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[98.138.229.135]) by gmr-mx.google.com with SMTP id h7si3460854icn.2.2011.11.27.20.58.51; Sun, 27 Nov 2011 20:58:52 -0800 (PST) Received-SPF: pass (google.com: best guess record for domain of kali9putra@yahoo.com designates 98.138.229.135 as permitted sender) client-ip=98.138.229.135; Received: from [98.138.90.52] by nm37.bullet.mail.ne1.yahoo.com with NNFMP; 28 Nov 2011 04:58:51 -0000 Received: from [98.138.89.170] by tm5.bullet.mail.ne1.yahoo.com with NNFMP; 28 Nov 2011 04:58:51 -0000 Received: from [127.0.0.1] by omp1026.mail.ne1.yahoo.com with NNFMP; 28 Nov 2011 04:58:51 -0000 X-Yahoo-Newman-Id: 614118.80176.bm@omp1026.mail.ne1.yahoo.com Received: (qmail 63009 invoked from network); 28 Nov 2011 04:58:51 -0000 X-Yahoo-Newman-Property: ymail-3 X-YMail-OSG: 9QEtHjYVM1mwPlMcN9HYJSIqY23KcytvK74CmJ.m8C26_EZ dvSE9.GEI_GgLWQIAqyAzDsgslKoss.Zsjets60R6tsFuwgH7w_24Iw8JtY. DVbBPI6dFkwen3_fQ6lggEfxPysuyfx2TpYlTso0MaAQiT75dx2nQ2mnq6Ic .4JbbLvCadqlASkNQ2WMnV8ZGJTqVup06RYYaKFCcm1IqW7ZgXN2bDD0X5uo Uc3Ccv_mviHTqaJkESk_2xUQgya_2wVph9_e1Rf678OUNFzXyYoQVQnTYtvi yRvHNYwi5zVlkNoBHhs0vzz4lYcu3esWL451c4lzekH04_0LEFL72knTlHJj BJyfMsnlievMTjUU5hz7ezigL7WCFmgO0a5tuSFg0cSR1PJTh_N4Sgk_ELV6 STLfd1eZ.L5gSrOTKAYMVy48OQOaM9gaJxHJ4Uh88IFHmWABr.JSKtY9z1GW pbGWkEm8mMNZGUmaqwDCamUIzVvuEQQDWZEDuFaelgnCE4u1SzJWnaDNwkQ- - X-Yahoo-SMTP: xvGyF4GswBCIFKGaxf5wSjlg3RF108g- Received: from [10.0.1.2] (kali9putra@99.92.108.41 with xymcookie) by smtp114-mob.biz.mail.ne1.yahoo.com with SMTP; 27 Nov 2011 20:58:50 -0800 PST References: <1322084099.71575.YahooMailRC@web81302.mail.mud.yahoo.com> <20111125034454.GI6112@gonzales> <1BC62F1A-FC02-415D-8863-BA795809DB95@yahoo.com> <20111127163616.GC3125@gonzales> In-Reply-To: <20111127163616.GC3125@gonzales> X-Apple-Yahoo-Original-Message-Folder: AAlojbanery Mime-Version: 1.0 (iPad Mail 8G4) Message-Id: X-Mailer: iPad Mail (8G4) From: "John E. Clifford" X-Apple-Yahoo-Replied-Msgid: 1_11966644_AHXHjkQAAPijTtJnCAYnvGGizA0 Subject: Re: [lojban] No title, since the subject will have changed by the time it gets there Date: Sun, 27 Nov 2011 22:59:57 -0600 To: "lojban@googlegroups.com" X-Original-Sender: kali9putra@yahoo.com X-Original-Authentication-Results: gmr-mx.google.com; spf=pass (google.com: best guess record for domain of kali9putra@yahoo.com designates 98.138.229.135 as permitted sender) smtp.mail=kali9putra@yahoo.com; dkim=pass (test mode) header.i=@yahoo.com Reply-To: lojban@googlegroups.com Precedence: list Mailing-list: list lojban@googlegroups.com; contact lojban+owners@googlegroups.com List-ID: X-Google-Group-Id: 1004133512417 List-Post: , List-Help: , List-Archive: Sender: lojban@googlegroups.com List-Subscribe: , List-Unsubscribe: , Content-Type: multipart/alternative; boundary=Apple-Mail-1-551816272 Content-Transfer-Encoding: 7bit X-Spam-Score: -0.7 (/) X-Spam_score: -0.7 X-Spam_score_int: -6 X-Spam_bar: / --Apple-Mail-1-551816272 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset=ISO-8859-1 Sent from my iPad On Nov 27, 2011, at 10:36 AM, Martin Bays wrote: > * Friday, 2011-11-25 at 12:38 -0600 - John E. Clifford : >=20 >> On Nov 24, 2011, at 9:44 PM, Martin Bays wrote: >>=20 >>> * Wednesday, 2011-11-23 at 13:34 -0800 - John E Clifford : >>>=20 >>>> 1. {zo'e}, as implicit in unfilled places, can't mean either "what I (= would=20 >>>> have) had in mind" or a particular quantifier, because there are too m= any cases=20 >>>> where it has to mean the other. >>>=20 >>> Pardon? >>=20 >> What is obscure here? =20 >=20 > It was just the English language being its usual annoyingly ambiguous > self - I couldn't disambiguate the relative scopes of "can", "not" and > "either". After your restatement, I'm interpreting it as > "not ((possible:...) or (possible:...))". I hope that is what was > intended! Well, literally, NMDpq, but that is equivalent. >=20 >> {zo'e} can't be either one of these because both of them occur as >> reasonable expansions into the blank space and being one would >> preclude being the other. Unless you mean that a particular >> quantifier is one thing I might have had in mind. But that creates >> the problem of mixing variables with names, which is not where we want >> to go, I think. >=20 > OK. So we need a meaning for {zo'e} which has each of these two as > special cases. This could be "particular quantifier over a domain > I (could) have in mind" - a special case being that it's quantification > over a singleton domain, so equivalent to it just being a constant > I (could) have in mind. >=20 But making it a quantifier makes it subject to quantifier rules. To be sur= e, if it is restricted to some single object, the difference between some a= nd all disappears. The problem is ensuring that the thing at the end of{po= i} is in fact a predicate with a single (and the right)referent. Actually,= the single requirement doesn't generally need to hold, since we have plura= l reference, presumably -- unless you want a single bunch, which you are pr= etty much sure to get. But, of course, the particular and universal quanti= fiers don't collapse under negation. In short, I don't think this works. >>>>=20 >=20 > There's the related