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## Re: [ontolog-forum] Foundation Ontology Primitives

 To: ian@xxxxxxxxxxxxxxxx, "[ontolog-forum]" Jawit Kien Tue, 2 Feb 2010 10:11:27 -0600 <9f9644bb1002020811y4830b52br8b5a06b8140ffb5a@xxxxxxxxxxxxxx>

On Tue, Feb 2, 2010 at 3:53 AM, Ian Bailey wrote:
Hi Pat (I won't qualify which Pat, as I'm pretty sure the other one has me
on his filter list),

In the thread that's been running, you indicate that you believe there are
some common ontic primitives that can be shared across multiple ontologies -
e.g. as a foundation. Do you have a clear idea of what these might be ?
(some examples would really help my understanding).

If I have an issue with this idea, it is one of pragmatism. As an
aspiration, a common foundation ontology is something I would really
support. There seems to me to be a couple of practical problems:

* Reaching consensus - ontology is a topic that attracts people with strong
opinions. Consensus requires compromise, and I don't see much of that going
on in this community. In my experience of developing ISO standards, it
usually requires a commercial reward to ensure true consensus is met - i.e.
they all (well, most) stop bickering if they can see some profitability in
not bickering. I don't think ontology is at the maturity level where huge
sums of money depend on its success or failure, so I fear consensus is going
to be nigh impossible.

* Metaphysical choices - most serious ontologists are aware of the
metaphysical choices (i.e. the ground rules) of their ontology. This
involves questions of how time is managed (3D, 4D Endurant, 4D Perdurant,
etc.), whether the ontology is extensional or intensional (both for class
membership, and how spatial/temporal extents are handled), and whether the
ontology is first order or higher order.

The first problem may be overcome by shear will - if enough people take a
positive approach and really want to get it done, it might happen. The
second issue is much more thorny, and that's why I asked what you thought
the primitives were. I once suggested a set of ontic categories on this
forum and was soundly thrashed for it, so I'd understand if you want to send
them off-list. Also, getting a clearer idea of what choices all the major
ontologies took might give us a better idea what common primitives are
possible. Again, if folks want to send that to me off-list, I'll collate it
all anonymously and post it back. God forbid we should actually use the
forum to discuss any real ontologies.

Cheers
--
Ian Bailey
www.modelfutures.com
ian@xxxxxxxxxxxxxxxx

In the hope that we can actually discuss some details about ontologies and
useful primitives, I submit the following table:

 Attribute Goal Possessor Cause Hypernym Purpose Co-Agent Location Size Color Manner Source Deep_Object Material Subclass Deep_Subject Means Synonym Domain Modifier Time Equivalent Part User

Table 1. Current set of semantic relation types in MindNet

This is from:

MindNet: acquiring and structuring semantic

information from text

Stephen D. Richardson, William B. Dolan, Lucy Vanderwende

Microsoft Research

One Microsoft Way

Redmond, WA  98052

U.S.A.

following some trail that John Sowa started me following recently.

This is related to MNEX at http://stratus.research.microsoft.com/mnex/Main.aspx

I note that Pat Cassidy has a list of "connectives" at http://micra.com/dictionary/tagset.txt

But realistically, making a list of connectors is not the the interesting issue, in my mind.

If you focus on making this list, as your first problem, you lose sight of the second problem,

which is more interesting on this list i.e.: what axioms/true statements/rules exist that implicitly

define these connectives?

What I mean is, can we create a list of axioms for Color, or Purpose similar to the axioms for

Part which are mentioned on the page: http://en.wikipedia.org/wiki/Mereology#Axioms_and_primitive_notions

i.e.:  (stealing from that page)

## Axioms and primitive notions

It is possible to formulate a "naive mereology" analogous to naive set theory. Doing so gives rise to paradoxes analogous to Russell's paradox. Let there be an object O such that every object that is not a proper part of itself is a proper part of O. Is O a proper part of itself? No, because no object is a proper part of itself; and yes, because it meets the specified requirement for inclusion as a proper part of O. (Every object is, of course, an improper part of itself. Another, though differently structured, paradox can be made using improper part instead of proper part; and another using improper or proper part.) Hence mereology requires an axiomatic formulation.

A mereological "system" is a first-order theory (with identity) whose universe of discourse consists of wholes and their respective parts, collectively called objects. Mereology is a collection of nested and non-nested axiomatic systems, not unlike the case with modal logic.

The treatment, terminology, and hierarchical organization below follow Casati and Varzi (1999: chpt. 3) closely. For a more recent treatment, correcting certain misconceptions, see Hovda (2008). Lower-case letters denote variables ranging over objects. Following each symbolic axiom or definition is the number of the corresponding formula in Casati and Varzi, written in bold.

A mereological system requires at least one primitive binary relation (dyadic predicate). The most conventional choice for such a relation is Parthood (also called "inclusion"), "x is a part of y", written Pxy. Nearly all systems require that Parthood partially order the universe. The following defined relations, required for the axioms below, follow immediately from Parthood alone:

$PPxy \leftrightarrow (Pxy \and \lnot Pyx).$ 3.3
An object lacking proper parts is an atom. The mereological universe consists of all objects we wish to think about, and all of their proper parts:
• Overlap: x and y overlap, written Oxy, if there exists an object z such that Pzx and Pzy both hold.
$Oxy \leftrightarrow \exists z[Pzx \and Pzy ].$ 3.1
The parts of z, the "overlap" or "product" of x and y, are precisely those objects that are parts of both x and y.
• Underlap: x and y underlap, written Uxy, if there exists an object z such that x and y are both parts of z.
$Uxy \leftrightarrow \exists z[Pxz \and Pyz ].$ 3.2

Overlap and Underlap are reflexive, symmetric, and intransitive.

