SUO: *Date 29 Apr 2002 -- Differential Logic
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Differential Logic
Questions from an off-list correspondent lead me to think that
a few words of motivation and orientation might be of use here.
For the time being we are working in and over the realm of "zeroth order logic" (ZOL),
also known as Peirce's alpha level, propositional calculus, or sentential logic. I am
getting where I prefer the name ZOL because it lets us focus on the abstract structure
that is held in common by the several studies of boolean functions, monadic predicates,
and sentential logic without getting all hung up on what some perceive as the diverse
ontologies of functions, predicates, and sentences, which is a more concrete matter
that we can well put off till later.
In this light, I am viewing "zeroth order logic" (ZOL) as
a logical analogue of analytic geometry or geometric algebra,
that is, a calculus for the study of figures in a certain space.
Pursuing the analogy, I am simply asking what would be a suitable
"differential extension" of ZOL, in the way that analytic geometry
is extended by the concepts and operations of differential geometry.
As to my overall strategy of approach, I probably ought to say this:
Any time we go looking for a suitable analogue of a familiar notion
in a strange domain the first thing we have to do is to distinguish
the definition, that may be generalizable, from the formula that is
used to compute it, which may not make sense in a different context.
The main idea is to analyze an arbitrary function in terms of linear functions.
The first step of that analysis requires us to look for a linear function that
approximates the local behavior of the given function.
What we mean by "local behavior" is this. Pick a location p, for instance, one that
is given by the coordinates a_1, ..., a_k. The value of the function f at this point
is f(a_1, ..., a_k). Consider points q that are various distances away from p, say,
given by the coordinate values a_1 + x_1, ..., a_k + x_k for various values of the
variables x_1, ..., x_k. The value of f at such q is f(a_1 + x_1, ..., a_k + x_k).
The "local function" Df_p corresponding to f at p is defined as follows:
Df_p (x_1, ..., x_k) = f(a_1 + x_1, ..., a_k + x_k) - f(a_1, ..., a_k).
Given an suitable "notion of approximation", one may ask if there is
a linear function (more generally, a morphism) that is in this sense
the nearest of all the linear functions to the local function Df_p.
If there is, then that linear function is called the "differential"
of f at p, notated df_p. (Some writers call this the "derivative").
So far in this discussion, I have gotten only as far as describing
the "local proposition" that corresponds to the conjunction &(x, y)
at the point p where x = 1 and y = 1.
All of these concepts of "small changes" and so on have to be defined in terms
of the topology that one has in mind. As it happens, one can go quite a ways
with this development using only algebraic concepts. The main catch in the
above problem specification comes with the "notion of approximation" that
will have to be given before we can go from "finite difference calculus"
to "differential calculus" proper. There are several way to do this.
One last piece of advice. You can't really count on any prior intuitions
in a setting like this, but have to just follow the import of the definitions
wherever they may lead. We are presently viewing boolean functions of the type
B^k -> B on analogy with real functions of the type R^k -> R. This means that we
are regarding propositions as "scalar potentials" -- "hills to climb" in AI lingo --
and what we are looking for is the analogue of the "gradient". How this relates
to the "directional derivative" along a "path" of the type B -> B^k (the logical
analogue of an arc, a curve, or a path of the type R -> R^k) is another story.
The following textbook gives a good discussion of the way
that the algebraic and analytic concepts are interrelated:
| Loomis & Sternberg, 'Advanced Calculus',
| Addison-Wesley, Reading, MA, 1968.
|
| see Section 3.6, "The Differential", pages 140-144.
Jon Awbrey
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