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Differential Logic and Dynamic Systems
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Posted by: Jon Awbrey
Differential Logic and Dynamic Systems
Note. A nicer presentation of this topic can now be found here.
Author: Jon Awbrey
Created: 16 Dec 1993
Relayed: 31 Oct 1994
Revised: 03 Jun 2003
0. Purpose
1. Review and Transition
2. A Functional Conception of Propositional Calculus
2.1. Qualitative Logic and Quantitative Analogy
2.2. Philosophy of Notation: Formal Terms and Flexible Types
2.3. Special Classes of Propositions
2.4. Basis Relativity and Type Ambiguity
2.5. The Analogy Between Real and Boolean Types
2.6. Theory of Control and Control of Theory
2.7. Propositions as Types and Higher Order Types
2.8. Reality at the Threshold of Logic
2.9. Tables of Propositional Forms
3. A Differential Extension of Propositional Calculus
3.1. Differential Propositions: The Qualitative Analogue of Differential Equations
3.2. An Interlude on the Path
3.3. The Extended Universe of Discourse
3.4. Intentional Propositions
3.5. Life on Easy Street
4. Back to the Beginning: Some Exemplary Universes
4.1. A One-Dimensional Universe
4.2. Example 1. A Square Rigging
4.3. Back to the Feature
4.4. Tacit Extensions
4.5. Example 2. Drives and Their Vicissitudes
5. Transformations of Discourse
5.1. Foreshadowing Transformations: Extensions and Projections of Discourse
5.1.1. Extension from 1 to 2 Dimensions
5.1.2. Extension from 2 to 4 Dimensions
5.2. Thematization of Functions: And a Declaration of Independence for Variables
5.2.1. Thematization: Venn Diagrams
5.2.2. Thematization: Truth Tables
5.3. Propositional Transformations
5.3.1. Alias and Alibi Transformations
5.3.2. Transformations of General Type
5.4. Analytic Expansions: Operators and Functors
5.4.1. Operators on Propositions and Transformations
5.4.2. Differential Analysis of Propositions and Transformations
5.4.2.1. The Secant Operator: $E$
5.4.2.2. The Radius Operator: $e$
5.4.2.3. The Phantom of the Operators: !h!
5.4.2.4. The Chord Operator: $D$
5.4.2.5. The Tangent Operator: $T$
5.5. Transformations of Type B^2 -> B^1
5.5.1. Analytic Expansion of Conjunction
5.5.1.1. Tacit Extension of Conjunction
5.5.1.2. Enlargement Map of Conjunction
5.5.1.3. Digression: Reflection on Use and Mention
5.5.1.4. Difference Map of Conjunction
5.5.1.5. Differential of Conjunction
5.5.1.6. Remainder of Conjunction
5.5.1.7. Summary of Conjunction
5.5.2. Analytic Series: Coordinate Method
5.5.3. Analytic Series: Recap
5.5.4. Terminological Interlude
5.5.5. End of Perfunctory Chatter: Time to Roll the Clip!
5.6. Taking Aim at Higher Dimensional Targets
5.7. Transformations of Type B^2 -> B^2
Epilogue, Enchoiry, Exodus
---------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 1
Differential Logic and Dynamic Systems
| Stand and unfold yourself.
|
| 'Hamlet', 1.1.2
Purpose
This series of reports develops a differential extension of
propositional calculus and applies it to a context of problems
arising in dynamic systems. The work pursued here is coordinated
with a parallel application that focuses on neural network systems,
but the dependencies are arranged to make the present series the
main and the more self-contained work, to serve as a conceptual
frame and a technical background for the network project.
1. Review and Transition
This note continues a previous discussion on the problem of dealing
with change and diversity in logic-based intelligent systems. For
ease of reference, I begin by summarizing essential material from
previous reports.
Table 1 outlines the notation that I use for propositional calculus.
Explained as briefly as possible, I am using only two basic kinds
of truth-functional connectives, both of variable k-ary scope.
- For the first, I use concatenation as a connective
to indicate the logical conjunction of k arguments.
- For the other, I use a bracket of the form ( , , , )
as a connective which says that exactly one of its k
arguments is false.
All other truth-functional connectives can be obtained in a very
efficient style of representation through combinations of these
two forms.
This treatment of propositional logic is derived from the work of
C.S. Peirce [P1, P2], who gave this approach an extensive development
in his graphical systems of predicate, relational, and modal logic [Rob].
More recently, these ideas were revived and supplemented in an alternative
interpretation by G. Spencer-Brown [SpB]. Both of these authors used other
forms of enclosure where I use parentheses, but the structural topologies of
expression and the functional varieties of interpretation are fundamentally
the same.
While working with expressions solely in propositional calculus, the use
of plain parentheses to represent logical connectives is simplest to write
and easiest to read for both human and machine parsers. In the present text
I preserve this form of expression in tables and set-off displays, but in
contexts where parentheses are needed for functional notation I will use
a distinctive font for logical operators. [Not available in Ascii.]
The briefest expression for logical truth is the empty word, usually denoted by
epsilon or lambda in formal languages, where it forms the identity element for
concatenation. To make it visible in this text, I denote it by the equivalent
expression "(())", or, especially if operating in an algebraic context, by a
simple "1". Also when working in an algebraic mode, I use the plus sign "+"
for exclusive disjunction. Thus, we may express the following paraphrases
of algebraic forms:
A + B = (A, B)
A + B + C = ((A, B), C) = (A, (B, C))
One should be careful to observe that these last two
expressions are not equivalent to the form (A, B, C).
Table 1. Syntax and Semantics of a Calculus for Propositional Logic
o-------------------o-------------------o-------------------o
| Expression` ` ` ` | Interpretation` ` | Other Notations ` |
o-------------------o-------------------o-------------------o
| " " ` ` ` ` ` ` ` | True. ` ` ` ` ` ` | 1 ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| ()` ` ` ` ` ` ` ` | False.` ` ` ` ` ` | 0 ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| A ` ` ` ` ` ` ` ` | A.` ` ` ` ` ` ` ` | A ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| (A) ` ` ` ` ` ` ` | Not A.` ` ` ` ` ` | A'` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ~A` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| A B C ` ` ` ` ` ` | A and B and C.` ` | A & B & C ` ` ` ` |
o-------------------o-------------------o-------------------o
| ((A)(B)(C)) ` ` ` | A or B or C.` ` ` | A v B v C ` ` ` ` |
o-------------------o-------------------o-------------------o
| (A (B)) ` ` ` ` ` | A implies B.` ` ` | A => B` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | If A then B.` ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| (A, B)` ` ` ` ` ` | A not equal to B. | A =/= B ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | A exclusive-or B. | A `+ `B ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| ((A, B))` ` ` ` ` | A is equal to B.` | A `= `B ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | A if & only if B. | A <=> B ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| (A, B, C) ` ` ` ` | Just one of ` ` ` | A'B C `v` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | A, B, C ` ` ` ` ` | A B'C `v` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | is false. ` ` ` ` | A B C'` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| ((A),(B),(C)) ` ` | Just one of ` ` ` | A B'C' v` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | A, B, C ` ` ` ` ` | A'B C' v` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | is true.` ` ` ` ` | A'B'C ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | Partition all ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | into A, B, C. ` ` | ` ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| ((A, B), C) ` ` ` | Oddly many of ` ` | A + B + C ` ` ` ` |
| (A, (B, C)) ` ` ` | A, B, C ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | are true. ` ` ` ` | A B C `v` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | A B'C' v` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | A'B C' v` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | A'B'C ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
| (Q, (A),(B),(C))` | Partition `Q` ` ` | Q'A'B'C' v` ` ` ` |
| ` ` ` ` ` ` ` ` ` | into A, B, C. ` ` | Q A B'C' v` ` ` ` |
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | Q A'B C' v` ` ` ` |
| ` ` ` ` ` ` ` ` ` | Genus Q comprises | Q A'B'C ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` | species A, B, C.` | ` ` ` ` ` ` ` ` ` |
o-------------------o-------------------o-------------------o
NB. The usage that one often sees, of a plus sign "+"
to represent inclusive disjunction, and the reference
to this operation as "boolean addition", is a misnomer
on at least two counts. Boole used the plus sign to
represent exclusive disjunction (at any rate, an
operation of aggregation restricted in its logical
interpretation to cases where the represented sets
are disjoint [Boo, 32]), as any mathematician with
a sensitivity to the ring and field properties of
algebra would do:
| The expression x + y seems indeed uninterpretable,
| unless it be assumed that the things represented
| by x and the things represented by y are entirely
| separate; that they embrace no individuals in
| common. [Boo, 66].
It was only later that Peirce and Jevons treated inclusive
disjunction as a fundamental operation, but these authors,
with a respect for the algebraic properties that were already
associated with the plus sign, used a variety of other symbols
for inclusive disjunction [Sty, 177, 189]. It seems to have
been Schroeder who later reassigned the plus sign to inclusive
disjunction [Sty, 208]. Additional information, discussion,
and references can be found in [Boo] and [Sty, 177-263].
Aside from these historical points, which never really
count against a current practice that has gained a life
of its own, this usage does have a further disadvantage
of cutting or confounding the lines of communication
between algebra and logic. For this reason, I am
forced to avoid it here.
Jon Awbrey
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Posted by: Jon Awbrey
DIF. Note 2
| Out of the dimness opposite equals advance . . . .
| Always substance and increase,
| Always a knit of identity . . . . always distinction . . . .
| always a breed of life.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 28]
2. A Functional Conception of Propositional Calculus
In the general case, we start with a set of logical features {a_1, ..., a_n}
that represent properties of objects or propositions about the world. In
concrete examples the features {a_i} commonly appear as capital letters
from an "alphabet" like {A, B, C, ...} or as meaningful words from a
linguistic "vocabulary" of codes. This language can be drawn from
any sources, whether natural, technical, or artificial in character
and interpretation. In the application to dynamic systems we tend
to use the letters {x_1, ... , x_n} as our coordinate propositions,
and to interpret them as denoting properties of a system's "state",
that is, as propositions about its location in configuration space.
