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From: Jacob Errington <nictytan@gmail.com>
Date: Fri, 21 Sep 2012 00:56:19 -0400
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Subject: [lojban] The jai-zenba distribution property
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coi

This post probably won't interest that many, but I felt like the broader
community be aware of this. This post makes use of the "functional"
interpretation of Lojban, where a lot of {nu} places are {ka} places, where
{ka} denotes a n-arity function (where n is the number of ce'u in the
function) that "becomes" a {du'u} abstraction (or possibly a {nu}
abstraction, which makes the whole concept more useful).

Today in IRC, latro'a mentioned having said something that struck us a
being rather intriguing:
#1 {.i .ei mi jai zenba lo ni ce'u cafne kei fai lo nu te vecnu zo'e}
(I personally believe that the {nu} there should be a {ka} by virtue of
{jai}, but I digress, and that really isn't the point of this post.)

By the definition of {jai}, #1 is equivalent to
#2 {.i .ei lo nu mi te vecnu zo'e cu zenba lo ni ce'u cafne}.
So far so good.

Assuming the functional definition of {zenba}, {lo ka ce'u zenba lo ni ce'u
broda cu ka ce'u goi ko'a zo'u lo ni ko'a broda cu zenba dubu} (where
{dubu} refers to the identity function {lo ka ce'u du makau}), we can
rewrite #2 as
#3 {.i .ei lo ni lo nu mi te vecnu zo'e cu cafne cu zenba dubu}

We can now re-apply a jai to the bridi {lo nu mi te vecnu zo'e cu cafne}
forming {mi jai cafne lo nu te vecnu}, such that
#4 {.i .ei lo ni mi jai cafne fai lo nu te vecnu cu zenba dubu}

Finally, we can undo the operation performed in 3, moving {lo ni mi jai
cafne fai lo nu te vecnu} into zenba2, extracting {mi} by replacing with
{ce'u}, and moving {mi} into zenba1, eliminating dubu, yielding
#5 {.i .ei mi zenba lo ni ce'u jai cafne fai lo nu te vecnu}

Formally: {.i ko'a zo'u lo ka ce'u zenba lo ni ce'u jai broda fai ko'a cu
ka ce'u jai zenba lo ni ce'u broda kei fai ko'a}

This same rule can apply to comparatives.
e.g. {.i mi do sejaisejai zmadu fai ko'a fai ko'a lo ni ce'u cafne} === {.i
mi do zmadu lo ni ce'u jai cafne fai ko'a} where ko'a can be for example,
something like {lo ka tavla fo la. lojban.}
Note on how I use multiple {jai}: when using two {jai} on the same level,
the first fai to appear in the bridi fills the jai closest to the selbri.

In conclusion, this means that #1 and #5 are the same in trivial cases like
this, where logical connection or bound variables aren't involved.

.i mi'e la tsani mu'o

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coi<div><br></div><div>This post probably won&#39;t interest that many, but=
 I felt like the broader community be aware of this. This post makes use of=
 the &quot;functional&quot; interpretation of Lojban, where a lot of {nu} p=
laces are {ka} places, where {ka} denotes a n-arity function (where n is th=
e number of ce&#39;u in the function) that &quot;becomes&quot; a {du&#39;u}=
 abstraction (or possibly a {nu} abstraction, which makes the whole concept=
 more useful).</div>

<div><br></div><div>Today in IRC, latro&#39;a mentioned having said somethi=
ng that struck us a being rather intriguing:</div><div>#1 {.i .ei mi jai ze=
nba lo ni ce&#39;u cafne kei fai lo nu te vecnu zo&#39;e}</div><div>(I pers=
onally believe that the {nu} there should be a {ka} by virtue of {jai}, but=
 I digress, and that really isn&#39;t the point of this post.)</div>

<div><br></div><div>By the definition of {jai}, #1 is equivalent to=A0</div=
><div>#2 {.i .ei lo nu mi te vecnu zo&#39;e cu zenba lo ni ce&#39;u cafne}.=
=A0</div><div>So far so good.</div><div><br></div><div>Assuming the functio=
nal definition of {zenba}, {lo ka ce&#39;u zenba lo ni ce&#39;u broda cu ka=
 ce&#39;u goi ko&#39;a zo&#39;u lo ni ko&#39;a broda cu zenba dubu} (where =
{dubu} refers to the identity function {lo ka ce&#39;u du makau}), we can r=
ewrite #2 as</div>

<div>#3 {.i .ei lo ni lo nu mi te vecnu zo&#39;e cu cafne cu zenba dubu}</d=
iv><div><br></div><div>We can now re-apply a jai to the bridi {lo nu mi te =
vecnu zo&#39;e cu cafne} forming {mi jai cafne lo nu te vecnu}, such that</=
div>

<div>#4 {.i .ei lo ni mi jai cafne fai lo nu te vecnu cu zenba dubu}</div><=
div><br></div><div>Finally, we can undo the operation performed in 3, movin=
g {lo ni mi jai cafne fai lo nu te vecnu} into zenba2, extracting {mi} by r=
eplacing with {ce&#39;u}, and moving {mi} into zenba1, eliminating dubu, yi=
elding</div>

<div>#5 {.i .ei mi zenba lo ni ce&#39;u jai cafne fai lo nu te vecnu}</div>=
<div><br></div><div>Formally: {.i ko&#39;a zo&#39;u lo ka ce&#39;u zenba lo=
 ni ce&#39;u jai broda fai ko&#39;a cu ka ce&#39;u jai zenba lo ni ce&#39;u=
 broda kei fai ko&#39;a}</div>

<div><br></div><div>This same rule can apply to comparatives.=A0</div><div>=
e.g. {.i mi do sejaisejai zmadu fai ko&#39;a fai ko&#39;a lo ni ce&#39;u ca=
fne} =3D=3D=3D {.i mi do zmadu lo ni ce&#39;u jai cafne fai ko&#39;a} where=
 ko&#39;a can be for example, something like {lo ka tavla fo la. lojban.}</=
div>

<div>Note on how I use multiple {jai}: when using two {jai} on the same lev=
el, the first fai to appear in the bridi fills the jai closest to the selbr=
i.</div><div><br></div><div>In conclusion, this means that #1 and #5 are th=
e same in trivial cases like this, where logical connection or bound variab=
les aren&#39;t involved.
</div><div><br></div><div>.i mi&#39;e la tsani mu&#39;o</div><div><br></div=
>

<p></p>

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