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Date: Sat, 8 Nov 2014 19:46:32 -0500
From: John Cowan <cowan@mercury.ccil.org>
To: Ozymandias Haynes <ozymandias.haynes@gmail.com>
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Subject: Re: [bpfk] {ro}, existential import and De Morgan
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Ozymandias Haynes scripsit:

> The way that "All unicorns are white." is represented in predicate logic =
is=20
> with the formula $$ \forall x : [ U(x) \rightarrow W(x) ] $$. =20

This is precisely the point that pc (and following him, I) disputed.
This first-order predicate logic (FOPL) translation is *not* semantically
identical to the natural-language (NL) claim (which the Aristotelian
formulation follows), precisely because the FOPL version does not have
existential import (EI), whereas the NL version does.  If you ask someone
"Do all unicorns fly?"  they do not normally reply "Yes"; they either say
"No" or reject the question metalinguistically.

Pc and I hold that there is good reason to provide Lojban expressions
of both the FOPL and the NL versions of the claim, since they are
semantically distinct.  This can be easily done by saying that "ro da"
without a following "poi" (unrestricted quantification) takes the FOPL
interpretation, whereas "ro da poi broda" (restricted quantification)
takes the NL interpretation.  This does not in any way restrict FOPL,
since FOPL has *only* unrestricted variables, not restricted ones.  So it
would be easy to say that "ro" has EI in restricted quantifications,
and lacks EI in unrestricted ones.

Pc's further insight, however, is that it is essentially harmless to
extend "ro" to have EI in all cases.  Given the sentence, "ro da zo'u
ganai da broda gi da brode", it is obvious that this does not entail
"da broda", since it is under negation, and negated claims can never
have EI.  However, it is safe to replace "ro da" with "so'u da", *except*
in the case of an entirely empty universe.  If we are willing to give
up the desire to make vacuous universal claims about empty universes,
we have no trouble taking "ro" to always have EI.

When I first heard this argument, I didn't accept it either.  It took pc
about an hour of intensive two-way conversation to convince me that this
view is both self-consistent and consistent with FOPL-as-we-know-it (apart
from empty universes), so I don't expect you to swallow it as a result of
a brief email.  Nevertheless, however counterintuitive to people who know
FOPL, it is I believe sound, and has desirable properties for ordinary
NL statements, while in no way inhibiting properly formulated FOPL Lojban.

--
John Cowan          http://www.ccil.org/~cowan        cowan@ccil.org
My confusion is rapidly waxing

