Re: quantifiers

[jorge@PHYAST.PITT.EDU]

I agree with most of the analysis by pc. I only add some minor comments
to some side issues.

>         Quantifiers in Lojban differ markedly from those in standard
> logic.  The quantifiers of logic always occur outside the core sentence
> and only in association with a variable, never -- like the quantifiers of
> Lojban -- attached to a designating term nor in the argument place of some
> predicate.

The fact that quantifiers appear in the argument place is purely superficial.
As you note later, to interpret them they have to be transposed to the prenex
anyway. It is nothing but a convenient notation. As for taking "designating
terms" as a special class of terms, I don't think that it is really
necessary, since nothing is lost by treating all terms alike.

> And, in most standard systems, quantifiers in their
> own right are not restricted to subsets within the universe of discourse;
> such restrictions are done sententially after the quantifier is expressed.

The same can be said of Lojban, if one looks at it from certain vantage point.
The difference is in the notation, just a superficial matter, not in what a
sentence expresses. That is, the sentence

                lo prenu cu blanu

goes into the logician's notation as

                Ex prenu(x) & blanu(x)

even when we are not explicitly using a variable in the Lojban version.


>   The only thing to recall is that in
> logic, as usually in Lojban, quantifier DA POI broda requires that there
> be broda, even when the quantifier attached is "all" or "at most" (unlike
> the cases where the restriction is in the sentence rather than the phrase,
> "for all x , if x is broda, then..." etc.).

What do you mean by {<quant> da poi broda} requires that there be broda?
Is there any difference in meaning betwen these two:

        ro da poi broda cu brode
        For all x that is a broda, x is a brode.

        ro da broda nagi'a brode
        For all x, if x is broda, then x is a brode.

I don't see any difference. Perhaps in the first there is more of a
connotation that there is at least one broda, but I don't see any logical
requirement that there be one.

>  The three objects are the
> set itself, {x : x broda} or just {broda}, the massification of the set,
> Mas ({broda}), and the average, Avg({broda }).  Each of these is a unique
> individual and so has no use of integral quantifiers.

I agree... to a point. The simple forms {lo'i broda}, {loi broda} and
{lo'e} broda} have no use of integral quantifiers, because indeed they
all start from a single set and take it as one whole.

However, when we consider {lu'i} and {lu'o} things change, because in
this case we are free to select subsets of a certain cardinality
from the original set, and that gives rise to many possible individual
sets and masses. Thus, there is nothing strange about {re lu'i ci lo plise},
two sets of three apples.

{lo'i broda} and {loi broda} are short forms of {lu'i ro lo broda} and
{lu'o ro lo broda}, and because of the {ro} there is only one of each.

[The above is not strictly true for {loi broda} because of the special
use of fractionators, and the fact that the default for {loi} is {pisu'o}.
But it does hold for {piro loi broda}.]

> But each is also
> atomic, a partless point in logic, and so has no use for fractional
> quantifiers either.

Right, except for the convemient conventional use that we assign them:

>  That is, 0.n of a mass of a set is the
> mass of 0.n of the set, which, in turn, is actually another set.

I think that's sensible. (And also quite useful.)

> (By the
> way, properties of masses are properties of the whole mass typically.  It
> is an interesting study to find what prpoerties of masses are also
> properties of submasses and which are not.  We know that "inhabits Africa"
> is one that covers both and presumably there are others, but "wins the
> game" does not seem to descend.  This problem is one version of the
> fallacies of composition and division and a more interesting version than
> the usually discussed one between members of a class and the class itself
> -- usually meaning the mass).

Hear, hear! Which properties masses share with its submasses depends on
the property, there is no general rule for all properties.

[...]
>         Details of some cases of various expansions still need to be
> worked out, as do more careful explanations of double descriptions like lo
> ci lo mu broda and of lua.  This is a tentative sketch explanation for
> comments and development, but it does seem to work, so far as it goes, for
> both Lojban and logic.

You don't say it explicitly, but does this mean that you give your blessing
to the thesis that {lo broda} behaves just as {da poi broda} from the
logical point of view?  (Of course, linguistic connotations could later
develop, but basically they mean the same thing.)

Jorge