BPFK Section: Inexact Numbers
wrote:
>
> pc:
> > --- Jorge LlambÃas wrote:
> > > I understand external PA as {PA fi'u ro},
> i.e.
> > > PA out of all the referents of the sumti.
> >
> > But, but, but ... if they are fractional, why
> are
> > the other fractions not fractional? I gather
> > that you want to distinguish between {pimu},
> {pa
> > fi'u re} (and {mu fi'u pano}) and probably
> > {nopimu}.
>
> I would say {nopimu} and {pimu} are the same,
> just
> as {pa fi'u re} is also {fi'u re}.
>
> > this is legitimate, but does seem to
> > need an explanation, since these are normally
> the
> > same.
>
> The need to differentiate arises from the
> reified
> groups and sets. With lo/le/la we refer to many
> things
> at once, with loi/lei/lai/lo'i/le'i/la'i we
> refer to
> a single thing (with many members).
Of course, just here — on logical grounds as
well as various parts of CLL — I disagree with
you: {lo, le, la} introduce distributive groups
as much as {loi le lai} introduce collective
groups. The only way that {lo} for example could
-- in CLL Lojban — stand fro a number of things
is if it were a covert quantifier (or
conjunction, of cours, for finite groups, {le}
and {la} in particular) but it seems to me that
you and others of your persuasion have regularly
denied that {lo} expressions are quantifiers
under the skin (note: quantifiers are still not
strictly reference but near enough in this case
to let pass). While one of your definitions for
{lo} does sound a lot like a quantifier, the
other (admittedly the poorer of the two) clearly
is not. Of course, without the quantifiers or
conjunctions, the predications with {lo} become
mysterious (thought admittedly, with the
quantifiers, outer quantifiers take some
finagling). By the way, when I say "quantifier"
I do not necessarily mean {su'o da} and the like:
such variable binding expressions in Lojban may
logically be more complex than "there is an x."
> Then we
> need ways
> to talk about a number of things as a fraction
> of the
> total number of things on the one hand, and
> about a
> fraction of a (possibly membered) thing on the
> other.
>
> Using the same expression for both would lead
> to
> unclear cases. For example, {pimu le selcmi} is
> half the set, i.e. a set with half of the
> members of
> le selcmi, whereas {pa fi'u re le selcmi} gives
> half of the sets. One out of every two of the
> sets.
> We can't conflate them.
Let's see. One of these assumes that {le selcmi}
is a single set (a group with a single member) --
of something or other — and therefore that any
fraction must be a fraction of that set (and
presumably another set with that fraction of the
cardinal of the set). The other assumes that {le
selcmi} is (a group of) several sets and a
fractional quantifier is therefore (a group of)
that fraction of several sets. On your
reasoning, if {le selcmi} were (a group of)
several sets, {pimu le selcmi} would be half of
one of these sets. Now, since this is {le} we
presumably know how many sets are involved here
and so no confusion would result from the
ambiguity. With {lo}, where the size is in
doubt, there would be a functioning ambiguity
with at least the {pimu} case. And, of course,
if le selcmi is single, what does {fi'u re lo
selcmi} mean?
As I said, it seems to me you need and want both
of these modes of expression for both (all three
or so?) cases. And that suggests that, to avoid
ambiguity, the expressions for these various
purposes cannot be the same — but not different
form of the quantifier as such. As noted, I like
the {piPA/ PA fu'i PA si'e} for pieces of an
individual, {piPA / PA fui PA} for subwhatevers
and quite explicit forms for a fraction of a
member of the whatever involved {piPA /PA fu'i PA
le pa lo ....}, which is about right for all the
times we will use this.