thorny issue of observer places - although {sance} > is just "x1 is a sound emitted/produced by x2", so trees are no issue, > {carmi} is "x1 is intense/bright/saturated/brilliant in property (ka) x2 > as received/measured by observer x3". Is a candle's light carmi (be fi > zo'e) when there's no-one around to see it? Or is it only carmi be fi > zi'o This just shows how hopelessly bolloxed the treatment of blanks is. We hav= e to have them to have a usable language, but, if we do, the clarity and pe= rhaps the logic slips away. Of course, part of the problem is the number o= f places on many predicates, inviting most of them to be blank most of the = time. If more things were add ons rather than left offs, there would be fe= wer problems of this sort, though probably more of some other kind. >=20 >>=20 >>>=20 >>>> {zo'e} should be stated when a fixed, though perhaps unspecified, >>>> referent is intended. >>>=20 >>> I think having a word which literally acts as if the place were unfille= d >>> is a useful enough feature that we shouldn't do away with it unless >>> necessary. >>>=20 >>> Perhaps we can use {lo du} for the meaning you suggest? >>=20 >> I think I am missing your point here. {zi'o} says the place is >> unfilled. {zo'e} says the place is filled but I'm but telling you by >> what. And what does {lo du} do? It is either the self-identical >> things, which provides no information, >=20 > Yes, that was the intention. So it would have the meaning you're > suggesting for {zo'e}, whether or not {zo'e} itself does. >=20 > (Except that it only works if the unfilled second place of {du} is > interpreted correctly... it would be nice to have a clearer way of > getting at the always-true unary predicate. Do we have one?) I don't see the advantage of this. If we have to glork (where is this word= from, by the way,? It seems to mean something like "grok", but I don't re= cognize the source.) the identity of the second member of the identity, we = have to identify the first one as well and then we are back to just {zo'e} = again. On the other hand, if this is just the self identity, then it refer= s to any bunch in the universe of discourse and again we have to glory the = right one. So it keeps coming back to "what I have in mind", which doesn't= deal with all the particular quantifier cases. >=20 >>>> While I'm at it, we should change {ce'u} over to a variable-binding >>>> operator so we can do abstractions right. >>>=20 >>> Pardon >>=20 >> Make it be lambda and put variables after it, so we can distinguish >> when two arguments are the same from when they are different. >=20 > We can already get that effect by using anaphora - {lo ka ce'u ri broda} > is unary, while {lo ka ce'u ce'u broda} is binary. I find using anaphoric pronouns to refer to variables to be very anti logic= al ;variables are their own anaphora. >=20 >>>>=20 >>>> 3. Bunches relate to predicates in a variety of ways, for none of whic= h does=20 >>>> Lojban have an explicit marker, though some can be inferred from other= factors=20 >>>> (quantifiers, modals -- though we are somewhat defective there as well= , or maybe=20 >>>> just more pragmatic or rhetorical devices -- I'm not sure what general= ization or=20 >>>> stereotype is). I don't have a complete list and am unsure about the = status of=20 >>>> some I do have, so some discussion would be welcome. >>>=20 >>> Right, this is the part of your approach I'm unhappy with. I'm loath to >>> give up the simple version of plural semantics, whereby a selbri is >>> interpreted in a given world just as a relation on the set of bunches. >>>=20 >> But as far as I can see, you are the one who has given that up. >> I certainly have not. I have nothing but bunches of broda all the way >> up (or down, as the case may be). I am deliberately avoiding the use >> of"set", since that raises other problem. So far as I am concerned, >> the domain of the functions can be just bunches (of bunches, if you >> like). >>=20 >>> Complicating this with your "modes of predication" (conjunctive, >>> disjunctive, collective, statistical...) seems to fit lojban ill, >>> precisely because lojban has no way to mark them. >>=20 >> So far as I can tell, this jumble just comes with plural reference, >> together with an attempt to realistically with how various things we >> say are related to the things we talk about. I don't suppose the list >> is complete yet, but that is only a practical problem. As for not >> fitting Lojban, Lojban was designed without plural reference (or >> L-sets) and so makes no allowance for them. What it does partly make >> allowance for is using (C-)sets to represent plurals. But just what >> that involved was never spelled out too clearly (and much of what was >> originally spelled out was lost in xorlo), so we cannot merely take it >> over for L-sets. Restoring some of that, or devising new conventions, >> can cover much of the difficulty and perhaps all, depending on how >> wide our convention net spreads. >=20 > Ah, so it looks like I have been misunderstanding you. I understood you > as having the truth value of a predication (in a world) depend on three > things - the predicate, the bunches