There several logical systems which are formally indistinguishable but which differ in metalanguage and informal chat. They also extend in different ways when they are built into something like Lojban, Mixing the different metalanguages or the different chat or the different extentions makes for results that are even more paradoxical than already happens in any one of the systems. For present purposes, lets just consider two relatively typical systems, plural quantification ala McKay and mereology ala Lesniewski (there are other plural quantifications and other part-whole theories). Plural quantification (and plural reference, which goes with it in McKay, though need not) starts with a domain of individuals but then specifies reference and evaluation not as functions but as relations, Thus,
"term A refers to" is a predicate which may be true of several thing simultaneously and similarly "variable X evaluates as". Mereology begins with wholes and introduces individuals, if at all, only as wholes which are their only parts. The mereological metalanguage is normal, using functions rather than relations. Talking about sets in either system is a bit misleading. In plural reference, there is no entity (in terms of the system) between individuals and the role they play, so no sets. In mereology, there is no separate type between individuals and what plays the role, so no (special class of ) individuals. In fairness to terminology, the wholes of mereology are sometimes called sets or some such thing, but are definitely not the same as the usual, Cantorian, sets. Similarly, in nontechnical discussions in English, it is almost impossible to talk about several things simultaneously with out introducing some
collective _expression_: "plurality" or "bunch" (my favorite) or ... . The fundamental internal relation in the two systems are "among" for plural reference (a relation between some individuals simultaneously and some individuals simultaneously) and "is part of" in mereology (between two wholes). A moment's thought will show these are going to behave exactly the same, if you understand the system (if you don't see they behave the same, then we need to go back to the two systems in more detail). All this being the case, it turns out that, for informal discussions, at least, it is most efficient to use one of those handy English expressions to talk about the system without specifying which version you are using. But, since there is a temptation to take these terms literally, it is probably best to avoid freighted expressions like "set" or "group" or "mass" (this is why I like the unfreighted "bunch").
But the problems with literality are obviously different in the two cases: in one you call into existence some intermediate type, in the other you call into existence some fundamental individuals. Lojban seems to split the difference, talking as though there were things of the intermediate type (you can't actually says some plural reference sorts of things in grammatical Lojban "A, B among A,B, C," for example) but sometimes talking as though there were ultimate individuals (counting what is in lo broda, for example). But at other times (or, perhaps, with some predicate but not others) it goes with indefinitely extensible partness (as when 'lo gerku' can refer to dabs of stuff on my bumper from running into a pack of dogs). I think any effort to resolve this kind of tension in general, rather than on each particular occasion, is going to fail, since we clearly want and need both approaches. So the quest for THE meaning of a
Lojban _expression_ -- other than its formal role -- seems a not very promising task. One of the nice things about that pedantic definition of 'lo broda', "the salient node of the upward semi-lattice ..." is that covers just about every kind of case.
From: guskant <gusni.kantu@gmail.com>
To: lojban@googlegroups.com
Sent: Saturday, February 8, 2014 9:34
AM
Subject: Re: [lojban] Individuals and xorlo
Le samedi 8 février 2014 19:57:39 UTC+9, lojbab a écrit :
On 2/6/2014 8:58 PM, guskant wrote:
>
>
> Le vendredi 7 février 2014 06:22:09 UTC+9, xorxes a écrit :
>
>
> On Thu, Feb 6, 2014 at 1:34 AM, guskant <gusni...@gmail.com> wrote:
>
>
> Although it will become out of topic, I have another suggestion
> related to the BPFK page of gadri.
>
> "Any term without an explicit outer quantifier is a constant"
> should be changed to
> "Any term without an explicit outer quantifier can be a constant",
> because an usual predicate logic has an axiom on a constant c
> that "F(c) {inaja} there is at least one (individual) x such
> that F(x)";
>
>
> That applies to singular constants, whereas unquantified terms need
> not be singular, but the version with plural quantifiers will still
> be valid.
>
>
>
> Actually, there is no explicit plural qiantifier in Lojban, though
> implicitly there are.
su'ore is the plural quantifier
That is a singular quantifier. I did not mean it by "plural quantifier". I meant that of plural logic.
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