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Martin Bays, On 23/09/2012 23:44: [Where are you from, Martin, btw? I notice you write _singularisation_ and _maths_, i.e. not American English. Ah, I just thought to google & find you in Oxford and Canada, which explains that.]
* Monday, 2012-09-17 at 01:22 +0100 - And Rosta<and.rosta@hidden.email>:I think the options for the extension are: 1a one feline thing, which looks like a single cat 1b one feline thing, which looks like a bunch of cats 2a many separate feline things, which look like single cats 2b many separate feline things, which look like bunches of cats l- gives 1a/1b; r-/s- give 2a/2b.
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The four options I listed are mutually exclusive. The four options are four alternative extensions. They don't coexist within the one extension.OK, good. Some things are becoming clearer. Firstly: contrary to what I had understood from previous descriptions, these myopic singularisations of cats are not elements of the UoD on a par with the cats they singularise.
I'm not sure what counts as "on a par with"...
I would like to formalise this by considering them to be in different "sorts", meaning that variables (and hence quantifiers) come with the information as to which level (for want of a less contentious word) they can take values in.
The problem with this is that in Type-A ontology, either everything is a Sort or nothing is. If what you call 'individuation' can be applied to a Sort to yield 'individuals' in the same UoD, then everything is a Sort. If individuation is something that translates one UoD into another, so that what is an Individual in one UoD is a Sort in another, then within a UoD everything is an individual.
Secondly: we have two basic operations - massification, which I think we can identify with mereological sum, and individuation. The difficulties arise from the tangled interactions of the two, from determining how the resulting objects manifest in the language(s), and from determining their properties and relations. I do believe that the only fruitful way to go about this kind of discussion is to try to develop a formalisation. So let me try to develop one, and you can tell me where it diverges from what your intuitions desire. But first, let me deal with this:So now I'm imagining something along the lines of taking the mereological sum, but then dividing up that sum into "individuals" according to some criterion, and assigning the m.s. those properties which hold "generically" of those pieces... but perhaps this is wildly inaccurate?It's not sufficient. For example, you wrote the email I'm responding to, but if I divided you up into individuals by some criterion that makes you many individuals, I wouldn't expect that the property of having written the email holds generically of the individuals. Yet still, you did write the email.And that's really because the myopic singularisation of those individuals did? Not a massification, say?I can't think of clear differences between those two -- I can't think of analogues of "Weighs X amount" and "has X heads". But I haven't given it much thought.Based on what you've said, I think "wrote the email" is an analogue! If no one of these individuals satisfies it, then the MS doesn't. Unless I've wholly misunderstood MSs. Meanwhile, the sum aka massification of those individuals is free to do things that no one of them does, writing emails included. In the below, I assume that we're actually in agreement on this.
I may have misunderstood what you meant by "hold 'generically' of those pieces". I was accepting that a property of one piece is a property of the whole, but I wasn't accepting that a property of the whole is a default property of a typical piece. (E.g. typical subtypes of Martin are mathematicians, but aren't writers of that earlier email.)
Attempt at formalisation begins. I'll try to keep it readable. We start with mereology, which just means that we have an "is part of" relation which acts like you'd expect from the name: if x is part of something which is part of z, then x is part of z; two things are equal if and only if each is part of the other. The sum operation is defined in terms of parthood - x(+)y is the least z such that x and y are part of z, meaning that if x and y are part of some z', then z is part of z'. We assume that such sums exist, also for sums of infinitely many objects. (So in mathematical terminology, which I'll avoid using non-parenthetically, we are talking about a join-complete semilattice.) Massification is precisely summation. Bunches are sums. L-sets are sums. Now what's individuation? Individuating a whole must mean deciding that certain parts are "individuals". No two individuals should share a common part, and the sum of the individuals should be the whole. (So individuation of x is a choice of semilattice map from the semilattice below x onto an atomic boolean algebra.)
