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Date: Wed, 28 May 2014 10:14:46 -0700
From: Martin Bays <mbays@sdf.org>
To: lojban@googlegroups.com
Cc: guskant <gusni.kantu@gmail.com>
Subject: Re: [lojban] Individuals and xorlo
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* Tuesday, 2014-05-27 at 19:21 -0700 - guskant <gusni.kantu@gmail.com>:

> Le mercredi 28 mai 2014 05:42:53 UTC+9, Martin Bays a =E9crit :
> > * Monday, 2014-05-26 at 23:12 -0700 - guskant <gusni...@gmail.com<javas=
cript:>>:=20
> > > Le mardi 27 mai 2014 11:53:50 UTC+9, Martin Bays a =E9crit :=20
> > > > * Monday, 2014-05-26 at 08:01 -0700 - guskant <gusni...@gmail.com<j=
avascript:>>:=20
> > > > > Le lundi 26 mai 2014 04:49:09 UTC+9, Martin Bays a =E9crit :=20
> > > > > > * Monday, 2014-05-19 at 06:04 -0700 - guskant <gusni...@gmail.c=
om<javascript:>>:=20
> > > > > > > Le mardi 8 avril 2014 10:09:19 UTC+9, guskant a =E9crit :=20
> > > > > > > > http://www.lojban.org/tiki/gadri%3A+an+unofficial+commentar=
y+from+a+logical+point+of+view&no_bl=3Dy=20
> > > > > S1- {ro mlatu cu jbena}.=20
> > > > > S2- {ro mlatu cu jbena zo'e zo'e zo'e}.=20
> > > > > S3- {roda zo'u ganai da mlatu gi da jbena zo'e zo'e zo'e},=20
> > > > > Ax ~M(x) v J(x,f(x),g(x),h(x)),=20
> > > > > S4- {roda su'oidexipa su'oidexire su'oidexici zo'u=20
> > > > > ganai da mlatu gi da jbena dexipa dexire dexici},=20
> > > > > Ax EY1 EY2 EY3 ~M(x) v J(x,Y1,Y2,Y3),=20
> > > > >
> > > > > 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.
> > > The interpretation of {zo'e} as Skolem function rather reinforces xor=
lo,=20
> > > and makes clear that the CLL-interpretation of gadri is problematic.=
=20
> > Although I don't actually consider myself qualified to pronounce on wha=
t=20
> > xorlo is, my understanding is that the intention and common=20
> > understanding of xorlo have {lo} and {zo'e} constant in the sense of=20
> > being outside the scope of any quantifier, except when absolutely force=
d=20
> > to be inside. So e.g. in {ro da broda lo brode}, the (plural) referent=
=20
> > of {lo brode} is constant with respect to {da} under xorlo, whereas it=
=20
> > is not in CLL-lojban.=20
>=20
> Regarding {zo'e} as the outmost constant in a prenex of a statement is a=
=20
> special case of {zo'e} as Skolem functions. As for the example
>=20
> {ro da broda lo brode},
> that is=20
> Ax B(x,f(x)),
>=20
> it says nothing about whether {lo brode} as a Skolem function f(x) is=20
> constant for all x or not. That is to say, xorlo allows both=20
> interpretations "EYAx B(x,Y)" and "AxEY B(x,Y)" as a statement before=20
> Skolemization, while CLL-lo restricts the interpretation to "AxEy B(x,y)"=
=20
> (small y is a singular variable).

Assuming I understand you correctly as wanting {lo broda se broda ro da}
to have only the "EYAx" interpretation, this is in direct conflict with
the gadri BPFK section, which says
"Any term without an explicit outer quantifier is a constant, i.e. not
a quantified term. This means that it refers to one or more individuals,
and changing the order in which the constant term appears with respect
to a negation or with respect to a quantified term will not change the
meaning of the sentence.".

> If xorlo did not allow this=20
> interpretation, CLL 7.7 must have been abandoned. As long as both xorlo a=
nd=20
> CLL 7.7 are kept true, a constant {zo'e} is not always out of bound=20
> variables.
>=20
> > > If a statement includes no universal quantifier after transformed int=
o=20
> > > prenex normal form, the statement can be Skolemized into a statement =
in=20
> > > which all Skolem functions are Skolem constants. xorlo can precisely=
=20
> > > express these constants. CLL-lo cannot.=20
> > >=20
> > > xorlo can make explicit the difference of meaning between S3 and S6.1=
=20
> > for=20
> > > any sumti in a simple way like S6. CLL-lo restricts the outer quantif=
ier=20
> > > according to sumti, and makes it difficult to express the difference =
of=20
> > > meaning between S3 and S6.1.=20
> > > > > S6- {cy zo'u ro mlatu cu jbena fo cy},=20
> > > > > S6.1- Ax ~M(x) v J(x,f(x),g(x),h),=20
> >
> > You're right, the semantics you're suggesting aren't really CLL-lo. But=
=20
> > they share the scope-sensitivity of CLL-lo; that's all I really meant.=
=20
>=20
> CLL-lo cannot express S3 precisely for the same reason above. S3 of xorlo=
=20
> says nothing about whether the Skolem functions f(x),g(x),h(x) are Skolem=
=20
> constants or not. In other words, S3 of xorlo does not say the order of=
=20
> bound variables of a statement before Skolemization. Regarding it as S4 i=
s=20
> the most general case. Any of the statements with prenex "EY2 Ax EY1 EY3"=
=20
> "EY1 EY2 Ax EY3" etc may be Skolemized into S3, because a Skolen function=
=20
> {zo'e} does not indicate whether it is a Skolem constant or not.
>=20
> On the other hand, according to CLL-lo, speaker must always select the=20
> order of Ax, EY1, EY2 and EY3 of a statement before Skolemization.

