ONT Re: Differential Logic
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DLOG. Note D10
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Reality at the Threshold of Logic
| 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.
|
| W.V. Quine, 'Mathematical Logic', [Qui, 7]
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.
NB. I'm trying to keep the Asciification of the text as simple as possible
this time around, so I will use emphasis bars !...! around characters in
a context-dependent way, sometimes for Gothic and sometimes for Greek.
Thus, I will rely on the reader's own recognizance to discriminate
between entities like a "Script X" alphabet !X! = {x_1, ..., x_n}
and a "Greek Chi" vector field !X!. Also, the original version
of the Table below used underlined variants of the characters
in the middle column, to suggest quantities rising above the
relevant threshold. In this copy, I use grave markers `...`
around the thresheld symbols instead of underscoring them.
Finally, there does not seem to be any way to avoid the
clash of symbols between the stars, that is, the '*'
that is used in algebra to denote the dual space of
linear functionals, and the '*' that is used in
formal language theory to denote the set of
all finite sequences over an alphabet.
Table 5. A Bridge Over Troubled Waters
o-------------------------o-------------------------o-------------------------o
| Linear Space | Liminal Space | Logical Space |
o-------------------------o-------------------------o-------------------------o
| | | |
| !X! | !`X`! | !A! |
| | | |
| {x_1, ..., x_n} | {`x`_1, ..., `x`_n} | {a_1, ..., a_n} |
| | | |
| cardinality n | cardinality n | cardinality n |
o-------------------------o-------------------------o-------------------------o
| | | |
| X_i | `X`_i | A_i |
| | | |
| <|x_i|> | {(`x`_i), `x`_i} | {(a_i), a_i} |
| | | |
| isomorphic to K | isomorphic to B | isomorphic to B |
o-------------------------o-------------------------o-------------------------o
| | | |
| X | `X` | A |
| | | |
| <|!X!|> | <|!`X`!|> | <|!A!|> |
| | | |
| <|x_1, ..., x_n|> | <|`x`_1, ..., `x`_n|> | <|a_1, ..., a_n|> |
| | | |
| {<x_1, ..., x_n>} | {<`x`_1, ..., `x`_n>} | {<a_1, ..., a_n>} |
| | | |
| X_1 x ... x X_n | `X`_1 x ... x `X`_n | A_1 x ... x A_n |
| | | |
| Prod_i X_i | Prod_i `X`_i | Prod_i A_i |
| | | |
| isomorphic to K^n | isomorphic to B^n | isomorphic to B^n |
o-------------------------o-------------------------o-------------------------o
| | | |
| X* | `X`* | A* |
| | | |
| (hom : X -> K) | (hom : `X` -> B) | (hom : A -> B) |
| | | |
| isomorphic to K^n | isomorphic to B^n | isomorphic to B^n |
o-------------------------o-------------------------o-------------------------o
| | | |
| X^ | `X`^ | A^ |
| | | |
| (X -> K) | (`X` -> B) | (A -> B) |
| | | |
| isomorphic to (K^n -> K)| isomorphic to (B^n -> B)| isomorphic to (B^n -> B)|
o-------------------------o-------------------------o-------------------------o
| | | |
| X% | `X`% | A% |
| | | |
| [!X!] | [!`X`!] | [!A!] |
| | | |
| [x_1, ..., x_n] | [`x`_1, ..., `x`_n] | [a_1, ..., a_n] |
| | | |
| (X, X^) | (`X`, `X`^) | (A, A^) |
| | | |
| (X +-> K) | (`X` +-> B) | (A +-> B) |
| | | |
| (X, (X -> K)) | (`X`, (`X` -> B)) | (A, (A -> B)) |
| | | |
| isomorphic to: | isomorphic to: | isomorphic to: |
| | | |
| (K^n, (K^n -> K) | (B^n, (B^n -> B) | (B^n, (B^n -> K) |
| | | |
| (K^n +-> K) | (B^n +-> B) | (B^n +-> B) |
| | | |
| [K^n] | [B^n] | [B^n] |
o-------------------------o-------------------------o-------------------------o
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.
I now proceed to explain these concepts in more detail.
The two most important ideas developed in the table are:
1. The idea of a universe of discourse, which includes both
a space of "points" and a space of "maps" on those points.
2. 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.
For the sake of concreteness, let us suppose that we start with a continuous
n-dimensional vector space like X = <|x_1, ..., x_n|> ~=~ R^n. The coordinate
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:
`x`_i : R^n -> B such that:
`x`_i (x) = 1 if x in L_i,
`x`_i (x) = 0 if otherwise.
In other notations that are sometimes used, the operator <chi>( ) or the
corner brackets |^...^| 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:
`x`_i (x) = <chi>(x in L_i) = |^ x in L_i ^| = L_i (x).
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 in 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".
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 [here, the grave accents] and the subscript as
combining to form a subtextual spelling for the i^th threshold map. This can
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.
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`!|> ~=~ B^n, and
a space of functions (regions, propositions), `X`^ ~=~ (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].
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. 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.
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
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