Bart wrote:
We would like to use logical syntax and axioms to
analyze natural phenomenon, and organize them into some formal language, in
order to interpret them using ontologies, in this case hierarchical. A
problem arises when the interpretations we create, contain cyclic sentences
which are inconsistent.
There is much bigger problem here, the semantic
one.
Nobody can describe natural phenomena with logical
syntax. Otherwise science would be redundant.
Broadly, there are two types of conceptual systems:
logical semantic systems (LOS), involving the socalled formal semantics,
and real semantic systems (RES), involving the real semantics, as
in:
LOS = conceptual language + designation rules
(symbolsconstructs correspondence);
RES = conceptual language + designation rules +
denotation rules (a symbolthing correspondence) and representation rules
(semantic assumptions, or hypothesis) (constructsreal world things
correspondence).
The last one is about a theoretical system
expressed by a particular scientific symbolism, which significance is the
subject of this scientific domain.
For example, if you describe a physical
particle (p), the logical description will give you the individual variable with
its mass m as a predicate, designated as the Newtonian mass function. But
you need the correspondence rules from constructs to physical things;
namely:
1. a denotation rule (reference), indicating the
physical referent p;
2. a representation rule (semantic assumption),
measuring the position of the particle x(p, u,t) re. to some reference
frame of units u at some moment of time t.
With the RES you also find out that m(p) represents
the inertia of particle (semantic axiom), and that it is subject to the
conservation laws and Newton's law of motion (factual axioms).
Bottom line:
To transform all semantic problems into syntactic
problems is a wishful dream, with missed even R. Carnap.
Wonder have we ever thread on Syntactic
reductionism?
Azamat Abdoullaev
 Original Message 
Sent: Sunday, April 19, 2009 2:27
PM
Subject: Re: [ontologforum]
websyllogismandworldview
Hi Bart 
You wrote
Problem: We would like to use logical syntax
and axioms to analyze natural phenomenon, and organize them into some
formal language, in order to interpret them using ontologies, in this case
hierarchical. A problem arises when the interpretations we create, contain
cyclic sentences which are inconsistent. If we were to introduce recursion
to our interpretations, it would be limited to partial recursion. The
task, then, would be to create an interpretation of the sentence which
is acyclic, and our sentence consistent. The interpretation would
be decidable, and our recursive function total.
Pending some
crisp examples, its likely that the problem is solved by the design of the
system that is online at the site below [1]. The theory basis for
handling what you call 'cyclic sentences' in the system is in the paper
[2]. I'll be glad to send you a scanned in copy if you
wish.
Cheers,  Adrian
[1] Backchain Iteration: Towards a
Practical Inference Method that is Simple Enough to be Proved
Terminating, Sound and Complete. Journal of Automated Reasoning,
11:122
[2] Internet Business Logic A Wiki and SOA Endpoint for
Executable Open Vocabulary English over SQL and RDF Online at www.reengineeringllc.com
Shared use is free
Adrian Walker Reengineering
On Sat, Apr 18, 2009 at 7:20 PM, Bart Gajderowicz <bgajdero@xxxxxxxxxx>
wrote:
CM
> After rereading Adrian's response and looking at yours again
> *carefully*, I see that I was not reading what you were
saying > correctly. I thought you were responding to John's
argument, not > Adrian's. Apologies, my
mistake.
Thanks Chris, I appreciate you spending the time
reading over the posts and commenting.
CM > I think my
other two comments still > apply, however. >
I think my
terminology was too informal, and it would benefit me to formalize things
better.
I addressed the individual points and my position regarding
this thread at the end, but first let me state my idea in a hopefully
more formal manner here.
Problem: We would like to use logical
syntax and axioms to analyze natural phenomenon, and organize them into
some formal language, in order to interpret them using ontologies, in
this case hierarchical. A problem arises when the interpretations we
create, contain cyclic sentences which are inconsistent. If we were to
introduce recursion to our interpretations, it would be limited to
partial recursion. The task, then, would be to create an
interpretation of the sentence which is acyclic, and our sentence
consistent. The interpretation would be decidable, and our recursive
function total.
Proposed solution: One such approach would be to
view our ontology as a set of functions with a domain and range. The
domain would contain an entire object, with all its attributes and
properties (known and possibly unknown), and the functions would
represent contexts (presumably known) with which that object can be
interpreted. The attributes could be variable values or structures
themselves. Properties would be atomic values associated with an
object. The range then would be the resulting view of our object in
different contexts. Any cyclic definition which exists in the domain,
would become acyclic in the range. One approach of this would be to
translate attributes, which may be cyclic structures, into either acyclic
structures or atomic properties, thus creating our stop condition for a
total recursive function.
Summary: Essentially, we would like our
range to be decidable, while the domain can be
undecidable.
Comments on the original thread: CW> If I can
restate: Your claim was that if, in a logical framework
> capable of expressing syllogisms, syllogisms are allowed
to be > recursive, then the resulting framework will be useless
because it > will have the expressive power of a (universal) Turing
machine. > John's response is that all programming languages have that
power and > are, obviously, useful. John's response therefore
seems to be a > counterexample to your claim.
I was
stating that John is correct.
BG>> My point was, apologies
for not stating it clearly, this would rid
>> of completeness, for example in
FOL.
CW> I'm afraid I still do not understand. It is
just a fact that FOL is
> complete. There's nothing you can do such that it
would no longer be > a fact.
I meant in an individual FOL
statement. If we allowed a predicate with free variables to produce
infinitely many conclusions with a finite set of values, that would
introduce incompleteness to that sentence, not FOL itself. This is one
problem of my configuration that I've found in FOL, that I'm stating, in
order to get input from the forum.
CM> Again a terminological
issue. What do you mean by "the
> unsatisfiability of Turing machines in programming
languages"? Every > (standard) programming language is capable
of expressing exactly the > functions calculated by a Turing
machine.
Sorry, not unsatisfiability but "undecidable", the
inability to reach the end state, ie the halting problem. My mistake in
choice of words, as: a) it conflicts with the logical term b) I
actually meant to say undecidable
BG>> You're correct.
I wasn't referring to the a program that calculates
>> any recursive function, satisfiable or not.
Here, a "proper recursive >> function" is one that does
terminates, and the calling program will >> terminate, or not
terminate for an "improper function" accordingly.
CW>
Well, I think your terminology here will lead you to be
> misunderstood. The idea of a *function* terminating
or not is a > category mistake; *programs* terminate (on this or that
input). If > program P expresses a function F, then P terminates
on input > representing the number n if and only if F is *defined* on
n; that is, > if and only if F(n) has a value. A recursive
function that is defined > on every natural number is said to be
*total*; one that is defined >only on a subset of the natural
numbers is said to be *partial*. I > think your "proper" and
"improper" terminology is problematic.
Again, poor choice of
words on my part. I was speaking in terms of programing, meaning
that a program can have a function which is recursive, but we wouldn't
necessarily call the entire program recursive, only that it has recursive
components. And yes, whether it terminates or not does depend on
the input. The input I'm currently talking about is an ontology's
hierarchy, with possibly cyclic definitions, and how the change of
context would possibly introduce a stop condition for a recursive/cyclic
traversing of that hierarchy.
 Bart Gajderowicz MSc Candidate, '10 Dept.
of Computer Science Ryerson University http://www.scs.ryerson.ca/~bgajdero_________________________________________________________________
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