1. INTRODUCTION

Be prepared for a very strange logic, closer to Newton than Boole, intended
for use in designing and building interactive multimedia systems, such as
interactive museums, computer games, adaptive educational tools, interactive
novels, etc.  An important aspect of such systems is their *dynamics*: as
user(s) interact, predicates may have different effects in different contexts,
and especially at different times.

The evolution of the ideas in these notes is rather complex, and much of the
history will be obscure to computer scientists.  The creation of structural
linguistics by the Swiss linguist Ferdinand de Saussure around the turn of the
century is probably a better starting point than most.  His work was of course
followed up by linguists, but more interestingly, also by the French
anthropologist Claude Levi-Strauss.  After that comes a whole school of
structuralist and poststructuralist thought in France, including (at least)
the disciplines of sociology and literature, as well as anthropology and
linguistics; often the term *semiotics* is used as an umbrella name for this
work (though that term is also used in many other more specialized ways).
Skipping over Roland Barthes and many others, we come to Algirdas Greimas, who
became interested in the dynamics of stories, drawing on a tradition that goes
back to the Russian folklorist Vladimir Propp, as well as on
poststructuralism.  His work was, to some extent, formalized by Jean Petitot,
using ideas from catastrophe theory in the form popularized by Rene Thom.
(Petitot's ideas are not yet available in English, so my notion of what he did
comes only from secondary sources.)

Finally we come to the work of Peter Boegh Andersen on (what he calls)
*computer semiotics*, which directly addresses the applications mentioned in
the first sentence of this section.  His two exciting papers, "Multimedia
Phase Spaces" and "Dynamic Logic," directly inspired these notes, which are
intended for a course on user interface design (CSE 271, Winter 1998).
Andersen's work is very innovative, and yet seems quite practical for a kind
of computer application that may become extremely important over the next few
years.  This work is also interesting for the connections it builds between
the sciences and the humanities.

This whole area of investigation is still at a very early stage; therefore
formalizations should be considered tentative, and are also likely to be
incomplete; in particular, there are very likely to be bugs in the following,
and there are certain to be gaps.

2. BASIC IDEAS

We represent "predicates" by real-valued "potential functions," which generate
"force fields" over a space X, defined to be their gradient vector fields.
These will be used to move "actants" (which may be people, cars, cats, icons,
windows, etc. -- of course, not real ones, but their representations on a
computer screen, in a movie or a text, etc.), represented as points in X, in
proportion to the gradient, in practice at discrete intervals of time.

To get started, lets restrict to X the real line (or an interval), so we have
real valued functions of a real variable, which we will assume are piecewise
continuously differentiable.  Lets also adopt the convention that a
"predicate" (or potential function) P has derivative denoted P: (sorry, this
is as close as i can get to Newton's notation for derivative in pure ascii),
which is piecewise defined.  It follows that adding a constant to P makes no
difference to the dynamics of actants under P, and so it makes sense to define
equality in this logic to indicate that two potential functions differ only by
an additive constant.

Note also that multiplying a potential function by a positive constant will
only change the time scale; lets write P ~ Q iff P(x) = c*Q(x) + d, for c any
positive constant and d an arbitrary constant.  More generally we will use ~~
to indicate equivalence under nonlinear change of time, i.e., multiplication
by a positive function, plus addition of an arbitrary constant.

Throughout these notes, it will be worthwhile to consider a simple "standard"
example with P(x) = (x - a)^2 and Q(x) = (x - b)^2 (where ^ indicates
superscript).  These predicates move actants towards their minima, at a and b,
respectively.  To be more concrete, let X be the real interval of visible pure
colors, and let a and b correspond to red and yellow, respectively.

Andersen has given (what i consider) a brilliant definition of negation,

     (not P): = -1/P:

This is easily shown to satisfy the double negation law

    not not P = P

(where we really do need equality defined only up to an additive constant).

Andersen also defined conjunction and disjunction as

     (P and Q)(x) = max{P(x),  Q(x)}

     (P or Q)(x)  = min{P(x),  Q(x)} .

These definitions are exactly the duals of those that are usual in fuzzy
logic.  However, these definitions are somewhat less convincing than that of
negation, because they can introduce new discontinuities, and moreover the
value of the gradient at each point only depends on one of the constituents,
not both.

3. DISJUNCTION

Some work I did long ago on fuzzy logic (*Synthese* 1968) suggested using
multiplication for conjunction.  So if we again dualize, this suggests
defining disjunction by

     (P or Q)(x) = P(x) * Q(x) .

This varies smoothly with both arguments.  Moreover, for our "standard"
example, P(x) = (x-a)^2 and Q(x) = (x-b)^2, we get valleys ("basins of
attraction" in the dynamical systems jargon) at a and b, with a ridge between
them, and high walls on each side -- which is exactly what disjunction
*should* do.  These assertions use the product rule,

    (P or Q): = P*Q: + Q*P: .

In our color example, P or Q is red or yellow.

Of course, this disjunction is associative and commutative, but not
idempotent, since in fact

    (P or P) = P^2 ,

which is an "intensification" of P, and in an interesting way, since in our
standard example, it flattens the potential function near a (since the squares
of numbers < 1 go down) while making the walls (outside a +/- 1) steeper.  If
we form the disjunction of even more copies of P, the resulting curve flattens
even more near the zero, and the walls get even steeper a bit away from the
zero, because

    (P or P or ... P) = P^n ,

for n copies of P.

4. CONJUNCTION

For a couple of days I was stumped by conjunction.  One idea was to believe de
Morgan, in which case a small exercise in the calculus gives

    P and Q = not(not P or not Q) = -(P * Q) ,

as a candidate for conjunction.  This isnt too bad for the standard example,
because there is a nice valley between a and b, with peaks on each side (at a
and b); but unfortunately, the sides go sharply down after that, not up.