which are its arguments, and the > mode(s) of predication. Now I'm understanding you as saying that it > depends only on the first two, with the mode(s) merely being a way of > describing how it is that the truth value is related to the truth values > of the various predications where the bunches are replaced by their > subbunches. Is that right? >=20 I'm not sure what this means, but it should mean something like "the truth = value of a predication depends, inter alia, on the way the subbunches of th= e bunch which is the argument relate to the predicate." Does the bunch hav= e the property because all of it's subbunches do or because of them do or = because none of them other than the whole do, or is predicate applied to th= e bunch in some "statistical" way, and so on. Clearly, the students wear g= reen ties in a way quite different from the way they surround a building or= come from several countries or live at home or have above average intellig= ence or are civil. > Assuming it is - what makes me uneasy is the shift from one mode to > another when our bunches get large enough. >=20 But then I don't see what size has to do with it. I suppose you mean the p= ractical problem of a number of children getting to know a great number of = mothers, let alone loving them. Well, if it can't be done, then the connect= ions are not both conjunctive, and maybe no form is true (which, of course,= is always a possibility). But there is no shift from mode to another exce= pt as the facts require, that is, not at some theoretical point but just at= the practical one.. > In {ro lo verba cu prami lo mamta}, if {lo mamta} is interpreted as > a few specific mothers, presumably prami acts distributively in x2 > giving the meaning that each of the children love each of those mothers. >=20 > But if we enlarge the bunch to the maximal bunch of mothers, it switches > to be (something like) disjunctive - now we're just saying that each > child loves some mother (or maybe just Mother, if that's somehow > different?), perhaps their own, and not that each loves every mother. Well, I didn't think that was what the original said in the first place, no= r would I (without a lot of contextual build up) have taken {lo manta} to r= efer too some maximal set of mothers. I might have if the beginning were {= ro verba}, but even then, my first response would be to read in an implicit= "his own". I would never read it to be about Mother in xorxes' sense, sin= ce I don't believe in what little I can make of that notion. >=20 > So where did this switch occur on the way from our little bunch to the > maximal bunch, and why? Does it have something to do with adding mothers > which don't exist? Does adding just one non-existent mother cause the > switch? >=20 >>> The alternative is to further complicate the domain - adding more >>> derived entities beyond bunches. The marking can then be done with gadr= i >>> and quantifiers. >>=20 >> I'm not sure what you mean here. I don't see how adding new entities >> (what, I wonder >=20 > Things like the kind Mother. >=20 >> ) will help with the modes of predication issue. A few >> nice adverbs seem to be the most natural way to proceed. >=20 > So this would be explicitly marking which mode of predication is meant > to be in use, hence giving joint information about the precise predicate > intended (when there's vagueness in that) and the bunches intended. So far as I can see, the predicates nor the bunches change, just the mode. >=20 >>>> 4. We need a way to sort out the official meaning (sense, a function = on worlds)=20 >>>> and the ordinary meaning, an area in in the web of other meanings (pr= obably not=20 >>>> a spot in the Platonic tetrahedron anymore). And then say which one w= e are=20 >>>> talking about. >>>=20 >>> Pardon? >>=20 >> Well, it seems to me that people (myself included) flop back and forth >> among "lion" represents a function from(or relation between) things >> and truth values and "lion' represents a function from worlds to a set >> of objects in that world and "lion" represents the property of being >> a large cat (genus Panthera) ... . Trying to satisfy the various >> conditions these impose is a problem, since they are very different >> . I think part of our problem is that we often are at cross purposes >> here. >=20 > I see the first as being what you get from the second after specifying > a world, and the third as being another way of looking at the second. >=20 Specifying a world gives a set, from which a function can be derived, but t= he usual approach is to take the function as given and the set derived. As= to whether a genus-and-species can be used to generate any of the function= or sets or conversely, the jury is either still out or hung. > Martin --=20 You received this message because you are subscribed to the Google Groups "= lojban" group. To post to this group, send email to lojban@googlegroups.com. To unsubscribe from this group, send email to lojban+unsubscribe@googlegrou= ps.com. For more options, visit this group at http://groups.google.com/group/lojban= ?hl=3Den. --Apple-Mail-1-551816272 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=ISO-8859-1