I assume the key thing is that there be criteria for individuation, but not that not sharing a common part is one of them. Individual national teritories and colours and bands might share a common part.
Let's call an "individuated set" a set of objects no two of which have any common part. Then individuated sets are precisely the things we get by individuating wholes. So the "for each" and "there exists" quantifiers make sense for individuated wholes - they quantify over the individuals. I think it's worth pointing out immediately that these are quite different from the "for all of" and "for some of" quantifiers, which when applied to a whole would quantify over *all* its parts, not just the special ones we've decided should be called "individuals". We must be careful not to confuse these two kinds of quantification. We can also expect two kinds of generic quantifier; I suggest to call them "for generic" and "for most of" respectively. Now, what about "myopic singularisation" and your 4 possibilities? One individuation of a whole x is that which takes x itself as the only individual. When x is the sum of all cats, this gives 1b above.
It's not clear to me that it gives 1b specifically, rather than 1a/b. The contrast between 1a/b on the one hand and 2a/b on the other has to do with discriminating between multiple instances (e.g. of felinity), while the 1/2a vs 1/2b contrast has to do with whether each instance of felinity is a single cat or a bunch of cats. If you have one instance of Martin, or one instance of Wine, then if there are intrinsic criteria for differentiating between multiple Martins or between multiple wines, then the instance might be a massification of multiple Martins or of multiple wines, but in the absence of intrinsic criteria, the a/b contrast is absent.
Another way to individuate the sum of all cats is to take the individual cats as the individuals. This leads immediately to 2a. We get 2b if we take a different individuation, e.g. taking the masses of male, female and intersex cats as our three individuals. Is that the kind of thing you meant?
Yes, or also three herds or heaps of cats.
To get 1a, it's clear that we need something new. Nothing we currently have in our universe can play the role of the object you describe in 1a.
If I were in your shoes, I'd be trying to start with 1a and derive the rest from it...
So we're forced to introduce a new operation - myopic singularisation. You seem to have this applying to individuated sets, although it's not clear to me that the individuatedness is important.
The term "myopic singularization" implies you start from individuated sets and derive from that the MS. But I myself see the MS as basic; it's a Type/Sort such that no differentiation is made among its subtypes. It's Martin not seen as a bunch of Martins; it's wine not seen as a bunch of wines; it's triangularity not seen as a bunch of different triangularities.
Questions regarding MSs: (i) how are their properties and relations related to those of the individuals they originated with? I continue to be tempted to translate use of MSs into generic quantification over individuals. However, the most obvious (to me) way of doing this, assuming l_ is meant to correspond to MS, breaks the scope-invariance: I would want to read "le Pe Qe" as meaning that Qe holds for individuals e satisfying P generically with respect to all currently bound variables, so assuming obvious individuations, "la rrna le rrne mlcake" would be true but "le rrne la rrna mlcake" false (where rrn_ is to be read "_ is a natural number", and mlcake is "a<e").
This is because for every natural number there's one that's bigger than it but not always one that's smaller than it, natural numbers being positive integers? To the extent that I can make sense of"la rrna le rrne mlcake", I don't think I'd judge it true but"le rrne la rrna mlcake" false.
But we could get the desired scope invariance by exporting a single joint-generic quantifier to the far left - "la Pa le Qe Rake" becoming "for generic (a,e) such that P(a) and Q(e), R(a,e)". That would merely render the two sentences above indeterminate. But it would give "la rrna le rrne [a is greater than e or a is less than e]". Do you want that to be true?
I've only the haziest notion of what the truth-conditions might be for this, but I'll offer a tentative Yes.
(ii) Can MSs be summed with each other? Can I take an MS of dogs and an MS of cats, and bunch the two together?
Yes, you can form a massification or a MS of them.
(iii) Relatedly: can we myopically singularise myopic singularisations? Do you want to leave open the possibility that any thing might be considered to be an MS of some other things?
Absolutely, yes. Though I see it the other way round, that everything is a Type and may be considered to have subtypes.--And.