~

+

Given a transformation <math>F : [u_1, \ldots, u_n] \to [x_1, \dots, x_k],\!</math> we often have call to consider a family of related transformations, all having the form:

−

[[~~Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]~~]

−

~~<center>'''Figure 35. Conjunction as Transformation'''</center></p~~>

−

===~~==Tacit Extension~~ of ~~Conjunction=====~~

+

{| align="center" cellpadding="8" width="90%"

+

| <math>F^\dagger : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \dots, \mathrm{d}x_k].\!</math>

+

|}

+

The operator <math>\eta</math> is introduced to deal with the simplest one of these maps:

−

{| ~~width~~="~~100%~~" cellpadding="~~0" cellspacing="0~~"

+

{| align="center" cellpadding="8" width="90%"

−

| width="40%" ~~|  ~~

+

| <math>\eta F : [u_1, \ldots, u_n, \mathrm{d}u_1, \ldots, \mathrm{d}u_n] \to [\mathrm{d}x_1, \ldots \mathrm{d}x_k],\!</math>

−

| ~~width="60%" |~~

−

~~I teach straying from me, yet who can stray from me?~~

−

~~I follow you whoever you are from the present hour; ~~

−

~~My words itch at your ears till you understand them.~~

−

|-

−

~~|  ~~

−

~~| align="right" | — Walt Whitman~~, ~~''Leaves of Grass''~~, [~~Whi~~, 83]

|}

−

~~Earlier I~~ defined the ~~tacit extension operators <math>\epsilon</math> ~~: ''X'' • → ''Y'' • as maps embedding each proposition of a given universe ''X'' • in a more generously given universe ''Y'' • containing ''X'' •. Of immediate interest are the tacit extensions <math>\epsilon</math> : ''U'' • → E''U'' •, that locate each proposition of ''U'' • in the enlarged context of E''U'' •. In its application to the propositional conjunction ''J'' = ''u'' ''v'' in [''u'', ''v''], the tacit extension operator <math>\epsilon</math> produces the proposition <math>\epsilon</math>''J'' in E''U'' • = [''u'', ''v'', d''u'', d''v'']. The extended proposition <math>\epsilon</math>''J'' may be computed according to the scheme in Table 36, in effect, doing nothing more than conjoining a tautology of [d''u'', d''v''] to ''J'' in ''U'' •<~~/sup~~>.

+

which is defined by the following equations:

+

−

~~~~

+

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:center; width:70%"

−

{| align="center" border="1" cellpadding="12" cellspacing="0" style="~~font-weight:bold;~~ text-align:~~left~~; width:96%"

−

~~|+ Table 36. Computation of <math>\epsilon</math>''J''~~

|

−

~~{| align="left" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"~~

+

<math>\begin{matrix}

−

~~| width="8%" |~~ <math>\~~epsilon</math>''J''~~

+

\mathrm{d}x_1

−

~~| width~~=~~"4%" |~~ =

+

& = & \boldsymbol\varepsilon F_1 (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)

−

~~| ''J''‹''u''~~, ~~''v''›~~

+

& = & F_1 (u_1, \ldots, u_n)

−

|-

+

\\[4pt]

−

~~| width="8%" |~~ &~~nbsp;~~

+

\cdots && \cdots && \cdots

−

~~| width="4%" | =~~

+

\\[4pt]

−

~~| ''u'' ''v''~~

+

\mathrm{d}x_k

−

|-

+

& = & \boldsymbol\varepsilon F_k (u_1, \ldots, u_n, ~ \mathrm{d}u_1, \ldots, \mathrm{d}u_n)

−

~~| width~~=~~"8%" |~~ &~~nbsp;~~

+

& = & F_k (u_1, \ldots, u_n)

−

~~| width="4%" | =~~

+

\end{matrix}</math>

−

~~| ''u'' ''v''~~ (d~~''u'')(~~d~~''v''~~) ~~|| +~~

−

~~| ''u'' ''v''~~ (~~d''u''~~) ~~d''v'' || +~~

−

~~| ''u'' ''v'' d''u'' (d''v'') || +~~

−

~~| ''u'' ''v'' d''u'' d''v''~~

|}

−

|-

+

−

|

+

−

{~~| align="left" border="0" cellpadding="4" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"~~

+

−

~~| width="8%"~~ | <math>\~~epsilon~~</math>''J''

+

In effect, the operator <math>\eta\!</math> is nothing but the stand-alone version of a procedure that is otherwise invoked subordinate to the work of the radius operator <math>\mathsf{e}.\!</math> Operating independently, <math>\eta\!</math> achieves precisely the same results that the second <math>\boldsymbol\varepsilon\!</math> in <math>(\boldsymbol\varepsilon, \boldsymbol\varepsilon)\!</math> accomplishes by working within the context of its ordered pair thematic frame. From this point on, because the use of <math>\boldsymbol\varepsilon\!</math> and <math>\eta\!</math> in this setting combines the aims of both the tacit and the thematic extensions, and because <math>\eta\!</math> reflects in regard to <math>\boldsymbol\varepsilon\!</math> little more than the application of a differential twist, a mere turn of phrase, we refer to <math>\eta\!</math> as the ''trope extension'' operator.

−

~~| width~~=~~"4%" |~~ =

+

−

~~| width~~=~~"24%" |~~ ''u'' ''~~v'' (d''u'')(d''v'')~~

+

=====The Chord Operator : '''D'''=====

−

| width="4%" ~~| +~~

+

−

~~| width~~="~~60%~~" ~~|  ~~

+

{| width="100%" cellpadding="0" cellspacing="0"

−

|-

−

~~| width~~="8%" ~~|  ~~

| width="4%" |  

−

| width="24%" | ~~''u'' ''v'' (d''u'') d''v''~~

+

| width="92%" |

−

~~| width="4%" | +~~

+

What difference would it practically make to any one if this notion rather than that notion were true? If no practical difference whatever can be traced, then the alternatives mean practically the same thing, and all dispute is idle.

−

~~| width="60%" |  ~~

−

|-

−

~~| width="8%"~~ ~~|  ~~

| width="4%" |  

−

~~| width="24%" | ''u'' ''v''  d''u'' (d''v'')~~

−

~~| width="4%" | +~~

−

~~| width="60%" |  ~~

|-

−

| ~~width~~="8%" ~~|  ~~

+

| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 45]

−

~~| width="4%" |  ~~

−

~~| width~~="~~24%~~" | ~~''u'' ''v''  d''u''~~&~~nbsp~~;~~ d~~''v''

−

~~| width="4%" |  ~~

−

~~| width="60%" |  ~~

−

|}

−

~~ ~~

−

~~The lower portion~~ of the ~~Table contains the dispositional features of <math>\epsilon</math>''J'' arranged in such~~ a ~~way~~ that the ~~variety~~ of ~~ordinary features spreads across the rows and the variety~~ of ~~differential features runs through the columns~~. ~~This organization serves to facilitate pattern matching in the remainder of our computations.~~ ~~Again, the tacit extension is usually so trivial~~ a ~~concern~~ that ~~we do not always bother to make~~ an ~~explicit note~~ of it, ~~taking it for granted that any function ''F'' that is being employed~~ in a ~~differential context~~ is ~~equivalent to <math>\epsilon</math>''F'', for a suitable <math>\epsilon</math>~~.

+

Next we discuss an operator that is always immanent in this form of analysis, and remains implicitly present in the entire proceeding. It may appear once as a record: a relic or revenant that reprises the reminders of an earlier stage of development. Or it may appear always as a resource: a reserve or redoubt that caches in advance an echo of what remains to be played out, cleared up, and requited in full at a future stage. And all of this remains true whether or not we recall the key at any time, and whether or not the subtending theme is recited explicitly at any stage of play.

−

~~Figures~~ 37-~~a through 37~~-~~d present several pictures~~ of the ~~proposition~~ ''J'' and ~~its tacit extension~~ <math>\~~epsilon~~</math>~~''J''.~~ ~~Notice in these Figures how~~ <math>\~~epsilon~~</math>~~''J'' in E''U''~~<~~sup~~>~~ •~~</~~sup~~> ~~visibly extends ''J'' in ''U''~~<~~sup~~>~~ •~~</~~sup~~>~~, by annexing~~ to ~~the indicated cells of ''J'' all~~ of the ~~arcs that exit from or flow out of them. In effect, this extension attaches to these cells all~~ of the ~~dispositions that spring from them, in other words, it attributes to these cells all~~ of ~~the conceivable changes that are their issue~~.

+

This is the operator that is referred to as <math>\mathsf{r}^0\!</math> in the initial stage of analysis (Figure 33-i) and that is expanded as <math>\mathsf{d}^1 + \mathsf{r}^1\!</math> in the subsequent step (Figure 33-ii). In congruence, but not quite harmony with our allusions of analogy that are not quite geometry, we call this the ''chord operator'' and denote it <math>\mathsf{D}.\!</math> In the more casual terms that are here introduced, <math>\mathsf{D}</math> is defined as the remainder of <math>\mathsf{E}\!</math> and <math>\mathsf{e}\!</math> and it assigns a due measure to each undertone of accord or discord that is struck between the note of enterprise <math>\mathsf{E}\!</math> and the bar of exigency <math>\mathsf{e}.\!</math>

−

+

The tension between these counterposed notions, in balance transient but regular in stridence, may be refracted along familiar lines, though never by any such fraction resolved. In this style we write <math>\mathsf{D} = (\boldsymbol\varepsilon, \mathrm{D}),\!</math> calling <math>\mathrm{D}\!</math> the ''difference operator'' and noting that it plays a role in this realm of mutable and diverse discourse that is analogous to the part taken by the discrete difference operator in the ordinary difference calculus. Finally, we should note that the chord <math>\mathsf{D}\!</math> is not one that need be lost at any stage of development. At the <math>m^\text{th}\!</math> stage of play it can always be reconstituted in the following form:

−

~~[[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]~~

−

<~~center>'''~~Figure 37-~~a. ~~Tacit Extension of ''J''  (Areal)'''~~</~~font~~><~~/center>

−

~~[[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]~~

−

~~<center>'''Figure 37-b. Tacit Extension of ''J''  (Bundle)'''</center>~~

−

+

{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:70%"

−