Systems vary in what relations they take as primitive and as defined. For example, in extensional mereologies (defined below), Parthood can be defined from Overlap as follows:

$Pxy \leftrightarrow \forall z[Ozx \rightarrow Ozy].$ 3.31

The axioms are:

M1, Reflexive: An object is a part of itself.
$\ Pxx.$ P.1
M2, Antisymmetric: If Pxy and Pyx both hold, then x and y are the same object.
$(Pxy \and Pyx) \rightarrow x = y.$ P.2
M3, Transitive: If Pxy and Pyz, then Pxz.
$(Pxy \and Pyz) \rightarrow Pxz.$ P.3
• M4, Weak Supplementation: If PPxy holds, there exists a z such that Pzy holds but Ozx does not.
$PPxy \rightarrow \exists z[Pzy \and \lnot Ozx].$ P.4
• M5, Strong Supplementation: Replace "PPxy holds" in M4 with "Pyx does not hold".
$\lnot Pyx \rightarrow \exists z[Pzy \and \lnot Ozx].$ P.5
• M5', Atomistic Supplementation: If Pxy does not hold, then there exists an atom z such that Pzx holds but Ozy does not.
$\lnot Pxy \rightarrow \exists z[Pzx \and \lnot Ozy \and \lnot \exists v [PPvz]].$ P.5'
• Top: There exists a "universal object", designated W, such that PxW holds for any x.
$\exists W \forall x [PxW].$ 3.20
Top is a theorem if M8 holds.
• Bottom: There exists an atomic "null object", designated N, such that PNx holds for any x.
$\exists N \forall x [PNx].$ 3.22
• M6, Sum: If Uxy holds, there exists a z, called the "sum" or "fusion" of x and y, such that the objects overlapping of z are just those objects which overlap either x or y.
$Uxy \rightarrow \exists z \forall v [Ovz \leftrightarrow (Ovx \or Ovy)].$ P.6
• M7, Product: If Oxy holds, there exists a z, called the "product" of x and y, such that the parts of z are just those objects which are parts of both x and y.
$Oxy \rightarrow \exists z \forall v [Pvz \leftrightarrow (Pvx \and Pvy)].$ P.7
If Oxy does not hold, x and y have no parts in common, and the product of x and y is undefined.
• M8, Unrestricted Fusion: Let φ(x) be a first-order formula in which x is a free variable. Then the fusion of all objects satisfying φ exists.
$\exists x [\phi(x)] \to \exists z \forall y [Oyz \leftrightarrow \exists x[\phi (x) \and Oyx]].$ P.8
M8 is also called "General Sum Principle", "Unrestricted Mereological Composition", or "Universalism". M8 corresponds to the principle of unrestricted comprehension of naive set theory, which gives rise to Russell's paradox. There is no mereological counterpart to this paradox simply because Parthood, unlike set membership, is reflexive.
• M8', Unique Fusion: The fusions whose existence M8 asserts are also unique. P.8'
• M9, Atomicity: All objects are either atoms or fusions of atoms.
$\exists y[Pyx \and \forall z[\lnot PPzy]].$ P.10


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 Current Thread Re: [ontolog-forum] Foundation Ontology Primitives, (continued) Re: [ontolog-forum] Foundation Ontology Primitives, John F. Sowa Re: [ontolog-forum] Foundation Ontology Primitives, Ian Bailey Re: [ontolog-forum] Foundation Ontology Primitives, Patrick Cassidy Re: [ontolog-forum] Foundation Ontology Primitives, Pat Hayes Re: [ontolog-forum] Foundation Ontology Primitives, Jawit Kien Re: [ontolog-forum] Foundation Ontology Primitives, John F. Sowa Re: [ontolog-forum] Foundation Ontology Primitives, Sjir Nijssen Re: [ontolog-forum] Foundation Ontology Primitives, Ronald Stamper Re: [ontolog-forum] Foundation Ontology Primitives, Rob Freeman Re: [ontolog-forum] Foundation Ontology Primitives, Duane Nickull Re: [ontolog-forum] Foundation Ontology Primitives, Jawit Kien <= Re: [ontolog-forum] Foundation Ontology Primitives, Patrick Cassidy Re: [ontolog-forum] Foundation Ontology Primitives, Pat Hayes Re: [ontolog-forum] Foundation Ontology Primitives, Christopher Menzel Re: [ontolog-forum] Foundation Ontology Primitives, Patrick Cassidy Re: [ontolog-forum] Foundation Ontology Primitives, Duane Nickull Re: [ontolog-forum] Foundation Ontology Primitives, Patrick Cassidy Re: [ontolog-forum] Foundation Ontology Primitives, Duane Nickull Re: [ontolog-forum] Foundation Ontology Primitives, Pat Hayes Re: [ontolog-forum] Foundation Ontology Primitives, Matthew West Re: [ontolog-forum] Foundation Ontology Primitives, Duane Nickull