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# Re: Thoughts on nonlinear language & the geometry of propositional thought

From: David G. Durand Monday, May 1, 2006, 19:04
```On 5/1/06, And Rosta <and.rosta@...> wrote:
> Irrespective of the expressive potential of the medium, propositional thought has its own
> intrinsic geometry, a geometry that resembles the structure of trees in
> Transformational Grammar. A proposition has the structure of a tree: the
> predicate is the mother node and its daughters are its arguments. Just as
> with a movement chain in TG, one node can be a daughter (i.e. argument) of
> many mothers (predicates). This geometry of propositional thought therefore
> defines the essence of the task facing any conlinguistic scheme that aims to
> map thoughts into a given expressive medium.
For mathematical logic, that is (almost) true. I say almost because a
single proposition being proved has many dependent justifications,
each propositions. But some of those propositions may be used many
times in justifying intermediate results in an argument. Thus there is
multiple inheritance in that kind of structure, in any case. So it's
more of a Directed Acyclic Graph (where propositions are allowed any
connections as long as there is no loop with something directly or
indirectly justifying itself). Almost all logics exclude statements
that are true because they are true -- though that is one way to
represent axioms.

Even while this is true for formal mathematics, in principle, there
are kinds of mathematical/propositional argument that don't work in
such an explicit dependency style: In category theory for instance,
the core of many arguments is a set of Items with arrows between them,
and a set of associated conditions (say that all paths between two
nodes are equivalent, or that paired arrows in opposing directions
commute (make no change if both followed in sequence), or that an
specified number (or infinite number) of arrows exist in a particular
part of the diagram.

There are propositional representations of these diagrams in the form
that you spoke of, and I elaborated on -- but they are essentially
unusably clumsy, as they add barriers to comprehending a theory that
is so abstract that even with the diagrams it is brain-bendingly hard.

Finally, it is not at all clear that humans avoid looping
justifications in daily life at all, and there are even mathematical
theories (Barwise's work on "non-well founded" set theory and logic,
for instance).

I tend to be skeptical of the notion that a semantic notation will
offer significant general benefits in language use, but I do think
that there are examples where visualizations offer some significant
benefits.

I also think that the poetic uses of directly linking word/concept
units are worth exploring even if there are no concrete advantages. In
that regard, I have found several of the non-linear experiments
intriguing.

This last point also addresses the fact that many of these nonlinear
systems are looking at structures _below the level_ of a proposition,
so that e.g. a single participant in a situation might have many links
to many other parts of the situation, with the set of propositions
represented by the picture potentially being infinite (A was the man
who patted the Dog B which was patted by the man A who patted the dog
B... etc.), depending on your definition of proposition, of course.
There are some interesting presentational problems here, as a
non-linear representation of Oliver Twist, say, that had only one node
for Oliver, would probably not be understandable -- though perhaps
someone can read a diagram like that in the time that it would have
taken to read a novel; I don't know.

--
-- David
```