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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)
Revision as of 02:48, 18 May 2008
, 02:48, 18 May 2008→Materials from "Dif Log Dyn Sys" for Reuse Here: add excerpt
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, which can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.
We have just gone through a lot of work, apparently doing nothing more substantial than spinning a complex spell of notational devices through a labyrinth of baffled spaces and baffling maps. The reason for doing this was to bind together and to constitute the intuitive concept of a universe of discourse into a coherent categorical object, the kind of thing, once grasped, which can be turned over in the mind and considered in all its manifold changes and facets. The effort invested in these preliminary measures is intended to pay off later, when we need to consider the state transformations and the time evolution of neural network systems.
==A Differential Extension of Propositional Calculus==
<blockquote>
<p>Fire over water:<br>
The image of the condition before transition.<br>
Thus the superior man is careful<br>
In the differentiation of things,<br>
So that each finds its place.</p>
<p>''I Ching'', Hexagram 64, [Wil, 249]</p>
</blockquote>
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.
===Differential Propositions : The Qualitative Analogues of Differential Equations===
In order to define the differential extension of a universe of discourse [<font face="lucida calligraphy">A</font>], the initial alphabet <font face="lucida calligraphy">A</font> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in [<font face="lucida calligraphy">A</font>]. Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in [<font face="lucida calligraphy">A</font>] may change or move with respect to the features that are noted in the initial alphabet.
Hence, let us define the corresponding ''differential alphabet'' or ''tangent alphabet'' as d<font face="lucida calligraphy">A</font> = {d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>}, in principle just an arbitrary alphabet of symbols, disjoint from the initial alphabet <font face="lucida calligraphy">A</font> = {''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>}, 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 d<font face="lucida calligraphy">A</font> is often conceived to be changeable from point to point of the underlying space ''A''. (For all we know, the state space ''A'' 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 <font face="lucida calligraphy">A</font> and d<font face="lucida calligraphy">A</font>.)
In the above terms, a typical tangent space of ''A'' at a point ''x'', frequently denoted as T<sub>''x''</sub>(''A''), can be characterized as having the generic construction d''A'' = 〈d<font face="lucida calligraphy">A</font>〉 = 〈d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>〉. Strictly speaking, the name ''cotangent space'' is probably more correct for this construction, but the fact that we take up spaces and their duals in pairs to form our universes of discourse allows our language to be pliable here.
Proceeding as we did before with the base space ''A'', we can analyze the individual tangent space at a point of ''A'' as a product of distinct and independent factors:
: d''A'' = ∏<sub>''i''</sub> d''A''<sub>''i''</sub> = d''A''<sub>1</sub> × … × d''A''<sub>''n''</sub>.
Here, d<font face="lucida calligraphy">A</font><sub>''i''</sub> is an alphabet of two symbols, d<font face="lucida calligraphy">A</font><sub>''i''</sub> = {(d''a''<sub>''i''</sub>), d''a''<sub>''i''</sub>}, where (d''a''<sub>''i''</sub>) is a symbol with the logical value of "not d''a''<sub>''i''</sub>". Each component d''A''<sub>''i''</sub> has the type '''B''', under the correspondence {(d''a''<sub>''i''</sub>), d''a''<sub>''i''</sub>} <math>\cong</math> {0, 1}. However, clarity is often served by acknowledging this differential usage with a superficially distinct type '''D''', whose intension may be indicated as follows:
: '''D''' = {(d''a''<sub>''i''</sub>), d''a''<sub>''i''</sub>} = {same, different} = {stay, change} = {stop, step}.
Viewed within a coordinate representation, spaces of type '''B'''<sup>''n''</sup> and '''D'''<sup>''n''</sup> 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===
<blockquote>
<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>
the point of its possible disappearance.</p>
<p>Michel Foucault, ''The Discourse on Language'', [Fou, 215]</p>
</blockquote>
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''' → ''X''. In this case, the set of paths ('''B''' → ''X'') is isomorphic to the cartesian square ''X''<sup>2</sup> = ''X'' × ''X'', or the set of ordered pairs from ''X''.
We may analyze ''X''<sup>2</sup> = {‹''u'', ''v''› : ''u'', ''v'' ∈ ''X''} into two parts, specifically, the pairs that lie on and off the diagonal:
: ''X''<sup>2</sup> = {‹''u'', ''v''› : ''u'' = ''v''} ∪ {‹''u'', ''v''› : ''u'' ≠ ''v''}
In symbolic terms, this partition may be expressed as:
: ''X''<sup>2</sup> <math>\cong</math> Diag(''X'') + 2 * Comb(''X'', 2),
where:
: Diag(''X'') = {‹''x'', ''x''› : ''x'' ∈ ''X''},
and where:
: Comb(''X'', ''k'') = "''X'' choose ''k''" = {''k''-sets from ''X''},
so that:
: Comb(''X'', 2) = {{''u'', ''v''} : ''u'', ''v'' ∈ ''X''}.
