RE: TECH: goat's legs; quantification and restriction

[I.Alexander.bra0122@OASIS.ICL.CO.UK]

doi lojbab.

Lot's of interesting points here.
I may ramble on a bit (surprise is optional :).

> The answer is that "da" is defined to be a quantified variable under the
> standards of quantification logic, and that means that quantifiers on "da"
> are exact number claims, since "da" is totally unrestricted.

1.  I'm familiar with existential and universal quantification,
and I've seen various authors use a number of extensions to these,
in an apparently ad hoc fashion.

You can do things the hard way:

        A(z in TypicalGoat: E(x, y: leg(z, x) & leg(z, y) & x != y))

which is one interpretation of

        There exist two legs on a typical goat.

or

        A(z in TypicalGoat: E(x: head(z, x) & A(y: head(z, y) => y = x)))

for

        The typical goat has exactly one head.

Or you can define new "quantifiers".  N(x:P(x)) is quite common for
"the number of objects 'x' satisfying predicate 'P'".  I've also
seen S(x:P(x)) for "the sum of the numbers 'x' satisfying 'P'" and
other mathematical operations.  I seem to remember someone using
U(x:P(x)) to mean "the unique 'x' such that P(x)" (which obviously
may be undefined), but I can't remember where.

I haven't however come across any more general treatment of numerical
quantification, and would be interested in finding out more about it.

2.  Is the existential quality of {da} part of {da} itself,
or simply a result of its default {su'o} quantifier?
I've always thought the "existential" nature of {da} a bit oddly
defined, when {roda} is a universal quantification.  Up till now,
I've treated this as more-or-less a convention - not an unnatural
one, but a convention nevertheless.  I haven't thought of universal
quantification as a variant of existential quantification.  I normally
interpret quantifications as they come, or occasionally turn them into
"There exists an x such that ...", but your insistence that numbers
quantifying "da" are exact enumerations suggests interpreting them
as "The number of 'x' such that ... is 'n'", which makes the use of
{ro} for universal quantification seem pretty obvious.  Still not
_totally_ comfortable with this, but we'll see ...

3.  It seems that {da} may be the crux of our problems.  When I replied
to And, it hadn't occurred to me to use {da} - the first use I can find
in the recent correspondence is from Mark.  If we change the {da} into
something else:

        lo'e kanba cu se tuple re bu'a

I don't think anyone can quarrel with it.  There _are_ predicates,
{pritu} for example, which make this true.  It may also work with
{zo'e} or {co'e}, although I'm not so sure about this.

Also -

> re tuple can be analyzed more or less as "re da poi tuple" though I'm never
> quite sure that this works under all logical analyses.

- another situation where inserting a {da poi} seemed to make a significant
difference arose in my comments on Nick's set theory piece, where

        .abu jo'e by. se cmima ro cmima be .abu .e .by

seems to expand to

        ge .abu jo'e by. se cmima ro cmima be .abu
        gi .abu jo'e by. se cmima ro cmima be by.

while

        .abu jo'e by. se cmima ro da poi cmima .abu .a by.

becomes

        .abu jo'e by. se cmima ro da poi ga ke'a cmima .abu gi ke'a cmima by.

so that inserting a {da poi} requires changing the logical connective.


> In the "ri'u ri" (which incidentally seems flawed in that I would
> read the phrase as "to the right of 'two somethings'", where
> 'something', being unrestricted, refers to anything in the universe) ...

I've already corrected {ri'u ri}.  And of course, to make the point
about _tense_, I should probably have used {ri'u [ku] ma'i vo'a}.
However, one of the sources of confusion in this discussion is deciding
when something is restricted and when not.  Is it not the case that my
'two somethings' were restricted to being goat's legs?


> In the version without the phrase, your predicate is ONLY that
> there are exactly two somethings that are legs.  This is NOT a
> universal in predicate logic terms - a universal statement is one
> that is quantified with "ro" ...

Sorry, I didn't intend "UNIVERSAL" to refer to quantification as such
(or at least only in the sense of {roroi fe'e roroi}),
but in contrast to specific.

Hmmm, that's not very clear.  There may be a better technical linguistic
term, but I can't think of it at the moment.  I've been trying to think
of some examples, and getting slightly bogged down in issues like the old
"{lo} = general, {le} = specific" one.