~~~~[[~~Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center~~]]~~~~

+

|

−

~~<center>'''Figure 37-c. Tacit Extension of ''J''~~&~~nbsp;~~&~~nbsp;(Compact)'''~~</~~font></center></p~~>

+

<math>\begin{array}{lll}

+

\mathsf{D}

+

& = & \mathsf{E} - \mathsf{e}

+

\\[6pt]

+

& = & \mathsf{r}^0

+

\\[6pt]

+

& = & \mathsf{d}^1 + \mathsf{r}^1

+

\\[6pt]

+

& = & \mathsf{d}^1 + \ldots + \mathsf{d}^m + \mathsf{r}^m

+

\\[6pt]

+

& = & \displaystyle \sum_{i=1}^m \mathsf{d}^i + \mathsf{r}^m

+

\end{array}</math>

+

|}

−

~~[[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]~~

−

~~<center>'''Figure 37-d. Tacit Extension of ''J''  (Digraph)'''</center>~~

−

The computational scheme that was shown in Table 36 treated ''J'' as a proposition in ''U'' • and formed <math>\epsilon</math>''J'' as a proposition in E''U'' •. When ''J'' is regarded as a mapping ''J'' : ''U'' • → ''X'' • then <math>\epsilon</math>''J'' must be obtained as a mapping <math>\epsilon</math>''J'' : E''U'' • → ''X'' •. By default, the tacit extension of the map ''J'' : [''u'', ''v''] → [''x''] is naturally taken to be a particular map, of the following form:

−

~~: <math>\epsilon</math>''J'' : [''u'', ''v'', d''u'', d''v''] → [''x''] &sube; [''x'', d''x'']~~

−

This is the map that looks like ''J'' when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that ''J'' already employs.

+

=====The Tangent Operator : '''T'''=====

−

But the choice of a particular thematic variable, for example ''x'' for ¢(''J''), is a shade more arbitrary than the initial choice of variable names {''u'', ''v''}. This means that the map I am calling the ''trope extension'', specifically:

−

~~: <math>\eta</math>''J'' ~~:~~ [''u'', ''v~~''~~, d~~''u''~~, d''v''] → [d''x''] &sube; [''x'', d''x'']~~

−

~~since it looks just the same as <math>\epsilon</math>''J'' in the way that its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.~~

−

These considerations have the practical consequence that all of our computations and illustrations of <math>\epsilon</math>''J'' perform the double duty of capturing an image of <math>\eta</math>''J'' as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta</math>''J'', because the exercise would be identical to the work already done for <math>\epsilon</math>''J''. Since the computations given for <math>\epsilon</math>''J'' are expressed solely in terms of the variables {''u'', ''v'', d''u'', d''v''}, these variables work equally well for finding <math>\eta</math>''J''. Furthermore, since each of the above Figures shows only how the level sets of <math>\epsilon</math>''J'' partition the extended source universe E''U'' • = [''u'', ''v'', d''u'', d''v''], all of them serve equally well as portraits of <math>\eta</math>''J''.

−

~~=====Enlargement Map of Conjunction~~=====

{| width="100%" cellpadding="0" cellspacing="0"

| width="4%" |  

| width="92%" |

−

~~No one could have established the existence~~ of ~~any details that might not just as well~~ have ~~existed in earlier times too;~~ ~~but all~~ the ~~relations between things had shifted slightly~~. ~~Ideas that had once been~~ of ~~lean account grew fat~~.

+

They take part in scenes of whose significance they have no inkling. They are merely tangent to curves of history the beginnings and ends and forms of which pass wholly beyond their ken. So we are tangent to the wider life of things.

| width="4%" |  

|-

−

| align="right" colspan="3" | — ~~Robert Musil~~, ''~~The Man Without Qualities~~'', [~~Mus~~, 62]

+

| align="right" colspan="3" | — William James, ''Pragmatism'', [Jam, 300]

|}

−

The ~~enlargement map E~~''J'' is ~~computed from~~ the ~~proposition ''J'' by making~~ a ~~particular class~~ of ~~formal substitutions for its variables~~, ~~in this case~~ ''u''~~ + ~~d~~''u'' for ''u'' and ''v'' + ~~d~~''v'' for ''v''~~, ~~and subsequently expanding~~ the ~~result in whatever way happens~~ to ~~be convenient for the end in view~~.

+

The operator tagged as <math>\mathsf{d}^1\!</math> in the analytic diagram (Figure 33) is called the ''tangent operator'' and is usually denoted in this text as <math>\mathsf{d}\!</math> or <math>\mathsf{T}.\!</math> Because it has the properties required to qualify as a functor, namely, preserving the identity element of the composition operation and the articulated form of every composition of transformations, it also earns the title of a ''tangent functor''. According to the custom adopted here, we dissect it as <math>\mathsf{T} = \mathsf{d} = (\boldsymbol\varepsilon, \mathrm{d}),\!</math> where <math>\mathrm{d}\!</math> is the operator that yields the first order differential <math>\mathrm{d}F\!</math> when applied to a transformation <math>F,\!</math> and whose name is legion.

−

~~Table~~ ~~38 shows~~ a ~~typical scheme~~ of ~~computation, following a systematic method~~ of ~~exploiting boolean expansions over selected variables,~~ and ~~ultimately developing~~ E~~''J'' over~~ the ~~cells of [''u''~~,~~ ''v'']~~. The ~~critical step~~ of ~~this procedure uses~~ the ~~facts that (0~~,~~ ''x'') ~~=~~ 0 + ''x'' = ''x'' and~~ (1,~~ ''x''~~)~~ = 1 + ''x'' = (''x'') for any boolean variable ''x''~~.

