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Date: Sun, 9 Nov 2014 01:19:50 -0800
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Subject: Re: [bpfk] {ro}, existential import and De Morgan
From: Ozymandias Haynes <ozymandias.haynes@gmail.com>
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je'e la djan io

> 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

Oh, there's no danger of that.  I'm not the swallowing type.

What I am here to do is engage in a substantive discussion about a
fine point of Lojban semantics.  I'm sure you're busy, and I imagine
you are very tired indeed of having conversations about something you
wrote seventeen years ago.  Particularly, as la mukti notes, on a
topic that's been discussed before.

I suspect that the point you're making with your dual translations fit
better in one of those previous conversations.

As I said in my first message, your dual translation of "All A is B"
is irrelevant to the point that la mukti carefully articulated in his
original post.  He was not translating any English sentences; he was
quoting directly from the CLL.  In examples 11.5 through 11.7, the
predicate logic negation theorem is applied to "ro da poi" statements.
The section is online here: http://lojban.github.io/cll/16/11/ or at
the bottom of page 405 in the physical book.

This is the text of the relevant passage:

"As explained in Section 9, when a prenex negation boundary expressed
by "naku" moves past a quantifier, the quantifier has to be inverted.
The same is true for "naku" in the bridi proper. ... Thus, the
following are equivalent to Example 11.4:

11.5) su'oda poi verba cu klama rode poi ckule naku
...
11.6) su'oda poi verba cu klama naku su'ode poi ckule
...
11.7) naku roda poi verba cu klama su'ode poi ckule"

la mukti and I are not arguing that the CLL does not assign
Aristotelian semantics to "ro da poi" statements and reserve the
predicate logic semantics solely for "ro da zo'u ganai" statements.
We are arguing that when you do that, the predicate logic negation
theorem does not hold for "ro da poi" statements.  The CLL says that
it does.

la mukti carefully walks through the reasoning in his post by
constructing a simpler sentence (but one still structurally similar)
and then evaluating its truth using the Aristotelian semantics given
in the CLL for "ro da poi" statements and which you are advocating in
this thread.  Then he evaluates its truth using the negation theorem.
The results contradict each other.

If the predicate logic negation theorem had simply been stated to
apply only to unrestricted universal quantification rather than "ro da
poi" clauses then there would be no contradiction.

I would like to know your thoughts specifically about la mukti's
construction, if you find the time.  Do you think the construction is
valid, and that a contradiction obtains?  If not, at what specific
point in his argument does the construction fail?