If I say "Some people eat meat" in English, that would normally mean,
and be interpreted as, a completely general statement about the way
things are, rather than referring to a specific event.  *That*'s what
I meant by "universal".  The way to say this in Lojban appears to be

        roroi fe'e roroi ku lo remna cu citka loi rectu

(I got myself into all sorts of confusion until I realised that
the order of the {ro} and (implicit) on-the-{lo} quantifiers
was crucial).

If I say simply

        lo remna cu citka loi rectu

it could presumably mean the same as above, or the same as

        lo remna caca'a citka loi rectu
        Some people are now actually [in the process of] eating some meat.

or any number of other things depending on the tense.

When we look at "specific"s, I can think of three senses in which
we can be specific, which I'm going to call "accidental",
"definitive" and "declarative".

A denotation is "accidentally" specific if it simply happens
to have a single referent.

        lo prenu poi ca tavla la kolin.

might refer to a single person at the time it was uttered,
but we can only determine this from real-world knowledge.

A denotation is "definitively" specific if the definitions
of the words of which it is composed constrain it to a
single referent.

        lo stedu be la djan.

is guaranteed to denote something unique, since someone called
"John" will only have one of that particular part of the body.
(I'll ignore the extremely unlikely cases where we are talking
about heads of several people named John, or John was born
with an unusual genetic defect.)

A denotation is "declaratively" specific if makes an overt
claim to be specific, by the use of words such as {le} and {ro}.

The question is whether using an explicit number places something
in this last category.  On the face of it this would appear to be
the case.  If the number is exact, and the gadri is veridical,
this appears to be as specific as you can get.  And yet ...

{pa kanba} appears to have a single referent, but we don't
seem to have any way of knowing its identity, at least not
unless we examine a potentially large amount of surrounding
context.

        re remna cu sarcu lo'enu co'a lanzu
        It takes two to start a family.

(The {lo'e}'s in there to preempt la nitcion. from suggesting
some of the more interesting possibilities :)

I can't quite make up my mind about this one.  Even if we agree
that the {re} is exact, we don't appear to have specified _which_
two people we mean.  Unless of course we were to choose a
_specific_ family-starting-event, but that we haven't done here.

(I'm still not sure of all the default quantifiers.  It feels
like the default quantifier on {lo'e} ought to be {ro}, in which
case the above doesn't appear to mean what I said, because the
order of the quantifiers is wrong.  We need to put the {ro}
out into a prenex

        roda po'u lo'enu co'a lanzu zo'u re remna cu sarcu da
)

Let's try again.

        re remna cu sarcu lonu sazri le stela
        It requires two people to operate the lock.

        re remna cu sarcu lonu co'a kalri le vorme
        It requires two people to open the door.

In the first case, it might well require the two specific
people who have one key each, whereas in the second, it might
simply be that one person on his own wouldn't be strong enough,
or there might be an elaborate mechanism with a lever too far
away from the door itself for a single person to operate.
Perhaps the first example needs a {le} to make it (declaratively)
specific.


More on {lo} and {le}:

Some of us have a tendency to use {lo} to mean "a", in the sense
of something which is _specific_ but which we haven't mentioned
before, and which we therefore have no expectation that you
know which we mean.  One of the main points of a recent discussion
about {le} was that it tended to embody an assumption that the
intended referent was expected to be inferrable from the total
(linguistic and extra-linguistic) context.  On the other hand,
Colin appears to be of the opinion that {le} is a more appropriate
_introductory_ gadri than {lo}, because it is specific.
({ro lo su'o broda} presumably has the same degree of _specificity_ as
{le broda}, without the "descriptive" connotations, but it's not clear
whether it means anything different from {ro broda}.)
I've seen Cowan do the same thing, and I think I'm beginning
to see the reasoning behind this.  Is the assumption that
the listener should know what is meant when {le} is used,
at the time it is used, inappropriate?


And finally
(at least for now - thinking about these issues has raised
a number of peripheral questions, some of which I expect I'll get
round to posting eventually),
I'm still not sure to what extent the potential scope of tense
information, which may have been elided, gives us leeway in
interpreting a jufra.  If by supplying different tenses, it would
be possible to produce a wide variety of meanings, is a statement
without any tense information not ambiguous amongst all of them?

mi'e .i,n.