+

Figure 34 illustrates a stage of analysis where we ignore everything but the tangent functor <math>\mathsf{T}\!</math> and attend to it chiefly as it bears on the first order differential <math>\mathrm{d}F\!</math> in the analytic expansion of <math>F.\!</math> In this situation we often refer to the extended universes <math>\mathrm{E}U^\bullet\!</math> and <math>\mathrm{E}X^\bullet\!</math> under the equivalent designations <math>\mathsf{T}U^\bullet\!</math> and <math>\mathsf{T}X^\bullet,\!</math> respectively. The purpose of the tangent functor <math>\mathsf{T}\!</math> is to extract the tangent map <math>\mathsf{T}F\!</math> at each point of <math>U^\bullet,\!</math> and the tangent map <math>\mathsf{T}F = (\boldsymbol\varepsilon, \mathrm{d})F\!</math> tells us not only what the transformation <math>F\!</math> is doing at each point of the universe <math>U^\bullet\!</math> but also what <math>F\!</math> is doing to states in the neighborhood of that point, approximately, linearly, and relatively speaking.

−

~~~~

+

{| align="center" border="0" cellpadding="10"

−

{| align="center" border="1" cellpadding="8" ~~cellspacing="0" style="font-weight:bold; text-align:left; width:96%"~~

−

~~|+ Table 38. Computation of E''J'' (Method 1)~~

|

−

~~{| align~~=~~"left" border~~=~~"0" cellpadding~~=~~"0" cellspacing~~=~~"0" style~~=~~"font-weight:bold; text-align:left; width:100%"~~

+

<pre>

−

| ~~width="8%"~~ | ~~E''J''~~

+

U% $T$ $T$U% $T$U%

−

| ~~width="4%"~~ | =

+

o------------------>o============o

−

| ~~width~~=~~"44~~%" | ''J''‹''u'' ~~+ d~~''u'', ''v'' ~~+ d~~''v~~''›~~

+

| | |

−

| width="44%" |  

+

| | |

+

| | |

+

| | |

+

F | | $T$F = | <!e!, d> F

+

| | |

+

| | |

+

| | |

+

v v v

+

o------------------>o============o

+

X% $T$ $T$X% $T$X%

+

Figure 34. Tangent Functor Diagram

+

</pre>

+

|}

+

* '''NB.''' There is one aspect of the preceding construction that remains especially problematic. Why did we define the operators <math>\mathrm{W}\!</math> in <math>\{ \eta, \mathrm{E}, \mathrm{D}, \mathrm{d}, \mathrm{r} \}\!</math> so that the ranges of their resulting maps all fall within the realms of differential quality, even fabricating a variant of the tacit extension operator to have that character? Clearly, not all of the operator maps <math>\mathrm{W}F\!</math> have equally good reasons for placing their values in differential stocks. The reason for it appears to be that, without doing this, we cannot justify the comparison and combination of their functional values in the various analytic steps. By default, only those values in the same functional component can be brought into algebraic modes of interaction. Up till now the only mechanism provided for their broader association has been a purely logical one, their common placement in a target universe of discourse, but the task of converting this logical circumstance into algebraic forms of application has not yet been taken up.

+

===Transformations of Type '''B'''2 → '''B'''1===

+

To study the effects of these analytic operators in the simplest possible setting, let us revert to a still more primitive case. Consider the singular proposition <math>J(u, v)= u\!\cdot\!v,\!</math> regarded either as the functional product of the maps <math>u\!</math> and <math>v\!</math> or as the logical conjunction of the features <math>u\!</math> and <math>v,\!</math> a map whose fiber of truth <math>J^{-1}(1)\!</math> picks out the single cell of that logical description in the universe of discourse <math>U^\bullet.\!</math> Thus <math>J,\!</math> or <math>u\!\cdot\!v,\!</math> may be treated as another name for the point whose coordinates are <math>(1, 1)\!</math> in <math>U^\bullet.\!</math>

+

====Analytic Expansion of Conjunction====

+

{| width="100%" cellpadding="0" cellspacing="0"

+

| width="4%" |  

+

| width="92%" |

+

In her sufferings she read a great deal and discovered that she had lost something, the possession of which she had previously not been much aware of: a soul.

+

What is that? It is easily defined negatively: it is simply what curls up and hides when there is any mention of algebraic series.

+

| width="4%" |  

|-

−

|  

+

| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 118]

+

|}

+

Figure 35 pictures the form of conjunction <math>J : \mathbb{B}^2 \to \mathbb{B}\!</math> as a transformation from the 2-dimensional universe <math>[u, v]\!</math> to the 1-dimensional universe <math>[x].\!</math> This is a subtle but significant change of viewpoint on the proposition, attaching an arbitrary but concrete quality to its functional value. Using the language introduced earlier, we can express this change by saying that the proposition <math>J : \langle u, v \rangle \to \mathbb{B}\!</math> is being recast into the thematized role of a transformation <math>J : [u, v] \to [x],\!</math> where the new variable <math>x\!</math> takes the part of a thematic variable <math>\check{J}.\!</math>

+

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"

+

| [[Image:Diff Log Dyn Sys -- Figure 35 -- A Conjunction Viewed as a Transformation.gif|center]]

|-

−

| ~~width~~="8%" | ~~ ~~

+

| height="20px" valign="top" | <math>\text{Figure 35.} ~~ \text{Conjunction as Transformation}\!</math>

−

| width="4%" | =

+

|}

−

| width="44%" | ~~(''u'', d''u'')(''v'', d''v'')~~

+

−

| width="44%" | ~~&nbsp~~;

+

=====Tacit Extension of Conjunction=====

+

{| width="100%" cellpadding="0" cellspacing="0"

+

| width="40%" |  

+

| width="60%" |

+

I teach straying from me, yet who can stray from me?

+

I follow you whoever you are from the present hour;

+

My words itch at your ears till you understand them.

|-

|  

−

|-

+

| align="right" | — Walt Whitman, ''Leaves of Grass'', [Whi, 83]

−

~~| width="8%" |  ~~

−

~~| width="4%" | =~~

−

~~| width="44%" |  ''u''  ''v''  ''J''‹1 + d''u'', 1 + d''v''›~~

−

~~| width="44%" | +~~

−

|-

−

~~| width="8%" |  ~~

−

~~| width="4%" |  ~~

−

~~| width="44%" |  ''u'' (''v'') ''J''‹1 + d''u'', 0 + d''v''›~~

−

~~| width="44%" | +~~

−

|-

−

~~| width="8%" |  ~~

−

~~| width="4%" |  ~~

−

~~| width="44%" | (''u'') ''v''  ''J''‹0 + d''u'', 1 + d''v''›~~

−

~~| width="44%" | +~~

−

|-

−

~~| width="8%" |  ~~

−

~~| width="4%" |  ~~

−

~~| width="44%" | (''u'')(''v'') ''J''‹0 + d''u'', 0 + d''v''›~~

−

~~| width="44%" |  ~~

−

|-

−

~~|  ~~

−

|-

−

~~| width="8%" |  ~~

−

~~| width="4%" | =~~

−

~~| width~~="~~44%~~" | &~~nbsp;''u''  ''v'' &nbsp~~;~~''J''‹(d''u'')~~,~~ (d''v'')›~~

−

~~| width="44%" | +~~

−

|-

−

~~| width="8%" |  ~~

−

~~| width="4%" |  ~~

−

~~| width="44%" |  ''u'' (''v'') ''J'~~'~~‹(d~~''u''),~~  d''v'' ›~~

−

~~| width="44%" | +~~

−

|-

−

~~| width="8%" |  ~~

−

~~| width="4%" |  ~~

−

~~| width="44%" | (''u'') ''v''  ''J''‹ d''u'' ~~, ~~(d''v'')›~~

−

~~| width="44%" | +~~

−

|-

−

~~| width="8%" |  ~~

−

~~| width="4%" |  ~~

−

~~| width="44%" | (''u'')(''v'') ''J''‹ d''u'' ,  d''v'' ›~~

−

~~| width="44%" |  ~~

|}

+

Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math> Of immediate interest are the tacit extensions <math>\boldsymbol\epsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math> In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math> The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table 36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math>

+

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:75%"

+

|+ style="height:30px" | <math>\text{Table 36.} ~~ \text{Computation of}~ \boldsymbol\varepsilon J\!</math>

+

|

+

<math>\begin{array}{*{9}{l}}

+

\boldsymbol\varepsilon J & = & J {}_{^\langle} u, v {}_{^\rangle}

+

\\[4pt]

+

& = & u \cdot v

+

\\[4pt]

+

& = & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}

+

& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{ } \mathrm{d}v \texttt{ }

+

& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{(} \mathrm{d}v \texttt{)}

+

& + & u \cdot v \cdot \texttt{ } \mathrm{d}u \texttt{ } \cdot \texttt{ } \mathrm{d}v \texttt{ }

+

\end{array}\!</math>

|-

|

−

{| align="~~left~~" border~~="0" cellpadding~~="0" cellspacing="0" style="~~font-weight:bold;~~ text-align:~~left~~; width:100%"

+

<math>\begin{array}{*{4}{l}}

−

| ~~width="8%" | E''~~J''

+

\boldsymbol\varepsilon J

−

~~| width="23%" | = ''u'' ''v'' (d''u'')(d''v'')~~

+

& = && u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{(} \mathrm{d}v \texttt{)}

−

~~| width="23%" |  ~~

+

\\[4pt]

−

~~| width="23%"~~ | ~~ ~~

+

&& + & u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \texttt{~} \mathrm{d}v \texttt{~}

−

~~| width="23%" |  ~~

+

\\[4pt]

+

&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{(} \mathrm{d}v \texttt{)}

+

\\[4pt]

+

&& + & u \cdot v \cdot \texttt{~} \mathrm{d}u \texttt{~} \cdot \texttt{~} \mathrm{d}v \texttt{~}

+

\end{array}\!</math>

+

|}

+

+

The lower portion of the Table contains the dispositional features of <math>\boldsymbol\varepsilon J\!</math> arranged in such a way that the variety of ordinary features spreads across the rows and the variety of differential features runs through the columns. This organization serves to facilitate pattern matching in the remainder of our computations. Again, the tacit extension is usually so trivial a concern that we do not always bother to make an explicit note of it, taking it for granted that any function <math>F\!</math> being employed in a differential context is equivalent to <math>\boldsymbol\varepsilon F\!</math> for a suitable <math>\boldsymbol\varepsilon.\!</math>

+

Figures 37-a through 37-d present several pictures of the proposition <math>J\!</math> and its tacit extension <math>\boldsymbol\varepsilon J.\!</math> Notice in these Figures how <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet\!</math> visibly extends <math>J\!</math> in <math>U^\bullet\!</math> by annexing to the indicated cells of <math>J\!</math> all the arcs that exit from or flow out of them. In effect, this extension attaches to these cells all the dispositions that spring from them, in other words, it attributes to these cells all the conceivable changes that are their issue.

+

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"

+

| [[Image:Diff Log Dyn Sys -- Figure 37-a -- Tacit Extension of J.gif|center]]

|-

−

| ~~width~~="8%" ~~|  ~~

+

| height="20px" valign="top" | <math>\text{Figure 37-a.} ~~ \text{Tacit Extension of}~ J ~\text{(Areal)}\!</math>

−

~~| width="23%" |  ~~

−

~~| width="23%" | + ''u'' (''v'') (d''u'') d''v''~~

−

~~| width="23%" |  ~~

−

~~| width="23%" |  ~~

−

|-

−

~~| width="8%" |  ~~

−

~~| width="23%" |  ~~

−

~~| width="23%" |  ~~

−

~~| width="23%" | + (''u'') ''v'' d''u'' (d''v'')~~

−

~~| width~~="~~23%~~" | ~~ ~~

−

|-

−

~~| width="8%" |  ~~

−

~~| width="23%" |  ~~

−

~~| width="23%" |  ~~

−

~~| width="23%" |  ~~

−

~~| width="23%" | + (''u'')~~(~~''v''~~) ~~d''u'' d''v''~~

|}

−

|}

−

~~ ~~

−

Table 39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.

+

−

