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Hello all, thanks to those who took the time to respond!
@ Alex - As an exercise the past few weeks, I've been trying to formalize semi-formal natural language claims made by various people. One such (concrete) example, is indeed Hilbert's geometry axioms, where he claims to only commit to points, lines and planes; yet entities such as linesegments, rays, space etc. seem to pop-up. For this particular example, I'm uncertain whether creating a thing called "linesegment" changes in any substantive way Hilbert's ontology.
The dilemma is exactly as you describe it, and while I am asking the question in general, it is motivated by several concrete examples.
Alex wrote:
in general if I have chance to define something (entity) using relation, then I define it.
Why do you opt for relations over a specifying a new entity (or are you doing both)? What's the reasoning behind your choice?
I used the word "implicit," because, as in the example, such a relation (in my mind) points to the existence of some unnamed entity (thing that we can quantify over).
@ Pat - Thanks for your response, this is along the lines of what i'm looking for.
Pat wrote:
Yes, exactly. Though in CL, there is less to choose between them as the first is also committing to the existence of the linesegment relation itself.
This is true, and in CL we can quantify over named relations, so perhaps this distinction is academic, though in TFOL such a distinction has practical implications. Additionally, this is a grey area of ontologies for me. For example, if you say
(forall (x) (entity x) <---> S (some sentences)
it's almost as though you are naming a set of axioms in the relation. It seems a skip/hop away from 2nd or higher order logic, yet it clearly isn't. In terms of mimicking natural language, and convenience in terms of referring to S in such a way, such a construction seems tremendously useful. Does it however introduce problems in reasoning time / complexity?
Pat wrote:
And what does OVER commitment mean, here?
Meaning I have now introduced some new thing that i need to account for explicitly - might it have some unintended consequences? For example, if I try to convert a relation to a function, if I have it undefined in some places, it would introduce an inconsistency to my ontology. Along the same lines, I'm trying to figure out whether this choice actually has practical ramifications beyond stylistic choices; it's a relatively opaque question for me at the moment.
Pat wrote:
Philosophers are often leery of admitting to the existence of things because they have a philosophical agenda to reduce everything to some small subclass of entities (for example, nominalists like Chris Partridge tend to think that only actual concrete physical things are really real), but when writing ontologies for practical use, we should not be guided by merely philosophical agendas.
Aside from some aesthetic appeal to simplicity / elegance is there a more grounded motivation? I imagine there are stronger reasons than simply a desire for a small subclass of entities. My initial (naive) impression is that a universe that has fewer things defined, is pragmatically, easier to reason with. Yet defining a relation would seem to introduce just as much complexity as a new entity which captures the same restrictions on things that already exist in the U of D.
Pat wrote:
I see very few practical problems arising from having ontologies commit to the existence of things fairly freely, and having the relevant things in ones ontology tends to make it a lot easier to say what you need to say.
Would you care to elaborate what these practical problems might be?
@Ravi - I'm not entirely sure what you mean, can you expand a bit?
I should clarify, when I wrote "behaviour", i mean to say when you write (forall (x) (entity x) <--> S , you are defining the "behaviour" of entity X. Choosing whether you do so explicitly by naming some new thing X, or implicitly, by defining relations which capture S, is the crux of the question here.
I would like to catalog the benefits / disadvantages of each approach. I intuitively prefer naming new things (both for convenience and explicitness), yet how much are we changing an ontology by adding these new constructs?
Cheers,
Ali On Sun, Feb 15, 2009 at 6:14 PM, Ravi Sharma <ravisharma@xxxxxxxxxxx> wrote:
Ali
A physicist or object oriented programmer would
both want the significant attributes as well as behavior (e.g. processes that
describe the entity) to be significant for understanding the entity. If
ontology is not tied to understanding the entity, then we might not get reality
representation of the entity in that ontology!
John,
Thank you kindly for your reply. I should say that I'm familiar with Tarski's
work as well, though on a digression, I thought he committed to points, and
nothing else, not vanishingly small spheres, i'll
have to look over his axioms again. Moreover, i'm
under the impression that Tarski and Hilbert's
geometries are mutually interpretable, though I haven't
come across a proof of this in logic, though many in geometric algebras exist.
The issue i'm grappling with
is the degree of ontological commitment one makes when picking a relation vs an
"entity" (something that we quantify over) to represent a concept.
