Hi Ferenc,

The statement you made below:

But, importantly, this assumption
cannot in fact be implemented in a

reasoning system, as higher-order
logics are semantically incomplete:

given any proposed reasoning method
for a second- or higher-order

logic, there will be logically valid
arguments whose validity cannot it

cannot demonstrated by that method.

There are many, many (in fact MOST
practical) applications which don’t require completeness. For example,
SQL has a ‘select’ variant that retrieves the top N rows of an
ordered list, thereby converting its query to a beam search. The purpose of
such quick look queries is to investigate feasibility of a solution.

Algorithm:

If the top 100 don’t have anything remotely
like the answer you’re seeking,

then mark this part of the search
UnSolvable;

else if the top 100 have one or more
acceptable answers,

then take the best one and work it
next;

else if the theory’s top 100 matches
have mixed results, then

begin

evaluate
each those results;

compare each
to results from the other theories returning the top 100 of their own;

choose the theory with the best <risk/reward/expense/schedule>
payoff to open next;

recurse;

end;

The most obvious example I am familiar
with is the time-sensitive data called a “Plot” which is used in
radar systems to mark an observation of a signal return with locality
information, timestamp, and query to identify which previously known object
should have reached that same locality by the same time, within a window. A
beam is sent out; it reflects off a craft; the reflection propagates to the
radar receiver; a Plot is sent to the tracking software to see if it can match
the query unambiguously against known tracks; if so, plot accepted and matched
against a known track, but if not, the Plot may be any combination of:

An old, weak track that had disappeared
for a while, but is now back;

A brand new track of a
previously unknown object;

A bounce from ground clutter
(steel-girded buildings, cars, dentures, …);

A completely new class of
object not previously known to this system;

An ephemeral signal from no
object likely to be of interest to the radar;

The theory and practice of real-time
systems is based on knowing precisely when it is “best” to throw
away some data, and when it is “best” to keep data and process it. Sometimes
data has to be dropped in nearly every practical system. That makes nearly
every practical system incomplete.

For example, EVERY web server and EVERY
database server is incomplete – there is a timeout associated with every
loop to allow for contact being broken at an arbitrary point and subsequent
recovery. So some queries will just not be returned EVER.

So the notion of incompleteness seems to
be almost archaic in modern systems. It is only important within a very
limited scope of an algorithm that must run restrictedly in its own precisely defined
environment without taking an learning approach to new conditions it encounters.

-Rich

Sincerely,

Rich Cooper

EnglishLogicKernel.com

Rich AT EnglishLogicKernel DOT com

**From:**
ontolog-forum-bounces@xxxxxxxxxxxxxxxx
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] **On Behalf Of **FERENC KOVACS

**Sent:** Sunday, December 06, 2009
12:26 PM

**To:** ontolog-forum@xxxxxxxxxxxxxxxx

**Subject:** Re: [ontolog-forum]
Ontologies as social mediators

On Dec 5, 2009, at 10:48 AM, doug foxvog wrote:

> Ferenc wrote:

>>

>> ... I have been trying to visualize the

>> condition where objects, properties and relations change theri
character

>> and they become one of the other two as a result of mental operations.

>> ... So jsut a few examples

>> One of these operations is abstraction

>> that helps you see the properties of an object and create a list of
such

>> properties.

>> Then by taking one of those properties as an OBJECT, you can go on to

>> define another set of properties.

>

> This is quite valid. The issue of "chang[ing] their
character" arises from

> the limitations of first order logic. Computational complexity
becomes

> greater when describing properties of properties and reasoning about them.

That is of course true in full higher-order systems but, as noted a

couple of times in this forum by Pat Hayes and others, a system does

not become higher-order in a sense that increases its complexity

simply in virtue of permitting reasoning about properties and

relations. The complexity of such reasoning increases (from the

semi-decidability of first-order logic to full undecidability) only

under a very strong assumption: that every subset of the domain of an

interpretation is (the extension of) a property; more generally, that

every set of n-tuples of objects in the domain is (the extension of)

an n-place relation. This is the assumption that, from a purely

theoretical perspective, separates first-order from higher-order

logic. But, importantly, this assumption cannot in fact be

implemented in a reasoning system, as higher-order logics are

semantically incomplete: given any proposed reasoning method for a

second- or higher-order logic, there will be logically valid argu

ments whose validity cannot it cannot demonstrated by that method.

> In higher-order logic, one can treat relations, classes, and statements

> "relation instances", and instances of classes and make
statements about

> them using relations which accept such classes. The Cyc reasoner has
been

> doing this since at least the mid-1990s.

As have many other higher-order systems (as I'm sure you know). For

the reasons above, however, the implemented reasoning methods for

these systems -- even those whose formal semantics are fully

higher-order -- are theoretically first-order (in the sense that they

are complete relative to a weaker semantics for the systems (so-called

"general semantics", based on the work of Henkin) that abandon the

strong assumption above). Indeed, some implementations of such

systems (the HOL System, for example) translate the entire system into

an explicit first-order theory to which standard first-order reasoning

methods can then be applied directly.

Chris Menzel

I am certainly not well educated in formal logic, but to me it appears to me
that

1) semantics (semantic analyis) should not
be constrained to syntax parsing for terminal symbols,

2) reasoning should not be confined to the use of syllogisms
in inferences, and

3) computability of valid conclusions by processing chained data base tables
should not be a problem provided that a proper GUI is available

4) without verbs as relations the representations of
concepts in various networks do not make sense or offer pracical use for
people who want to go from one domain to another using a common
interface/template that could be provided by relations other than those in use
today. You may want to zoom in and out from your 2D representations as opposed
to moving in the plane. And to that end you need to be able to change scale in
an orderly fashion