~~~~

+

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"

−

{| align="center" border="~~1" cellpadding="12~~" cellspacing="0" style="~~font-weight:bold;~~ text-align:~~left~~; width:96%"

+

| [[Image:Diff Log Dyn Sys -- Figure 37-b -- Tacit Extension of J.gif|center]]

−

|~~+ Table 39. Computation of E''J'' (Method 2)~~

−

|

−

~~{| align="left" border="0" cellpadding="0" cellspacing="0" style="font~~-~~weight:bold; text~~-~~align:left; width:100%"~~

−

~~| width="8%" | E''J''~~

−

~~| colspan="2" | = ‹''u'' + d''u''› <math>\cdot</math> ‹''v'' + d''v''›~~

−

~~| width="23%" |  ~~

−

~~| width="23%" |  ~~

−

|-

−

~~|  ~~

−

|-

−

~~| width="8%" |  ~~

−

~~| colspan="2" | = ''u'' ''v'' + ''u'' d''v'' + ''v'' d''u'' + d''u'' d''v''~~

−

~~| width="23%" |  ~~

−

~~| width="23%" |  ~~

−

|-

−

| ~~ ~~

|-

−

| ~~width~~="8%" ~~| E''J''~~

+

| height="20px" valign="top" | <math>\text{Figure 37-b.} ~~ \text{Tacit Extension of}~ J ~\text{(Bundle)}\!</math>

−

~~| width~~="~~23%~~" | ~~= ''u'' ''v'' (d''u'')(d''v'')~~

−

~~| width="23%" | + ''u'' (''v'') (d''u'') d''v''~~

−

~~| width="23%" | + (''u'') ''v'' d''u'' (d''v'')~~

−

~~| width="23%" | + (''u'')~~(~~''v''~~) ~~d''u'' d''v''~~

|}

−

|}

−

~~ ~~

−

~~Figures 40-a through 40-d present several views of the enlarged proposition E''J''.~~

+

−

~~ ~~

+

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"

−

~~~~[[Image:Diff Log Dyn Sys -- Figure 40-a -- ~~Enlargement~~ of J.gif|center]]~~~~

+

| [[Image:Diff Log Dyn Sys -- Figure 37-c -- Tacit Extension of J.gif|center]]

−

~~<center>~~'''~~Figure 40-a. ~~Enlargement~~ of ''J~~''  ~~(~~Areal~~)~~'''~~</~~font></center></p>

+

|-

+

| height="20px" valign="top" | <math>\text{Figure 37-c.} ~~ \text{Tacit Extension of}~ J ~\text{(Compact)}\!</math>

+

|}

−

~~[[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]~~

−

~~<center>'''Figure 40-b. Enlargement of ''J''  (Bundle)'''</center>~~

−

~~ ~~

+

{| align="center" border="0" cellspacing="10" style="text-align:center; width:100%"

−

~~~~[[Image:Diff Log Dyn Sys -- Figure 40-c -- ~~Enlargement~~ of J.gif|center]]~~~~

+

| [[Image:Diff Log Dyn Sys -- Figure 37-d -- Tacit Extension of J.gif|center]]

−

~~<center>~~'''~~Figure 40-c. ~~Enlargement~~ of ''J~~''  ~~(~~Compact~~)~~'''~~</~~font></center></p>

+

|-

+

| height="20px" valign="top" | <math>\text{Figure 37-d.} ~~ \text{Tacit Extension of}~ J ~\text{(Digraph)}\!</math>