Because I have to consider non-deterministic systems from the outset,
I often use the word "state" in a loose sense, to denote the position
or configuration component of a contemplated state vector, whether or
not it ever gets a deterministic completion.
The set of logical features {a_1, ..., a_n} provides a basis for generating
an n-dimensional "universe of discourse" that I denote as [a_1, ..., a_n].
It is useful to consider each universe of discourse as a unified categorical
object that incorporates both the set of points <|a_1, ... , a_n|> and the set
of propositions f : <|a_1, ..., a_n|> -> B that are implicit with the ordinary
picture of a venn diagram on n features. Thus, we may regard the universe of
discourse [a_1, ..., a_n] as an ordered pair having the type (B^n, (B^n - >B)),
and we may abbreviate this last type designation as (B^n +-> B), or even more
succinctly as [B^n]. (NB. I am using "<| ... |>" as "generator brackets".)
Table 2 exhibits the scheme of notation I use to formalize the domain
of propositional calculus, corresponding to the logical content of
truth tables and venn diagrams. Although it overworks the square
brackets a bit, I also use either one of the equivalent notations
[n] or #n# to denote the data type of a finite set on n elements.
Table 2. Fundamental Notations for Propositional Calculus
o---------o-------------------o-------------------o-------------------o
| Symbol` | Notation` ` ` ` ` | Description ` ` ` | Type` ` ` ` ` ` ` |
o---------o-------------------o-------------------o-------------------o
| !A! ` ` | {a_1, ..., a_n} ` | Alphabet` ` ` ` ` | [n] `= `#n# ` ` ` |
o---------o-------------------o-------------------o-------------------o
| `A_i` ` | {(a_i), a_i}` ` ` | Dimension i ` ` ` | `B` ` ` ` ` ` ` ` |
o---------o-------------------o-------------------o-------------------o
| `A` ` ` | <|!A!|> ` ` ` ` ` | Set of cells, ` ` | `B^n` ` ` ` ` ` ` |
| ` ` ` ` | <|a_i, ..., a_n|> | coordinate tuples,| ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | {<a_i, ..., a_n>} | interpretations,` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | A_1 x ... x A_n ` | points, or vectors| ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | Prod_i A_i` ` ` ` | in the universe ` | ` ` ` ` ` ` ` ` ` |
o---------o-------------------o-------------------o-------------------o
| `A* ` ` | (hom : A -> B)` ` | Linear functions` | (B^n)*` = `B^n` ` |
o---------o-------------------o-------------------o-------------------o
| `A^ ` ` | (A -> B)` ` ` ` ` | Boolean functions | `B^n -> B ` ` ` ` |
o---------o-------------------o-------------------o-------------------o
| `A% ` ` | [!A!] ` ` ` ` ` ` | Universe of Disc. | (B^n, (B^n -> B)) |
| ` ` ` ` | (A, A^) ` ` ` ` ` | based on features | (B^n +-> B) ` ` ` |
| ` ` ` ` | (A +-> B) ` ` ` ` | {a_1, ..., a_n} ` | [B^n] ` ` ` ` ` ` |
| ` ` ` ` | (A, (A -> B)) ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | [a_1, ..., a_n] ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o-------------------o-------------------o-------------------o
Jon Awbrey
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Posted by: Jon Awbrey
DIF. Note 3
2.1. Qualitative Logic and Quantitative Analogy
| Logical, however, is used in a third sense, which is at once more
| vital and more practical; to denote, namely, the systematic care,
| negative and positive, taken to safeguard reflection so that it
| may yield the best results under the given conditions.
|
| John Dewey, 'How We Think', [Dew, 56]
These concepts and notations can now be explained in greater detail.
In order to begin as simply as possible, I distinguish two levels of
analysis and set out initially on the easier path. On the first level
of analysis, I take spaces like B, B^n, and (B^n->B) at face value and
treat them as the primary objects of interest. On the second level of
analysis, I use these spaces as coordinate charts for talking about
points and functions in more fundamental spaces.
A pair of spaces, of types B^n and (B^n->B), give typical expression
to everything that we commonly associate with the ordinary picture of
a venn diagram. The dimension, n, counts the number of "circles" or
simple closed curves that are inscribed in the universe of discourse,
corresponding to its relevant logical features or basic propositions.
Elements of type B^n correspond to what are often called propositional
"interpretations" in logic, that is, the different assignments of truth
values to sentence letters. Relative to a given universe of discourse,
these interpretations are visualized as its "cells", in other words,
the smallest enclosed areas or undivided regions of the venn diagram.
The functions f : B^n -> B correspond to the different ways of shading
the venn diagram to indicate arbitrary propositions, regions, or sets.
Regions included under a shading indicate the "models", and regions
excluded represent the "non-models" of a proposition. To recognize
and formalize the natural cohesion of these two layers of concepts
into a single universe of discourse, I introduce the type notations
[B^n] = B^n +-> B to stand for the pair of types (B^n, (B^n -> B)).
The resulting "stereotype" serves to frame the universe of discourse
as a unified categorical object, and makes it subject to prescribed
sets of evaluations and transformations (categorical morphisms or
"arrows") that affect the universe of discourse as an integrated
whole.
Most of the time we can serve the algebraic, geometric, and logical interests
of our study without worrying about their occasional conflicts and incidental
divergences. The conventions and definitions already set down will continue
to cover most of the algebraic and functional aspects of our discussion, but
to handle the logical and qualitative aspects we will need to add a few more.
In general, abstract sets may be denoted by gothic, greek, or script capital
variants of A, B, C, and so on, with elements denoted by a corresponding set
of subscripted letters in plain lower case, for example, !A! = {a_i}. Most
of the time, a set such as !A! = {a_i} will be employed as the "alphabet" of
a formal language. These alphabet letters serve to name the logical features
(properties or proposition) that generate a particular universe of discourse.
When we want to discuss the particular features of a universe of discourse,
beyond the abstract designation of a type like (B^n +-> B), then we may use
the following notations. If !A! = {a_1, ..., a_n} is an alphabet of logical
features, then A = <|!A!|> = <|a_1, ..., a_n|> is the set of interpretations,
A^ = (A -> B) is the set of propositions, and A% = [!A!] = [a_1, ..., a_n] is
the combination of these interpretations and propositions into the universe of
discourse that is based on the features {a_1, ..., a_n}.
As always, especially in concrete examples, these rules may be dropped whenever
necessary, reverting to a free assortment of feature labels. However, when we
need to talk about the logical aspects of a space that is already named as a
vector space, it will be necessary to make special provisions. At any rate,
these elaborations can be deferred until actually needed.
Jon Awbrey
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Posted by: Jon Awbrey
DIF. Note 4
2.2. Philosophy of Notation: Formal Terms and Flexible Types
| Where number is irrelevant, regimented mathematical technique has
| hitherto tended to be lacking. Thus it is that the progress of
| natural science has depended so largely upon the discernment
| of measurable quantity of one sort or another.
|
| W.V. Quine, 'Mathematical Logic', [Qui, 7]
For much of our discussion propositions and boolean functions are treated as the
same formal objects, or as different interpretations of the same formal calculus.
This rule of has exceptions, though. There is a distinctively logical interest
in the use of propositional calculus that is not exhausted by its functional
interpretation. It is part of our task in this study to deal with these
uniquely logical characteristics as they present themselves both in our
subject matter and in our formal calculus. Just to provide a hint of
what's at stake: In logic, as opposed to the more imaginative realms
of mathematics, we consider it a good thing to always know what we are
talking about. Where mathematics encourages tolerance for uninterpreted
symbols as intermediate terms, logic exerts a keener effort to interpret
directly each oblique carrier of meaning, no matter how slight, and to
unfold the complicities of every indirection in the flow of information.
Translated into functional terms, this means that we want to maintain a
continual, immediate, and persistent sense of both the converse relation
f^(-1) c B x B^n, or what is the same thing, f^(-1) : B -> Pow(B^n), and
the "fibers" or inverse images f^(-1)(0) and f^(-1)(1), associated with
each boolean function f : B^n -> B that we use. In practical terms, the
desired implementation of a propositional interpreter should incorporate
our intuitive recognition that the induced partition of the functional
domain into level sets f^(-1)(b), for b in B, is part and parcel of
understanding the denotative uses of each propositional function f.
Jon Awbrey
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Posted by: Jon Awbrey
DIF. Note 5
2.3. Special Classes of Propositions
It is important to remember that the coordinate propositions {a_i},
besides being projection maps a_i : B^n -> B, are propositions on
an equal footing with all others, even though employed as a basis
in a particular moment. This set of n propositions may sometimes
be referred to as the "basic" or "simple" propositions that found
the universe of discourse. As typical and collective notations,
we may use the forms {a_i : B^n -> B} = (B^n -i-> B) = (B^n -:> B)
to indicate the adoption of a set of a_i as a basis for discourse.
Among the 2^2^n propositions or functions in (B^n -> B) are several
fundamental sets of 2^n propositions each that take on special forms
with respect to a given basis !A! = {a_i}. Three of these forms are
especially common, the "linear", the "positive", and the "singular"
propositions. Each set is naturally parameterized by the coordinate
vectors in B^n and falls into n+1 ranks, with a binomial coefficient
C(n, k) giving the number of propositions that have rank or weight k.
The "linear" propositions, {hom : B^n -> B} = (B^n -h-> B) = (B^n ++> B),
may be expressed as sums of the following form:
Sum_i e_i = e_1 + ... + e_n where e_i = a_i or e_i = 0.