Sent from my iPad
On Nov 27, 2011, at 10:36 AM, Martin Bays <mbays@sdf.org> wrote:

* Friday, 2011-11-25 at 12:38 -0600 - John E. Clifford= <kali9putra@yahoo.com>:<= /span>

On Nov 24, 2011,= at 9:44 PM, Martin Bays <mbays@sdf.org= > wrote:

* Wednesday, 2011-11-23 at 13:34 -0800 - John E Clifford <kali9putra@yahoo.com>:
1. {zo'e}, as implicit in unfi= lled places, can't mean either "what I (would
have) had in mind" or a particular quantifier= , because there are too many cases
where it has to mean the other.
<= /blockquote>

Pardon?

What is obscure here?  
=
It was just the English language being its usual annoyingly ambig= uous
self - I couldn't disambiguate the relative scopes of = "can", "not" and
"either". After your restatement, I'm inte= rpreting it as
"not ((possible:...) or (possible:...))". I = hope that is what was
intended!

Well, literally, NMDpq, but that is equivalent.

{z= o'e} can't be either one of these because both of them occur as
<= /blockquote>
reasonable expansions into the = blank space and being one would
preclude being the other.  Unless you mean that a particul= ar
quantifier is one= thing I might have had in mind.  But that creates
the problem of mixing variables with na= mes, which is not where we want
to go, I think.

O= K. So we need a meaning for {zo'e} which has each of these two asspecial cases. This could be "particular quantifier over a domain
I (could) have in mind" - a special case being that it's quan= tification
over a singleton domain, so equivalent to it jus= t being a constant
I (could) have in mind.
=
But making it a quantifier makes it subj= ect to quantifier rules.  To be sure, if it is restricted to some sing= le object, the difference between some and all disappears.  The proble= m is ensuring that the thing at the end of{poi} is in fact a predicate with= a single (and the right)referent.  Actually, the single requirement d= oesn't generally need to hold, since we have plural reference, presumably -= - unless you want a single bunch, which you are pretty much sure to get. &n= bsp;But, of course, the particular and universal quantifiers don't collapse= under negation.  In short, I don't think this works.