We can now use the features in d<font face="lucida calligraphy">X</font> = {d''x''<sub>''i''</sub>} = {d''x''<sub>1</sub>, …, d''x''<sub>''n''</sub>} to classify the paths of ('''B''' → ''X'') by way of the pairs in ''X''<sup>2</sup>. If ''X'' <math>\cong</math> '''B'''<sup>''n''</sup> then a path in ''X'' has the form ''q'' : ('''B''' → '''B'''<sup>''n''</sup>) <math>\cong</math> '''B'''<sup>''n''</sup> × '''B'''<sup>''n''</sup> <math>\cong</math> '''B'''<sup>2''n''</sup> <math>\cong</math> ('''B'''<sup>2</sup>)<sup>''n''</sup>. Intuitively, we want to map this ('''B'''<sup>2</sup>)<sup>''n''</sup> onto ''D''<sup>''n''</sup> by mapping each component '''B'''<sup>2</sup> onto a copy of '''D'''. But in our current situation "'''D'''" is just a name we give, or an accidental quality we attribute, to coefficient values in '''B''' when they are attached to features in d<font face="lucida calligraphy">X</font>.
Therefore, define d''x''<sub>''i''</sub> : ''X''<sup>2</sup> → '''B''' such that:
:{| cellpadding=2
| d''x''<sub>''i''</sub>(‹''u'', ''v''›)
| =
| <font face=system>(</font> ''x''<sub>''i''</sub>(''u'') , ''x''<sub>''i''</sub>(''v'') <font face=system>)</font>
|-
|
| =
| ''x''<sub>''i''</sub>(''u'') + ''x''<sub>''i''</sub>(''v'')
|-
|
| =
| ''x''<sub>''i''</sub>(''v'') – ''x''<sub>''i''</sub>(''u'').
|}
In the above transcription, the operator bracket of the form "<font face=system>( … , … )</font>" is a ''cactus lobe'', signifying ''just one false'', in this case among two boolean variables, while "+" is boolean addition in the proper sense of addition in GF(2), and is thus equivalent to "–", in the sense of adding the additive inverse.
The above definition is equivalent to defining d''x''<sub>''i''</sub> : (''B'' → ''X'') → '''B''' such that:
:{| cellpadding=2
| d''x''<sub>''i''</sub>(''q'')
| =
| <font face=system>(</font> ''x''<sub>''i''</sub>(''q''<sub>0</sub>) , ''x''<sub>''i''</sub>(''q''<sub>1</sub>) <font face=system>)</font>
|-
|
| =
| ''x''<sub>''i''</sub>(''q''<sub>0</sub>) + ''x''<sub>''i''</sub>(''q''<sub>1</sub>)
|-
|
| =
| ''x''<sub>''i''</sub>(''q''<sub>1</sub>) – ''x''<sub>''i''</sub>(''q''<sub>0</sub>),
|}
where ''q''<sub>''b''</sub> = ''q''(''b''), for each ''b'' in '''B'''. Thus, the proposition d''x''<sub>''i''</sub> is true of the path ''q'' = ‹''u'', ''v''› exactly if the terms of ''q'', the endpoints ''u'' and ''v'', lie on different sides of the question ''x''<sub>''i''</sub>.
Now we can use the language of features in 〈d<font face="lucida calligraphy">X</font>〉, indeed the whole calculus of propositions in [d<font face="lucida calligraphy">X</font>], to classify paths and sets of paths. In other words, the paths can be taken as models of the propositions ''g'' : d''X'' → '''B'''. For example, the paths corresponding to ''Diag''(''X'') fall under the description <font face=system>(</font>d''x''<sub>1</sub><font face=system>)</font>…<font face=system>(</font>d''x''<sub>''n''</sub><font face=system>)</font>, which says that nothing changes among the set of features {''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>}.
