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Re: [ontolog-forum] Foundation Ontology Primitives

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>, Jawit Kien <jawit.kien@xxxxxxxxx>
Cc: ian@xxxxxxxxxxxxxxxx
From: Pat Hayes <phayes@xxxxxxx>
Date: Thu, 4 Feb 2010 22:20:58 -0600
Message-id: <EA32E418-4D42-4D05-868E-2282B3B33321@xxxxxxx>

On Feb 2, 2010, at 10:11 AM, Jawit Kien wrote:

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.

But the obvious conclusion is even more obvious here than it was for Russell: there is absolutely no reason to suppose that such an object O exists. In fact, the 'paradox' is best viewed as a reductio argument that it cannot possibly exist. In other words, M8, the unrestricted fusion axiom, is nonsensical. IMO, it is nonsensical upon inspection, at least for any notions of 'part' and 'object' that are the slightest use in actual reasoning. Consider for example the 'object' consisting of all livers of Chinese males over 40 years old. Is this what anyone would be inclined to call an 'object'? 

If M8 were restricted to parts which were all under parts of a single underlap, then it would make more sense: and this would be exactly analogous to a common idea in set theory, that comprehension *restricted to subsets of a set* makes sense, whereas unrestricted comprehension does not. 

Pat Hayes

(Every object is, of course, animproper 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 andy.
  • 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 toRussell'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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