Re: quantifiers

[jorge@PHYAST.PITT.EDU]

pc:
> 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.

Well, why not make that the reading in Lojban as well, if it's the usual one
in modern logic? It certainly is the most convenient one, if we want to make
manipulations of things such as negation easy.

> In fact,
> if there are no Ss, then "All S are P" is true on this version (false
> antecedent means true conditional).

Exactly. And I believe that should be the reading of {ro broda cu brode}.

> 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.

I really don't see the benefit of this convention, but if that's how it
is defined, then that's that. It certainly makes logical transformations
complicated. In a way, it is like saying that {ro lo broda} really is
{ro lo su'o broda} (each of the at least one broda that there are) rather
than {ro lo ro broda} (each of all the broda that there are, which
might be none).

> 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.

It's probably not a big deal, since empty descriptions are extremely rare
anyway, but I find it a bit of a blemish that it be so. Since the distinction
could be made with the inner-quantifier/cardinality-marker anyway, it
seems a pity that the simpler logical form (with no existential import)
gets the more complicated notation.

> 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.)

I'm flattered. Now all we need is for Lojbab to accept it  :)

> 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).

The problem for me with "all" is that it often has massification (or
collectivisation) import, and I find the individual vs mass distinction
to be far more important to keep straight in Lojban. (In logic,
collectivization seems to be ignored, not even mentioned, at the basic
level, but in Lojban it can't be.) That's why I prefer to use "each".

Jorge