+

|}

−

+

The computational scheme shown in Table 36 treated <math>J\!</math> as a proposition in <math>U^\bullet\!</math> and formed <math>\boldsymbol\varepsilon J\!</math> as a proposition in <math>\mathrm{E}U^\bullet.\!</math> When <math>J\!</math> is regarded as a mapping <math>J : U^\bullet \to X^\bullet\!</math> then <math>\boldsymbol\varepsilon J\!</math> must be obtained as a mapping <math>\boldsymbol\varepsilon J : \mathrm{E}U^\bullet \to X^\bullet.\!</math> By default, the tacit extension of the map <math>J : [u, v] \to [x]\!</math> is naturally taken to be a particular map, one that has the following form:

−

~~[[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of~~ J.~~gif|center]]~~

−

<~~center~~><~~font size="+1"~~>~~'''Figure 40-d. Enlargement of ''~~J~~''  (Digraph)'''~~</~~font~~></~~center~~>

−

An intuitive reading of the proposition E''J'' becomes available at this point, and may be useful. Recall that propositions in the extended universe E''U'' • express the ''dispositions'' of system and the constraints that are placed on them. In other words, a differential proposition in E''U'' •<~~/sup~~> ~~can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand E''~~J~~'' as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of ''J''~~, ~~that is~~, ~~the region of the universe where ''J'' is true. This interpretation is visibly clear in the Figures above~~, ~~and appeals~~ to ~~the imagination in a satisfying way~~, but it has the added benefit of giving fresh meaning to the original name of the shift operator E. Namely, E''J'' can be read as a proposition that ''enlarges'' on the meaning of ''J'', in the sense of explaining its practical bearings and clarifying what it means in terms of the available options for differential action and the consequential effects that result from each choice.

+

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

+

| <math>\boldsymbol\varepsilon J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [x] \subseteq [x, \mathrm{d}x].\!</math>

+

|}

−

~~Treated this way,~~ the ~~enlargement E''~~J~~'' has strong ties to~~ the ~~normal use~~ of ~~''J'', no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating~~ the ~~models of ''J'',~~ in ~~effect, pointing to~~ the ~~interpretive elements in its fiber of truth ''J''~~<~~sup~~>~~–1~~</~~sup~~>~~(1). It is this kind of ''use'' that is often compared with the ''mention'' of a proposition, and thereby hangs a tale~~.

+

This is the map that looks like <math>J\!</math> when painted in the frame of the extended source universe and that takes the same thematic variable in the extended target universe as the one that <math>J\!</math> already employs.

−

~~=====Digression~~ : ~~Reflection on Use and Mention=====~~

+

But the choice of a particular thematic variable, for example <math>x\!</math> for <math>\check{J},\!</math> is a shade more arbitrary than the initial choice of variable names <math>\{ u, v \}.\!</math> This means that the map we are calling the ''trope extension'', namely:

−

{| ~~width~~="~~100%~~" cellpadding="~~0" cellspacing=~~"0"

+

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

−

| width="4%" ~~|  ~~

+

| <math>\eta J : [u, v, \mathrm{d}u, \mathrm{d}v] \to [\mathrm{d}x] \subseteq [x, \mathrm{d}x],\!</math>

−

| ~~width="92%" |~~

−

Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked — almost as one might turn a stone over to see what its hidden side is like or what is covered by it.

−

~~| width="4%" |  ~~

−

|-

−

~~| align="right" colspan="3" | — John Dewey~~, ~~''How We Think''~~, [~~Dew~~, 57]

|}

−

~~The contrast drawn~~ in ~~logic between~~ the ''~~use~~'' ~~and~~ the ''~~mention~~'' of ~~a proposition corresponds to~~ the ~~difference that we observe in functional terms between using "~~''J''~~ "~~ to ~~indicate~~ the ~~region~~ ''J''<~~sup~~>~~–1~~</~~sup~~>~~(1) and using "~~''J'' ~~" to indicate the function~~ ''J''~~. You may think that one of these uses ought to be proscribed~~, ~~and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name "~~''J'' ~~" is used as a sign of the function~~ ''J'', ~~and if the function~~ ''J'' ~~has its use in signifying something else~~, ~~as would constantly be~~ the ~~case when some future theory~~ of ~~signs has given a functional meaning to every sign whatsoever, then is not "~~''J'' ~~" by transitivity a sign of the thing itself? There are~~, ~~of course~~, ~~two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable~~, ~~indeed~~, ~~not many. Not every feature~~ of ~~a feature is a feature~~ of ~~the featuree. Otherwise we have an inference like the following: If a buffalo is white, and white is a color, then a buffalo is a color. But a buffalo is not, only buff is~~.

+

since it looks just the same as <math>\boldsymbol\varepsilon J\!</math> in the way that its fibers paint the source domain, belongs just as fully to the family of tacit extensions, generically considered.

+

These considerations have the practical consequence that all of our computations and illustrations of <math>\epsilon</math>''J'' perform the double duty of capturing an image of <math>\eta</math>''J'' as well. In other words, we are saved the work of carrying out calculations and drawing figures for the trope extension <math>\eta</math>''J'', because the exercise would be identical to the work already done for <math>\epsilon</math>''J''. Since the computations given for <math>\epsilon</math>''J'' are expressed solely in terms of the variables {''u'', ''v'', d''u'', d''v''}, these variables work equally well for finding <math>\eta</math>''J''. Furthermore, since each of the above Figures shows only how the level sets of <math>\epsilon</math>''J'' partition the extended source universe E''U'' • = [''u'', ''v'', d''u'', d''v''], all of them serve equally well as portraits of <math>\eta</math>''J''.

−

The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.

+

=====Enlargement Map of Conjunction=====

{| width="100%" cellpadding="0" cellspacing="0"

| width="4%" |  

| width="92%" |

−

~~The well-known capacity~~ that ~~thoughts have —~~ as ~~doctors~~ have ~~discovered &mdash~~; for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the ~~individual being with other people and~~ things; ~~but unfortunately what gives them their power~~ of ~~healing seems to be the same as what diminishes the quality of personal experience in them~~.

+

No one could have established the existence of any details that might not just as well have existed in earlier times too; but all the relations between things had shifted slightly. Ideas that had once been of lean account grew fat.

| width="4%" |  

|-

−

| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, ~~130~~]

+

| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 62]

|}

−

~~=====Difference Map~~ of ~~Conjunction=====~~

+

The enlargement map E''J'' is computed from the proposition ''J'' by making a particular class of formal substitutions for its variables, in this case ''u'' + d''u'' for ''u'' and ''v'' + d''v'' for ''v'', and subsequently expanding the result in whatever way happens to be convenient for the end in view.

−

~~{| width="100%" cellpadding="0" cellspacing="0"~~

+

Table 38 shows a typical scheme of computation, following a systematic method of exploiting boolean expansions over selected variables, and ultimately developing E''J'' over the cells of [''u'', ''v'']. The critical step of this procedure uses the facts that (0, ''x'') = 0 + ''x'' = ''x'' and (1, ''x'') = 1 + ''x'' = (''x'') for any boolean variable ''x''.

−

~~| width="4%" |~~  

−

~~| width="92%" |~~

−

~~"It doesn't matter what one does", the Man Without Qualities said to himself~~, ~~shrugging his shoulders. "In~~ a ~~tangle~~ of ~~forces like this it doesn~~'~~t make a scrap~~ of ~~difference~~." ~~He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity~~ of ~~contact. And then~~, ~~striding through his adjacent dressing-room~~, ~~he passed a punching-ball that hung there~~; ~~he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.~~

−

~~| width~~=~~"4%" |~~  

−

|-

−

~~| align="right" colspan~~=~~"3" |~~ &~~mdash~~; ~~Robert Musil,~~ ''~~The Man Without Qualities~~''~~, [Mus, 8]~~

−

|}

−

~~With the tacit extension map~~ <~~math~~>\epsilon</math>''J'' and the enlargement map E''J'' well in place, the difference map D''J'' can be computed along the lines displayed in Table 41, ending up, in this instance, with an expansion of D''J'' over the cells of [''u'', ''v''].

