Monday, September 12, 2011

More about Malezieu’s principle

As I have defined an Ockham set or oset, an oset is nothing different from its elements. A term referring to an oset (‘that dozen of eggs’) is referring to all its elements in just the way that a grammatically plural term (‘those 12 eggs’) is referring to them. It just happens to be grammatically singular, and should not be confused with a genuine singular term referring to a thing. It is not a thing, but a number of things (except in the limiting case of the singleton oset, perhaps, which is identical with its only member).

Malezieu’s principle is that if the things exist, then the oset exists. If a exists (singular) and b exists (singular), then a and b exist (plural) , and the oset O (i.e. a and b) exists also. If some or all of the elements do not exist, or cease to exist then the oset does not exist, or ceases to exist, also. The reference of ‘that dozen eggs’ fails as soon as even one egg is broken. This contrasts with set theory, where we have to postulate the existence of a set containing the members.

Thus there cannot be two possible worlds, each of which contains a and b, such that the oset of a and b exists in one possible world, but not the other. This has the following corollary

(M) If an oset exists in one possible world but not another, it follows that at least one of its elements is in the first world, that is not in the other.
This leads to the problem of the infinite universe where there is no oset corresponding to the infinitely many elements. If we allow the possibility at all, it follows from (M) above that there cannot be two possible worlds where any element one is identical with some element in the other, but where the oset of all of them exists in one, but not the other. Either such a world is impossible, or every infinite world is like this. If it is impossible, then Oxinf is not independent – if every finite oset excludes at least one thing, unlike set theory this guarantees the existence of infinite oset. But if it is possible, this rules out infinite osets.

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6 Comments:

Blogger David Brightly said...

Hello Ed,

As we have seen, I've been having trouble following this latest series of posts. At their core there seems to be at the very least a claim that could perhaps be put as follows:

In a language whose plural referring terms take the form 'a and b and c...and z' (*) for some finite number of individual proper names a,b,..,z, a referring term can refer to at most a finite number of individuals. As a corollary, applied to an infinite domain such as the natural numbers, such a language has no way to refer to all the individuals in the domain.

Have I at least go that much right? But as this result seems pretty obvious are you in fact making some bigger claim and can you help me further in seeing what it is?

* We could get fussy about the exact syntax of referring terms but that would be otiose. The point being that atomic referring terms all refer finitely and no finite syntactic construction can yield a term that refers inifinitely. This is analogous to the observation that in ZFC, without an axiom of infinity, no amount of finite guddling with finite sets yields an infinite set.

11:03 am  
Blogger Edward Ockham said...

>>In a language whose plural referring terms take the form 'a and b and c...and z' (*) for some finite number of individual proper names a,b,..,z, a referring term can refer to at most a finite number of individuals.

If the proper name string is finite in length, then only a finite number of individuals can be referred to, yet. But (a) the string could be infinite and (b) in any case we have other resources for plural reference. For example ‘those men’ or ‘that band of men’. There is no reason, on the face of it, why we shouldn’t be able be able to refer to all of the elements of an infinite domain.

>> But as this result seems pretty obvious are you in fact making some bigger claim and can you help me further in seeing what it is?

It’s not obvious, for the reasons pointed out above. For example, on the face of it, it seems we can use terms like ‘the natural numbers’. I.e. all of them. What I have tried to show is that if a domain containing infinitely many things, but no corresponding things to ‘those things’ is possible at all, then all such domains are the same. I.e. if such a domain is possible at all, then no possible domain contains an infinite oset. Or such a domain is impossible, in which case a domain with infinitely elements necessarily contains on oset containing all of them. So we have a stark and nasty choice. Either infinite osets are impossible. Or they are necessary. Both of these alternatives contrast with ordinary set theory.

>>This is analogous to the observation that in ZFC, without an axiom of infinity, no amount of finite guddling with finite sets yields an infinite set.

Quite the opposite, if I am right. If the problematic case I have described (i.e. infinitely elements, no set of them) is impossible, then we don't need such an axiom. We simply postulate that every finite set omits some element. This guarantees the existence of an infinite oset. Quite unlike standard set theory.

11:26 am  
Blogger David Brightly said...

>> ...if a domain containing infinitely many things, but no corresponding things to ‘those things’ is possible at all...

Sorry, Ed, but I can't make sense of this at all. If language L contains the referring phrase 'those things' and we apply it to an infinite domain like the natural numbers we can choose what we like as the referent of 'those things' but the natural choice is the entire domain. I just don't see how the description 'a domain containing infinitely many things, but no corresponding things to ‘those things’ falls into the realm of possibility at all. It isn't, as it were, 'properly formed'. And the possible/impossible dichotomy doesn't apply.

12:17 pm  
Blogger David Brightly said...

Just to clarify what I mean by 'ill-formed' here: in discussions of possibility language 'floats above' the possible worlds. The possible worlds are independent of language. What you are suggesting as a 'possibility' involves an interaction between language and world, viz, 'those things' refering to all the things or not. You are quantifying not over possible worlds but over possible world-language interactions. It's this I have trouble making sense of. But then I have trouble with possible worlds talk anyway!

12:36 pm  
Blogger Edward Ockham said...

I covered that in this post

(Ockinf) For any X's, there is some y such that y is not one of the X's.

Assuming we are happy with the plural quantification over Xs, this seems perfectly well-formed. In the same post, I anticipated the objection that surely, in all cases, we can refer to all the things there are. That is not the question. The question is whether Ockinf above is impossible. Does it contain an actual contradiction? If it does, it is well hidden – Belette, who is a pretty astute commenter, thinks Ockinf is obviously and boringly true.

12:42 pm  
Blogger Edward Ockham said...

>>The possible worlds are independent of language. What you are suggesting as a 'possibility' involves an interaction between language and world, viz, 'those things' refering to all the things or not.

I think my 'lasso' analogy was a mistake. It suggested that while 'every thing' can lasso every thing, 'any things' may not be flexible enough to get round all things, in the case of an infinite domain. As though all the things were really there, but language couldn't get to them.

No. 'All things' means all things, just as 'every thing' means every thing. The question is the ontological one of whether it is logically possible there may not be such things as all the things there are.

12:46 pm  

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