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(Excuse slowness to respond; I assure you that this was due to
miscellaneous pressures of life, not metaphysical pain!)
* Tuesday, 2012-09-18 at 04:40 +0100 - And Rosta <and.rosta@hidden.email>:
> Martin Bays, On 18/09/2012 02:37:
> > * 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.
> >
> > What do you mean by "the extension"? The extension of mlt_ in the UoD?
>
> Yes.
>
> > So you're going back to having the UoD somehow contort to adapt to the
> > choice of quantifier? If so, how does that work? If not, then what?
>
> The four options I listed are what you get by applying different
> criteria for individuation and separating one individual from another.
> Either that's four ways of construing the same UoD (which is my take
> on things) or there's a four-way choice of UoDs, or...
>
> >>> [reordered]
> >>> Then there's the complementary question: if these myopic
> >>> singularisations satisfy mlt_, then (one would naively expect) they'll
> >>> end up in massifications. How does that work, or is it just disallowed?
> >>
> >> If you mean they'll end up in a massification of mlt, it's not one of the options.
> >
> > OK. So why not? What can go in such a massification?
>
> The extension. 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 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.
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.
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.)
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.
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?
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.
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. In any case, the
result is a new kind of thing. In particular, it is illegal to take the
sum of a cat and a myopic singularisation of cats.
We might want to say that 1b was really a ("trivial") myopic
singularisation.
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").
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?
(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?
(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?
Hoping this is getting us somewhere, and apologising for making you read
things which look more than a little like maths,
Martin
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