TECH: Properties problems solved?

[John Cowan <cowan@SNARK.THYRSUS.COM>]

Having read the comments of Nick, Bob Chassell, lojbab/Nora, Gerald Koenig,
and Jim Carter (who is the Mark Hubey of the Loglan Project :-) :-) ),
and having reread Quine (>Methods of Logic<, 4th ed.) some light begins
to dawn in my fevered brain.

There seem to be two main differences in form between standard logic and
logic a la Lojban.  One has been brought up peripherally, and I think I
will knock it off before proceeding to the meat of this posting.  In standard
logic notation, quantifiers have short scope:  in "(Ex) Fx . Gx", the scope of
the binding of "x" is "Fx" only.  In Lojban, however, the scope extends to
the next prenex binding the same variable; or the next cmavo that unbinds
quantifiers, either "ni'o" or "da'o"; or the "tu'u" (explicit or implicit) that
terminates a prenexed "tu'e...tu'u" group.

A consequence of this view is that Lojban is a "fully alphatized" language;
that is, there can be no rebinding of variables in inner scopes that have
already been used in outer scopes.  Every scope starts with the prenex
that contains it and extends to the end-of-all-scopes point.
Each variable in such a set of nested scopes must have a distinct name.

The other, more important, difference is that in standard logic variables
are not implicitly quantified.  In Lojban logic, "Fx" means the same as
"(Ex) Fx"; that is, "da prenu" is the same as "su'o da zo'u da prenu".
What, then, is the meaning of "Fx" in standard logic?  It is what Quine
calls an "open sentence", and others have called a "propositional function".
It is neither true nor false, but rather true >of< something.  Thus, if
"F" is short for "is a baseball", then "Fx" is true if x is a baseball.
This is precisely what I was groping for when I talked of "certain abstract
objects, namely those which can be applied to objects to produce truth
values."

I therefore am now in a position to propose a resolution of the property
problem by convention.  Let us hold that any da-series variables which
first appear within a "ka...kei" property abstractions are taken to be
free variables in the logician's sense; they are not quantified, but serve
as mere placeholders.  Thus, the translation of "Fx" is now
"le ka da broda", or "de" if "da" is already in use, etc.  Now we can
distinguish clearly between:

1)      la djan. zmadu la djordj. le ka da prami mi
        John exceeds George in-the property-of (x loves me).
        John loves me more than George loves me.

and

2)      la djan. zmadu la djordj. le ka mi prami da
        John exceeds George in-the property-of (I love x).
        I love John more than I love George.

Since "da" can be ellipsized like any other sumti, it is legal to omit it
and in these contexts will probably cause no confusion.  However, oddities
like:

3)      le ka [zo'e] binxo lo bitmu da
        the property-of ( [something] becomes a-wall under-conditions x)
        the property of being the conditions under which something not
                specified, but understood from context, becomes a wall

can be clearly and precisely distinguished from the more normal

4)      le ka [da] binxo lo bitmu [zo'e]
        the property-of ( x becomes a wall [under-some-conditions] )
        the property of becoming a wall (under some unspecified conditions).

I believe that using this convention eliminates any need for a "kau" suffix
here, leaving "kau" to its original role for indicating indirect questions;
it seems clear that this also satisfies Nora's desire for a conservative
solution, being a mere change of interpretation in (presumably rare)
sentences with explicit unbound "da/de/di" within "ka...nei".

I think, but I am not sure, that "ka...nei" without unbound "da" continues
to provide what JCB, Bob Chassell, Gerald Koenig (insofar as I understand
him), and Jim Carter want as regards "essentialism".  And of course, Lojban
remains an elliptical rather than an auto-replicating language, where the
values of missing sumti must be glorked from context.

--
John Cowan      cowan@snark.thyrsus.com         ...!uunet!lock60!snark!cowan
                        e'osai ko sarji la lojban.