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TECH: lambda and "ka" revisited
After reading Quine's >Word and Object< a(over pc's strenuous objections)
and talking to pc on the phone, I now believe I understand what "ka"
was menat to be about and how it can be intrepreted, and why a general
"lambda" mechanism is needed.
The objects described by "le ka ... kei" expressions in Lojban are
essentially those described in W&O \S 34 as intensions. (Note the "s".)
For example, "le ka gerku" is "being a dog", or doghood; "le ka cukta" is
"being a book", or bookness. Both of these can be though of as monadic
intensions, what are traditionally called attributes. Brotherhood, though,
is a dyadic intension, traditionally called a relation (not to be confused
with the mathematician's sense of "relation" as "set of ordered pairs.")
In general, we have no problem understanding these "le ka"
constructs: the relation "brotherhood" is "le ka bruna" and the related
attribute "being a brother to John/brotherhood with John" is "le ka bruna la
djan." If a sumti is left out, it is one of those that is used to
specify the parameters of the intension.
But consider "le ka gerku" again. In fact, "gerku" is a 2-place selbri
in Lojban! So must we say that "le ka gerku" is the relation between
a dog and its breed? By introducing a specific method of filling the
place which cannot be confused with "zo'e" (simple omission) we can distinguish
the cases of "doghood" and "dog-to-breed relationship".
Higher numbers of places cause worse problems. Thus for "dunda" we have
the attribute of being a giver, the attribute of being a receiver of a gift,
the attribute of being a gift, the relation between giver and receiver
(for some specific gift), the relation between giver and gift (for some
specific receiver), the relation between receiver and gift (for some specific
giver), and the triadic relation among all three, for seven cases in all!
"le ka dunda" can mean whichever of these is wanted, for "zo'e" is blessedly
unspecified, but there must be some way of telling them apart.
Quine's notation is "x[x is a dog]" and "xy[x is a dog of breed y]". Here
x and y are regular logical variables, and this construct binds them.
That would lead to something like:
1) le ka xe'u da xe'u de zo'u da gerku de
which is dreadfully verbose. pc thought that:
2) le ka xe'u da de zo'u da gerku de
would be better, or even
3) le ka da de xe'u da gerku de
where "xe'u" is of selma'o ZOhU. (In Example 1 it belongs to PA, and
in Example 2 to a new selma'o -- this requires a small grammar change.)
The corresponding cases for "x[x is a dog] are just:
4) le ka xe'u da zo'u da gerku [zo'e]
5) le ka xe'u da zo'u da gerku [zo'e]
6) le ka da xe'u da gerku [zo'e]
Jorge has proposed overloading "ke'a", but Lojban Central doesn't like
that, because the rules for subscripting become harder to keep track of
when you have abstractions within relative clauses within abstractions,
so we believe a new cmavo is required.
Note that Examples 1 and 4 can be shortened to:
7) le ka xe'u da gerku xe'u de
8) le ka xe'u da gerku [zo'e]
and of course there's no problem with leaving out "xe'u da/de" whenever
context makes it clear what's wanted, but context can't do the whole
9) mi ckaji le ka dunda
is fine if I have the property of being a giver, but
10) mi ckaji le ka se dunda
11) mi ckaji le ka dunda xe'u da
is downright false!
We can't just have a new variable "xe'u" as a sumti by itself, because
the bound variable may need recycling:
12) do ckaji le ka ga xe'u da xirma gi lo xirma cu citka da
You have-as-property the attribute-of
either (you are-a-horse) or (some horse(s) eat you)
You are either a horse or horse-fodder.
(Of course, the colloquial of #12 can be rendered without dragging in
properties, just using a general term, but it illustrates the point:
the attribute "being either a horse or eaten by a horse" is as legitimate
as any other.)
A few trailing notes: Quine wants to equate properties with sets for
his limited purposes, but even he points out that that won't do for
idioms of propositional attitude: the set of animals with hearts is
equal to the set of animals with kidneys, but the property "having a heart"
is not the same as the property "having a kidney".
Quine also says that propositions are 0-adic intensions, but in Lojban
we use a different abstractor: "du'u". This allows the default assumption
to be that "ka" denotes a monadic intension even without "xe'u da"
explicitly appearing anywhere (it would tend to be assumed in the x1
place), leaving "du'u" for the 0-adic case. Further, "du'u" provides an
x2 place (of the abstraction, not the abstract bridi) which is a textual
representation: it relates propositions to sentences.
So my recommendation is to add "xe'u" as a new kind of PA, semantically
only used by itself to quantify da-series logical variables. It could
also be used in the form "xe'u broda" if my proposal that "PA broda"
= "PA da poi broda" is accepted, to make implicit dummy variables of restricted
13) le ka xe'u da bruna xe'u nakni
is the relationship "brotherhood" in the narrow sense, where each
relatand is constrained to being male: in
14) le ka xe'u da bruna xe'u de
we just have the relationship between a male and his sibling of either sex.
Comments urgently requested!
John Cowan email@example.com
e'osai ko sarji la lojban.