Re: quantifiers

[Gerald Koenig <jlk@NETCOM.COM>]

pc said:
>
>Yes, veion, we have been here before from various other angles.  I am only
>trying to get some systematic understanding of where it is we are, using
>the reasonably well-established concepts and notation of formal logic
>(which is, after all, supposed to underlie Lojban) as a tool.  It does
>seem that what we are finding is that the underlay is the logic of Russell
>and, latterly, Quine, where referring gives way to "being the value of a
>bound variable."  It works - - no task goes undone, and expressions map in
>natural ways and yet the explanation sounds very odd.  Linguists who know
>logic (McCawley always pops up here) and logicians (well, philosophers --
>Strawson, e.g.) who know (weeeell) linguistics find the diffe rences
>disturbing, even when the mechanics work out right.  It does seem odd to
>be able to talk about everything but never about any particular thing,
>since we tend to take the latter kind of talk as basic and talk of
>generalities as an abstraction from talk about individuals.  But, as JCB
>used to point out, this late neolithic language Loglan is not bound by the
>prejudices of earlier-lithic languages (as witness the "basic" vocabulary,
>largely unavailable to even our grandparents -- mine anyway).

I think that Quine straddled the issue of whether individuals or
abstractions are fundamental to logic pretty well.  I below quote myself
quoting Quine in the "any" discussions:


Quine quote of the day:

"The bulk of logical reasoning takes place on a level which does not
presuppose abstract entities. ...I consider it a defect in an
all-purpose formulation of the theory of reference if it represents us
as referring to abstract entities from the very beginning rather than
only where there is a real purpose in such reference."

djer




>
>xorxes:
>>I thought {ro} never had existential import. You are saying that it does
>>in {ro broda} but it does not in {ro da poi broda}. I guess that if it's
>>defined like that then that's that, but I really don't see the point of
>>complicating {ro} in such a way. Is it just to copy the behaviour of the
>>English "every"? It doesn't seem to be a very good reason. A much better
>>translation for {ro} in any case is "each". Does "each" have existential
>>import in English? It is very unsettling to find out that {ro} changes
>>meaning with context.
>
>In logic, the universal quantifier always has existential import; that is,
>the domain of discourse is never empty (except in one special field, which
>turns out to rest on the error of assuming that the domain of discourse
>must be the class of existents).  The question is only about what set is
>said to be non-empty and comes to the fore in translating categorical A
>sentences, All S are P.  If that is read, as it usually is in modern
>logic, as "for all x, if Sx then Px", then only the domain of discourse is
>said to be non-empty, but the set of Ss is not guaranteed to be.  In fact,
>if there are no Ss, then "All S are P" is true on this version (false
>antecedent means true conditional).  On the other hand, if "all S are P is
>read as a restricted quantifier to S, "for all x that are S, Px,"  the
>universal covers only the Ss and thus claims that there are some of them,
>since whatever the range of the universal, it is non-empty.  So, if there
>are no Ss, the second version is false.  These two versions correspond
>exactly (indeed, the Lojban was designed to represent directly) the two
>sentences, ro da broda nagi'a brode (actually the fully sentential version
>of this, ro da zo'u da broda [whatever the sentential version of the
>predicative form is] da brode) and ro da poi broda cu brode (or, again, ro
>da poi broda zo'u da brode), respectively.  ro does not change its
>meaning; only the kind of thing its claims that there are changes, from
>unrestricted to restricted. The odd "universal true even when there are no
>Ss " is a feature of the conditional, not of the universal.  Notice, the
>"there are S version" has several forms: ro broda = ro lo broda = ro da
>poi broda.  (I trust xorxes' grammar so much that I have revised my
>understanding of all this mess to accomodate his insistence on just this
>equation and others related to it.  So, I did not say RECENTLY that some
>of these have existential import and some don't, all of these are the same
>and requires that there be brodas.) By the way, "each" does have
>existential import in English and, indeed, requires that there be more
>than one of the critters and that we can line 'em up ("every" is
>etymologically "ever each", "line 'em up and run through the whole lot one
>by one").  "All" is about as neutral as English gets, since "any" clearly
>does not have existential import (an etymological curiosity).  See
>Vendler's article or the the appropriate section in the Dictionary of
>Philosophy.
>
>xorxes:
>> > > >         ro da poi broda cu brode
>> > > >         ro da broda nagi'a brode
>> > >
>> > > If there are no brodas, the first is false while the second is true,
>> > > regardless of what brode is.
>
>>And the explanation for why the first was false was that {ro da poi broda}
>>was supposed to be Ax e {broda}.
>
>Actually, I said
>>Restricted quantifier sentence are false when restricted to an empty set
>> (existential import).
>That does, unfortunately, use set talk (a logician's habit), but the set
>is inessential: a restricted quantifier sentence is false when there are
>no things of the sort to which it is restricted.  I would have written
>(and did somewhere, I thought) this as Ax such that x broda, not with
>set-membership notation.
>
>xorxes:
>>Talking about sets is a convenience, and when the set
>>exists it makes little difference whether you use it or not to explain
>>the meaning of {lo broda}. But when there is no corresponding set,
>>the expression is still meaningful
>Agreed.  The point of using set talk is to find a
>coherent way of dealing with all the descriptors and a way that was
>different from the purely quantifier versions, i.e., to match in logic the
>differences in Lojban structure.  So, we will say these are th e classes,
>not the sets, if we want to stick to the pattern, which is the point of
>this exercise. But whether we use the sets or not, or even classes or not,
>the quantifiers remain and that is the crux of non-referring part.
>
>        Since, as I have said, we can always manage equipollence between
>the two systems, I don't suppose I can give contrasting cases.  If I say
>it one way or the other, they describe the same situation, are true or
>false together and necessarily so.  But suppo se that in fact there is
>exactly one thing of a certain sort, S, and I want to say that it has a
>further proerty, P.  Then "Every S is P" and "The S is P" turn out to be
>equipollent, yet one of them is overtly a general claim, about everything
>or at least every S and the other is a singular claim that refers to a
>particular thing and says something about it.  The first does say
>something about that particular thing, but only as falling under a general
>category, without referring to that thing at all.  If there were several
>Ss and one of them were John, then "All Ss are P" would say something
>about John without referring to him, and it would certasinly not mean the
>same thing as "John is a P."  But the cardinality of a class (so long as
>it is not a wrong-sized class) should not change the significance of a
>quantified sentence where the quantifier is restricted to that class, so
>something is different between the two even when they come out to describe
>the same state of affairs.  Since all the descriptors t hat can have them
>have real quantifiers, they all fail to refer to the members of the sets
>they are restricted to, for "refers to" is just what the "the S" and
>"John"  sentences have that the quantified sentences lack (well, don't
>have, anyhow).  I think (looking at this) that this aamounts to saying
>that referring expressions refer and others don't and so the claim is just
>that le, lo, and la are not referring expressions, however much they may
>act like them at a practical level.  I do not know whether t here might be
>interesting Whorfian effects from this odd difference (it looks like just
>the sort of thing that Whorf would have thought to be source for such
>effects -- if he thought there were such things -- since he seemed to
>think that Hopi time refere nces that worked functionally as near as you
>like to SAE but were based on a totally different "metaphysic" was the
>mark of a profound difference).
>