I imagine each choice of representation has its own strengths and
weaknesses, though i'm not entirely
sure what these would be.
My question to the community is -- have you come across scenarios where
it is more advantageous to commit to the existence of an entity as opposed to
capturing the "behaviour" of an entity, implicitly via a relation?
Why / why not?
Ali
On Sat, Feb 14, 2009 at 10:37 AM, John F. Sowa <sowa@xxxxxxxxxxx> wrote:
Ali,
Although I used a theory with a single axiom and predicate to
illustrate the idea, it is better to consider an entire theory
(defined by the total collection of axioms) to determine the
ontological commitment.
> For example, take the notion of linesegment in (or extending)
> Hilbert's geometry
formalization. One might be tempted to
> implement it is as strictly a relation between 2 (or 3) points
> say in ontology O1 - i.e. (linsegment x y z) where (x,y,z) are
> all points. Another, might in ontology O2, be tempted to define
> a new entity "linesegment" which consists of points -i.e.
> (linesegment XY x y). Is one making a stronger ontological
> commitment than the other?
Since Hilbert's axioms
already specify lines, points, and a lot
more, I suspect that your axioms (assuming that they are consistent
with Hilbert's) would be a rather
straightforward application that
wouldn't add much, if anything to
the ontological commitment.
For a more radical example, I suggest Tarski's
version of solid
geometry, in which the only primitives are spheres of arbitrary
finite size:
Tarski, Alfred (1929) "Foundations of the geometry of
solids,"
in Tarski (1982) _Logic, Semantics, Metamathematics_, Second
edition, Hackett Publishing Co., Indianapolis, pp. 24-29.
In that short paper, Tarski used a version of mereology instead
of set theory. That made a much smaller commitment right at
the beginning, since mereology does not have the generative
capacity of set theory -- i.e., it doesn't
support the option
of building complex mathematical structures from iterations of
the empty set -- {}, {{}}, {{},{{}}}, {{},{{}},{{{}}}}...
For a brief summary, see the paragraph below.
For pretty pictures inspired by Tarski's
geometry, see
http://frot.org/t/tarski/
Note that the paper is only 6 pages long. That is not long enough
to build up all Euclidean geometry. What Tarski did was to build
the foundation and demonstrate that it had a great deal of power.
Finally, he defined 'point' as the limit of a nest of spheres.
Then he showed that the axioms for Euclidean geometry could be
defined in terms of those points:
1. The only ontological commitment is to finite spheres.
2. Points, straight lines, and planes don't
"truly" exist on
the same level as spheres. They are abstractions defined as
limiting cases of infinite series of spheres.
Since all physical structures are made of tiny atoms (or particles
even smaller than atoms), truly straight physical lines, planes, and
solids never occur in nature, and they're
impossible to construct.
Therefore, all the constructs of point-based Euclidean geometry
are "imaginary" or "fictitious" structures that cannot
exist
physically.
That was Whitehead's motivation for
his system of "extensive
abstraction". He independently developed an approach that was
even more general than Tarski's
because he started with arbitrary
four-dimensional blobs. But I recommend Tarski's
approach for
an initial study, since his paper is only 6 pages long.
John
________________________________________________________________________
Source: http://en.wikipedia.org/wiki/Alfred_Tarski
In the 1920s and 30s, Tarski often taught high school geometry. In 1929,
he showed that much of Euclidean solid geometry could be recast as a
first order theory whose individuals are spheres, a primitive notion, a
single primitive binary relation "is contained in," and two axioms
that,
among other things, imply that containment partially orders the spheres.
Relaxing the requirement that all individuals be spheres yields a
formalization of mereology far easier to exposit that Lesniewski's
variant. Starting in 1926, Tarski devised an original axiomatization for
plane Euclidean geometry, one considerably more concise than Hilbert's.
Tarski's axiomatization is a
first-order theory devoid of set theory,
whose individuals are points, and having only two primitive relations.
In 1930, he proved this theory decidable because it can be mapped into
another theory he had already proved decidable, namely his first-order
theory of the real numbers. Near the end of his life, Tarski wrote a
very long letter, published as Tarski and Givant (1999), summarizing his
work on geometry.
--
(•`'·.¸(`'·.¸(•)¸.·'´)¸.·'´•)
.,.,
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