-Oz

On Sat, Nov 8, 2014 at 4:46 PM, John Cowan <cowan@mercury.ccil.org> wrote:
> Ozymandias Haynes scripsit:
>
>> The way that "All unicorns are white." is represented in predicate logic=
 is
>> with the formula $$ \forall x : [ U(x) \rightarrow W(x) ] $$.
>
> 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 (apar=
t
> 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 p=
lug
>> it into an online renderer. \forall is the universal quantifier, x is th=
e
>> bound variable, \rightarrow is implication, and U and W are functions
>> corresponding to 'x is a unicorn' and 'x is white' resp.).  As John says=
,
>> one way to translate this into Lojban is "ro da zo'u ganai da pavyseljir=
na
>> gi da blabi".  This is irrelevant to la mukti's construction, however.  =
He
>> did not use that Lojban sentence in his example, he used one that's
>> formally equivalent to da with poi.  The negation theorem is stated in i=
ts
>> full generality in the CLL and not only on sentences of the form above.
>>  Indeed, using that implication form as a definition of "ro da poi X" is
>> precisely what is needed to fit with the negation theorem and with
>> predicate logic, and those are precisely the semantics that I am advocat=
ing.
>>
>> It=E2=80=99s easy to see that these sentences are consistent with the ne=
gation
>> theorem.  Recall that a logical implication is a function of statements;
>> it's truth value depends only on the truth value of the statements it ac=
ts
>> on.  An IF (...) THEN (...) statement is defined to be false when the fi=
rst
>> argument, called the antecedent, is true and the second argument, called
>> the consequent, is false.  All other pairs of arguments result in true.
>>
>> Under our assumption that nothing satisfies pavyseljirna, "ro da zo'u ga=
nai
>> da pavyseljirna gi da blabi" is true because for every value of da, the
>> antecedent is false.  Therefore "naku ro da zo'u ganai da pavyseljirna g=
i
>> da blabi" is false.  According to the negation theorem "su'o da naku zo'=
u
>> ganai da pavyseljirna gi da blabi" must also be false.  This says that
>> there must an object which falsifies the implication, and as I said in t=
he
>> last paragraph this can only happen when the antecedent is true and the
>> consequent false.  The antecedent claims that x is a unicorn, so a true
>> antecedent would contradict our assumption about unicorns.  Of course th=
e
>> particular functions we chose, unicorns and white, are not important; al=
l
>> statements of this form are consistent with the negation theorem.
>>
>> So if we wanted to keep the importing semantics, how would negation have=
 to
>> work?  We first rewrite "ro da poi P" in the importing sense as a formul=
a
>> in predicate logic to manipulate it symbolically, then translate it back
>> into Lojban.  This still uses the implication, but includes the addition=
al
>> restriction that something must satisfy P.  We therefore represent "naku=
 ro
>> da poi P zo=E2=80=99u Q" as $$ \neg \forall x \exists y : P(y) \land [P(=
x)
>> \rightarrow Q(x)]) $$.  Applying the theorem to the formula, we get $$
>> \exists x \forall y : \neg (P(y) \land [P(x) \rightarrow Q(x)]) $$ which=
 is
>> equivalent by another elementary theorem to $$ \exists x \forall y : \ne=
g
>> P(y) \lor \neg (P(x) \rightarrow Q(x)) $$ which can be translated back i=
nto
>> Lojban as =E2=80=9Cro da su=E2=80=99o de zo=E2=80=99u de P inajanai gana=
i da P gi da Q=E2=80=9D.  Notice in
>> particular that there are now two sumti involved.  This is because in th=
e
>> importing sense there are really two different claims being made and eac=
h
>> use their own variable.  I played with this for about half an hour tonig=
ht
>> and couldn=E2=80=99t find an equivalent form that resulted in more elega=
nt Lojban;
>> perhaps an importing advocate can do better.
>>
>> That=E2=80=99s one of four cases; three others are treated similarly, an=
d then
>> negation dragging across unrestricted da operates according to the norma=
l
>> rules.  Imagine trying to move naku around in an ordinary sentence under
>> these rules!
>>
>> I don=E2=80=99t know what pc said to John but it is simply not true that=
 the
>> Aristotelian sense of =E2=80=9CAll P are Q=E2=80=9D is compatible with p=
redicate logic.  On
>> page 54 of Hilbert and Ackermann=E2=80=99s classic _Principles of Mathem=
atical
>> Logic_ appears the following:
>>
>> =E2=80=9CAccording to Aristotle the sentence =E2=80=98All A is B=E2=80=
=99 is valid only when there
>> are objects which are A.  Our deviation from Aristotle in this respect i=
s
>> justified by the mathematical applications of logic, in which the
>> Aristotelian interpretation would not be useful.=E2=80=9D
>>
>> Its possible that there is some confusion over an elementary theorem whi=
ch
>> states $$ \forall x : P(x) $$ implies $$ \exists x : P(x) $$.  If we loo=
k
>> closely at that we see that, in John=E2=80=99s words, the quantification=
 there
>> corresponds to Lojban=E2=80=99s unrestricted logical variables; restrict=
ed logical
>> variables must first be rewritten as pure formulae, as I did above, befo=
re
>> applying the theorem.
>>
>> mi=E2=80=99e az
>>
>>
>> On Sunday, October 19, 2014 10:08:14 AM UTC-7, John Cowan wrote:
>> >
>> > Alex Burka scripsit:
>> >
>> > > Ok, so just to clarify what you were correcting, with importing {ro}
>> > > you would say {ro broda cu brode} and {ro da poi broda cu brode} are
>> > > the same thing and require {su'o da broda}, while {ro da ganai broda
>> > > gi brode} is different and just requires a non-empty universe?
>> >
>> > Right.  The difference is between restricted and unrestricted
>> > quantification.
>> >
>> > --
>> > John Cowan          http://www.ccil.org/~cowan        co...@ccil.org
>> > <javascript:>
>> > Lope de Vega: "It wonders me I can speak at all.  Some caitiff rogue
>> > did rudely yerk me on the knob, wherefrom my wits yet wander."
>> > An Englishman: "Ay, belike a filchman to the nab'll leave you
>> > crank for a spell." --Harry Turtledove, Ruled Britannia
>> >
>
>
> --
> 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
>
> --
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