T= here's the related thorny issue of observer places - although {sance}
is just "x1 is a sound emitted/produced by x2", so trees are no = issue,
{carmi} is "x1 is intense/bright/saturated/brilliant= in property (ka) x2
as received/measured by observer x3". = Is a candle's light carmi (be fi
zo'e) when there's no-one = around to see it? Or is it only carmi be fi
zi'o
=

This just shows how hopelessly bolloxed t= he treatment of blanks is.  We have to have them to have a usable lang= uage, but, if we do, the clarity and perhaps the logic slips away.  Of= course, part of the problem is the number of places on many predicates, in= viting most of them to be blank most of the time.  If more things were= add ons rather than left offs, there would be fewer problems of this sort,= though probably more of some other kind.



{zo'e} should be stated when a fixed, though perhaps unspec= ified,
referent i= s intended.

I think hav= ing a word which literally acts as if the place were unfilled
= is a useful enough feature that we shouldn't do away with it unless
necessary.

Perhaps= we can use {lo du} for the meaning you suggest?

I think I am missing your point here. {zi'o} says th= e place is
unfilled.=  {zo'e} says the place is filled but I'm but telling you by
what.  And what does {lo= du} do?  It is either the self-identical
things, which provides no information,

Yes, that was the intention. So it wou= ld have the meaning you're
suggesting for {zo'e}, whether o= r not {zo'e} itself does.

(Except that it = only works if the unfilled second place of {du} is
interpre= ted correctly... it would be nice to have a clearer way of
= getting at the always-true unary predicate. Do we have one?)

I don't see the advantage of this.  If w= e have to glork (where is this word from, by the way,?  It seems to me= an something like "grok", but I don't recognize the source.) the identity o= f the second member of the identity, we have to identify the first one as w= ell and then we are back to just {zo'e} again.  On the other hand, if = this is just the self identity, then it refers to any bunch in the universe= of discourse and again we have to glory the right one.  So it keeps c= oming back to "what I have in mind", which doesn't deal with all the partic= ular quantifier cases.

<= blockquote type=3D"cite">
While I'm at it, we should change {ce'u} over to a variable-binding=
operator so we c= an do abstractions right.
=

Pardon

Make it be lambd= a and put variables after it, so we can distinguish
=
when two arguments are the same from when t= hey are different.

We can alr= eady get that effect by using anaphora - {lo ka ce'u ri broda}
is unary, while {lo ka ce'u ce'u broda} is binary.

I find using anaphoric pronouns to refer to variabl= es to be very anti logical ;variables are their own anaphora.=


3. Bunches relate to predicates in a varie= ty of ways, for none of which does
Lojban have an explicit marker, though some can be infer= red from other factors
(quantifiers, modals -- though we are somewhat defective there as we= ll, or maybe
jus= t more pragmatic or rhetorical devices -- I'm not sure what generalization = or
stereotype is= ).  I don't have a complete list and am unsure about the status of
some I do have, so = some discussion would be welcome.
Right, this is the part of your approach I'm unhappy with. I'm l= oath to
give up the simple version of plural semantics, = whereby a selbri is
interpreted in a given world just as= a relation on the set of bunches.

But as far as I can see, yo= u are the one who has given that up.
I certainly have not.  I have nothing but bunches of = broda all the way
up= (or down, as the case may be).  I am deliberately avoiding the use
of"set", since that ra= ises other problem.  So far as I am concerned,
=
the domain of the functions can be just bun= ches (of bunches, if you
<= span>like).
<= br>
C= omplicating this with your "modes of predication" (conjunctive,
<= /blockquote>
disjunctive, collective, statistical...) seems to fit lojban ill,
precisely because lojban has no way to mark them.<= br>

