pc answers

["John E. Clifford" <pcliffje@crl.com>]

Some questions have come to me about what logic says or how logic would
handle certain problems.  Here are a few short answers. A.  Rebinding a
quantified variable.  In logic, it is bad form but perfectly legal to put
a quantifier on x (say) in the scope of another quantifier on x (it is
part of the usual test to check that students can figure what is bound by
what quantifier).  The two quantifiers are totally independent, that is,
the whole is interpreted as if one of the quantifiers was on x and the
other on y (say) -- except that there can be no occurrences of of the
outer bound variable in the scope of the inner binder).  Notice that logic
can get away with this because the scopes of the quantifiers are totally
determined.
        Logical quantifiers are all also singular at heart, the apparently
plural ones, like "there are three...," are abbreviations whose behavior,
including instantiation, is governed by the underlying unabbreviated
complex . Thus a subselection of such a plural grouping would require a
new quantifier and a new mention of the predication defining the original
grouping: "There are three Fs and two Fs are Gs" for "There are three Fs
and two of them are Gs".  Some of these can be collapsed a bit, but the
rules for those collapses do not fit clean patterns, so far as I can see.
        In Lojban, where the scope of a quantifier does not have a natural
bound (which suggests, by the way, that they are not really quantifiers in
the strict sense but reference registers), using the logic system would
mean that the second quantification simply superseded the first and that
is often not desirable, since we may want to go on with the original
identification after the aside of the second quantifier.  So, Lojban has
to use a new variable for each genuine change of quantifier and that
leaves a second quantifier on an already bound variable (and all variables
are already bound the second time they appear) open to interpretation.
Subselection seems like the natural way to go.  But, like the logic
system, the subselection supersedes the original selection, so that a
third quantifier is a subselection of the subselection, not of the
original selection, which is irretrievable.  If you think you are going to
want the original back, it is better to make the subselections explicitly
(with "member of" of whatever) rather than with just altered quantifiers.
Alternatively, we might consider a device like we once (at least -- God
knows what is going on now) had for recapturing time axes (also probably
registers, by the way) after some of the hairier shifts, essentially a
device to mark popping back so- and-so many pages of the history of shifts
(or, for variables, of subselection quantifiers).  Like most of these
clever devices which rely on memory, this one would probably not work.
        As for how all this would come out in logic (ignoring the paucity
of quantifiers in standard logic by allowing an "almost all" and a "most"
and the accompanying restrictive forms, which are impossible in really
standard logic) the basic part is (almost-all dog x ) x has teeth and
(most dog-with-teeth y) y bites ... As set up here, x cannot go into the
... since that is not in the scope of the quantifier on x, which ends with
the bit about having teeth, which is all that the original claimed for
dogs: almost all have teeth.  We could fix that, let us suppose, in the
interest in getting on with the problem.  In any case, if we put y in the
..., we would pretty clearly have that most dogs with teeth bite
themselves, i.e., that each relevant toothed dog bites itself.  To get it
biting all or some or somewhere in between of other toothed dogs would
take another quantifier on -- depending upon what you want -- toothed dogs
or toothed dogs that bite something/some toothed dog/whatever.  Or just
dogs again.  Putting x in would (assuming we had the scope problem solved)
amount to saying that almost all dogs have teeth and are bitten by most
dogs with teeth.

