Alas, &= ;, I think, still wants to have a language. Combinatory logic and its= ilk (all the other versions of computable functions) notoriously don't giv= e languages that one can speak. So there is an added constraint and o= ne which, I think, pretty much means that the fullest form of an &lang = isn't possible. As long as there is only one term, there are no= (well, very few and easily surmounted) problems. But the second term= , maybe even if it can be specified from the first, does create inevitable = discontinuities in a linear structure, and there is no obvious way to bridg= e those gaps but by anaphoric forms of some sort. As noted, the fact = that the structures can be set up to give each term a canonical introductio= n does simplify these repetitions by avoiding needing to refer to buried occurrences in some way. On the other hand, talking about com= binatory logic and the like does call to mind languages without terms at al= l, only predicates, which are feasible (and might actually be useful for te= sting SWH), but are a long way from FOL and the clarity that is said to res= ide there (so far as I know --I've never seen anyone work on the issue).

From: .arpis. <rpglover64+jbobau@gmail.com>
= To: lojban@googlegroups.co= m
Sent: Friday, July = 13, 2012 1:05 PM
Subject:<= /b> Re: [lojban] &Lang

O= n Fri, Jul 13, 2012 at 10:31 AM, And Rosta <and.rosta@gmail.com> wrote:
.arpis., On 13/07/2012 00:59:

I'm not sure I understand the predicate-argument structure well
enough to do so. Could you give an example of how it would look with
normal variables and explain the meaning of the example?

An example (without wanting to be distracted by irrelevant details):

"Da djuno lo du'u de prami da gi'e de nelci da"
djuno(da, &(prami(de, da), nelci(de, da)))

Task 1: notate this (linearly) so that "da" and "de" each appear only once.=

And then:

Task 2: Improve the system so that speakers don't have to hold in memory th= e arbitrary correspondence between variable & phonological form.

Will De Bruijn indices deal with Task 1? If they do, I'd be thrilled to hav= e learnt summat new.

Ah... No. De Bruijn indices d= o nothing about factoring out common use sites. You might want to look into= the SKI (or SK) calculus; it is capable of expressing everything that the = lambda calculus is.

What you really want is to define a predicate out of djuno, du'u, prami= , and nelci that has the right meaning and never use "da" or "de" at all, b= ecause if they appear only once and in obvious positions, they are superflu= ous.

Here's how I would go about deriving such a predicate, in case it helps= :

Recognize that \ko'a. \ko'e. gi'e (prami ko'a ko'e) (nelci ko'a ko= 'e) is a useful abstraction
This abstraction can be generalized to
\conn. \b1. \b2. \ko'a. \ko'e. conn (b1 ko'a ko'e) (b2 ko'a ko'e) which red= uces to SK as follows
\conn. \b1. \b2. \ko'a. \ko'e. (conn (b1 ko'a ko'e))    = ;            &n= bsp;            = ;    (b2 ko'a ko'e)
\conn. \b1. \b2. \ko'a. S (\ko'e. conn (b1 ko'a ko'e))   &nb= sp;            =    (\ko'e. (b2 ko'a) ko'e)
\conn. \b1. \b2. \ko'a. S (S (\ko'e. conn) (\ko'e. (b1 ko'a) ko'e)) &n= bsp;          (b2 ko'a)
\co= nn. \b1. \b2. \ko'a. S (S (K conn) (b1 ko'a))   &nb= sp; &= nbsp;       (b2 ko'a)

\conn. \b1. \b2. \ko'a. (S (S (K conn) (b1 ko'a)))   &nb= sp;            =             &nb= sp; (b2 ko'a)
\conn. \b1. \b2. S (\ko'a. S (S (K conn) (b1 ko'a)))    = ;             (= \ko'a. b2 ko'a)
\conn. \b1. \b2. S (S (\ko'a. S) (\ko'a. S (K conn) (b1 ko'a)))  =           b2
\conn. \b1. \b2. S (S S (S (\ko'a. S (K conn)) (\ko'a. b1 ko'a))) &nbs= p;       b2
\conn. \b1. \b2. S (S S (S (S (\ko'a. S) (\ko'a. K conn)) b1)) = ;        b2
\conn. \b1. \b2. S (S S S S S (K conn) b1) b2

And now I'm bored, but= I think that this was illustrative.
Other things you may want to look a= t: combinatory logic, concatenative programming (or stack-based programming= ), point-free style (or tacit programming), and the J programming language.=

--And.

On Thu, Jul 12, 2012 at 6:44 PM, And Rosta <and.rosta@gmail.com <mailto:and.rosta@gmail.com>> wrote:

.arpis., On 12/07/2012 23:09:

The common way (speaking as a computer scientis= t) of avoiding variable repetition (and variable naming issues) in formal m= ethods is to use De Bruijn indices (https://en.wikipedia.org/ wiki/De_Bru= ijn_index <https://en.wikipedia.org/wiki/De_Bru= ijn_index>), but this fails the requirement of fitting in human memo= ry.

Can you give an example of how it would work if applied to a = predicate--argument structure? I can't follow the wikipedia article enough = to see how it would work (-- setting aside the memory limitation problem yo= u mention).

On Thu, Jul 12, 2012 at 2:31 PM, And Rosta <= and.rosta@gmail.com <mailto:and.rosta@gmail.com> <mailto:= and.rosta@gmail.com <mailto:and.rosta@gmail.com>>> wro= te:

Let me try harder to explai= n:

Take a variety of predicate= logic consisting of predicates, quantifiers and variables. In normal predi= cate logic notation, if a given variable is an argument of more than one pr= edicate, it gets repeated. And every bound variable is notated at least twi= ce, once where it is shown bound by quantifier and once where it is argumen= t of a predicate. But this is one and the same variable; the repetition is = a mere notational device necessary to linearize the string.

If the notation could be tw= o dimensional, then you wouldn't need to write variables at all. For simpli= city's sake, I'll describe a notation for only predicates and variables: Use a 2 dimensional grid, i= nfinite in both dimensions.
Each 'row' corresponds to a= variable.
Each 'column' corresponds t= o a predicate.
Predicates are notated by s= ets of symbols, one symbol for each argument place. Each argument place sym= bol is placed on the appropriate row for the variable that fills the argume= nt place.

That's the basic data struc= ture for predicate--argument structure. A 2-dimensional notation can notate= it without redundancy. But spoken language is 1-dimensional. Is there a wa= y of linearizing predicate-argument structure ergonomically, in such a way = that it is not so verbose or so taxing on the memory that the advantages of= its logical explicitness and unambiguousness are not outweighed?

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