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Re: [lojban] Individuals and xorlo



nice!


On Monday, May 26, 2014 10:01 AM, guskant <gusni.kantu@gmail.com> wrote:




Le lundi 26 mai 2014 04:49:09 UTC+9, Martin Bays a écrit :
* Monday, 2014-05-19 at 06:04 -0700 - guskant <gusni...@gmail.com>:

> Le mardi 8 avril 2014 10:09:19 UTC+9, guskant a écrit :
> I have finished English translation of my commentary on gadri from a
> logical point of view:
> http://www.lojban.org/tiki/ gadri%3A+an+unofficial+ commentary+from+a+logical+ point+of+view&no_bl=y
>
> Any comments or questions will be appreciated.

Not that I really want to drag myself into discussing these things
again, but I happened to see this and thought I should recall
a complication. Apologies if I'm repeating something hidden in this
thread.

Saying that {zo'e} and {lo broda} introduce "constants" isn't really
enough to explain how they work, because of cases where a description
includes a bound variable, e.g.
    {ro da poi verba cu prami lo rirni be da} .

If I recall correctly, there was the other year some agreement that {lo
rirni be da} there should be taken to introduce a contextually specified
function from children to parents (like a Skolem function), which in
some sense salvages the idea of constancy (it's constantly
a non-constant function!). But anyway, I think it's something that needs
to be explained in any theory of gadri.

I note that the "double circle" example in the text
    su'o da zo'u loi re lo'i ro mokca noi sepli py noi mokca ku'o da cu
        relcuktai
    Two sets of points that are equidistant from a point P is a double
        circle.
is of this form, but doesn't give the intended meaning under the above
interpretation (or any other that I can think of).

Martin

Thank you for the question. Here is my answer. I will add this topic to the commentary.

Generally, all {zo'e} in a statement that contains one or more bound variable(s), no matter if they are explicit or not, must be Skolem functions. If they were not, the official interpretation (CLL 7.7) of implicit {zo'e} should have been modified.

For example, we may freely say:

S1- {ro mlatu cu jbena}.

According to CLL 7.7, it has the same meaning as

S2- {ro mlatu cu jbena zo'e zo'e zo'e}.
(I omit x2 of {mlatu} for simplicity.)

Unless all cats in this universe of discourse were born to common parents at the same time at the same place, these {zo'e} are not constants but Skolem functions f(x) g(x) h(x) respectively:

S3- {roda zo'u ganai da mlatu gi da jbena zo'e zo'e zo'e},
that is
Ax ~M(x) v J(x,f(x),g(x),h(x)),
where x corresponds to {da}, and is a singular variable bound by a universal quantifier A,
~ is negation,
v is OR,
M and J are predicates.

S3 is a Skolemized form of a statement

S4- {roda su'oidexipa su'oidexire su'oidexici zo'u 
ganai da mlatu gi da jbena dexipa dexire dexici}, 
that is
Ax EY1 EY2 EY3 ~M(x) v J(x,Y1,Y2,Y3),
where Y1 Y2 Y3 are plural variables bound by existential quantifiers E.

In Skolemizing S4 into S3, {su'oidexipa}, {su'oidexire} and {su'oidexici} of S4 are replaced by {zo'e}s that are respectively equal to f(x), g(x) and h(x). If {zo'e} were not Skolem functions, we should have abandoned the interpretation "S1 = S2" so that the omitted sumti could have been bound variables. (It would not be the case if we accepted the idea in "Section 4.3.1. If zo'e could be a bound plural variable" of my commentary, but it is another story.)

If we want to make explicit that a Skolem function {zo'e} is a Skolem plural constant (that is, the referent of {zo'e} does not vary according to {da}), we should say the corresponding plural variable earlier than {roda} in the prenex of the statement before Skolemization.
For example, in order to mean that {zo'e} at x4 of {jbena} refers to the Earth that is common to all cats, the statement before Skolemization should be

S5- {su'oidexici roda su'oidexipa su'oidexire zo'u
ganai da mlatu gi da jbena dexipa dexire dexici},
that is
EY3 Ax EY1 EY2 ~M(x) v J(x,Y1,Y2,Y3).

By skolemizing S5, we obtain a statement that is
S6.1- Ax ~M(x) v J(x,f(x),g(x),h),
where h is a Skolem plural constant: h does not depend on x because EY3 of S5 was said earlier than Ax in the prenex. 

Lojban _expression_ of S6.1 might not officially be explained, but I would profit the property that Lojban prenex can include constants:

S6- {cy zo'u ro mlatu cu jbena fo cy},
which is the same as
{cy roda poi mlatu zo'u da jbena fo cy}
and
{cy roda zo'u ganai da mlatu gi da jbena fo cy}.

In S6, I used {cy} instead of {zo'e} for the constant, otherwise we could not distinguish which {zo'e} was on the prenex.


Although it might be off-topic, the following thread on the order of tagged sumti and its scope suggests me of an idea:
https://groups.google.com/d/topic/lojban/PhZD1fO64jc/discussion

I suggest that not only the scope of tagged sumti but also that of terbri sumti reflects their order. For example, I suggest considering that S6.1 and S6 are the same as 

S7- {fo cy fa ro mlatu cu jbena}.




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