On Feb 4, 2010, at 10:20 PM, Pat Hayes wrote:
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'?
Of course that's right, Pat, and I in fact started a msg that made more or less the same point. But a charitable interpretation here is that the authors' purpose was simply to point out that a "naive" mereology might be tempted to adopt a mereological version of the unrestricted comprehension schema (from which, of course, the above paradox follows) and hence that mereology, just like set theory, needs to be rigorously axiomatized.