The "positive" propositions, {pos : B^n -> B} = (B^n -p-> B) = (B^n >=> B),
may be expressed as products of the following form:
Prod_i e_i = e_1 * ... * e_n where e_i = a_i or e_i = 1.
The "singular" propositions, {x : B^n -> B} = (B^n -s-> B) = (B^n ::> B),
may be expressed as products of the following form:
Prod_i e_i = e_1 * ... * e_n where e_i = a_i or e_i = (a_i).
In each case the rank k ranges from 0 to n and counts the number of
positive appearances of coordinate propositions a_i in the resulting
expression. For example, for n = 3, the linear proposition of rank 0
is "0", the positive proposition of rank 0 is "1", and the singular
proposition of rank 0 is "(a_1)(a_2)(a_3)".
The coordinate projections or simple propositions a_i : B^n -> B are both
linear and positive. So these two kinds of propositions, the linear or the
positive, may be viewed as two different ways of generalizing the class of
simple projections. The linear and the positive propositions are generated
by taking boolean sums and products, respectively, over selected subsets of
the basic propositions in {a_i}. Therefore, each set of functions can be
parameterized by the subsets J of the basic index set I = {1, ..., n}.
Let us define A_J as the subset of A that is given by {a_i : i in J}.
Then we may comprehend the action of the linear and the positive
propositions in the following terms:
- The linear proposition hom_J : B^n -> B evaluates each cell x of B^n
by looking at x's coefficients with respect to the features that hom_J
"likes", namely those in A_J, and then adds them up in B. Thus, hom_J (x)
computes the parity of the number of features that x has in A_J, yielding
one for odd and zero for even. Expressed in this idiom, hom_J (x) = 1
says that x seems "odd" (or "oddly true") to A_J, whereas hom_J (x) = 0
says that x seems "even" (or "evenly true") to A_J, so long as we recall
that "zero times" is evenly often, too.
- The positive proposition pos_J : B^n -> B evaluates each cell x of B^n
by looking at x's coefficients with regard to the features that pos_J
"likes", namely those in A_J, and then takes their product in B. Thus,
pos_J (x) assesses the unanimity of the multitude of features that x has
in A_J, yielding one for all and aught for else. In these consensual or
contractual terms, pos_J (x) = 1 means that x is "AOK" or congruent with
all of the conditions of A_J, while pos_J (x) = 0 means that x defaults
or dissents from some condition of A_J.
Jon Awbrey
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Posted by: Jon Awbrey
DIF. Note 6
2.4. Basis Relativity and Type Ambiguity
Finally, two things are important to keep in mind with regard to the
simplicity, linearity, positivity, and singularity of propositions.
First, all of these properties are relative to a particular basis.
For example, a singular proposition with respect to a basis !A!
will not remain singular if A is extended by a number of new
and independent features. Even if we stick to the original
set of pairwise options {a_i} |_| {(a_i)} to select a new
basis, the sets of linear and positive propositions are
determined by the choice of simple propositions, and
this determination is tantamount to the conventional
choice of a cell as origin.
Second, the singular propositions B ::> B, picking out as they do a single cell
or a coordinate tuple of B^n, become the carriers or the vehicles of a certain
type-ambiguity that vacillates between the dual forms B^n and (B^n ::> B) and
infects the whole hierarchy of types built on them. In plainer language, the
terms that signify the interpretations x : B^n and the singular propositions
x : B^n ::> B are fully equivalent in information, and this means that every
token of the type B^n can be reinterpreted as an appearance of the subtype
B^n ::> B. And vice versa, the two types can be exchanged with each other
everywhere that they turn up. In practical terms, this allows the use of
singular propositions as a way of denoting points, forming an alternative
to coordinate tuples. For example, relative to the universe of discourse
[a_1, a_2, a_3] the singular proposition a_1 a_2 a_3 : B^3 ::> B could be
explicitly retyped as a_1 a_2 a_3 : B^3 to indicate the point <1, 1, 1>,
but in most cases the proper interpretation could be gathered from context.
Both notations remain dependent on a particular basis, but the code that is
generated under the singular option has the advantage in its self-commenting
features, in other words, it constantly reminds us of its basis in the process
of denoting points. When the time comes to put a multiplicity of different bases
into play, and to search for objects and properties that remain invariant under the
transformations between them, this infinitesimal potential advantage may well evolve
into an overwhelming practical necessity.
Jon Awbrey
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Posted by: Jon Awbrey
DIF. Note 7
2.5. The Analogy Between Real and Boolean Types
| Measurement consists in correlating our subject matter with the series of
| real numbers; and such correlations are desirable because, once they are
| set up, all the well-worked theory of numerical mathematics lies ready at
| hand as a tool for our further reasoning.
|
| W.V. Quine, 'Mathematical Logic', [Qui, 7]
There are two further reasons why I am spending so much time on a careful
treatment of types, and they both have to do with our being able to take
full computational advantage of certain dimensions of flexibility in the
types that apply to terms. First, the domains of differential geometry
and logic programming are connected by analogies between real and boolean
types of the same pattern. Second, the types involved in these patterns
have important isomorphisms connecting them that apply on both the real
and the boolean sides of the picture.
Amazingly enough, these isomorphisms are themselves schematized by the
axioms and theorems of propositional logic. This fact is known as the
"propositions as types" analogy or the Curry-Howard isomorphism [How].
In another formulation it says that terms are to types as proofs are
to propositions. This principle seems to have more implications for
our subject than I can fully comprehend at present, though I sense
that they must be crucial. (Cf. [LaS, 42-46] and [SeH] for a good
discussion and further references.) To anticipate the bearing of
these issues on our immediate topic, Table 3 sketches a partial
overview of the Real to Boolean analogy that may serve to
illustrate the paradigm that I have in mind.
Table 3. Analogy of Real and Boolean Types
o-------------------------o-------------------------o-------------------------o
| ` ` `Real Domain R` ` ` | ` ` ` ` ` <-> ` ` ` ` ` | ` `Boolean Domain B ` ` |
o-------------------------o-------------------------o-------------------------o
| ` ` ` ` ` R^n ` ` ` ` ` | ` ` ` Basic Space ` ` ` | ` ` ` ` ` B^n ` ` ` ` ` |
o-------------------------o-------------------------o-------------------------o
| ` ` ` `R^n -> R ` ` ` ` | ` ` Function Space` ` ` | ` ` ` `B^n -> B ` ` ` ` |
o-------------------------o-------------------------o-------------------------o
| ` ` (R^n -> R) -> R ` ` | ` ` Tangent Vector` ` ` | ` ` (B^n -> B) -> B ` ` |
o-------------------------o-------------------------o-------------------------o
| R^n -> ((R^n -> R) -> R)| ` ` `Vector Field ` ` ` | B^n -> ((B^n -> B) -> B)|
o-------------------------o-------------------------o-------------------------o
| (R^n x (R^n -> R)) -> R | ` ` ` ` `ditto` ` ` ` ` | (B^n x (B^n -> B)) -> B |
o-------------------------o-------------------------o-------------------------o
| ((R^n -> R) x R^n) -> R | ` ` ` ` `ditto` ` ` ` ` | ((B^n -> B) x B^n) -> B |
o-------------------------o-------------------------o-------------------------o
| (R^n -> R) -> (R^n -> R)| ` ` ` Derivation` ` ` ` | (B^n -> B) -> (B^n -> B)|
o-------------------------o-------------------------o-------------------------o
| ` ` ` `R^n -> R^m ` ` ` | `Basic Transformation ` | ` ` ` `B^n -> B^m ` ` ` |
o-------------------------o-------------------------o-------------------------o
| (R^n -> R) -> (R^m -> R)| Function Transformation | (B^n -> B) -> (B^m -> B)|
o-------------------------o-------------------------o-------------------------o
| ` ` ` ` ` ... ` ` ` ` ` | ` ` ` ` ` ... ` ` ` ` ` | ` ` ` ` ` ... ` ` ` ` ` |
o-------------------------o-------------------------o-------------------------o
The Table exhibits a sample of likely parallels between the real and boolean domains.
The central column gives a selection of terminology that I borrow from typical usage
in differential geometry and extend in its meaning to the logical side of the Table.
These are the varieties of spaces that come up when we turn to analyzing the dynamics
of processes that followe their courses through the states of an arbitrary space X.
Moreover, when it becomes necessary to approach situations of overwhelming dynamic
complexity in a succession of qualitative reaches, then the methods of logic that
are afforded by the boolean domains, with their declarative means of synthesis
and deductive modes of analysis, supply a natural battery of tools for the task.
It is usually expedient to take these spaces two at a time, in dual pairs of the
form X and (X -> K). In general, one creates pairs of type schemas by replacing
any space X with its dual (X -> K), for example, pairing the type X -> Y with the
type (X -> K) -> (Y -> K), and X x Y with (X -> K) x (Y -> K). Here, I am using
the word "dual" in its larger sense to mean all of the functionals, not just the
linear ones. Given any function f : X -> K, the "converse" or inverse relation
corresponding to f is denoted as f^(-1), and the subsets of X that are defined
by f^(-1)(k), taken over k in K, are called the "fibers" or the "level sets"
of the function f.
Jon Awbrey
---------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 8
2.6. Theory of Control and Control of Theory
| You will hardly know who I am or what I mean,
| But I shall be good health to you nevertheless,
| And filter and fibre your blood.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 88]
In the boolean context, a function f : X -> B is tantamount to a "proposition"
about elements of X, and the elements of X constitute the "interpretations" of
that proposition. The fiber f^(-1)(1) comprises the set of "models" of f, or
examples of elements in X satisfying the proposition f. The fiber f^(-1)(0)
collects the complementary set of "anti-models", or the exceptions to the
proposition f that exist in X. Of course, the space of functions (X -> B)
is isomorphic to the set of all subsets of X, called the "power set" of X
and often denoted as Pow(X) or 2^X.