+

−

~~~~

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"

−

{| align="center" border="1" cellpadding="4" cellspacing="0" style="~~font-weight:bold;~~ text-align:left; width:96%"

+

|+ style="height:30px" | <math>\text{Table 38.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 1)}\!</math>

−

|+ Table 41. Computation of ~~D''~~J'' (Method 1)

|

−

{~~| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"~~

+

<math>\begin{array}{*{9}{l}}

−

~~| width="8%" | D''J''~~

+

\mathrm{E}J & = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}

−

~~| width="4%" | =~~

+

\\[4pt]

−

~~| width="42%" |~~ E''J''

+

& = & \texttt{(} u, \mathrm{d}u \texttt{)} \cdot \texttt{(} v, \mathrm{d}v \texttt{)}

−

~~| width~~=~~"4%" |~~ +

+

\\[4pt]

−

~~| width="42%" | <math>~~\~~epsilon</math>''J''~~

+

& = & \texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(1 + \mathrm{d}u, 1 + \mathrm{d}v)}

−

|-

+

& + & \texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(1 + \mathrm{d}u, \mathrm{d}v)}

−

~~| width~~=~~"8%" |~~ &~~nbsp;~~

+

& + & \texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, 1 + \mathrm{d}v)}

−

~~| width="4%" | =~~

+

& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}

−

~~| width~~=~~"42%" | ''J''‹''~~u'' + d''u'', ''v'' + d''v~~''›~~

+

\\[4pt]

−

~~| width="4%" |~~ +

+

& = &

−

~~| width="42%" | ''J''‹''~~u'', ''v~~''›~~

+

\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)})}

−

|-

+

& + &

−

~~| width~~=~~"8%" |~~ &~~nbsp;~~

+

\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot J_{(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v)}

−

~~| width="4%" | =~~

+

& + &

−

~~| width="42%" |~~ (''u'', d''u'')(~~''v''~~, d''v'')

+

\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot J_{(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)})}

−

~~| width="4%" |~~ +

+

& + &

−

~~| width="42%" | ''~~u~~'' ''~~v''

+

\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot J_{(\mathrm{d}u, \mathrm{d}v)}

−

|}

+

\end{array}\!</math>

|-

|

−

{~~| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"~~

+

<math>\begin{array}{*{9}{l}}

−

~~| width="8%" | D''~~J''

+

\mathrm{E}J

−

~~| width="3%" | =~~

+

& = &

−

~~| width="20%" align=center | 0~~

+

\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}

−

~~| width="23%" |~~ &~~nbsp;~~

+

\\[4pt]

−

~~| width~~=~~"23%" |~~ &~~nbsp;~~

+

&&& + &

−

~~| width="23%" |  ~~

+

\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }

−

|-

+

\\[4pt]

−

~~| width="8%" |  ~~

+

&&&&& + &

−

~~| width="3%" | +~~

+

\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}

−

~~| width="20%" | ''~~u~~'' ''~~v'' (d''u'') d''v''

+

\\[4pt]

−

~~| width="23%" |~~ + ''u'' (''v'')(d''u'') d''v''

+

&&&&&&& + &

−

~~| width="23%" |~~ &~~nbsp;~~

+

\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }

−

~~| width="23%" |~~ &~~nbsp;~~

+

\end{array}</math>

−

|-

−

~~| width="8%" |~~ &~~nbsp;~~

−

~~| width="3%" | +~~

−

~~| width="20%" | ''u'' ''v''~~&~~nbsp;~~&~~nbsp;d''u'' (d''v'')~~

−

~~| width="23%" |~~ &~~nbsp;~~

−

~~| width="23%" | +~~ (''u'') ''v'' d''u'' (d''v'')

−

~~| width="23%" |~~ &~~nbsp;~~

−

|-

−

~~| width="8%" |~~ &~~nbsp;~~

−

~~| width="3%" | +~~

−

~~| width="20%" | ''u'' ''v''~~&~~nbsp;~~&~~nbsp;d''u''~~&~~nbsp;~~&~~nbsp;d''v''~~

−

~~| width="23%" |~~ &~~nbsp;~~

−

~~| width="23%" |~~ &~~nbsp;~~

−

~~| width="23%" | +~~ (''u'')(''v'') d''u'' d''v''

|}

−

|-

+

+

Table 39 exhibits another method that happens to work quickly in this particular case, using distributive laws to multiply things out in an algebraic manner, arranging the notations of feature and fluxion according to a scale of simple character and degree. Proceeding this way leads through an intermediate step which, in chiming the changes of ordinary calculus, should take on a familiar ring. Consequential properties of exclusive disjunction then carry us on to the concluding line.

+

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:80%"

+

|+ style="height:30px" | <math>\text{Table 39.} ~~ \text{Computation of}~ \mathrm{E}J ~\text{(Method 2)}\!</math>

|

−

{~~| align~~=~~"left" border~~=~~"0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"~~

+

<math>\begin{array}{*{9}{c}}

−

~~| width="8%" | D''~~J''

+

\mathrm{E}J

−

~~| width="3%" |~~ =

+

& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)

−

~~| width="23%" | ''~~u~~'' ''~~v~~'' (~~(d''u'')(d''v~~'')~~)

+

\\[6pt]

−

~~| width="22%" |~~ + ''u'' (''v'')(d''u'') d''v''

+

& = & u \cdot v

−

~~| width="22%" |~~ + (''u'') ''v'' d''u'' (d''v'')

+

& + & u \cdot \mathrm{d}v

−

~~| width="22%" |~~ + (''u'')(''v'') d''u'' d''v''

+

& + & \mathrm{d}u \cdot v

+

& + & \mathrm{d}u \cdot \mathrm{d}v

+

\\[6pt]

+

\mathrm{E}J

+

& = &

+

\texttt{ } u \texttt{ } \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}

+

& + &

+

\texttt{ } u \texttt{ } \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \texttt{ } \mathrm{d}v \texttt{ }

+

& + &

+

\texttt{(} u \texttt{)} \texttt{ } v \texttt{ } \cdot \texttt{ } \mathrm{d}u \texttt{ } \texttt{(} \mathrm{d}v \texttt{)}

+

& + &

+

\texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot \texttt{ } \mathrm{d}u \texttt{ }~\texttt{ } \mathrm{d}v \texttt{ }

+

\end{array}\!</math>

|}

−

|}

−

~~ ~~

−

~~Alternatively,~~ the ~~difference map D~~''J'' ~~can be expanded over the cells~~ of [d''u'', d''v''~~] to arrive at~~ the ~~formulation shown~~ in ~~Table~~ 42. ~~The same development would~~ be ~~obtained from~~ the ~~previous Table by collecting~~ terms ~~in an alternate manner~~, ~~along~~ the ~~rows rather than~~ the ~~columns~~ of the ~~middle portion~~ of the ~~Table~~.

+

+

Figures 40-a through 40-d present several views of the enlarged proposition E''J''.

+

+

[[Image:Diff Log Dyn Sys -- Figure 40-a -- Enlargement of J.gif|center]]

+

<center>'''Figure 40-a. Enlargement of ''J''  (Areal)'''</center>

+

+

[[Image:Diff Log Dyn Sys -- Figure 40-b -- Enlargement of J.gif|center]]

+

<center>'''Figure 40-b. Enlargement of ''J''  (Bundle)'''</center>

+

+

[[Image:Diff Log Dyn Sys -- Figure 40-c -- Enlargement of J.gif|center]]

+

<center>'''Figure 40-c. Enlargement of ''J''  (Compact)'''</center>

+

+

[[Image:Diff Log Dyn Sys -- Figure 40-d -- Enlargement of J.gif|center]]

+

<center>'''Figure 40-d. Enlargement of ''J''  (Digraph)'''</center>

+

An intuitive reading of the proposition E''J'' becomes available at this point, and may be useful. Recall that propositions in the extended universe E''U'' • express the ''dispositions'' of system and the constraints that are placed on them. In other words, a differential proposition in E''U'' • can be read as referring to various changes that a system might undergo in and from its various states. In particular, we can understand E''J'' as a statement that tells us what changes need to be made with regard to each state in the universe of discourse in order to reach the ''truth'' of ''J'', that is, the region of the universe where ''J'' is true. This interpretation is visibly clear in the Figures above, and appeals to the imagination in a satisfying way, but it has the added benefit of giving fresh meaning to the original name of the shift operator E. Namely, E''J'' can be read as a proposition that ''enlarges'' on the meaning of ''J'', in the sense of explaining its practical bearings and clarifying what it means in terms of the available options for differential action and the consequential effects that result from each choice.

+

Treated this way, the enlargement E''J'' has strong ties to the normal use of ''J'', no matter whether it is understood as a proposition or a function, namely, to act as a figurative device for indicating the models of ''J'', in effect, pointing to the interpretive elements in its fiber of truth ''J''–1(1). It is this kind of ''use'' that is often compared with the ''mention'' of a proposition, and thereby hangs a tale.

+

=====Digression : Reflection on Use and Mention=====

−

