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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

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===Reality at the Threshold of Logic===

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But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device. To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.

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W.V. Quine, ''Mathematical Logic'', [Qui, 7]

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</blockquote>

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Table 5 accumulates an array of notation that I hope will not be too distracting. Some of it is rarely needed, but has been filled in for the sake of completeness. Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"

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|+ '''Table 5. A Bridge Over Troubled Waters'''

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|- style="background:paleturquoise"

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! Linear Space

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! Liminal Space

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! Logical Space

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|-

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|

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X

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{''x''1, …, ''x''''n''}

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cardinality ''n''

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|

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X

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{''x''1, …, ''x''''n''}

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cardinality ''n''

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|

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A

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{''a''1, …, ''a''''n''}

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cardinality ''n''

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|-

+

|

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''X''''i''

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〈''x''''i''〉

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isomorphic to '''K'''

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|

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''X''''i''

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{(''x''''i''), ''x''''i''}

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isomorphic to '''B'''

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|

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''A''''i''

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{(''a''''i''), ''a''''i''}

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isomorphic to '''B'''

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|-

+

|

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''X''

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〈X〉

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〈''x''1, …, ''x''''n''〉

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{‹''x''1, …, ''x''''n''›}

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''X''1 × … × ''X''''n''

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∏''i'' ''X''''i''

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isomorphic to '''K'''''n''

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|

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''X''

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〈X〉

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〈''x''1, …, ''x''''n''〉

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{‹''x''1, …, ''x''''n''›}

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''X''1 × … × ''X''''n''

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∏''i'' ''X''''i''

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isomorphic to '''B'''''n''

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|

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''A''

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〈A〉

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〈''a''1, …, ''a''''n''〉

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{‹''a''1, …, ''a''''n''›}

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''A''1 × … × ''A''''n''

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∏''i'' ''A''''i''

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isomorphic to '''B'''''n''

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|-

+

|

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''X''*

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(hom : ''X'' → '''K''')

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isomorphic to '''K'''''n''

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|

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''X''*

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(hom : ''X'' → '''B''')

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isomorphic to '''B'''''n''

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|

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''A''*

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(hom : ''A'' → '''B''')

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isomorphic to '''B'''''n''

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|-

+

|

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''X''^

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(''X'' → '''K''')

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isomorphic to:

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('''K'''''n'' → '''K''')

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|

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''X''^

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(''X'' → '''B''')

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isomorphic to:

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('''B'''''n'' → '''B''')

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|

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''A''^

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(''A'' → '''B''')

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isomorphic to:

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('''B'''''n'' → '''B''')

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|-

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|

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''X''•

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[X]

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[''x''1, …, ''x''''n'']

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(''X'', ''X''^)

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(''X'' +→ '''K''')

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(''X'', (''X'' → '''K'''))

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isomorphic to:

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('''K'''''n'', ('''K'''''n'' → '''K'''))

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('''K'''''n'' +→ '''K''')

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['''K'''''n'']

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|

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''X''•

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[X]

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[''x''1, …, ''x''''n'']

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(''X'', ''X''^)

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(''X'' +→ '''B''')

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(''X'', (''X'' → '''B'''))

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isomorphic to:

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('''B'''''n'', ('''B'''''n'' → '''B'''))

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('''B'''''n'' +→ '''B''')

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['''B'''''n'']

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|

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''A''•

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[A]

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[''a''1, …, ''a''''n'']

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(''A'', ''A''^)

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(''A'' +→ '''B''')

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(''A'', (''A'' → '''B'''))

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isomorphic to:

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('''B'''''n'', ('''B'''''n'' → '''B'''))

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('''B'''''n'' +→ '''B''')

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['''B'''''n'']

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|}

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The left side of the Table collects mostly standard notation for an ''n''-dimensional vector space over a field '''K'''. The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus. (I plan to use this notation in the logical analysis of neural network systems.) The middle column of the table is designed as a transitional step from the case of an arbitrary field '''K''', with a special interest in the continuous line '''R''', to the qualitative and discrete situations that are instanced and typified by '''B'''.

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I now proceed to explain these concepts in more detail. The two most important ideas developed in the table are:

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* The idea of a universe of discourse, which includes both a space of ''points'' and a space of ''maps'' on those points.

