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==A Differential Extension of Propositional Calculus==

==A Differential Extension of Propositional Calculus==

<p>Fire over water:<br>

Fire over water:<br>

The image of the condition before transition.<br>

The image of the condition before transition.<br>

Thus the superior man is careful<br>

Thus the superior man is careful<br>

In the differentiation of things,<br>

In the differentiation of things,<br>

So that each finds its place.</p>

So that each finds its place.

 

<p>''I Ching'', Hexagram 64, [Wil, 249]</p>

| align="right" | &mdash; ''I Ching'', Hexagram 64, [Wil, 249]

This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage.  I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems.  The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors.  The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.

This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage.  I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems.  The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors.  The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.

===An Interlude on the Path===

===An Interlude on the Path===

<p>There would have been no beginnings:<br>

instead, speech would proceed from me,<br>

while I stood in its path - a slender gap -<br>

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

the point of its possible disappearance.</p>

 

<p>Michel Foucault, ''The Discourse on Language'', [Fou, 215]</p>

| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]

It may help to get a sense of the relation between '''B''' and '''D''' by considering the ''path classifier'' (or equivalence class of curves) approach to tangent vectors.  As if by reflex, the thought of physical motion makes us cross over to a universe marked by the nominal character [<font face="lucida calligraphy">X</font>].  Given the boolean value system, a path in the space ''X'' = 〈<font face="lucida calligraphy">X</font>〉 is a map ''q'' : '''B''' &rarr; ''X''.  In this case, the set of paths ('''B''' &rarr; ''X'') is isomorphic to the cartesian square ''X''² = ''X'' &times; ''X'', or the set of ordered pairs from ''X''.

It may help to get a sense of the relation between '''B''' and '''D''' by considering the ''path classifier'' (or equivalence class of curves) approach to tangent vectors.  As if by reflex, the thought of physical motion makes us cross over to a universe marked by the nominal character [<font face="lucida calligraphy">X</font>].  Given the boolean value system, a path in the space ''X'' = 〈<font face="lucida calligraphy">X</font>〉 is a map ''q'' : '''B''' &rarr; ''X''.  In this case, the set of paths ('''B''' &rarr; ''X'') is isomorphic to the cartesian square ''X''² = ''X'' &times; ''X'', or the set of ordered pairs from ''X''.

===The Extended Universe of Discourse===

===The Extended Universe of Discourse===

<p>At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.</p>

 

<p>Michel Foucault, ''The Discourse on Language'', [Fou, 215]</p>

At the moment of speaking, I would like to have perceived a nameless voice, long preceding me, leaving me merely to enmesh myself in it, taking up its cadence, and to lodge myself, when no one was looking, in its interstices as if it had paused an instant, in suspense, to beckon to me.

| align="right" colspan="3" | &mdash; Michel Foucault, ''The Discourse on Language'', [Fou, 215]

Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' E<font face="lucida calligraphy">A</font> as:

Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' E<font face="lucida calligraphy">A</font> as:

===Intentional Propositions===

===Intentional Propositions===

<p>Do you guess I have some intricate purpose?<br>

Do you guess I have some intricate purpose?<br>

Well I have . . . . for the April rain has, and the mica on<br>

Well I have . . . . for the April rain has, and the mica on<br>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.</p>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the side of a rock has.

 

<p>Walt Whitman, ''Leaves of Grass'', [Whi, 45]</p>

| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 45]

In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion.  These moments have reference to typical instances and relative intervals, not actual or absolute times.  For example, to discuss ''velocities'' (first order rates of change) we need

In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion.  These moments have reference to typical instances and relative intervals, not actual or absolute times.  For example, to discuss ''velocities'' (first order rates of change) we need

===Life on Easy Street===

===Life on Easy Street===

<p>Failing to fetch me at first keep encouraged,<br>

Failing to fetch me at first keep encouraged,<br>

Missing me one place search another,<br>

Missing me one place search another,<br>

I stop some where waiting for you</p>

I stop some where waiting for you

 

<p>Walt Whitman, ''Leaves of Grass'', [Whi, 88]</p>

| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]

The finite character of the extended universe [E<font face="lucida calligraphy">A</font>] makes the problem of solving differential propositions relatively straightforward, at least,

The finite character of the extended universe [E<font face="lucida calligraphy">A</font>] makes the problem of solving differential propositions relatively straightforward, at least,