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 ''X'' which 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===
<blockquote>
<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>
</blockquote>
Next, we define the so-called ''extended alphabet'' or ''bundled alphabet'' E<font face="lucida calligraphy">A</font> as:
: E<font face="lucida calligraphy">A</font> = <font face="lucida calligraphy">A</font> ∪ d<font face="lucida calligraphy">A</font> = {''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>, d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>}
This supplies enough material to construct the ''differential extension'' E''A'', or the ''tangent bundle'' over the initial space ''A'', in the following fashion:
:{| cellpadding=2
| E''A''
| =
| ''A'' × d''A''
|-
|
| =
| 〈E<font face="lucida calligraphy">A</font>〉
|-
|
| =
| 〈<font face="lucida calligraphy">A</font> ∪ d<font face="lucida calligraphy">A</font>〉
|-
|
| =
| 〈''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>, d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>〉,
|}
thus giving E''A'' the type '''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>.
Finally, the tangent universe E''A''<sup> •</sup> = [E<font face="lucida calligraphy">A</font>] is constituted from the totality of points and maps, or interpretations and propositions, that are based on the extended set of features E<font face="lucida calligraphy">A</font>:
: E''A''<sup> •</sup> = [E<font face="lucida calligraphy">A</font>] = [''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>, d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>],
thus giving the tangent universe E''A''<sup> •</sup> the type
('''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> +→ '''B''') = ('''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup> → '''B''')).
A proposition in the tangent universe [E<font face="lucida calligraphy">A</font>] is called a ''differential proposition'' and forms the analogue of a system of differential equations, constraints, or relations in ordinary calculus.
With these constructions, to be specific, the differential extension E''A'' and the differential proposition ''h'' : E''A'' → '''B''', we have arrived, in concept at least, at one of the major subgoals of this study. At this juncture, I pause by way of summary to set another Table with the current crop of mathematical produce (Table 8).
<font face="courier new">
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 8. Notation for the Differential Extension of Propositional Calculus'''
|- style="background:paleturquoise"
! Symbol
! Notation
! Description
! Type
|-
| d<font face="lucida calligraphy">A<font>
| {d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>}
|
Alphabet of<br>
differential<br>
features
| [''n''] = '''n'''
|-
| d''A''<sub>''i''</sub>
| {(d''a''<sub>''i''</sub>), d''a''<sub>''i''</sub>}
|
Differential<br>
dimension ''i''
| '''D'''
|-
| d''A''
|
〈d<font face="lucida calligraphy">A</font>〉<br>
〈d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>〉<br>
{‹d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>›}<br>
d''A''<sub>1</sub> × … × d''A''<sub>''n''</sub><br>
∏<sub>''i''</sub> d''A''<sub>''i''</sub>
|
Tangent space<br>
at a point:<br>
Set of changes,<br>
motions, steps,<br>
tangent vectors<br>
at a point
| '''D'''<sup>''n''</sup>
|-
| d''A''*
| (hom : d''A'' → '''B''')
|
Linear functions<br>
on d''A''
| ('''D'''<sup>''n''</sup>)* = '''D'''<sup>''n''</sup>
|-
| d''A''^
| (d''A'' → '''B''')
|
Boolean functions<br>
on d''A''
| '''D'''<sup>''n''</sup> → '''B'''
|-
| d''A''<sup>•</sup>
|
[d<font face="lucida calligraphy">A</font>]<br>
(d''A'', d''A''^)<br>
(d''A'' +→ '''B''')<br>
(d''A'', (d''A'' → '''B'''))<br>
[d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>]
|
Tangent universe<br>
at a point of ''A''<sup>•</sup>,<br>
based on the<br>
tangent features<br>
{d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>}
|
('''D'''<sup>''n''</sup>, ('''D'''<sup>''n''</sup> → '''B'''))<br>
('''D'''<sup>''n''</sup> +→ '''B''')<br>
['''D'''<sup>''n''</sup>]
|}
</font><br>
The adjectives ''differential'' or ''tangent'' are systematically attached to every construct based on the differential alphabet d<font face="lucida calligraphy">A</font>, taken by itself. Strictly speaking, we probably ought to call d<font face="lucida calligraphy">A</font> the set of ''cotangent'' features derived from <font face="lucida calligraphy">A</font>, but the only time this distinction really seems to matter is when we need to distinguish the tangent vectors as maps of type ('''B'''<sup>''n''</sup> → '''B''') → '''B''' from cotangent vectors as elements of type '''D'''<sup>''n''</sup>. In like fashion, having defined E<font face="lucida calligraphy">A</font> = <font face="lucida calligraphy">A</font> ∪ d<font face="lucida calligraphy">A</font>, we can systematically attach the adjective ''extended'' or the substantive ''bundle'' to all of the constructs associated with this full complement of 2''n'' features.