> between the dollar signs is LaTeX markup; if you can't read it you can pl=
ug=20
> it into an online renderer. \forall is the universal quantifier, x is the=
=20
> bound variable, \rightarrow is implication, and U and W are functions=20
> corresponding to 'x is a unicorn' and 'x is white' resp.).  As John says,=
=20
> one way to translate this into Lojban is "ro da zo'u ganai da pavyseljirn=
a=20
> gi da blabi".  This is irrelevant to la mukti's construction, however.  H=
e=20
> did not use that Lojban sentence in his example, he used one that's=20
> formally equivalent to da with poi.  The negation theorem is stated in it=
s=20
> full generality in the CLL and not only on sentences of the form above.=
=20
>  Indeed, using that implication form as a definition of "ro da poi X" is=
=20
> precisely what is needed to fit with the negation theorem and with=20
> predicate logic, and those are precisely the semantics that I am advocati=
ng.
>=20
> It=E2=80=99s easy to see that these sentences are consistent with the neg=
ation=20
> theorem.  Recall that a logical implication is a function of statements;=
=20
> it's truth value depends only on the truth value of the statements it act=
s=20
> on.  An IF (...) THEN (...) statement is defined to be false when the fir=
st=20
> argument, called the antecedent, is true and the second argument, called=
=20
> the consequent, is false.  All other pairs of arguments result in true.
>=20
> Under our assumption that nothing satisfies pavyseljirna, "ro da zo'u gan=
ai=20
> da pavyseljirna gi da blabi" is true because for every value of da, the=
=20
> antecedent is false.  Therefore "naku ro da zo'u ganai da pavyseljirna gi=
=20
> da blabi" is false.  According to the negation theorem "su'o da naku zo'u=
=20
> ganai da pavyseljirna gi da blabi" must also be false.  This says that=20
> there must an object which falsifies the implication, and as I said in th=
e=20
> last paragraph this can only happen when the antecedent is true and the=
=20
> consequent false.  The antecedent claims that x is a unicorn, so a true=
=20
> antecedent would contradict our assumption about unicorns.  Of course the=
=20
> particular functions we chose, unicorns and white, are not important; all=
=20
> statements of this form are consistent with the negation theorem.
>=20
> So if we wanted to keep the importing semantics, how would negation have =
to=20
> work?  We first rewrite "ro da poi P" in the importing sense as a formula=
=20
> in predicate logic to manipulate it symbolically, then translate it back=
=20
> into Lojban.  This still uses the implication, but includes the additiona=
l=20
> restriction that something must satisfy P.  We therefore represent "naku =
ro=20
> da poi P zo=E2=80=99u Q" as $$ \neg \forall x \exists y : P(y) \land [P(x=
)=20
> \rightarrow Q(x)]) $$.  Applying the theorem to the formula, we get $$=20
> \exists x \forall y : \neg (P(y) \land [P(x) \rightarrow Q(x)]) $$ which =
is=20
> equivalent by another elementary theorem to $$ \exists x \forall y : \neg=
=20
> P(y) \lor \neg (P(x) \rightarrow Q(x)) $$ which can be translated back in=
to=20
> Lojban as =E2=80=9Cro da su=E2=80=99o de zo=E2=80=99u de P inajanai ganai=
 da P gi da Q=E2=80=9D.  Notice in=20
> particular that there are now two sumti involved.  This is because in the=
=20
> importing sense there are really two different claims being made and each=
=20
> use their own variable.  I played with this for about half an hour tonigh=
t=20
> and couldn=E2=80=99t find an equivalent form that resulted in more elegan=
t Lojban;=20
> perhaps an importing advocate can do better.
>=20
> That=E2=80=99s one of four cases; three others are treated similarly, and=
 then=20
> negation dragging across unrestricted da operates according to the normal=
=20
> rules.  Imagine trying to move naku around in an ordinary sentence under=
=20
> these rules!
>=20
> I don=E2=80=99t know what pc said to John but it is simply not true that =
the=20
> Aristotelian sense of =E2=80=9CAll P are Q=E2=80=9D is compatible with pr=
edicate logic.  On=20
> page 54 of Hilbert and Ackermann=E2=80=99s classic _Principles of Mathema=
tical=20
> Logic_ appears the following:
>=20
> =E2=80=9CAccording to Aristotle the sentence =E2=80=98All A is B=E2=80=99=
 is valid only when there=20
> are objects which are A.  Our deviation from Aristotle in this respect is=
=20
> justified by the mathematical applications of logic, in which the=20
> Aristotelian interpretation would not be useful.=E2=80=9D
>=20
> Its possible that there is some confusion over an elementary theorem whic=
h=20
> states $$ \forall x : P(x) $$ implies $$ \exists x : P(x) $$.  If we look=
=20
> closely at that we see that, in John=E2=80=99s words, the quantification =
there=20
> corresponds to Lojban=E2=80=99s unrestricted logical variables; restricte=
d logical=20
> variables must first be rewritten as pure formulae, as I did above, befor=
e=20
> applying the theorem.
>=20
> mi=E2=80=99e az
>=20
>=20
> On Sunday, October 19, 2014 10:08:14 AM UTC-7, John Cowan wrote:
> >
> > Alex Burka scripsit:=20
> >
> > > Ok, so just to clarify what you were correcting, with importing {ro}=
=20
> > > you would say {ro broda cu brode} and {ro da poi broda cu brode} are=
=20
> > > the same thing and require {su'o da broda}, while {ro da ganai broda=
=20
> > > gi brode} is different and just requires a non-empty universe?=20
> >
> > Right.  The difference is between restricted and unrestricted=20
> > quantification.=20
> >
> > --=20
> > John Cowan          http://www.ccil.org/~cowan        co...@ccil.org=20
> > <javascript:>=20
> > Lope de Vega: "It wonders me I can speak at all.  Some caitiff rogue=20
> > did rudely yerk me on the knob, wherefrom my wits yet wander."=20
> > An Englishman: "Ay, belike a filchman to the nab'll leave you=20
> > crank for a spell." --Harry Turtledove, Ruled Britannia=20
> >


--=20
John Cowan          http://www.ccil.org/~cowan        cowan@ccil.org
If you have ever wondered if you are in hell, it has been said, then
you are on a well-traveled road of spiritual inquiry.  If you are
absolutely sure you are in hell, however, then you must be on the Cross
Bronx Expressway.  --Alan Feuer, New York Times, 2002-09-20

--=20
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