The operation of replacing X by (X -> B) in a type schema corresponds
to a certain shift of attitude towards the space X, in which one passes
from a focus on the ostensibly individual elements of X to a concern with
the states of information and uncertainty that one possesses about objects
and situations in X. The conceptual obstacles in the path of this transition
can be smoothed over by using singular functions (X ::> B) as stepping stones.
First of all, it's an easy step from an element x of type B^n to the equivalent
information of a singular proposition x : X ::> B, and then only a small jump of
generalization remains to reach the type of an arbitrary proposition f : X -> B,
perhaps understood to indicate a relaxed constraint on the singularity of points
or a neighborhood circumscribing the original x. I have frequently discovered
this to be a useful transformation, communicating between the "objective" and
the "intentional" perspectives, in spite perhaps of the open objection that
this distinction is transient in the mean time and ultimately superficial.
It is hoped that this measure of flexibility, allowing us to stretch a point
into a proposition, can be useful in the examination of inquiry driven systems,
where the differences between empirical, intentional, and theoretical propositions
constitute the discrepancies and the distributions that drive experimental activity.
I can give this model of inquiry a cybernetic cast by realizing that theory change
and theory evolution, as well as the choice and the evaluation of experiments, are
actions that are taken by a system or its agent in response to the differences
that are detected between observational contents and theoretical coverage.
All of the above notwithstanding, there are several points that distinguish
these two tasks, namely, the "theory of control" and the "control of theory",
features that are often obscured by too much precipitation in the quickness
with which we understand their similarities. In the control of uncertainty
through inquiry, some of the actuators that we need to be concerned with are
axiom changers and theory modifiers, operators with the power to compile and
to revise the theories that generate expectations and predictions, effectors
that form and edit our grammars for the languages of observational data, and
agencies that rework the proposed model to fit the actual sequences of events
and the realized relationships of values that are observed in the environment.
Moreover, when steps must be taken to carry out an experimental action, there
must be something about the particular shape of our uncertainty that guides us
in choosing what directions to explore, and this impression is more than likely
influenced by previous accumulations of experience. Thus it must be anticipated
that much of what goes into scientific progress, or any sustainable effort toward
a goal of knowledge, is necessarily predicated on long term observation and modal
expectations, not only on the more local or short term prediction and correction.
Jon Awbrey
---------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 9
2.7. Propositions as Types and Higher Order Types
The arrangement of types collected in Table 3 can
serve as a good introduction to several ideas about
"higher order propositional expressions" (HOPE's) and
also about the "propositions as types" (PAT) isomorphism.
Table 3. Analogy of Real and Boolean Types
o-------------------------o-------------------------o-------------------------o
| ` ` `Real Domain R` ` ` | ` ` ` ` ` <-> ` ` ` ` ` | ` `Boolean Domain B ` ` |
o-------------------------o-------------------------o-------------------------o
| ` ` ` ` ` R^n ` ` ` ` ` | ` ` ` Basic Space ` ` ` | ` ` ` ` ` B^n ` ` ` ` ` |
o-------------------------o-------------------------o-------------------------o
| ` ` ` `R^n -> R ` ` ` ` | ` ` Function Space` ` ` | ` ` ` `B^n -> B ` ` ` ` |
o-------------------------o-------------------------o-------------------------o
| ` ` (R^n -> R) -> R ` ` | ` ` Tangent Vector` ` ` | ` ` (B^n -> B) -> B ` ` |
o-------------------------o-------------------------o-------------------------o
| R^n -> ((R^n -> R) -> R)| ` ` `Vector Field ` ` ` | B^n -> ((B^n -> B) -> B)|
o-------------------------o-------------------------o-------------------------o
| (R^n x (R^n -> R)) -> R | ` ` ` ` `ditto` ` ` ` ` | (B^n x (B^n -> B)) -> B |
o-------------------------o-------------------------o-------------------------o
| ((R^n -> R) x R^n) -> R | ` ` ` ` `ditto` ` ` ` ` | ((B^n -> B) x B^n) -> B |
o-------------------------o-------------------------o-------------------------o
| (R^n -> R) -> (R^n -> R)| ` ` ` Derivation` ` ` ` | (B^n -> B) -> (B^n -> B)|
o-------------------------o-------------------------o-------------------------o
| ` ` ` `R^n -> R^m ` ` ` | `Basic Transformation ` | ` ` ` `B^n -> B^m ` ` ` |
o-------------------------o-------------------------o-------------------------o
| (R^n -> R) -> (R^m -> R)| Function Transformation | (B^n -> B) -> (B^m -> B)|
o-------------------------o-------------------------o-------------------------o
| ` ` ` ` ` ... ` ` ` ` ` | ` ` ` ` ` ... ` ` ` ` ` | ` ` ` ` ` ... ` ` ` ` ` |
o-------------------------o-------------------------o-------------------------o
First, observe that the type of a "Tangent Vector at a Point",
also known as a "directional derivative" at that point, has the
form (K^n -> K) -> K, where K is the chosen ground field, in the
present case either R or B. At a point in a space of type K^n,
a directional derivative operator !q! takes a function on that
space, an f of type (K^n -> K), and maps it to a ground field
value of type K. This value is known as the "derivative" of
f in the direction !q! [Che46, 76-77]. In the boolean case,
!q! : (B^n -> B) -> B has the form of a proposition about
propositions, in other words, a proposition of the next
higher type.
Next, by way of illustrating the propositions as types theme, consider
a proposition of the form X => (Y => Z). One knows from propositional
calculus that this is logically equivalent to a proposition of the form
(X & Y) => Z. But this equivalence should remind us of the functional
isomorphism that exists between a construction of the type X -> (Y -> Z)
and a construction of the type (X x Y) -> Z. The propositions as types
analogy permits us to take a functional type like this and, under the
right conditions, replace the functional arrows "->" and products "x"
with the respective logical arrows "=>" and products "&". Accordingly,
viewing the result as a proposition, we can employ axioms and theorems
of propositional calculus to suggest appropriate isomorphisms among the
categorical and functional constructions.
Finally, examine the middle four rows of Table 3. These display
a series of isomorphic types that stretch from the categories that
are labeled "Vector Field" to those that are labeled "Derivation".
A "vector field", also known as an "infinitesimal transformation",
associates a tangent vector at a point with each point of a space.
In symbols, a vector field is a function !X! : X -> |_| !X!_x that
assigns to each point x of the space X a tangent vector !X!_x to X
at that point [Che46, 82-83]. If X is of type K^n, then !X! is of
type K^n -> ((K^n -> K) -> K). This has the pattern X -> (Y -> Z),
with X = K^n, Y = (K^n -> K), and Z = K.
Applying the propositions as types analogy, one can follow this pattern
through a series of metamorphoses from the type of a vector field to the
type of a derivation, as traced out in Table 4. Observe how the function
f : X -> K, associated with the place of Y in the pattern, moves through
its paces from the second to the first position. In this way, the vector
field !X!, initially viewed as attaching each tangent vector !X!_x to the
site x where it acts in X, now comes to be seen as acting on each scalar
potential f : X -> K like a generalized species of differentiation,
producing another function !X!f : X -> K of the same type.
Table 4. An Equivalence Based on the Propositions as Types Analogy
o------------------------o------------------------o--------------------------o
| ` ` ` `Pattern` ` ` ` `| ` ` `Construction ` ` `| ` ` ` ` Instance` ` ` ` `|
o------------------------o------------------------o--------------------------o
| ` ` `X -> (Y -> Z)` ` `| ` ` `Vector Field ` ` `| K^n -> ((K^n -> K) -> K) |
o------------------------o------------------------o--------------------------o
| ` ` (X x Y) `-> Z ` ` `| ` ` ` ` ` ` ` ` ` ` ` `| (K^n x (K^n -> K)) -> K `|
o------------------------o------------------------o--------------------------o
| ` ` (Y x X) `-> Z ` ` `| ` ` ` ` ` ` ` ` ` ` ` `| ((K^n -> K) x K^n) -> K `|
o------------------------o------------------------o--------------------------o
| ` ` `Y -> (X -> Z)` ` `| ` ` ` Derivation` ` ` `| (K^n -> K) -> (K^n -> K) |
o------------------------o------------------------o--------------------------o
Jon Awbrey
---------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 10
2.8. 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 dots or full stops
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:
- The idea of a universe of discourse, which includes both
a space of "points" and a space of "maps" on those points.
- 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
---------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 11
2.9. Tables of Propositional Forms
| To the scientist longing for non-quantitative techniques, then,
| mathematical logic brings hope. It provides explicit techniques
| for manipulating the most basic ingredients of discourse.
|
| W.V. Quine, 'Mathematical Logic', [Qui, 7-8]
To prepare for the next phase of discussion, Tables 6 and 7 collect
and summarize all of the propositional forms on one and two variables.
These propositional forms are represented over bases of boolean variables
as complete sets of boolean-valued functions. Adjacent to their names and
specifications are listed what are roughly the simplest expressions in the
"cactus language", the particular syntax for propositional calculus that
I use in formal and computational contexts. For the sake of orientation,
the English paraphrases and the more common notations are listed in the
last two columns. As simple and circumscribed as these low-dimensional
universes may appear to be, a careful exploration of their differential
extensions will involve us in complexities sufficient to demand our
attention for some time to come.
Propositional forms on one variable correspond to boolean functions f : B^1 -> B.
In Table 6 these functions are listed in a variant form of truth table, one which
rotates the axes of the usual arrangement. Each function f_i is indexed by the
string of values that it takes on the points of the universe X% = [x] ~=~ B^1.
The binary index generated in this way is converted to its decimal equivalent,
and these are used as conventional names for the f_i, as shown in the first
column of the Table. In their own right the 2^1 points of the universe X%
are coordinated as a space of type B^1, this in light of the universe X%
being a functional domain where the coordinate projection x takes on
its values in B.