However, this logic is so strange that i dont see any reason to suppose we
*should* believe de Morgan.  Walking home one evening, an extremely simple and
very reasonable definition dawned on me that I should have seen in the first
place,

     P and Q = P + Q .

For the standard test case, this has derivative

     2x - a - b

which has just one zero, at (a+b)/2, which is exactly where we want it to be.

In our color example, P and Q is red and yellow, which will have minimum at
orange.

It is interesting to note that since the slope is the sum of the slopes of the
components, the valley will be steeper than that of either component.  In
paticular, we do NOT have

     P and P = P ,

because P and P is an "intensified" version of P; instead

     P and P = 2 * P .

This isnt as unreasonable it might seem at first; e.g., think of examples like

     more and more

     ever and ever

     higher and higher

     Ja and Ja (Ja is Rastafarian for "God" and "Ja and Ja" means something
                like "God most high")

which show that natural language conjunction really isnt like mathematical
logic.  In the color example, "red and red" gives a narrower range of color
frequencies centered at red.

It is pleasant that we get one distributive law for and/or,

     P or (Q and R) = (P or Q) and (P or R) .

However, we dont get the other distributive law, of which i still dont know
quite what to make.

5. TRUTH CONSTANTS

If we define a "truth constant" to be a function whose derivative is zero,
then there is just one, since additive constants dont matter; the constant
function roughly speaking means "anywhere" or in modern slang "whatever"; lets
denote it A.  It has no dynamics -- an actant just rests wherever it happens
to be.

In the color example, A just means "any color at all".

This notion of constant isnt very interesting, so lets redefine a truth
constant to be a function whose derivative is a constant; this makes sense
because only the derivative affects dynamics.  The result is that our strange
logic gets more truth constants than you might expect, and none of them are
much very like the "true" or "false" of traditional mathematical logic (or
Boolean algebra).

The function with derivitive -1 (or any negative value) can be thought of as
"less" or "down" or maybe "bottom", since actants are always drawn towards
minus infinity, while +1 (or any positive value) can be thought of as "more"
or "up" or maybe "top".  Lets write M for the +1 case and L for the -1 case.

In the color example, L takes us off the scale towards infrared, while M takes
us off scale towards ultraviolet.

We now get some strange "laws of thought":

     (P or M)(x) = P(x) * x

     (P or L)(x) = - P(x) * x

     A or P ~ P

(so A behaves rather like false here.)  Plus we have

     not L = M

     not M = L

which seems good.  We can do similar calculations for conjunction,

     P and M ~~ P

provided P is always positive (since P(x) + x = P(x)(1 + x/P(x)) and (1 +
x/P(x)) is always positive.  (More generally, if P has some isolated zeroes,
then "P and M" may have isolated singularities that are not shared by P.)

Similarly, we can get

     P and L ~~ P

provided P(x) is greater than or equal to x + c everywhere, for some constant
c.  And of course (quite easily),

     P and A = P

(so A behaves like true here.)

The strangest "truth value" arises from the desire to have the system closed
under the various logical operations; in particular, we want "not A" to have
some value: but because this requires dividing by zero everywhere, the result
can only be the totally undefined function.  Lets denote it by U; it means
"annihilation" -- it throws actants out of the game entirely; there is no
place at all where they can have a well defined gradient.  The following hold
because doing airthmetic with undefined values cannot ever yield defined
values.

     not A = U

     not U = U

     P and U = U

     P or U = U

If we allow U then we might think undefined values could be allowed in other
functions wherever we like; this would certainly give more truth constants,
though they would all be rather boring; but i think undefined values should
only be allowed for proper singularities in otherwise continuous functions.
(It could be argued that we shouldnt allow U at all.)

In the color example, U mean no color at all.

It should be emphasized that this is a *complete* classification of all the
possible constants in our logic.

6. Miscellaneous

Our notion of equality is a *congruence* for conjunction and disjunction, in
the sense that if P1 = P2 and Q1 = Q2, then P1 + P2 = Q1 + Q2 and P1 * P2 = Q1
* Q2; equality is also a congruence for negation, in that P1 = P2 implies not
P1 = not P2.  However, ~ and ~~ are not congrences for conjunction and
disjunction, which means that they have to be handled carefully.  For example,
L ~ 2L and of course M ~ M, but L + M ~ 2L + M is not true, since the left and
right sides are A and L, respectively.  However, ~ is a congruence for
negation.  The relation ~~ is even less of a congruence than ~ is, and not
even negation works.

We gave been using at least 3 different oppositions; in particular, Andersen's
negation is composed from two other opositions, namely arithmatic negation and
reciprocal, and we have already seen arithmatic negation appear in some
formulae.  This observation is interesting because of Greimas's famous
semiotic square, each vertical side of which is an application of one kind of
negation, and each horizontal of a second:

              P ---------- P'
              |            |
              |            |
              |            |
              |            |
             -P --------- -P'

Thinking a little more about the equivalence ~, any function whose derivative
is always positive is equivalent to M, and any whose derivative is always
negative is equivalent to L.  Catastrophe theory provides a more sophisticated
approach to this kind of equivalence (which i no longer remember!) whereby
functions just have to have the same pattern of derivatives, and something
like this might help us out in the future.

7. Notes

It seems to me that we can do pretty much the same things for multivariate
functions, using partial derivatives etc, and all results should be pretty
much the same.

There are almost certainly bugs in the above, both in calculations and
interpretations; also some other interesting calculations just havent been
done yet.  Thanks to Prof. Andersen for catching some bugs and unclarities.

Question: would it be better to use something like =~ for our equality
relation on potential functions?

Question: changes of time scale should really concern the *arguments* of a
potential function, not its values.  Is this worth bothering about?