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→‎A Differential Extension of Propositional Calculus: update

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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.

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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>

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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>

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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.

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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.

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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:

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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:

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

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| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.\!</math>

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| <math>\mathrm{d}A ~=~ \prod_{i=1}^n \mathrm{d}A_i ~=~ \mathrm{d}A_1 \times \ldots \times \mathrm{d}A_n.</math>

|}

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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:

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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:

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

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| <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>

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| <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>

|}

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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.

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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===

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There would have been no beginnings: instead, speech would proceed from me, while I stood in its path – a slender gap – the point of its possible disappearance.

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There would have been no beginnings:  instead, speech would proceed from me, while I stood in its path – a slender gap – the point of its possible disappearance.

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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>

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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>

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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:

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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:

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

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

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

|}

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

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

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

|}

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|}

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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:

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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:

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|}

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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>

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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:

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|}

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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.

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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.

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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:

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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:

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|}

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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>

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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>

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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>

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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>

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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.

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Finally, a few words of explanation may be in order.  If this concept of a path appears to be described in a roundabout fashion, it is because I am trying to avoid using any assumption of vector space properties for the space <math>X</math> that contains its range.  In many ways the treatment is still unsatisfactory, but improvements will have to wait for the introduction of substitution operators acting on singular propositions.

===The Extended Universe of Discourse===

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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:

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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:

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

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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:

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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:

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

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

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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.

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With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),\!</math> we have arrived, in main outline, at one of the major subgoals of this study. Table 8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.

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With these constructions, the differential extension <math>\mathrm{E}A</math> and the space of differential propositions <math>(\mathrm{E}A \to \mathbb{B}),</math> we have arrived, in main outline, at one of the major subgoals of this study. Table 8 summarizes the concepts that have been introduced for working with differentially extended universes of discourse.

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

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|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}\!</math>

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|+ style="height:30px" | <math>\text{Table 8.} ~~ \text{Differential Extension : Basic Notation}</math>

|- style="height:40px; background:ghostwhite"

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

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| <math>\text{Symbol}</math>

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

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| <math>\text{Notation}</math>

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

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| <math>\text{Description}</math>

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

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| <math>\text{Type}</math>

|-

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

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

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| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}\!</math>

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| <math>\{ {}^{\backprime\backprime} \mathrm{d}a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} \mathrm{d}a_n {}^{\prime\prime} \}</math>

|

<math>\begin{matrix}

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\text{differential symbols}

\end{matrix}</math>

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| <math>[n] = \mathbf{n}\!</math>

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| <math>[n] = \mathbf{n}</math>

|-

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

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

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| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}\!</math>

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| <math>\{ \mathrm{d}a_1, \ldots, \mathrm{d}a_n \}</math>

|

<math>\begin{matrix}

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\text{differential features}

\end{matrix}</math>

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| <math>[n] = \mathbf{n}\!</math>

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| <math>[n] = \mathbf{n}</math>

|-

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

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

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

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

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| <math>\text{Differential dimension}~ i\!</math>

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| <math>\text{Differential dimension}~ i</math>

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| <math>\mathbb{D}\!</math>

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| <math>\mathbb{D}</math>

|-

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

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

|

<math>\begin{matrix}

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\text{at a point}

\end{matrix}</math>

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

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

|-

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

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

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| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})\!</math>

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| <math>(\mathrm{hom} : \mathrm{d}A \to \mathbb{B})</math>

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| <math>\text{Linear functions on}~ \mathrm{d}A\!</math>

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| <math>\text{Linear functions on}~ \mathrm{d}A</math>

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| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n\!</math>

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| <math>(\mathbb{D}^n)^* \cong \mathbb{D}^n</math>

|-

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

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

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| <math>(\mathrm{d}A \to \mathbb{B})\!</math>

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| <math>(\mathrm{d}A \to \mathbb{B})</math>

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| <math>\text{Boolean functions on}~ \mathrm{d}A\!</math>

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| <math>\text{Boolean functions on}~ \mathrm{d}A</math>

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

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

|-

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

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

|

<math>\begin{matrix}

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The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself. Strictly speaking, we probably ought to call <math>\mathrm{d}\mathcal{A}</math> the set of ''cotangent'' features derived from <math>\mathcal{A},</math> but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> from cotangent vectors as elements of type <math>\mathbb{D}^n.</math> In like fashion, having defined <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A},</math> we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of <math>{2n}\!</math> features.

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The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet <math>\mathrm{d}\mathfrak{A},</math> taken by itself. Strictly speaking, we probably ought to call <math>\mathrm{d}\mathcal{A}</math> the set of ''cotangent'' features derived from <math>\mathcal{A},</math> but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type <math>(\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> from cotangent vectors as elements of type <math>\mathbb{D}^n.</math> In like fashion, having defined <math>\mathrm{E}\mathcal{A} = \mathcal{A} \cup \mathrm{d}\mathcal{A},</math> we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of <math>{2n}</math> features.

It eventually becomes necessary to extend the initial alphabet even further, to allow for the discussion of higher order differential expressions. Table 9 provides a suggestion of how these further extensions can be carried out.

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{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"

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|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}\!</math>

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|+ style="height:30px" | <math>\text{Table 9.} ~~ \text{Higher Order Differential Features}</math>

|

<math>\begin{array}{lllll}

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{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:75%"

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|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}\!</math>

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|+ style="height:30px" | <math>\text{Table 10.} ~~ \text{A Realm of Intentional Features}</math>

|

<math>\begin{array}{lllll}

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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>

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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^\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:

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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.

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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.

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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.

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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.

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The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\mathrm{E}A.</math> Finding all the models of <math>q,\!</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.

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The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least, in principle. The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)</math> in <math>\mathrm{E}A.</math> Finding all the models of <math>q,</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,</math> can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely. While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.

In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.

Jon Awbrey

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Differential Logic and Dynamic Systems (view source)

Revision as of 21:56, 14 January 2021

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