[Inquiry] Re: Differential Logic

[Jon Awbrey]

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DLOG.  Note D16

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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!    |
|                                       |                                        |
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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 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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