B.  Typical/stereotypical/average.  _lo'e_ and its kin (have I got the
right ones out of this tangle?) are among the many ways Lojban has of
dealing with the sets of things.  In English and most usually familiar
languages, plural nouns refer to these sets in a variety of ways, not
clearly distinguished: as sets, collectively, distributively, and
statistically, to name a few of the most common.  Take (a classic)
"Chicagoans drink more beer than New Yorkers"  This one at least can't be
about sets, since sets don't drink beer.  It is pretty unlikely
distributively, since there is almost certainly a New Yorker who drinks
more beer than some Chicagoan (although there is no guarantee that the
distribution is strictly universal all around, it may be just "most" or
some such).  The likely cases are collectively (the tunnage of beer drunk
by Chicagoans exceeds that by New Yorkers, the summations of all the
individual drinks of all the individuals) or statistically (the average
Chicagoan -- probably a straightforward mean, total tunnage divided by
population -- drinks more beer than the average New Yorker).  Notice that
the ambiguity of the English is serious since the last two cases very
likely have different truth values; even if the average Chicagoan drinks a
lot more beer than the average New Yorker, there are enough more New
Yorkers to make their collective drinking more.
        Lojban disambiguates (that is a large part of what Lojban is
about, after all) this situation by using at least four different forms: a
descriptor for sets, quantifier + variable + poi + predicate for
distributions, mass descriptions for collectives, and descriptor _lo'e_
for statistical claims (without, of course, claiming that the survey has
actually been done.  I sometimes wonder whether, if the absen ce of the
survey becomes too severe whether we ought not shift to _le'e_(?), "what I
take to be typical,"  but I guess that is meant to be "typical of what I
take to be" instead). As a result, _lo'e_broda_ has (as lojbab has said
often) all the essential properties of a broda and all the other relevant
properties in middling degrees (mean for most numerical ones, median for
scalars, modal for the rest - -- probably with some local corrections,
especially where different types interact, e.g., income and wealth).  For
each class, the _lo'e_ is unique but not, generally, concrete, not, in
fact, a member of the set.  However, it may be, as a sumti, the value of a
bound variable (but it does not have to be: we can restrict our universe
of discourse to just to the members of the set and still use _lo'e_ as a
convenient manner of speaking, an abbreviation for a very long
discussion).  If the _lo'e_ remna_ is allowed into the universe, than it
seems picky to keep its head out, since it does, of course, have exactly
one head.  Notice that this head need not be _lo'e_stedu_ or even
_lo'e_remna_stedu_ and, of course, need not be the head of any member of
the set of remna -- it is a less clearly defined statistical abstraction.
Indeed, most philosophically inclined discssors of this issue have said
that _lo'e_ constructions have no meaning in isolation but only as part of
the whole sentence in which they occur (an argument against having them in
the universe) and _lo_stedu_be_lo'e_remna_ probably should share in that
contextualization (maybe what xorxes means by insisting that the head has
to be a _lo'e_ sort of thing, too, since it is not strictly a _lo'e_, as
noted above).  It might indeed be best not to get even that much of
concession and say simply _lo'e_remna_cu_pamei_ se_stedu_ rather than
introducing sumti at all.

C. The relation between descriptors and quantifiers (though I cannot now
find this one to get the exact context).  In the interesting sense, none.
In logic, descriptors refer to monads, singular and atomic, so neither
multiple nor fractional quantifiers make sense in the structure
(quantifier)(description).  The sentence that a descriptor converts into a
term may contain quantifiers or it may contain variables bound by
quantifiers outside the description.  One descriptor (of the four or so
that have some frequency in logic) is totally definable in terms of
quantifiers, a mere abbreviation, and another can be used to define the
standard quantifiers ("all" and "some") completely.  But the sort of thing
we want to do with the external quantifiers on descriptions in Lojban is
not feasible in that way in logic.
        The corresponding move in logic (or the nearest thing to it)
requires that the descriptors, regardless of being logical monads, refer
to sets or masses or whaever (preferably by an explicit metapredicate in
the description) and then need an appropriate predicate of constituency:
"is a member of", "is a subset of", "is a component of", "is a submass of"
and so on.  The quantifier in question would then bind the subject term of
this association -- as well as what is to be said about this thing -- and
the descriptor would fall into the second place.  If the quantification
gave a new non-monadic form, a second quantifier-and-predicate might be
needed.  The results is similar in spirit, if not in form, to the
complexes which xorxes and and toss back and forth.  I skip over the exact
form that various Lojban sentences might take because I was hard pressed
to find one where there was agreement what it meant in the relevant ways.

        Nobody asked, but.  Remember that quantifiers are about the
universe of discourse and _zaste_ is about (relative) reality.  The two
need bear no relation to one another.  You can leave the default reality
but talk only about Narnia or you can set reality on Narnia but talk only
about the things in the world around you.  In both of these cases, nothing
exists: _ro_da_nalzaste_ is true.  The usual situation is that there is
some overlap between reality and universe of discourse, so that
_da_nalzaste_ is true but so is _da_zaste_.  To be sure, in science we
want to be sure that our universe is reality or a subpart of it, at least
that the universe does not include embarassing things from outside reality
that defy the laws of reality.
pc>|83