As far as I can see, none of your examples is an example of an infinite proposition (in the sense I was using the term). They are only examples that quantify over infinite domains or domains that contain some infinite entities but none of the propositions themselves is infinite. Can you honestly claim that you can entertain your alleged proposition N? I don’t think you (or any finite being for that matter) can.

I don’t think your answer helps either. That is not what I was asking. I do not doubt that we can use a number of linguistic devices to refer to a proposition (including names and definite descriptions) but in order to have any propositional attitude towards it we need to be able to entertain it.

My challenge is that, even if N existed, no finite being would be able to entertain it because N would be an “infinite” proposition. according to you, there would be nothing paradoxical in my knowing that N, right? But I have no idea of what I would know if where I to know that N. I guess I would know that you don’t know that you don’t know that you don’t know … But I don’t know what that means (what proposition it expresses)> Moreover, what is it that you don’t know according to N (or that you don’t know that you don’t know or that …)? Well, that you don’t know that you don’t know that you don’t know that… I have no idea of what it is for me or for you to know, believe or even entertain such an infinite proposition. Do you?

Now, if I am right, then, if N exists, it is true and necessarily so, but that reasoning is no way for you (or me) to know that N because in order to know that N you (or I) would have to be able to entertain the proposition N and as I argued I think neither of us can.

I believe that I’m in New York. Plausibly, that belief is a part, or at least a constituent, of my mind. And I’m a constituent of the proposition that belief has as its content. But there’s nothing paradoxical or metaphysically dubious in any of this. This is all to say that constitution plays by very different rules to parthood.

I don’t think it’s wrong to say that if I didn’t know N, I wouldn’t know N. If you try to give a reductive account of this in terms of other things, it will be hard. But that’s just on a par with anti-reductionism about all sorts of things.

If you want to model it, it’s going to be just like non well founded sets, as Dustin has been saying, so sets like {{{…}}}. That ‘set’ is a member of itself, and in a sense it is infinite, though it doesn’t look overly problematic.

One related point. That a can’t be a proper part of a seems reasonable enough. But the principle you need here is that a can’t be one among many constituents of a, and that’s much more contentious I think. The ‘set’ {{{…}}} doesn’t have itself as a part, but it does have itself as its only member. And that makes, I think, a big metaphysical difference.

The question is what is the proposition Brian does not know and your saying that is this proposition doesn’t help. Let me put the question differently: ‘What would Brian know were he to know that N?’. It seems that he would know that he doesn’t know that he doesn’t know that he doesn’t know that… My question is what is it that he doesn’t know he doesn’t know that he doesn’t know that he doesn’t know …?

What is an infinite proposition?

Some further worries about N. First of all, what is it exactly that you don’t know according to N? You don’t know that you don’t know that you don’t know… what?

Second, is N an infinite proposition? And if so can we even entertain it?