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* The idea of passing from a more complex universe to a simpler universe by a process of ''thresholding'' each dimension of variation down to a single bit of information.

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For the sake of concreteness, let us suppose that we start with a continuous ''n''-dimensional vector space like ''X'' = 〈''x''1, …, ''x''''n''〉 <math>\cong</math> '''R'''''n''. The coordinate

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system X = {''x''''i''} is a set of maps ''x''''i'' : '''R'''''n'' → '''R''', also known as the coordinate projections. Given a "dataset" of points ''x'' in '''R'''''n'', there are numerous ways of sensibly reducing the data down to one bit for each dimension. One strategy that is general enough for our present purposes is as follows. For each ''i'' we choose an ''n''-ary relation ''L''''i'' on '''R''', that is, a subset of '''R'''''n'', and then we define the ''i''th threshold map, or ''limen'' ''x''''i'' as follows:

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: ''x''''i'' : '''R'''''n'' → '''B''' such that:

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: ''x''''i''(''x'') = 1 if ''x'' ∈ ''L''''i'',

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: ''x''''i''(''x'') = 0 if otherwise.

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In other notations that are sometimes used, the operator <math>\chi (\ )</math> or the corner brackets <math>\lceil \ldots \rceil</math> can be used to denote a ''characteristic function'', that is, a mapping from statements to their truth values, given as elements of '''B'''. Finally, it is not uncommon to use the name of the relation itself as a predicate that maps ''n''-tuples into truth values. In each of these notations, the above definition could be expressed as follows:

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: ''x''''i''(''x'') = <math>\chi (x \in L_i)</math> = <math>\lceil x \in L_i \rceil</math> = ''L''''i''(''x'').

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Notice that, as defined here, there need be no actual relation between the ''n''-dimensional subsets {''L''''i''} and the coordinate axes corresponding to {''x''''i''}, aside from the circumstance that the two sets have the same cardinality. In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, ''L''''i'' is bounded by some hyperplane that intersects the ''i''th axis at a unique threshold value ''r''''i'' ∈ '''R'''. Often, the hyperplane is chosen normal to the axis. In recognition of this motive, let us make the following convention. When the set ''L''''i'' has points on the ''i''th axis, that is, points of the form ‹0, …, 0, ''r''''i'', 0, …, 0› where only the ''x''''i'' coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation. In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''. For a knowledge based system ''X'', this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.

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States of knowledge about the location of a system or about the distribution of a population of systems in a state space ''X'' = '''R'''''n'' can now be expressed by taking the set X = {''x''''i''} as a basis of logical features. In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the ''i''th threshold map. This can

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help to remind us that the ''threshold operator''  )''i'' acts on ''x'' by setting up a kind of a "hurdle" for it. In this interpretation, the coordinate proposition ''x''''i'' asserts that the representative point ''x'' resides ''above'' the ''i''th threshold.

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Primitive assertions of the form ''x''''i''(''x'') can then be negated and joined by means of propositional connectives in the usual ways to provide information about the state ''x'' of a contemplated system or a statistical ensemble of systems. Parentheses "( )" may be used to indicate negation. Eventually one discovers the usefulness of the ''k''-ary ''just one false'' operators of the form "( , , , )", as treated in earlier reports. This much tackle generates a space of points (cells, interpretations), ''X'' = 〈X〉 <math>\cong</math> '''B'''''n'', and

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a space of functions (regions, propositions), ''X''^ <math>\cong</math> ('''B'''''n'' → '''B'''). Together these form a new universe of discourse ''X'' • = [X] of the type ('''B'''''n'', ('''B'''''n'' → '''B''')), which we may abbreviate as '''B'''''n'' +→ '''B''', or most succinctly as ['''B'''''n''].

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The square brackets have been chosen to recall the rectangular frame of a venn diagram. In thinking about a universe of discourse it is a good idea to keep this picture in mind, where we constantly think of the elementary cells ''x'', the defining features ''x''''i'', and the potential shadings ''f'' : ''X'' → '''B''', all at the same time, remaining aware of the arbitrariness of the way that we choose to inscribe our distinctions in the medium of a continuous space.

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Finally, let ''X''* denote the space of linear functions, (hom : ''X'' → '''K'''), which in the finite case has the same dimensionality as ''X'', and let the same notation be extended across the table.

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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.

Jon Awbrey

12,080

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Directory talk:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

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