Eventually we may want to extend our basic alphabet even further, to allow for discussion of higher order differential expressions. For those who want to run ahead, and would like to play through, I submit the following gamut of notation (Table 9).
<font face="courier new">
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 9. Higher Order Differential Features'''
| width=50% |
{| cellpadding="4" style="background:lightcyan"
| <font face="lucida calligraphy">A</font>
| =
| d<sup>0</sup><font face="lucida calligraphy">A</font>
| =
| {''a''<sub>1</sub>,
| …,
| ''a''<sub>''n''</sub>}
|-
| d<font face="lucida calligraphy">A</font>
| =
| d<sup>1</sup><font face="lucida calligraphy">A</font>
| =
| {d''a''<sub>1</sub>,
| …,
| d''a''<sub>''n''</sub>}
|-
|
|
| d<sup>''k''</sup><font face="lucida calligraphy">A</font>
| =
| {d<sup>''k''</sup>''a''<sub>''1''</sub>,
| …,
| d<sup>''k''</sup>''a''<sub>''n''</sub>}
|-
| d<sup>*</sup><font face="lucida calligraphy">A</font>
| =
| {d<sup>0</sup><font face="lucida calligraphy">A</font>,
| …,
| d<sup>''k''</sup><font face="lucida calligraphy">A</font>,
| …}
|}
| width=50% |
{| cellpadding="4" style="background:lightcyan"
| E<sup>0</sup><font face="lucida calligraphy">A</font>
| =
| d<sup>0</sup><font face="lucida calligraphy">A</font>
|-
| E<sup>1</sup><font face="lucida calligraphy">A</font>
| =
| d<sup>0</sup><font face="lucida calligraphy">A</font> ∪ d<sup>1</sup><font face="lucida calligraphy">A</font>
|-
| E<sup>''k''</sup><font face="lucida calligraphy">A</font>
| =
| d<sup>0</sup><font face="lucida calligraphy">A</font> ∪ … ∪ d<sup>''k''</sup><font face="lucida calligraphy">A</font>
|-
| E<sup>∞</sup><font face="lucida calligraphy">A</font>
| =
| ∪ d<sup>*</sup><font face="lucida calligraphy">A</font>
|}
|}
</font><br>
===Intentional Propositions===
<blockquote>
<p>Do you guess I have some intricate purpose?<br>
Well I have . . . . for the April rain has, and the mica on<br>
the side of a rock has.</p>
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 45]</p>
</blockquote>
In order to analyze the behavior of a system at successive moments in time, while staying within the limitations of propositional logic, it is necessary to create independent alphabets of logical features for each moment of time that we contemplate using in our discussion. These moments have reference to typical instances and relative intervals, not actual or absolute times. For example, to discuss ''velocities'' (first order rates of change) we need
to consider points of time in pairs. There are a number of natural ways of doing this. Given an initial alphabet, we could use its symbols as a lexical basis to generate successive alphabets of compound symbols, say, with temporal markers appended as suffixes.
As a standard way of dealing with these situations, I produce the following scheme of notation, which extends any alphabet of logical features through as many temporal moments as a particular order of analysis may demand. The lexical operators p<sup>''k''</sup> and Q<sup>''k''</sup> are convenient in many contexts where the accumulation of prime symbols and union symbols would otherwise be cumbersome.
<font face="courier new">
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
|+ '''Table 10. A Realm of Intentional Features'''
| width=50% |
{| cellpadding="4" style="background:lightcyan"
| p<sup>0</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font>
| =
| {''a''<sub>1</sub> ,
| …,
| ''a''<sub>''n''</sub> }
|-
| p<sup>1</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font>′
| =
| {''a''<sub>1</sub>′,
| …,
| ''a''<sub>''n''</sub>′}
|-
| p<sup>2</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font>″
| =
| {''a''<sub>1</sub>″,
| …,
| ''a''<sub>''n''</sub>″}
|-
| ...
|
|
|
| ...
|-
| p<sup>''k''</sup><font face="lucida calligraphy">A</font>
| =
|
|
| {p<sup>''k''</sup>''a''<sub>1</sub>,
| …,
| p<sup>''k''</sup>''a''<sub>''n''</sub>}
|}
| width=50% |
{| cellpadding="4" style="background:lightcyan"
| Q<sup>0</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font>
|-
| Q<sup>1</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font> ∪ <font face="lucida calligraphy">A</font>′
|-
| Q<sup>2</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font> ∪ <font face="lucida calligraphy">A</font>′ ∪ <font face="lucida calligraphy">A</font>″
|-
| ...
