Syncategorematicity is considered as a specific context-dependency.
After some historical remarks concerning the distinction between
categorematic and syncategorematic expressions, a proposal is made as
to how define syncategorematic expressions in the framework of Boolean
semantics. A definition of syncategorematicity based on the notion of
an atom of denotational algebra is proposed and various examples,
among which are categorially polyvalent modifiers and some vague
adjectives, are briefly discussed.
http://lola8.unideb.hu/resume/zuber.htm
The main sense of the word "syncategorematic" as applied to
expressions was roughly this semantic sense (see Kretzmann 1982, pp.
212 ff.). Buridan and other late medieval logicians proposed that
categorematic expressions constitute the "matter" of sentences while
the syncategorematic expressions constitute their "form" (see the text
quoted by Bocheński 1956, §26.11). (In a somewhat different, earlier,
grammatical sense of the word, syncategorematic expressions were said
to be those that cannot be used as subjects or predicates in
categorical propositions; see Kretzmann 1982, pp. 211-2.) The idea of
syncategorematicity is somewhat imprecise, but there are serious
doubts that it can serve to characterize the idea of a logical
expression, whatever this may be. Most prepositions and adverbs are
presumably syncategorematic, but they are also presumably non-logical
expressions. Conversely, predicates such as "are identical", "is
identical with itself", "is both identical and not identical with
itself", etc., which are resolutely treated as logical in recent
logic, are presumably categorematic. (They are of course categorematic
in the grammatical sense, in which prepositions and adverbs are
equally clearly syncategorematic.)
http://plato.stanford.edu/entries/logical-truth/
In addition, an Italian scientist at T-13, Walter Fontana, was
developing what he called an "algorithmic chemistry," in which a form
of logical calculus provided a grammar under whose rules emergent
behavior might form.
-Artificial Life
Chaos has created special techniques of using computers and special
kinds of graphic images, pictures that capture a fantastic and
delicate structure underlying complexity. The new science has spawned
its own language, an elegant shop talk of fractals and bifurcations,
intermittencies and periodicities, folded-towel diffeorphisms and
smooth noodle maps. These are the new elements of motion, just as, in
traditional physics, quarks and gluons are the new elements of matter.
To some physicists chaos is a science of process rather than state, of
becoming rather than being.
-Chaos
ARTIFICIAL LIFE - The Quest for a New Creation
Steven Levy
Pantheon Books New York
Copyright (c) 1992 by Steven Levy
http://www.amazon.com/exec/obidos/tg/detail/-/0679743898/
- Traveling Salesman, Tractability; Number of Operations Increases
Exponentially by Adding
So far we have looked at tears in the very fabric of mathematics.
There are, however, other, more practical, defects. It may be possible
for us to solve a problem in principle but, even with a computer,
impossible to do it in a realistic time frame. This is the issue of
algorithmic "complexity," as opposed to computability, and it concerns
the amount of time required to solve a problem using a Turing machine.
While the work of Godel, Turing, Church, and Chaitin highlighted the
issue of computability, algorithmic complexity is very much a workaday
issue. It turns out that among computable problems, certain classes
are much more difficult to solve than others. The number of
calculations, expressed in terms such as "floating point" operations
(the number of operations performed by an algorithm), indicates the
amount of work needed to solve a given problem.
The algorithms used to describe computable problems can be divided
into two classes, based on the length of time it takes to find the
solution to a problem as a function of some number N that measures its
size. The good news is for problems that axe polynomial (i.e., an
algebraic power of N, e.g., N squared, N cubed, etc.), when they are
said to be tractable--the length of time required to crack them does
not become unbounded as the size increases. Problems solvable in
polynomial time are said to be in the class P. Mathematicians and
computer scientists blanch when the time required to solve a problem
increases in an exponential fashion (something to the power of N).
These problems are called intractable because the time required to
solve them rapidly spirals out of control. Even the raw power of a
computer has little effect. These problems, which are not solvable in
polynomial time, are said to be in the class NP.
Probably the most famous example of an NP problem is the traveling
salesman problem. This is the mathematical expression of the dilemma
faced by a salesman who has to visit N cities once only in such a way
as to minimize the total distance traveled: he has a penny-pinching
boss and has to keep fuel costs as low as possible.
The problem is easy to formulate but, try as they might, no computer
scientist or mathematician has come up with a well-behaved
deterministic algorithm (i.e., one that is not random and allows only
one outcome for any given set of circumstances) that can find
solutions to it on a computer in polynomial time. For a handful of
cities and roads it may be easy to determine the salesman's solution
because not that many options exist. If the number of cities is five,
say, a computer could easily calculate the twelve possibilities. With
ten cities, there are 181,440 possibilities. However, even with the
number-crunching power of the fastest available machine, the time
required to solve the problem rapidly spirals out of control. For just
twenty-five cities the number of possible journeys is so immense that
a computer evaluating a million possibilities per second would take
9.8 billion years--around two-thirds of the age of the universe--to
search through them all.
A large number of other real-world problems are known to lie in this
category, many of them concerned with similar optimization problems.
For the owner of a printed circuit-board factory, for example, the
function that needs to be maximized is manufacturing efficiency. For a
pharmaceutical company, the function that must be maximized is the
snugness of the fit of a drug molecule within a target protein found
in the body. Frequently, elegant analytical mathematics is unable to
provide us with a simple way of locating these optima, since hard
optimization tasks are intractable (NP) problems.
But how do we know for sure that a given problem belongs in the NP
class? Just as schoolchildren will always maintain that many
mathematical problems are impossible to solve, the labeling of a
problem as NP could say more about a mathematician's incompetence than
anything about the problem itself. In fact, this question is one of
the foremost open problems of contemporary mathematics and computer
science. Little progress has been made to prove the conjecture that NP
problems possess no solutions available by conventional--deterministic--
algorithms in polynomial time. However, even without such a proof, at
the very least the belief (supported by algorithmic experimentation)
that a problem is of the NP variety implies that a significant
breakthrough will be needed to solve it.
For those who want to reproduce the workings of the world within
computers, these NP problems are at first sight rather depressing.
Remarkably, however, as we will discuss in later chapters, nature has
provided us with tools to tackle them.
Frontiers of Complexity - The Search for Order in a Chaotic World
Peter Coveney and Roger Highfield - 1995
http://www.amazon.com/exec/obidos/ASIN/0449910814/