Table 6. Propositional Forms on One Variable
o---------o---------o---------o----------o------------------o-----------o
| L_1 ` ` | L_2 ` ` | L_3 ` ` | L_4 ` ` `| L_5 ` ` ` ` ` ` `| L_6 ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| Decimal | Binary `| Vector `| Cactus` `| English ` ` ` ` `| Ordinary` |
o---------o---------o---------o----------o------------------o-----------o
| ` ` ` ` | ` ` ` x : ` 1 0 ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
o---------o---------o---------o----------o------------------o-----------o
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_0 ` ` | `f_00 ` | ` 0 0 ` | ` ( ) ` `| false ` ` ` ` ` `| ` ` 0 ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_1 ` ` | `f_01 ` | ` 0 1 ` | ` (x) ` `| not x ` ` ` ` ` `| ` `~x ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_2 ` ` | `f_10 ` | ` 1 0 ` | ` `x` ` `| x ` ` ` ` ` ` ` `| ` ` x ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_3 ` ` | `f_11 ` | ` 1 1 ` | `(( ))` `| true` ` ` ` ` ` `| ` ` 1 ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
o---------o---------o---------o----------o------------------o-----------o
Propositional forms on two variables correspond to boolean functions f : B^2 -> B.
In Table 7 each function f_i is indexed by the values that it takes on the points
of the universe X% = [x, y] ~=~ B^2. Converting the binary index thus generated
to a decimal equivalent, we obtain the functional nicknames that are listed in
the first column. The 2^2 points of the universe X% are coordinated as a space
of type B^2, as indicated under the heading of the Table, where the coordinate
projections x and y run through the various combinations of their values in B.
Table 7. Propositional Forms On Two Variables
o---------o---------o---------o----------o------------------o-----------o
| L_1 ` ` | L_2 ` ` | L_3 ` ` | L_4 ` ` `| L_5 ` ` ` ` ` ` `| L_6 ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| Decimal | Binary` | Vector` | Cactus` `| English ` ` ` ` `| Ordinary` |
o---------o---------o---------o----------o------------------o-----------o
| ` ` ` ` | ` ` ` x : 1 1 0 0 | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| ` ` ` ` | ` ` ` y : 1 0 1 0 | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
o---------o---------o---------o----------o------------------o-----------o
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_0 ` ` | f_0000` | 0 0 0 0 | ` `() ` `| false ` ` ` ` ` `| ` ` 0 ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_1 ` ` | f_0001` | 0 0 0 1 | `(x)(y) `| neither x nor y `| `~x & ~y` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_2 ` ` | f_0010` | 0 0 1 0 | `(x) y` `| y and not x ` ` `| `~x & `y` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_3 ` ` | f_0011` | 0 0 1 1 | `(x)` ` `| not x ` ` ` ` ` `| `~x ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_4 ` ` | f_0100` | 0 1 0 0 | ` x (y) `| x and not y ` ` `| ` x & ~y` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_5 ` ` | f_0101` | 0 1 0 1 | ` ` (y) `| not y ` ` ` ` ` `| ` ` ` ~y` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_6 ` ` | f_0110` | 0 1 1 0 | `(x, y) `| x not equal to y`| ` x + `y` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_7 ` ` | f_0111` | 0 1 1 1 | `(x `y) `| not both x and y`| `~x v ~y` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_8 ` ` | f_1000` | 1 0 0 0 | ` x `y` `| x and y ` ` ` ` `| ` x & `y` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_9 ` ` | f_1001` | 1 0 0 1 | ((x, y))`| x equal to y` ` `| ` x = `y` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_10` ` | f_1010` | 1 0 1 0 | ` ` `y` `| y ` ` ` ` ` ` ` `| ` ` ` `y` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_11` ` | f_1011` | 1 0 1 1 | `(x (y))`| not x without y `| ` x => y` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_12` ` | f_1100` | 1 1 0 0 | ` x ` ` `| x ` ` ` ` ` ` ` `| ` x ` ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_13` ` | f_1101` | 1 1 0 1 | ((x) y) `| not y without x `| ` x <= y` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_14` ` | f_1110` | 1 1 1 0 | ((x)(y))`| x or y` ` ` ` ` `| ` x v `y` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
| f_15` ` | f_1111` | 1 1 1 1 | ` (())` `| true` ` ` ` ` ` `| ` ` 1 ` ` |
| ` ` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` `| ` ` ` ` ` ` ` ` `| ` ` ` ` ` |
o---------o---------o---------o----------o------------------o-----------o
Jon Awbrey
---------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 12
| Fire over water:
| The image of the condition before transition.
| Thus the superior man is careful
| In the differentiation of things,
| So that each finds its place.
|
| 'I Ching', Hexagram 64, [Wil, 249]
3. A Differential Extension of Propositional Calculus
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.
Jon Awbrey
---------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 13
3.1. Differential Propositions: The Qualitative Analogue of Differential Equations
In order to define the differential extension of a universe of discourse [!A!],
the initial alphabet !A! must be extended to include a collection of symbols for
"differential features", or basic "changes" that are capable of occurring in [!A!].
Intuitively, these symbols may be construed as denoting primitive features of change,
qualitative attributes of motion, or propositions about how things or points in [!A!]
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!A! = {da_1, ... , da_n}, in principle, just an arbitrary alphabet of symbols,
disjoint from the initial alphaber !A! = {a_1, ..., a_n}, 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!A! 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 !A! and d!A!.)
In the above terms, a typical "tangent space of A at a point x", frequently
denoted as T_x (A), can be characterized as having the generic construction
dA = <|d!A!|> = <|da_1, ..., da_n|>. 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:
- dA = Prod_i dA_i = dA_1 x ... x dA_n.
Here, dA_i is an alphabet of two symbols, dA_i = {(da_i), da_i},
where (da_i) is a symbol with the logical value of "not da_i".
Each component dA_i has the type B, under the correspondence
{(da_i), da_i} ~=~ {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 = {(da_i), da_i} = {same, different} = {stay, change} = {stop, step}.
Viewed within a coordinate representation, spaces of type B^n and D^n 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.
Jon Awbrey
----------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 14
3.2. An Interlude on the Path
| There would have been no beginnings:
| instead, speech would proceed from me,
| while I stood in its path - a slender gap -
| the point of its possible disappearance.
|
| Michel Foucault, 'The Discourse on Language', [Fou, 215]
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 [!X!]. Given the boolean value system, a path in
the space X = <|!X!|> is a map q : B -> X. In this case, the set of paths (B -> X)
is isomorphic to the cartesian square X^2 = X x X, or the set of ordered pairs from X.
We may analyze X^2 = {<u, v> : u, v in X} into two parts,
specifically, the pairs that lie on and off the diagonal:
- X^2 = {<u, v> : u = v} |_| {<u, v> : u =/= v}.
In symbolic terms, this partition may be expressed as:
- X^2 ~=~ Diag(X) + 2 * Comb(X, 2),
where:
- Diag(X) = {<x, x> : x in X},
and where:
- Comb(X, k) = "X choose k" = {k-sets from X},
so that:
- Comb(X, 2) = {{u, v} : u, v in X}.
We can now use the features in d!X! = {dx_1, ... , dx_n} to classify the paths
of (B -> X) by way of the pairs in X^2. If X ~=~ B^n then a path in X has the
form q : (B -> B^n) ~=~ B^n x B^n ~=~ B^2n ~=~ (B^2)^n. Intuitively, we want
to map this (B^2)^n onto D^n by mapping each component B^2 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!X!.
Therefore, define dx_i : X^2 -> B such that:
- dx_i (<u, v>)
= (| x_i (u) , x_i (v) |)
= x_i (u) + x_i (v)
= x_i (v) - x_i (u).
NB. In the above transcription, "(| ... , ... |)" 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 thus equivalent to "-", in the sense of adding the additive inverse.
The above definition is equivalent to defining dx_i : (B -> X) -> B such that:
- dx_i (q)
= (| x_i (q_0) , x_i (q_1) |)
= x_i (q_0) + x_i (q_1)
= x_i (q_1) - x_i (q_0),
where q_b = q(b), for each b in B. Thus, the proposition dx_i
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_i.
Now we can use the language of features in <|d!X!|>, indeed the whole calculus
of propositions in [d!X!], to classify paths and sets of paths. In other words,
the paths can be taken as models of the propositions g : dX -> B. For example,
the paths corresponding to Diag(X) fall under the description (dx_1)...(dx_n),
which says that nothing changes among the set of features {x_1, ..., x_n}.
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.
Jon Awbrey
---------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 15
3.3. The Extended Universe of Discourse
| 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.
|
| Michel Foucault, 'The Discourse on Language', [Fou, 215]
Next, we define the so-called "extended alphabet" or "bundled alphabet" E!A! as:
- E!A! = !A! |_| d!A! = {a_1, ..., a_n, da_1, ..., da_n}.
This supplies enough material to construct the "differential extension" EA,
or the "tangent bundle" over the initial space A, in the following fashion:
- EA
= A x dA = <|E!A!|> = <|!A! |_| d!A!|>
= <|a_1, ..., a_n, da_1, ..., da_n|>,
thus giving EA the type B^n x D^n.
Finally, the tangent universe EA% = [E!A!] is constituted from the totality
of points and maps, or interpretations and propositions, which are based on
the extended set of features E!A!:
- EA% = [E!A!] = [a_1, ..., a_n, da_1, ..., da_n],
thus giving the tangent universe E!A! the type
(B^n x D^n +-> B) = (B^n x D^n, (B^n x D^n -> B).
A proposition in the tangent universe [E!A!] 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 EA
and the differential proposition h : EA -> 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).