|
| ...
|-
| Q<sup>''k''</sup><font face="lucida calligraphy">A</font>
| =
| <font face="lucida calligraphy">A</font> ∪ <font face="lucida calligraphy">A</font>′ ∪ … ∪ p<sup>''k''</sup><font face="lucida calligraphy">A</font>
|}
|}
</font><br>
The resulting augmentations of our logical basis found a series of discursive universes that may be called the ''intentional extension'' of propositional calculus. The pattern of this extension is analogous to that of the differential extension, which was developed in terms of the operators d<sup>''k''</sup> and E<sup>''k''</sup>, and there is an obvious and natural relation between these two extensions that falls within our purview to explore. In contexts displaying this regular pattern, where a series of domains stretches up from an anchoring domain ''X'' through an indefinite number of higher reaches, I refer to a particular collection of domains based on ''X'' as a ''realm'' of ''X'', and when the succession exhibits a temporal aspect, as a ''reign'' of ''X''.
For the purposes of this discussion, let us define an ''intentional proposition'' as a proposition in the universe of discourse Q''X''<sup> •</sup> = [Q<font face="lucida calligraphy">X</font>], in other words, a map ''q'' : Q''X'' → '''B'''. The sense of this definition may be seen if we consider the following facts. First, the equivalence Q''X'' = ''X'' × ''X''′ motivates the following chain of isomorphisms between spaces:
:{| cellpadding=2
| (Q''X'' → '''B''')
| <math>\cong</math>
| (''X'' × ''X''′ → '''B''')
|-
|
| <math>\cong</math>
| (''X'' → (''X''′ → '''B'''))
|-
|
| <math>\cong</math>
| (''X''′ → (''X'' → '''B''')).
|}
Viewed in this light, an intentional proposition ''q'' may be rephrased as a map ''q'' : ''X'' × ''X''′ → '''B''', which judges the juxtaposition of states in ''X'' from one moment to the next. Alternatively, ''q'' may be parsed in two stages in two different ways, as ''q'' : ''X'' → (''X''′ → '''B''') and as ''q'' : ''X''′ → (''X'' → '''B'''), which associate to each point of ''X'' or ''X''′ a proposition about states in ''X''′ or ''X'', 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 q indicates a method for the systematic selection of local goals. As a general form of description, we may refer to a map of the type ''q'' : Q<sup>''i''</sup>''X'' → '''B''' as an "''i''<sup>th</sup> order intentional proposition". Naturally, when we speak of intentional propositions without qualification, we usually mean first order intentions.
Many different realms of discourse have the same structure as the extensions that have been indicated here. From a strictly logical point of view, each new layer of terms is composed of independent logical variables that are no different in kind from those that go before, and each further course of logical atoms is treated like so many individual, but otherwise indifferent bricks by the prototype computer program that I use as a propositional interpreter. Thus, the names that I use to single out the differential and the intentional extensions, and the lexical paradigms that I follow to construct them, are meant to suggest the interpretations that I have in mind, but they can only hint at the extra meanings that human communicators may pack into their terms and inflections.
As applied here, the word ''intentional'' is drawn from common use and may have little bearing on its technical use in other, more properly philosophical, contexts. I am merely using the complex of intentional concepts - aims, ends, goals, objectives, purposes, and so on - metaphorically to flesh out and vividly to represent any situation where one needs to contemplate a system in multiple aspects of state and destination, that is, its being in certain states and at the same time acting as if headed through certain states. If confusion arises, more neutral words like ''conative'', ''contingent'', ''discretionary'', ''experimental'', ''kinetic'', ''progressive'', ''tentative'', or ''trial'' would probably serve as well.
===Life on Easy Street===
<blockquote>
<p>Failing to fetch me at first keep encouraged,<br>
Missing me one place search another,<br>
I stop some where waiting for you</p>
<p>Walt Whitman, ''Leaves of Grass'', [Whi, 88]</p>
</blockquote>
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,
in principle. The solution set of the differential proposition ''q'' : E''A'' → '''B''' is the set of models ''q''<sup>–1</sup>(1) in E''A''. Finding all of the models of ''q'', the extended interpretations in E''A'' that satisfy ''q'', can be carried out by a finite search. Being in possession of complete algorithms for propositional calculus 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 of [E<font face="lucida calligraphy">A</font>] 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, my focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications. In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus. But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.
==Back to the Beginning : Some Exemplary Universes==
==Back to the Beginning : Some Exemplary Universes==