~~~~

+

{| width="100%" cellpadding="0" cellspacing="0"

−

{| ~~align~~="~~center" border="1" cellpadding="12" cellspacing="0" style="font-weight:bold; text-align:left; width:96~~%"

+

| width="4%" |  

−

~~|+ Table 42. Computation of D''J'' (Method 2)~~

+

| width="92%" |

−

|

+

Reflection is turning a topic over in various aspects and in various lights so that nothing significant about it shall be overlooked — almost as one might turn a stone over to see what its hidden side is like or what is covered by it.

−

~~{| align="left" border="0~~" cellpadding="0" cellspacing="0~~" style="font-weight:bold; text-align:left; width:100%~~"

+

| width="4%" |  

−

| width="8%" | ~~D''J''~~

−

| width="4%" | =

−

~~| width="16%" | <math>\epsilon</math>''J''~~

−

| width="4%" | +

−

~~| colspan="5" | E''J''~~

|-

−

| ~~width~~="8%" ~~|  ~~

+

| align="right" colspan="3" | — John Dewey, ''How We Think'', [Dew, 57]

−

~~| width="4%" | =~~

−

~~| width="16%" | ''J''‹''u'', ''v''›~~

−

~~| width="4%" | +~~

−

| colspan="5" ~~| ''J''‹''u'' + d''u'', ''v'' + d''v''›~~

−

|-

−

~~| width="8%"~~ | &~~nbsp~~;

−

~~| width="4%" | =~~

−

~~| width="16%" | ''u'' ''v''~~

−

~~| width="4%" | +~~

−

~~| colspan="5" | (''u''~~, ~~d''u'')(~~''v'', ~~d''v'')~~

−

|-

−

~~| width="8%" |  ~~

−

~~| width="4%" | =~~

−

~~| width="16%" | 0~~

−

~~| width="4%" | +~~

−

~~| width="16%" | ''u'' d''v''~~

−

~~| width="4%" | +~~

−

~~| width="16%" | ''v'' d''u''~~

−

~~| width="4%" | +~~

−

~~| width="28%" | d''u'' d''v''~~

−

|-

−

~~| width="8%" | D''J''~~

−

~~| width="4%" | =~~

−

~~| width="16%" | 0~~

−

~~| width="4%" | +~~

−

~~| width="16%" | ''u'' (d''u'') d''v''~~

−

~~| width="4%" | +~~

−

~~| width="16%" | ''v'' d''u'' (d''v'')~~

−

~~| width="4%" | +~~

−

~~| width="28%" | ((''u''~~, ~~''v'')) d''u'' d''v''~~

−

|}

−

~~ ~~

−

~~Even more simply,~~ the ~~same result is reached by matching up~~ the ~~propositional coefficients~~ of <~~math~~>~~\epsilon~~</~~math~~>''J'' ~~and E~~''J'' ~~along~~ the ~~cells of [d~~''u'', d''v''] and ~~adding~~ the ~~pairs under boolean sums (~~that is, "~~mod 2~~", ~~where 1~~ + 1 = 0), ~~as shown~~ in ~~Table~~ 43.

+

The contrast drawn in logic between the ''use'' and the ''mention'' of a proposition corresponds to the difference that we observe in functional terms between using "''J'' " to indicate the region ''J''–1(1) and using "''J'' " to indicate the function ''J''. You may think that one of these uses ought to be proscribed, and logicians are quick to prescribe against their confusion. But there seems to be no likelihood in practice that their interactions can be avoided. If the name "''J'' " is used as a sign of the function ''J'', and if the function ''J'' has its use in signifying something else, as would constantly be the case when some future theory of signs has given a functional meaning to every sign whatsoever, then is not "''J'' " by transitivity a sign of the thing itself? There are, of course, two answers to this question. Not every act of signifying or referring need be transitive. Not every warrant or guarantee or certificate is automatically transferable, indeed, not many. Not every feature of a feature is a feature of the featuree. Otherwise we have an inference like the following: If a buffalo is white, and white is a color, then a buffalo is a color. But a buffalo is not, only buff is.

+

The logical or pragmatic distinction between use and mention is cogent and necessary, and so is the analogous functional distinction between determining a value and determining what determines that value, but so are the normal techniques that we use to make these distinctions apply flexibly in practice. The way that the hue and cry about use and mention is raised in logical discussions, you might be led to think that this single dimension of choices embraces the only kinds of use worth mentioning and the only kinds of mention worth using. It will constitute the expeditionary and taxonomic tasks of that future theory of signs to explore and to classify the many other constellations and dimensions of use and mention that are yet to be opened up by the generative potential of full-fledged sign relations.

+

{| width="100%" cellpadding="0" cellspacing="0"

+

| width="4%" |  

+

| width="92%" |

+

The well-known capacity that thoughts have — as doctors have discovered — for dissolving and dispersing those hard lumps of deep, ingrowing, morbidly entangled conflict that arise out of gloomy regions of the self probably rests on nothing other than their social and worldly nature, which links the individual being with other people and things; but unfortunately what gives them their power of healing seems to be the same as what diminishes the quality of personal experience in them.

+

| width="4%" |  

+

|-

+

| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 130]

+

|}

+

=====Difference Map of Conjunction=====

+

{| width="100%" cellpadding="0" cellspacing="0"

+

| width="4%" |  

+

| width="92%" |

+

"It doesn't matter what one does", the Man Without Qualities said to himself, shrugging his shoulders. "In a tangle of forces like this it doesn't make a scrap of difference." He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.

+

| width="4%" |  

+

|-

+

| align="right" colspan="3" | — Robert Musil, ''The Man Without Qualities'', [Mus, 8]

+

|}

+

With the tacit extension map <math>\epsilon</math>''J'' and the enlargement map E''J'' well in place, the difference map D''J'' can be computed along the lines displayed in Table 41, ending up, in this instance, with an expansion of D''J'' over the cells of [''u'', ''v''].

