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<math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math><br>
<math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math><br>
<math>[\mathbb{B}^n]</math>
<math>[\mathbb{B}^n]</math>
|}<br>
'''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'''
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'''
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 5).
Table 5 summarizes the basic notations that are needed to describe the (first order) differential extensions of propositional calculi in a corresponding manner.
Table 5 summarizes the basic notations that are needed to describe the (first order) differential extensions of propositional calculi in a corresponding manner.
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
<math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br>
<math>[\mathbb{D}^n]</math>
<math>[\mathbb{D}^n]</math>
|}
|}<br>
'''…'''
'''…'''
