Re: quantifiers

["John E. Clifford" <pcliffje@CRL.COM>]

On Wed, 26 Jul 1995 jorge@PHYAST.PITT.EDU wrote:
>
> > 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.

Actually, Ex e {prenu} x blanu.  whether that is the same or not is open
to dispute.  I keep them separate for now.

> >   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.

If there are no brodas, the first is false while the second is true,
regardless of what brode is.  Restricted quantifier sentence are false
when restricted to an empty set (existential import).  Conditionals with
false antecedents (and hence also their universal generalizations) are
automatically true.

> 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}.]

Thanx.  As I said, I did not deal with these doubly descriptored forms,
since I do not really understand them yet.  I must say that what you say
makes precious little logical sense, since in logic the set would be
fundamental and here it is getting pretty unclear what, if anything, is
basic. As noted, the fractionators make no sense literally here; does the
lu'o locution move in he right direction? It is not clear from your
description.
> 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.)
Let's see: lo broda cu brode= k{broda}<>0 & Ex e {broda}, x brode, which,
given he general principle that Ax(x e {broda} <=> x broda) should be
equipollent (at least) to Ex st x broda, x brode, which is da poi broda
cu brode (via da poi broda zo'u da brode).  Whether this will work with
second or later terms in a matrix is less clear and there may be reasons
to question each transition here, but at least the first case seems to work.

Note again that there is still no way given to directly refer to an
individual in all these forms.
pc>|83