Table 8. Notation for the Differential Extension of Propositional Calculus
o---------------------------------------------------------------------o
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o-------------------o-------------------o-------------------o
| Symbol` | Notation` ` ` ` ` | Description ` ` ` | Type` ` ` ` ` ` ` |
o---------o-------------------o-------------------o-------------------o
| d!A!` ` | {da_1, ..., da_n} | Alphabet of ` ` ` | [n] `= `#n# ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | differential` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | features` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o-------------------o-------------------o-------------------o
| dA_i` ` | {(da_i), da_i}` ` | Differential` ` ` | `D` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | dimension i ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o-------------------o-------------------o-------------------o
| dA` ` ` | <|d!A!|>` ` ` ` ` | Tangent space ` ` | `D^n` ` ` ` ` ` ` |
| ` ` ` ` | <|da_i,...,da_n|> | at a point: ` ` ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | {<da_i,...,da_n>} | Set of changes, ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | dA_1 x ... x dA_n | motions, steps, ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | Prod_i dA_i ` ` ` | tangent vectors ` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | at a point` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o-------------------o-------------------o-------------------o
| dA* ` ` | (hom : dA -> B) ` | Linear functions` | (D^n)* `~=~ `D^n` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | on dA ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o-------------------o-------------------o-------------------o
| dA^ ` ` | (dA -> B) ` ` ` ` | Boolean functions | `D^n -> B ` ` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | on dA ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o-------------------o-------------------o-------------------o
| dA% ` ` | [d!A!]` ` ` ` ` ` | Tangent universe` | (D^n, (D^n -> B)) |
| ` ` ` ` | (dA, dA^) ` ` ` ` | at a point of A%, | (D^n +-> B) ` ` ` |
| ` ` ` ` | (dA +-> B)` ` ` ` | based on the` ` ` | [D^n] ` ` ` ` ` ` |
| ` ` ` ` | (dA, (dA -> B)) ` | tangent features` | ` ` ` ` ` ` ` ` ` |
| ` ` ` ` | [da_1, ..., da_n] | {da_1, ..., da_n} | ` ` ` ` ` ` ` ` ` |
o---------o-------------------o-------------------o-------------------o
The adjectives "differential" or "tangent" are systematically attached to
every construct based on the differential alphabet d!A!, taken by itself.
Strictly speaking, we probably ought to call d!A! the set of "cotangent"
features derived from !A!, but the only time this distinction really
seems to matter is when we need to distinguish the tangent vectors
as maps of type (B^n -> B) -> B from cotangent vectors as elements
of type D^n. In like fashion, having defined E!A! = !A! |_| d!A!,
we can systematically attach the adjective "extended" or the
substantive "bundle" to all the constructs associated with
this full complement of 2n 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).
Table 9. Higher Order Differential Features
o----------------------------------------o----------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| !A! ` = d^0.!A! = {a_1, ..., a_n} ` ` `| E^0.!A! `= d^0.!A!` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| d!A! `= d^1.!A! = {da_1, ..., da_n} ` `| E^1.!A! `= d^0.!A! |_| d^1.!A!` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` d^k.!A! = {d^k.a_1,...,d^k.a_n}| E^k.!A! `= d^0.!A! |_| ... |_| d^k.!A!`|
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| d*!A! = {d^0.!A!, ..., d^k.!A!, ...}` `| E^oo.!A! = d*!A!` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
o----------------------------------------o----------------------------------------o
Jon Awbrey
---------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 16
3.4. Intentional Propositions
| Do you guess I have some intricate purpose?
| Well I have . . . . for the April rain has, and the mica on
| the side of a rock has.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 45]
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^k and Q^k are convenient in many contexts
where the accumulation of prime symbols and union symbols
would otherwise be cumbersome.
Table 10. A Realm of Intentional Features
o---------------------------------------o----------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| p^0.!A! `= `!A! `= `{a_1, ..., a_n} ` | Q^0.!A! `= `!A! ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| p^1.!A! `= `!A!' = `{a_1', ..., a_n'} | Q^1.!A! `= `!A! |_| !A!'` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| p^2.!A! `= `!A!" = `{a_1", ..., a_n"} | Q^2.!A! `= `!A! |_| !A!' |_| !A!" ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| ... ` ` ` ` ... ` ` ... ` ` ` ` ` ` ` | ... ` ` ` ` ... ` ` ` ` ` ` ` ` ` ` ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| p^k.!A! `= `{p^k.a_1, ..., p^k.a_n} ` | Q^k.!A! `= `!A! |_| ... |_| p^k.!A! ` `|
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
o---------------------------------------o----------------------------------------o
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^k and E^k, 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, "a reign of X".
For the purposes of this discussion, let us define an "intentional proposition"
as a proposition in the universe of discourse QX% = [Q!X!], in other words,
a map q : QX -> B. The sense of this definition may be seen if we consider
the following facts. First, the equivalence QX = X x X' motivates the
following chain of isomorphisms between spaces:
- (QX -> B) ~=~ (X x X' -> B) ~=~ (X -> (X' -> B)) ~=~ (X' -> (X -> B)).
Viewed in this light, an intentional proposition q may be rephrased as a map
q : X 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 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^i.X -> B as an "i^th 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.
Jon Awbrey
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Posted by: Jon Awbrey
DIF. Note 17
3.5. Life on Easy Street
| Failing to fetch me at first keep encouraged,
| Missing me one place search another,
| I stop some where waiting for you
|
| Walt Whitman, 'Leaves of Grass', [Whi, 88]
The finite character of the extended universe [E!A!] makes the problem
of solving differential propositions relatively straightforward, at least,
in principle. The solution set of the differential proposition q : EA -> B
is the set of models (q^(-1))(1) in EA. Finding all of the models of q, the
extended interpretations in EA 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!A!] 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.
Jon Awbrey
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Posted by: Jon Awbrey
DIF. Note 18
| I would have preferred to be enveloped in words,
| borne way beyond all possible beginnings.
|
| Michel Foucault, 'The Discourse on Language', [Fou, 215]
4. Back to the Beginning: Some Exemplary Universes
To anchor our understanding of differential logic, let us look at how the
various concepts apply in the simplest possible concrete cases, where the
initial dimension is only 1 or 2. In spite of the obvious simplicity of
these cases, it is possible to observe how central difficulties of the
subject begin to arise already at this stage.
Jon Awbrey
---------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 19
4.1. A One-Dimensional Universe
| There was never any more inception than there is now,
| Nor any more youth or age than there is now;
| And will never be any more perfection than there is now,
| Nor any more heaven or hell than there is now.
|
| Walt Whitman, Leaves of Grass, [Whi, 28]
Let !X! = {x_1} = {A} be an alphabet that represents one boolean variable or
a single logical feature. In this example I am using the capital letter "A"
in a more usual informal way, to name a feature and not a space, at variance
with my formerly stated formal conventions. At any rate, the basis element
A = x_1 may be interpreted as a simple proposition or a coordinate projection
A = x_1 : B^1 -:> B. The space X = <|A|> = {(A), A} of points (cells, vectors,
interpretations) has cardinality 2^n = 2^1 = 2 and is isomorphic to B = {0, 1}.
Moreover, X may be identified with the set of singular propositions {x : B ::> B}.
The space of linear propositions X* = {hom : B ++> B} = {0, A} is algebraically
dual to X and also has cardinality 2. Here, "0" is interpreted as denoting the
constant function 0 : B -> B, amounting to the linear proposition of rank 0,
while A is the linear proposition of rank 1. Last but not least we have the
positive propositions {pos : B oo> B} = {A, 1}, of rank 1 and 0, respectively,
where "1" is understood as denoting the constant function 1 : B -> B. In sum,
there are 2^2^n = 2^2^1 = 4 propositions altogether in the universe of discourse,
comprising the set X^ = {f : X -> B} = {0, (A), A, 1} ~=~ (B -> B).
The first order differential extension of !X! is E!X! = {x_1, dx_1} = {A, dA}.
If the feature "A" is understood as applying to some object or state, then the
feature "dA" may be interpreted as an attribute of the same object or state that
says that it is changing "significantly" with respect to the property A, or that
it has an "escape velocity" with respect to the state A. In practice, differential
features acquire their logical meaning through a class of "temporal inference rules".
For example, relative to a frame of observation that is left implicit for now,
one is permitted to make the following sorts of inference: from the fact that
A and dA are true at a given moment one may infer that (A) will be true in the
next moment of observation. Altogether in the present instance, there is the
fourfold scheme of inference that is shown below:
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `From `(A) & (dA) `infer` (A) `next.` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `From `(A) &` dA` `infer` `A` `next.` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `From ` A `& (dA) `infer` `A` `next.` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` `From ` A `&` dA` `infer` (A) `next.` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
It might be thought that we need to bring in an independent time variable
at this point, but an insight of fundamental importance appears to be that
the idea of process is more basic than the notion of time. A time variable
is actually a reference to a "clock", that is, a canonical or a convenient
process that is established or accepted as a standard of measurement, but
in essence no different than any other process. This raises the question
of how different subsystems in a more global process can be brought into
comparison, and what it means for one process to serve the function of
a local standard for others. But these inquiries only wrap up puzzles
in further riddles, and are obviously too involved to be handled at
our current level of approximation.
| The clock indicates the moment . . . . but what does
| eternity indicate?
|
| Walt Whitman, 'Leaves of Grass', [Whi, 79]
Observe that the secular inference rules, used by themselves,
involve a loss of information, since nothing in them can tell
us whether the momenta {(dA), dA} are preserved or changed in
the next instance. In order to know this, we would have to
determine d^2.A, and so on, pursuing an infinite regress.
Ultimately, in order to rest with a finitely determinate
system, it is necessary to make an infinite assumption,
for example, that d^k.A = 0 for all k greater than some
fixed value M. Another way to escape the regress is
through the provision of a dynamic law, in typical
form making higher order differentials dependent
on lower degrees and estates.