So far as I can tell, this jumble= just comes with plural reference,
together with an attempt to realistically with how various = things we
say are re= lated to the things we talk about.  I don't suppose the list
is complete yet, but that is = only a practical problem.  As for not
fitting Lojban, Lojban was designed without plural r= eference (or
L-sets)= and so makes no allowance for them.  What it does partly make<= br>
allowance for is using (C-)= sets to represent plurals.  But just what
that involved was never spelled out too clearly = (and much of what was
originally spelled out was lost in xorlo), so we cannot merely take it
over for L-sets.  = ;Restoring some of that, or devising new conventions,
can cover much of the difficulty and perh= aps all, depending on how
= wide our convention net spreads.
=
Ah, so it looks like I have been misunderstanding you. I understo= od you
as having the truth value of a predication (in a wor= ld) depend on three
things - the predicate, the bunches whi= ch are its arguments, and the
mode(s) of predication. Now I= 'm understanding you as saying that it
depends only on the = first two, with the mode(s) merely being a way of
describin= g how it is that the truth value is related to the truth values
<= span>of the various predications where the bunches are replaced by their
subbunches. Is that right?

<= /blockquote>I'm not sure what this means, but it should mean something like= "the truth value of a predication depends, inter alia, on the way the subb= unches of the bunch which is the argument relate to the predicate."  D= oes the bunch have the property because all of it's  subbunches do or = because of them do or because none of them other than the whole do, or is p= redicate applied to the bunch in some "statistical" way, and so on.  C= learly, the students wear green ties in a way quite different from the way = they surround a building or come from several countries or live at home or = have above average intelligence or are civil.

Assuming it is - what makes me uneasy is t= he shift from one mode to
another when our bunches get larg= e enough.

But then I don= 't see what size has to do with it.  I suppose you mean the practical = problem of a number of children getting to know a great number of mothers, = let alone loving them. Well, if it can't be done, then the connections are = not both conjunctive, and maybe no form is true (which, of course, is alway= s a possibility).  But there is no shift from mode to another except a= s the facts require, that is, not at some theoretical point but just at the= practical one..

In {ro lo ve= rba cu prami lo mamta}, if {lo mamta} is interpreted as
a f= ew specific mothers, presumably prami acts distributively in x2
<= span>giving the meaning that each of the children love each of those mother= s.

But if we enlarge the bunch to the maxi= mal bunch of mothers, it switches
to be (something like) di= sjunctive - now we're just saying that each
child loves som= e mother (or maybe just Mother, if that's somehow
different= ?), perhaps their own, and not that each loves every mother.

Well, I didn't think that was what the origin= al said in the first place, nor would I (without a lot of contextual build = up) have taken {lo manta} to refer too some maximal set of mothers.  I= might have if the beginning were {ro verba}, but even then, my first respo= nse would be to read in an implicit "his own".  I would never read it = to be about Mother in xorxes' sense, since I don't believe in what little I= can make of that notion.
So where did this switch occur on the way from our little bunch to = the
maximal bunch, and why? Does it have something to do wi= th adding mothers
which don't exist? Does adding just one n= on-existent mother cause the
switch?

The alternat= ive is to further complicate the domain - adding more
de= rived entities beyond bunches. The marking can then be done with gadri
and quantifiers.

I'm not sure what you mean here.  I don't see how adding new en= tities
(what, I wond= er

Things like the kind Mothe= r.

) will help w= ith the modes of predication issue.  A few
nice adverbs seem to be the most natural way to= proceed.

So this would be ex= plicitly marking which mode of predication is meant
to be i= n use, hence giving joint information about the precise predicateintended (when there's vagueness in that) and the bunches intended.<= /span>

So far as I  can see, the = predicates nor the bunches change, just the mode.

4.  We need a way to sort out the o= fficial meaning (sense, a function on worlds)
and the ordinary meaning, an area in in the w= eb of  other meanings (probably not
a spot in the Platonic tetrahedron anymore).  = ;And then say which one we are
talking about.

<= /blockquote>
Pardon?

Well, it s= eems to me that people (myself included) flop back and forth
among "lion" represents a function= from(or relation between) things
and truth values and "lion' represents a function from worl= ds to a set
of objec= ts in that world and "lion" represents the property of being
a large cat (genus Panthera) ... .=  Trying to satisfy the various
conditions these impose is a problem, since they are very = different
.  I = think part of our problem is that we often are at cross purposes
=
here.
<= span>
I see the first as being what you get from the second= after specifying
a world, and the third as being another w= ay of looking at the second.

= Specifying a world gives a set, from which a function can be derived, but t= he usual approach is to take the function as given and the set derived. &nb= sp;As to whether a genus-and-species can be used to generate any of the fun= ction or sets or conversely, the jury is either still out or hung.
Martin
=

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