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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. | | 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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| − | '''An Interlude on the Path'''
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| − | 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''.
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| − | 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:
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| − | : ''X''<sup>2</sup> = {‹''u'', ''v''› : ''u'' = ''v''} ∪ {‹''u'', ''v''› : ''u'' ≠ ''v''}
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| − | In symbolic terms, this partition may be expressed as:
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| − | : ''X''<sup>2</sup> <math>\cong</math> Diag(''X'') + 2 * Comb(''X'', 2),
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| − | where:
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| − | : Diag(''X'') = {‹''x'', ''x''› : ''x'' ∈ ''X''},
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| − | and where:
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| − | : Comb(''X'', ''k'') = "''X'' choose ''k''" = {''k''-sets from ''X''},
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| − | so that:
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| − | : Comb(''X'', 2) = {{''u'', ''v''} : ''u'', ''v'' ∈ ''X''}.
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| − | 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>.
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| − | Therefore, define d''x''<sub>''i''</sub> : ''X''<sup>2</sup> → '''B''' such that:
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| − | :{| cellpadding=2
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| − | | d''x''<sub>''i''</sub>(‹''u'', ''v''›)
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| − | | =
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| − | | <font face=system>(</font> ''x''<sub>''i''</sub>(''u'') , ''x''<sub>''i''</sub>(''v'') <font face=system>)</font>
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| − | |-
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| − | | =
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| − | | ''x''<sub>''i''</sub>(''u'') + ''x''<sub>''i''</sub>(''v'')
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| − | |-
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| − | | =
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| − | | ''x''<sub>''i''</sub>(''v'') – ''x''<sub>''i''</sub>(''u'').
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| − | |}
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| − | 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.
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| − | The above definition is equivalent to defining d''x''<sub>''i''</sub> : (''B'' → ''X'') → '''B''' such that:
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| − | :{| cellpadding=2
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| − | | d''x''<sub>''i''</sub>(''q'')
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| − | | =
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| − | | <font face=system>(</font> ''x''<sub>''i''</sub>(''q''<sub>0</sub>) , ''x''<sub>''i''</sub>(''q''<sub>1</sub>) <font face=system>)</font>
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| − | |-
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| − | | =
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| − | | ''x''<sub>''i''</sub>(''q''<sub>0</sub>) + ''x''<sub>''i''</sub>(''q''<sub>1</sub>)
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| − | |-
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| − | | =
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| − | | ''x''<sub>''i''</sub>(''q''<sub>1</sub>) – ''x''<sub>''i''</sub>(''q''<sub>0</sub>),
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| − | |}
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| − | 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>.
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| − | 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>}.
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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 ''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.
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| | '''The Extended Universe of Discourse''' | | '''The Extended Universe of Discourse''' |