Jon Awbrey
---------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 20
4.2. Example 1. A Square Rigging
| Urge and urge and urge,
| Always the procreant urge of the world.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 28]
By way of example, suppose that we are given the initial condition A = dA and
the law d^2.A = (A). Then, since "A = dA" <=> "A dA or (A)(dA)", we may infer
two possible trajectories, as displayed in Table 11. In either of these cases,
the state A (dA)(d^2.A) is a stable attractor or a terminal condition for both
starting points.
Table 11. A Pair of Commodious Trajectories
o---------o-------------------o-------------------o
| Time` ` | Trajectory 1` ` ` | Trajectory 2` ` ` |
o---------o-------------------o-------------------o
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| 0 ` ` ` | `A` `dA `(d^2.A)` | (A) (dA) `d^2.A ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| 1 ` ` ` | (A) `dA ` d^2.A ` | (A) `dA ` d^2.A ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| 2 ` ` ` | `A` (dA) (d^2.A)` | `A` (dA) (d^2.A)` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| 3 ` ` ` | `A` (dA) (d^2.A)` | `A` (dA) (d^2.A)` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
| 4 ` ` ` | `"` ` " ` `"` ` ` | `"` ` " ` `"` ` ` |
| ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
o---------o-------------------o-------------------o
Because the initial space X = <|A|> is one-dimensional, we can easily fit
the second order extension E^2.X = <|A, dA, d^2.A|> within the compass of
a single venn diagram, charting the couple of converging trajectories as
shown in Figure 12.
o-------------------------------------------------o
| E^2.X ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o-------------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` `/` ` ` ` ` ` ` `\` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` / ` ` ` `A` ` ` ` \ ` ` ` ` ` ` ` |
| ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` `\` ` ` ` ` ` ` |
| ` ` ` ` ` ` / ` ` ` ` ->- ` ` ` ` \ ` ` ` ` ` ` |
| ` ` ` ` ` `o` ` ` ` `/` `\` ` ` ` `o` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` `\` `/` ` ` ` `|` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` -o- ` ` ` ` `|` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` ` `^` ` ` ` ` `|` ` ` ` ` ` |
| ` ` ` `o---o---------o | o---------o---o` ` ` ` |
| ` ` ` / ` ` \ ` ` ` ` \|/ ` ` ` ` / ` ` \ ` ` ` |
| ` ` `/` ` ` `\` ` o ` `|` ` ` ` `/` ` ` `\` ` ` |
| ` ` / ` ` ` ` \ ` | ` /|\ ` ` ` / ` ` ` ` \ ` ` |
| ` `/` ` ` ` ` `\` | `/`|`\` ` `/` ` ` ` ` `\` ` |
| ` o ` ` ` ` ` ` o-|-o--|--o---o ` ` ` ` ` ` o ` |
| ` | ` ` ` ` ` ` ` | | `|` | ` ` ` ` ` ` ` ` | ` |
| ` | ` ` ` ` ` ` ` ---->o<----o` ` ` ` ` ` ` | ` |
| ` | ` ` ` ` ` ` ` ` | ` ` | ` ` ` ` ` ` ` ` | ` |
| ` o ` ` ` dA` ` ` ` o ` ` o ` ` `d^2.A` ` ` o ` |
| ` `\` ` ` ` ` ` ` ` `\` `/` ` ` ` ` ` ` ` `/` ` |
| ` ` \ ` ` ` ` ` ` ` ` \ / ` ` ` ` ` ` ` ` / ` ` |
| ` ` `\` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` `/` ` ` |
| ` ` ` \ ` ` ` ` ` ` ` / \ ` ` ` ` ` ` ` / ` ` ` |
| ` ` ` `o-------------o` `o-------------o` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 12. The Anchor
If we eliminate from view the regions of E^2.X that are ruled out
by the dynamic law d^2.A = (A), then what remains is the quotient
structure that is shown in Figure 13. This picture makes it easy
to see that the dynamically allowable portion of the universe is
partitioned between the properties A and d^2.A. As it happens,
this fact might have been expressed "right off the bat" by an
equivalent formulation of the differential law, one that uses
the exclusive disjunction to state the law as (A, d^2.A).
o-------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ->- ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `/` `\` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `\` `/` ` ` ` ` |
| ` ` ` ` ` ` ` ` o-------------o ` -o- ` ` ` ` ` |
| ` ` ` ` ` ` ` `/` ` ` ` ` ` ` `\` ^ ` ` ` ` ` ` |
| ` ` ` ` ` ` ` / ` ` ` dA` ` ` ` \/` ` ` ` `A` ` |
| ` ` ` ` ` ` `/` ` ` ` ` ` ` ` ` /\` ` ` ` ` ` ` |
| ` ` ` ` ` ` / ` ` ` ` ` ` ` ` `/` \ ` ` ` ` ` ` |
| ` ` ` ` ` `o` ` o ` ` ` ` ` ` / ` `o` ` ` ` ` ` |
| ` ` ` ` ` `|` ` `\` ` ` ` ` `/` ` `|` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` \ ` ` ` ` / ` ` `|` ` ` ` ` ` |
o------------|-------\-------/-------|------------o
| ` ` ` ` ` `|` ` ` ` \ ` ` / ` ` ` `|` ` ` ` ` ` |
| ` ` ` ` ` `|` ` ` ` `\` `/` ` ` ` `|` ` ` ` ` ` |
| ` ` ` ` ` `o` ` ` ` ` v / ` ` ` ` `o` ` ` ` ` ` |
| ` ` ` ` ` ` \ ` ` ` ` `o` ` ` ` ` / ` ` ` ` ` ` |
| ` ` ` ` ` ` `\` ` ` ` `^` ` ` ` `/` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` \ ` ` ` `|` ` ` ` / ` ` ` `d^2.A` |
| ` ` ` ` ` ` ` `\` ` ` `|` ` ` `/` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` o------|------o ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `|` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` `o` ` ` ` ` ` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o-------------------------------------------------o
Figure 13. The Tiller
What we have achieved in this example is to give a differential description of
a simple dynamic process. In effect, we did this by embedding a directed graph,
which can be taken to represent the state transitions of a finite automaton, in
a dynamically allotted quotient structure that is created from a boolean lattice
or an n-cube by nullifying all of the regions that the dynamics outlaws. With
growth in the dimensions of our contemplated universes, it becomes essential,
both for human comprehension and for computer implementation, that the dynamic
structures of interest to us be represented not actually, by acquaintance, but
virtually, by description. In our present study, we are using the language of
propositional calculus to express the relevant descriptions, and to comprehend
the structure that is implicit in the subsets of a n-cube without necessarily
being forced to actualize all of its points.
One of the reasons for engaging in this kind of extremely reduced, but explicitly
controlled case study is to throw light on the general study of languages, formal
and natural, in their full array of syntactic, semantic, and pragmatic aspects.
Propositional calculus is one of the last points of departure where we can view
these three aspects interacting in a non-trivial way without being immediately
and totally overwhelmed by the complexity they generate. Often this complexity
causes investigators of formal and natural languages to adopt the strategy of
focusing on a single aspect and to abandon all hope of understanding the whole,
whether it's the still living natural language or the dynamics of inquiry that
lies crystallized in formal logic.
From the perspective that I find most useful here, a language is a syntactic
system that is designed or evolved in part to express a set of descriptions.
When the explicit symbols of a language have extensions in its object world
that are actually infinite, or when the implicit categories and generative
devices of a linguistic theory have extensions in its subject matter that
are potentially infinite, then the finite characters of terms, statements,
arguments, grammars, logics, and rhetorics force an excess of intension
to reside in all these symbols and functions, across the spectrum from
the object language to the metalinguistic uses. In the aphorism from
W. von Humboldt that Chomsky often cites, for example, in [Cho86, 30]
and [Cho93, 49], language requires "the infinite use of finite means".
This is necessarily true when the extensions are infinite, when the
referential symbols and grammatical categories of a language possess
infinite sets of models and instances. But it also voices a practical
truth when the extensions, though finite at every stage, tend to grow
at exponential rates.
This consequence of dealing with extensions that are "practically infinite"
becomes crucial when one tries to build neural network systems that learn,
since the learning competence of any intelligent system is limited to the
objects and domains that it is able to represent. If we want to design
systems that operate intelligently with the full deck of propositions
dealt by intact universes of discourse, then we must supply them with
succinct representations and efficient transformations in this domain.
Furthermore, in the project of constructing inquiry driven systems,
we find ourselves forced to contemplate the level of generality
that is embodied in propositions, because the dynamic evolution
of these systems is driven by the measurable discrepancies that
occur among their expectations, intentions, and observations,
and because each of these subsystems or components of knowledge
constitutes a propositional modality that can take on the fully
generic character of an empirical summary or an axiomatic theory.
A compression scheme by any other name is a symbolic representation,
and this is what the differential extension of propositional calculus,
through all of its many universes of discourse, is intended to supply.
Why is this particular program of mental calisthenics worth carrying out
in general? By providing a uniform logical medium for describing dynamic
systems we can make the task of understanding complex systems much easier,
both in looking for invariant representations of individual cases and in
finding points of comparison among diverse structures that would otherwise
appear as isolated systems. All of this goes to facilitate the search for
compact knowledge and to adapt what is learned from individual cases to
the general realm.
Jon Awbrey
---------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 21
4.3. Back to the Feature
| I guess it must be the flag of my disposition, out of hopeful
| green stuff woven.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 31]
Let us assume that the sense intended for differential features is well enough
established in the intuition, for now, that I may continue with outlining the
structure of the differential extension [E!X!] = [A, dA]. Over the extended
alphabet E!X! = {x_1, dx_1} = {A, dA}, of cardinality 2^n = 2, we generate
the set of points, EX, of cardinality 2^2n = 4, that bears the following
chain of equivalent descriptions:
- EX
= <|A, dA|>
= {(A), A} x {(dA), dA}
= {(A)(dA), (A) dA, A (dA), A dA}.