The statement "there exists a function f(x) such that for all x,
P(x,f(x))" is logically equivalent to "for all x, there exists y such
that P(x,y)". Now it might be that f is a constant function, which
corresponds to there being a uniform witness for the "exists y". The two
presentations are equivalent. So if Skolemisation were the only
difference between xor-lo and CLL-lo, there'd be no difference.

> > > {su'o da zo'u loi re lo'i ro mokca noi sepli py noi mokca ku'o da cu=
=20
> > > relcuktai},=20
> > >=20
> > > the quantifier in the prenex is not universal A but existential E: it=
 is=20
> > > not a Skolemized form.=20
> > > It is expressed in predicate logic as=20
> > >=20
> > > Ex R(m,p,x),=20
> > > where x is a singular variable bound by an existential quantifier E,=
=20
> > > R is a predicate,=20
> > > m and p are constants.=20
> > >=20
> > > Because this statement contains no other outer quantifier, it is a=20
> > prenex=20
> > > normal form that contains no universal quantifier. It is therefore=20
> > > Skolemized into=20
> > >=20
> > > {loi re lo'i ro mokca noi sepli py noi mokca ku'o zo'e cu relcuktai},=
=20
> > > that is=20
> > > R(m,p,z),=20
> > > where z is a Skolem constant.=20
> > >=20
> > > There is no problem for interpreting it as "two sets of points that a=
re=20
> > > equidistant from a point P is a double circle."=20
> >
> > But you seem to have jumped the existential through the {re} quantifier=
=2E=20
> > The radii are meant to be allowed to be different for the two circles,=
=20
> > but in the original sentence the radii are quantified with outermost=20
> > scope.=20
> >
> > I was also confused because the english reads like a definition, wherea=
s=20
> > the lojban has no hint of that (and I'm not sure that adding a {ca'e}=
=20
> > would do it).=20
>=20
> {re} in this example is an inner quantifier, and it does not affect the=
=20
> order of outer quantifier.=20

True.

> {zo'e} in the statement {loi re lo'i ro mokca noi sepli py noi mokca ku'o=
=20
> zo'e cu relcuktai} is a plural constant. Precisely saying, this {zo'e} is=
=20
> {lo re zo'e} in this context. This statement does not make clear if each=
=20
> individual of the referent of {zo'e} distributively satisfies {sepli}, bu=
t=20
> such an interpretation is allowed. I used rather a bound singular variabl=
e=20
> {su'oda} in the original example because I wanted to make explicit that t=
he=20
> radii distributively satisfy {sepli}.

I still don't see how to make sense of the original sentence. But never
mind.

Martin

> When I created the example, I did not consider Skolem functions, but
> if I wanted to make scopes of the arguments explicit, I should have
> been said
>=20
> {py lu'a loi re lo'i ro mokca su'o da lo'i ro mokca zo'u loi re lo'i ro=
=20
> mokca noi sepli py noi mokca ku'o da cu relcuktai},
>=20
> where I added {lu'a} in order to draw each of {lo se gunma} in the=20
> loi-sumti. This trick allows inner quantification to behave as if outer=
=20
> quantification in the prenex.
>=20
> However, I don't think such a precision by prenex is not necessary for an=
=20
> example of repeating inner quantification.
>=20
> As a summary, xorlo can express the scopes of arguments without outer=20
> quantifier unambiguously as well as ambiguously, while CLL-lo must always=
=20
> do unambiguously.=20
>=20
> If we take the interpretation like S7, the scopes of the outmost terbri=
=20
> sumti of a statement become unambiguous also in xorlo, though I think thi=
s=20
> idea should be at most a plausible interpretation, not a restriction. In=
=20
> general, there are many cases where the order of arguments out of prenex =
is=20
> restricted by grammar, like that example of relcuktai. xorlo allows Lojba=
n=20
> users to select the most likely interpretation among some possible ones,=
=20
> while CLL-lo definitely requires prenex even for such a simple example. T=
he=20
> idea of xorlo made the language closer to natural expressions, while it=
=20
> reserves also the unambiguity of logic in expressions with prenex.

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