When we discussed this at great length a dozen years ago, the arguments mustered -- which I can't reconstruct from memory -- led to the clear conclusion that {ro} (given its undisputed properties) means "however many there are", i.e. a cardinal number whose value can be zero, but this did not mean that there should not be another word meaning an existential import universal quantifier.
So there are two or three different and separate arguments here, all confounding each other:
1. What does ro mean, and does it have EI? (A question settled a dozen years ago.)
2. Should there be a non-EI universal quantifier?
3. Should there be an EI universal quantifier? This is the question John seems to be addressing.
Furthermore, an additional separate question would be
4. In any bpfk revision of the CLL specification, which meaning should be paired with the phonological form /ro/?
--And.
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 (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 plug
> it into an online renderer. \forall is the universal quantifier, x is the
> 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 pavyseljirna
> 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 its
> 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 advocating.
>
> It’s easy to see that these sentences are consistent with the negation
> 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 acts
> on. An IF (...) THEN (...) statement is defined to be false when the first
> 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 ganai
> 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 gi
> 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 the
> 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 the
> particular functions we chose, unicorns and white, are not important; all
> 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 formula
> in predicate logic to manipulate it symbolically, then translate it back
> into Lojban. This still uses the implication, but includes the additional
> restriction that something must satisfy P. We therefore represent "naku ro
> da poi P zo’u 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 $$ \exists x \forall y : \neg
> P(y) \lor \neg (P(x) \rightarrow Q(x)) $$ which can be translated back into
> Lojban as “ro da su’o de zo’u de P inajanai ganai da P gi da Q”. Notice in
> particular that there are now two sumti involved. This is because in the
> importing sense there are really two different claims being made and each
> use their own variable. I played with this for about half an hour tonight
> and couldn’t find an equivalent form that resulted in more elegant Lojban;
> perhaps an importing advocate can do better.
>
> That’s one of four cases; three others are treated similarly, and then
> negation dragging across unrestricted da operates according to the normal
> rules. Imagine trying to move naku around in an ordinary sentence under
> these rules!
>
> I don’t know what pc said to John but it is simply not true that the
> Aristotelian sense of “All P are Q” is compatible with predicate logic. On
> page 54 of Hilbert and Ackermann’s classic _Principles of Mathematical
> Logic_ appears the following:
>
> “According to Aristotle the sentence ‘All A is B’ is valid only when there
> are objects which are A. Our deviation from Aristotle in this respect is
> justified by the mathematical applications of logic, in which the
> Aristotelian interpretation would not be useful.”
>
> Its possible that there is some confusion over an elementary theorem which
> states $$ \forall x : P(x) $$ implies $$ \exists x : P(x) $$. If we look
> closely at that we see that, in John’s words, the quantification there
> corresponds to Lojban’s unrestricted logical variables; restricted logical
> variables must first be rewritten as pure formulae, as I did above, before
> applying the theorem.
>
> mi’e 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
--
You received this message because you are subscribed to the Google Groups "BPFK" group.
To unsubscribe from this group and stop receiving emails from it, send an email to bpfk-list+unsubscribe@googlegroups.com.
To post to this group, send email to bpfk-list@googlegroups.com.
Visit this group at http://groups.google.com/group/bpfk-list.
For more options, visit https://groups.google.com/d/optout.