Re: pc answers

[Gerald Koenig <jlk@netcom.com>]

Jorge has convinced me that his translation of:

ci remna ku so gerku zo'u ra pencu ri, as:

For each of exactly three humans, there are 9 dogs that the human
touches. ( Not necessarily the same ones for each human. )

is quite correct using the existing lojban grammar rules.
This is the case where there are up to 27 dogs.

My predicate calculus formula:

E^!3(x) (remna (x)  E^!9(y)(gerku(y)  & pencu(x,y)))

declares that there are exactly 3 humans and exactly 9 dogs, for the
scope of the entire sentence.  How to say it in lojban?  I get:

ro lo ci remna ku ro lo ci gerku zo'u ra pencu ri

These sentences don't have to be in topic-comment form. They give the
desired two distinct interpretions of <ci nanmu pencu ci gerku>, a 9 dog
and a 27 dog interpretation.

1. ro lo ci nanmu cu pencu ro lo ci gerku,  (9 dogs) and

2. ci lo nanmu cu pencu ci lo gerku, (27 dogs)  which is the same as

ci nanmu cu pencu ci gerku,   by the quirk in the grammar.

An interesting point to me is that merely by exchanging ci and lo in
front of a gismu we are changing the underlying number explication from
the standard predicate calculus formulation to the lojban system of
selection from a larger set defined by the predicate. Or at least that
is my assumption. I assume that in the case of an exact numerical claim
with exactly N objects in the universe of discourse, there is no call
to make a selection from a larger set defined by the predicate. To
select N objects from N objects seems nonsensical to me.  Perhaps the
FOL (first order logic) system of making numerical claims has been
abandoned altogether. In that case there may be no particular
relationship between lojban and logic, it is simply a new language with
other interesting features.

It was believed prior to about 1900 that there were no problems with
the intuitive idea that there is a set defined by every predicate
asserting a property, such as gerku(x). This assumption led to
Russell's paradox and the need for new foundations of set theory on an
axiomatic basis. Sooner or later I would guess that we will have to
deal with problems arising from this same assumption, which is built
into the lojban number grammar.



>From the sumti paper:
______________________________________________________________
Using exact numbers as inner quantifiers in lo-series descriptions is
dangerous, because you are stating that exactly that many things exist
which really fit the description.  So examples like

7.7)    [su'o] lo ci gerku cu blabi
        [some-of] those-which-really-are three dogs are-white

are semantically anomalous; Example 7.7 claims that some dog (or dogs) is
white, but also that there are just three dogs in the universe!

Nevertheless, inner quantifiers are permitted on "lo" descriptors for
consistency's sake, and may occasionally be useful.


8. Indefinite Descriptions

By a quirk of Lojban syntax, it is possible to omit the descriptor "lo"
from a description like that of Example 7.5; namely, one which has an
explicit outer quantifier but no explicit inner quantifier.  The following
example:

8.1)    ci gerku [ku] cu blabi
        Three dogs are white.

is exactly equivalent in meaning to Example 7.5.  Even though the descriptor
is not present, the elidable terminator "ku" may still be used.
___________________________

djer