The space EX may be assigned the mnemonic type B x D, which is really
no different than B x B = B^2. An individual element of EX may be
regarded as a "disposition at a point" or a "situated direction",
in effect, a singular mode of change occurring at a single point
in the universe of discourse. In applications, the modality of
this change can be interpreted in various ways, for example,
as an expectation, an intention, or an observation with
respect the behavior of a system.
To complete the construction of the extended universe of discourse
EX% = [x_1, dx_1] = [A, dA], one must add the set of differential
propositions EX^ = {g : EX -> B} ~=~ (B x D -> B) to the set of
dispositions in EX. There are 2^2^2n = 16 propositions in EX^,
as detailed in Table 14.
Table 14. Differential Propositions
o-------o--------o---------o-----------o-------------------o----------o
| ` ` ` | ` ` `A : 1 1 0 0 | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| ` ` ` | ` ` dA : 1 0 1 0 | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
o-------o--------o---------o-----------o-------------------o----------o
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| f_0 ` | g_0 ` `| 0 0 0 0 | ` `() ` ` | False ` ` ` ` ` ` | ` `0` ` `|
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
o-------o--------o---------o-----------o-------------------o----------o
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| ` ` ` | g_1 ` `| 0 0 0 1 | `(A)(dA)` | Neither A nor dA` | ~A & ~dA |
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| ` ` ` | g_2 ` `| 0 0 1 0 | `(A) dA ` | Not A but dA` ` ` | ~A & `dA |
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| ` ` ` | g_4 ` `| 0 1 0 0 | ` A (dA)` | A but not dA` ` ` | `A & ~dA |
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| ` ` ` | g_8 ` `| 1 0 0 0 | ` A `dA ` | A and dA` ` ` ` ` | `A & `dA |
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
o-------o--------o---------o-----------o-------------------o----------o
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| f_1 ` | g_3 ` `| 0 0 1 1 | `(A)` ` ` | Not A ` ` ` ` ` ` | ~A` ` ` `|
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| f_2 ` | g_12` `| 1 1 0 0 | ` A ` ` ` | A ` ` ` ` ` ` ` ` | `A` ` ` `|
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
o-------o--------o---------o-----------o-------------------o----------o
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| ` ` ` | g_6 ` `| 0 1 1 0 | `(A, dA)` | A not equal to dA | `A + dA `|
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| ` ` ` | g_9 ` `| 1 0 0 1 | ((A, dA)) | A equal to dA ` ` | `A = dA `|
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
o-------o--------o---------o-----------o-------------------o----------o
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| ` ` ` | g_5 ` `| 0 1 0 1 | ` ` (dA)` | Not dA` ` ` ` ` ` | ` ` `~dA |
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| ` ` ` | g_10` `| 1 0 1 0 | ` ` `dA ` | dA` ` ` ` ` ` ` ` | ` ` ` dA |
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
o-------o--------o---------o-----------o-------------------o----------o
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| ` ` ` | g_7 ` `| 0 1 1 1 | `(A `dA)` | Not both A and dA | ~A v ~dA |
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| ` ` ` | g_11` `| 1 0 1 1 | `(A (dA)) | Not A without dA` | `A => dA |
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| ` ` ` | g_13` `| 1 1 0 1 | ((A) dA)` | Not dA without A` | `A <= dA |
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| ` ` ` | g_14` `| 1 1 1 0 | ((A)(dA)) | A or dA ` ` ` ` ` | `A v `dA |
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
o-------o--------o---------o-----------o-------------------o----------o
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
| f_3 ` | g_15` `| 1 1 1 1 | ` (())` ` | True` ` ` ` ` ` ` | ` `1` ` `|
| ` ` ` | ` ` ` `| ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` ` ` ` | ` ` ` ` `|
o-------o--------o---------o-----------o-------------------o----------o
Aside from changing the names of variables and shuffling the order of rows,
this Table follows the format that was used previously for boolean functions
of two variables. `The rows are grouped to reflect natural similarity classes
among the propositions. `In a future discussion, these classes will be given
additional explanation and motivation as the orbits of a certain transformation
group acting on the set of 16 propositions. Notice that four of the propositions,
in their logical expressions, resemble those given in the table for X^. Thus the
first set of propositions {f_i} is automatically embedded in the present set {g_j},
and the corresponding inclusions are indicated at the far left margin of the table.
Jon Awbrey
---------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 22
4.4. Tacit Extensions
| I would really like to have slipped imperceptibly into this lecture, as
| into all the others I shall be delivering, perhaps over the years ahead.
|
| Michel Foucault, 'The Discourse on Language', [Fou, 215]
Strictly speaking, however, there is a subtle distinction in type between the
function f_i : X -> B and the corresponding function g_j : EX -> B, even though
they share the same logical expression. Being human, we insist on preserving all
the aesthetic delights afforded by the abstractly unified form of the "cake" while
giving up none of the diverse contents that its substantive consummation can provide.
In short, we want to maintain the logical equivalence of expressions that represent
the same proposition, while appreciating the full diversity of that proposition's
functional and typical representatives. Both perspectives, and all the levels of
abstraction extending through them, have their reasons, as will develop in time.
Because this special circumstance points up an important general theme,
it is a good idea to discuss it more carefully. Whenever there arises
a situation like this, where one alphabet !X! is a subset of another
alphabet !Y!, then we say that any proposition f : <|!X!|> -> B has
a "tacit extension" to a proposition !e!f : <|!Y!|> -> B, and that
the space (<|!X!|> -> B) has an "automatic embedding" within the
space (<|!Y!|> -> B). The extension is defined in such a way
that !e!f puts the same constraint on the variables of X that
are contained in Y as the proposition f initially did, while
it puts no constraint on the variables of Y outside of X,
in effect, conjoining the two constraints.
If the variables in question are indexed as !X! = {x_1, ..., x_n}
and !Y! = {x_1, ..., x_n, ..., x_n+k}, then the definition of the
tacit extension from !X! to !Y! may be expressed in the form of
an equation:
- !e!f(x_1, ..., x_n, ..., x_n+k) = f(x_1, ..., x_n).
On formal occasions, such as the present context of definition,
the tacit extension from !X! to !Y! is explicitly symbolized by
the operator !e! : (<|!X!|> -> B) -> (<|!Y!|> -> B), where the
appropriate alphabets !X! and !Y! are understood from context,
but normally one may leave the "!e!" silent.
Let's explore what this means for the present Example.
Here, !X! = {A} and !Y! = E!X! = {A, dA}. For each of
the propositions f_i over X, specifically, those whose
expression e_i lies in the collection {0, (A), A, 1},
the tacit extension !e!f of f to EX can be phrased as
a logical conjunction of two factors, f_i = e_i.!t!,
where !t! is a logical tautology that uses all the
variables of !Y! - !X!. Working in these terms,
the tacit extensions !e!f of f to EX may be
explicated as shown in Table 15.
Table 15. Tacit Extensions of [A] to [A, dA]
o---------------------------------------------------------------------o
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `0` ` = ` ` `0` . ((dA), dA)` ` ` ` = ` ` ` ` ` ` `0` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` (A) ` = ` ` (A) . ((dA), dA)` ` ` ` = ` ` `(A)(dA) + (A) dA ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `A` ` = ` ` `A` . ((dA), dA)` ` ` ` = ` ` ` A (dA) +` A `dA ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
| ` `1` ` = ` ` `1` . ((dA), dA)` ` ` ` = ` ` ` ` ` ` `1` ` ` ` ` ` ` |
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
o---------------------------------------------------------------------o
In its effect on the singular propositions over X, this analysis has an
interesting interpretation. The tacit extension takes us from thinking
about a particular state, like A or (A), to considering the collection
of outcomes, the outgoing changes or the singular dispositions, that
spring from that state.
Jon Awbrey
-----------------------------------------------------------------------------------
Posted by: Jon Awbrey
DIF. Note 23
4.5. Example 2. Drives and Their Vicissitudes
| I open my scuttle at night and see the far-sprinkled systems,
| And all I see, multiplied as high as I can cipher, edge but
| the rim of the farther systems.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 81]
Before we leave the one-feature case let's look at a more substantial example,
one that illustrates a general class of curves that can be charted through
the extended feature spaces and that provides an opportunity to discuss
a number of important themes concerning their structure and dynamics.
Again, let !X! = {x_1} = {A}. In the discussion that follows I will consider
a class of trajectories having the property that d^k.A = 0 for all k greater
than some fixed m, and I indulge in the use of some picturesque terms that
describe salient classes of such curves. Given the finite order condition,
there is a highest order non-zero difference d^m.A exhibited at each point
in the course of any determinate trajectory that one may wish to consider.
With respect to any point of the corresponding orbit or curve let us call
this highest order differential feature d^m.A the "drive" at that point.
Curves of constant drive d^m.A are then referred to as "m^th gear curves".
Scholium. The fact that a difference calculus can be developed
for boolean functions is well known [Fuji], [Koh, sec. 8-4] and
was probably familiar to Boole, who was an expert in difference
equations before he turned to logic. And of course there is the
strange but true story of how the Turin machines of the 1840's
prefigured the Turing machines of the 1940's [Men, 225-297].
At the very outset of general purpose, mechanized computing
we find that the motive power driving the Analytical Engine
of Babbage, the kernel of an idea behind all of his wheels,
was exactly his notion that difference operations, suitably
trained, can serve as universal joints for any conceivable
computation [M&M], [Mel, ch. 4].
Given this language, the particular Example that I take up here can be described
as the family of 4^th gear curves through E^4.X = <|A, dA, d^2.A, d^3.A, d^4.A|>.
These are the trajectories generated subject to the dynamic law d^4.A = 1, where
it is understood in such a statement that all higher order differences are equal
to 0. Since d^4.A and all higher d^k.A are fixed, the temporal or transitional<