Saturday, September 10, 2011

Ockham sets: preliminaries

Judging by the comments from 'Belette', we need some preliminary remarks about Ockham sets. Let's begin with plural reference. In English (and Latin, and probably most natural languages) we have plural terms as well as singular terms. We can form plurals either by concatenating singular terms together, as in 'Peter and Paul'. Or we can use an ordinary plural referring term, as in 'those [two] apostles'. Both of these take verbs in the plural form. Thus "Peter and Paul are apostles", "those apostles are preachers". Finally, we can form a collective noun of the form 'an X [of] Ys'. Thus 'a dozen apostles', 'a pair of shoes' and so on. These nouns are grammatically singular, and usually take verbs in the singular. Thus "a pair of shoes is in the cupboard", "one dozen eggs is a good thing to buy". (Although intuitions differ. Do we say that a number of people are in the room? Or is in the room? But this is merely grammatical accident, and has no relevance to logic, I think).

It is a fundamental assumption of Ockham set theory that the singular form of the collective noun is a grammatical feature only, not a logical one, and that we can assert identities using all three forms of plural noun. Thus
1. Peter and Paul are these apostles.
2. These apostles are a couple of apostles
3. This couple of apostles is [or are] Peter and Paul.
The equivalent to a mathematical set containing more than one member, say {Peter, Paul}, is the reference of the expression formed by concatenating the two proper names with 'and'. Thus 'Peter and Paul'. The equivalent to a set defined by comprehension, e.g {x: apostle(x)}, would be the reference of the plural referring term '[all] the apostles'.

There is no real equivalent to the singleton set {Peter}, because the comma in set notation corresponds to the 'and' in Ockham notation, and once we remove the 'and' from 'Peter and Paul', we are left with 'Peter'. Thus in Ockham set theory, {Peter} = Peter, if that makes any sense. Remember that the curly brace is set theoretical notation, and has no equivalent in natural language. Ockham set theory is meant to capture our logical intuitions about the behaviour of plural and collective nouns in ordinary language, not some invented language like set theoretical notation with modern predicate logic.

There cannot be an empty set in Ockham. Once we remove the name 'Peter' from the expression " 'Peter' ", we are left with nothing at all. An Ockham set is identical with its members, and it is impossible that something is identical with nothing.

Set membership is signified in Ockham by the relational term 'one of'. This can be defined in terms of the primitive 'and' mentioned above, as follows.
a is one of Xs if (def) for some Ys, Xs = a and Ys
For example, let the Xs be Peter and Paul and John. Then there are clearly Ys, namely Paul and John, such that
Xs = Peter and Ys.
I.e. Peter is one of those three persons, Peter and Paul and John.

Belette also asked for a definition of 'infinity'. Let's try starting with 'finity' first, by the recursive definition.
1. Any one thing is finite in number
2. If any Xs are finite in number, then those Xs and any one y are also finite in number.
Thus Peter is finite in number, hence Peter and Paul are finite in number, hence Peter and Paul and John are finite in number, and so on.

Then I say that there are 'finitely many things' when there exists an oset O such that O is finite in number, and such that every thing is one of O. And finally, there are 'infinitely many things' when it is not the case that there are finitely many things. This could happen in two ways. In the first, as I have defined it in the previous post. If every oset is such that at least one thing is not one of it, there cannot be an oset O such that every thing is one of O, and so our condition for finitude fails. Or there can be an oset such that it is not finite, and no finite oset contains everything, and so our condition for finitude fails again. I shall argue that of these two ways in which there could be infinitely many things, only the first is possible.


William M. Connolley said...

(minor comment: OK, I accept your defn of infinite. I think I'd be happier if you just said "finite" rather than "finite in number" (I don't understand what the "in number" gains you) but that it only words).

And now hopping between posts, which may be confusing, but then so would be swapping comments between posts:

> it runs counter to our natural assumption that we can talk about or refer to 'all the things there are'

"running counter to natural assumption" is a weak-to-nonexistent argument. Relativity and QM run counter to most peoples assumptions. Furthermore, could we so refer, even in principle? There are uncounted billions of neutrons in any given distant star: I can't possibly try to refer to each of them by name or number.

William M. Connolley said...

> it seems possible that there are infinitely many things, without there being an Ockham set that includes all those things

That seems true, but I think it is in danger of being trivially true: you are in a position analogous to set theory without the axiom of infinity. without that, std set theory doesn't have infinite sets either.

Edward Ockham said...

>>That seems true, but I think it is in danger of being trivially true

I think it is true also, but it can hardly be trivially true as I have been arguing about this with people like Tom McKay for a long time.

But best to wait for the next post.