+

−

~~~~

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"

−

{| align="center" border="1" cellpadding="8" cellspacing="0" style="~~font-weight:bold;~~ text-align:left; width:96%"

+

|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 1)}\!</math>

−

|+ Table 43. Computation of D''J'' (Method 3)

|

−

{~~| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"~~

+

<math>\begin{array}{*{9}{l}}

−

~~| width="6%" |~~ D''J''

+

\mathrm{D}J

−

~~| width="3%" |~~ =

+

& = & \mathrm{E}J

−

~~| width="20%" | <math>~~\~~epsilon</math>''~~J''

+

& + & \boldsymbol\varepsilon J

−

~~| width~~=~~"3%" |~~ +

+

\\[6pt]

−

~~| width~~=~~"20%" | E''J''~~

+

& = & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}

−

~~| width="48%" |~~ &~~nbsp;~~

+

& + & J_{(u, v)}

−

|}

+

\\[6pt]

+

& = & \texttt{(} u, \mathrm{d}u \texttt{)} \cdot \texttt{(} v, \mathrm{d}v \texttt{)}

+

& + & u \cdot v

+

\end{array}</math>

|-

|

−

~~{| align="left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold; text-align:left; width:100%"~~

+

<math>\begin{array}{*{9}{l}}

−

~~| width="6%" |~~ <math>\~~epsilon</math>''~~J''

+

\mathrm{D}J

−

~~| width="23%" |~~ =&~~nbsp;''~~u''&~~nbsp;''~~v~~'' ~~(d''u'')(d''v~~'')~~

+

& = &

−

~~| width="23%" |~~ +&~~nbsp;''~~u~~''  ''~~v~~'' ~~(d''u'')&~~nbsp;~~d''v''

+

u \cdot v \cdot \qquad 0

−

~~| width="23%" | +~~&~~nbsp;~~&~~nbsp;''u''~~&~~nbsp;~~&~~nbsp;''~~v~~'' ~~d''u~~'' ~~(d''v'')

+

\\[6pt]

−

~~| width="25%" |~~ +&~~nbsp~~;&~~nbsp;''u''~~&~~nbsp;~~&~~nbsp;''v''~~&~~nbsp;~~&~~nbsp;~~d''u~~'' ~~d''v''

+

& + &

+

u \cdot v \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v

+

& + &

+

u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v

+

\\[6pt]

+

& + &

+

u \cdot v \cdot \texttt{~} \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}

+

&&& + &

+

\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}

+

\\[6pt]

+

& + &

+

u \cdot v \cdot \texttt{~} \mathrm{d}u \;\cdot\; \mathrm{d}v \texttt{~}

+

&&&&& + &

+

\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}

+

\end{array}</math>

|-

−

| ~~width="6%" | E''~~J''

+

|

−

~~| width~~=~~"23%" | = ''~~u~~'' ''~~v'' (d''u'')(d''v'')

+

<math>\begin{array}{*{9}{l}}

−

~~| width="23%" |~~ + ''u'' (''v'')(d''u'') d''v''

+

\mathrm{D}J

−

| width="~~23%~~" | + (''u'') ''v'' d''u'' (d''v'')

+

& = &

−

~~| width~~=~~"25%" |~~ + (''u'')(''v'') d''u'' d''v''

+

u \cdot v \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}

+

& + &

+

u \cdot \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \cdot \mathrm{d}v

+

& + &

+

\texttt{(} u \texttt{)} \cdot v \cdot \mathrm{d}u \cdot \texttt{(} \mathrm{d}v \texttt{)}

+

& + &

+

\texttt{(} u \texttt{)} \cdot \texttt{(} v \texttt{)} \cdot \mathrm{d}u \cdot \mathrm{d}v \texttt{~}

+

\end{array}</math>

+

|}

+

+

Alternatively, the difference map D''J'' can be expanded over the cells of [d''u'', d''v''] to arrive at the formulation shown in Table 42. The same development would be obtained from the previous Table by collecting terms in an alternate manner, along the rows rather than the columns of the middle portion of the Table.

+

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"

+

|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 2)}\!</math>

+

|

+

<math>\begin{array}{*{9}{l}}

+

\mathrm{D}J

+

& = & \boldsymbol\varepsilon J

+

& + & \mathrm{E}J

+

\\[6pt]

+

& = & J_{(u, v)}

+

& + & J_{(u + \mathrm{d}u, v + \mathrm{d}v)}

+

\\[6pt]

+

& = & u \cdot v

+

& + & \texttt{(} u, \mathrm{d}u \texttt{)} \cdot \texttt{(} v, \mathrm{d}v \texttt{)}

+

\\[6pt]

+

& = & 0

+

& + & u \cdot \mathrm{d}v

+

& + & v \cdot \mathrm{d}u

+

& + & \mathrm{d}u \cdot \mathrm{d}v

+

\\[6pt]

+

\mathrm{D}J

+

& = & 0

+

& + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v

+

& + & v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}

+

& + & \texttt{((} u, v \texttt{))} \cdot \mathrm{d}u \cdot \mathrm{d}v

+

\end{array}</math>

|}

+

+

Even more simply, the same result is reached by matching up the propositional coefficients of <math>\epsilon</math>''J'' and E''J'' along the cells of [d''u'', d''v''] and adding the pairs under boolean sums (that is, "mod 2", where 1 + 1 = 0), as shown in Table 43.

+

+

{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:left; width:90%"

+

|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Computation of}~ \mathrm{D}J ~\text{(Method 3)}\!</math>

+

|

+

<math>\begin{array}{*{5}{l}}

+

\mathrm{D}J & = & \boldsymbol\varepsilon J & + & \mathrm{E}J

+

\end{array}</math>

|-

|

−

{~~| align~~=~~"left" border="0" cellpadding="0" cellspacing="0" style="font-weight:bold~~; ~~text-align:left~~; ~~width:100%"~~

+

<math>\begin{array}{*{9}{l}}

−

| ~~width="6%"~~ | D''J''

+

\boldsymbol\varepsilon J

−

~~| width="23%" |~~ = 0 ~~<math>~~\cdot~~</math>~~ (d''u'')(d''v'')

+

& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}

−

~~| width="23%" |~~ + ''u~~'' <math>~~\cdot~~</math>~~ (d''u'') d''v''

+

& + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v

−

~~| width="23%" |~~ + ''v~~'' <math>~~\cdot~~</math>~~ d''u'' (d''v'')

+

& + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}

−

~~| width="25%" |~~ + ((''u'', ''v'')) d''u'' d''v''

+

& + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v

+

\\[6pt]

+

\mathrm{E}J

+

& = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}

+

& + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v

+

& + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}

+

& + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v

+

\end{array}</math>

+

|-

+

|

+

<math>\begin{array}{*{9}{l}}

+

\mathrm{D}J

+

& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}

+

& + & ~~ u ~ \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v

+

& + & ~~~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}

+

& + & \texttt{((} u, v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v

+

\end{array}\!</math>

|}

−

|}

+

−

~~~~

+

The difference map D''J'' can also be given a ''dispositional'' interpretation. First, recall that <math>\epsilon</math>''J'' exhibits the dispositions to change from anywhere in ''J'' to anywhere at all, and E''J'' enumerates the dispositions to change from anywhere at all to anywhere in ''J''. Next, observe that each of these classes of dispositions may be divided in accordance with the case of ''J'' versus (''J'') that applies to their points of departure and destination, as shown below. Then, since the dispositions corresponding to <math>\epsilon</math>''J'' and E''J'' have in common the dispositions to preserve ''J'', their symmetric difference (<math>\epsilon</math>''J'', E''J'') is made up of all the remaining dispositions, which are in fact disposed to cross the boundary of ''J'' in one direction or the other. In other words, we may conclude that D''J'' expresses the collective disposition to make a definite change with respect to ''J'', no matter what value it holds in the current state of affairs.

−

+

+

{| align="center" border="1" cellpadding="8" cellspacing="0" style="font-weight:bold; text-align:left; width:96%"

|

Line 4,736: Line 4,887:

|}

−

+

+

Figures 44-a through 44-d illustrate the difference proposition D''J''.

Line 7,730: Line 7,883:

* '''Nota Bene.''' The original Figure 70-b lost some of its labeling in a succession of platform metamorphoses over the years, so I have included an Ascii version below to indicate where the missing labels go.

+

{| align="center" border="0" cellpadding="10"

+

|

<pre>

o-----------------------o o-----------------------o o-----------------------o

Line 7,908: Line 8,063:

Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>

</pre>

+

|}

==Epilogue, Enchoiry, Exodus==

Jon Awbrey

12,080

edits