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Re: [lojban] Individuals and xorlo





Le mercredi 19 février 2014 05:38:50 UTC+9, xorxes a écrit :



On Tue, Feb 18, 2014 at 10:13 AM, guskant <gusni...@gmail.com> wrote:

Le mardi 18 février 2014 07:41:04 UTC+9, xorxes a écrit :

lo PA broda := zo'e noi ke'a PA mei gi'e broda


I prefer that definition to the current one because the system of counting is clearer than {zilkancu}, though atomicity is still not required for {PA mei}.

Atomicity is not strictly required for the definition, but it's kind of implicit. If atomicity is false, then "su'o N mei" is always true. They are just a series of tautological predicates. And "N mei" is always false for any finite N, a series of contradictory predicates. So we _can_ define "PA mei" in the absence of atomicity, but actually using those predicates for anything meaningful requires atomicity. In the absence of atoms, anything at all satisfies su'o N mei and consequently nothing at all satisfies N mei.

If we really need atomicity for {lo PA broda}, we could add a condition of individual for {lo pa broda}:
{lo pa broda} =ca'e {zo'e noi ro'oi da poi ke'a xi pa me ke'a xi re zo'u ke'a xi re me da gi'e broda}

However, I think atomicity is not necessary for a definition of inner quantifier. 

I agree it's not necessary for the definition, but the use of a finite inner quantifier presupposes individuals.




I don't yet understand how the definitions on {PA mei} could suggest implicit atomicity.

The definitions on the topic are:

(D1) ko'a su'o N mei := su'oi da poi me ko'a ku'o su'oi de poi me ko'a zo'u ge da su'o N-1 mei gi de na me da

(I interpret here {su'oi da su'oi de zo'u de na me da} = {su'oi da su'oi de naku zo'u de me da} in accordance with your suggestion http://www.lojban.org/tiki/scope+of+na , not {naku su'oi da su'oi de zo'u de me da} of CLL.)

(D2) ko'a N mei  := ko'a su'o N mei gi'e nai su'o N+1 mei 
(D3) lo PA broda := zo'e noi ke'a PA mei gi'e broda

Actually, (D2) fails on N=1:
ko'a pa mei 
= ko'a su'o pa mei gi'e nai su'o re mei 
= ge 
su'oi da poi me ko'a ku'o su'oi de poi me ko'a zo'u 
ge da su'o no mei gi de naku me da -----(S1)
gi 
naku su'oi da poi me ko'a ku'o su'oi de poi me ko'a zo'u 
ge da su'o pa mei gi de naku me da -----(S2)

Consider (S1): because {su'oi da no mei} is false, (S1) actually says only {su'oi da poi me ko'a zo'u ge da su'o pa mei gi de naku me da}. It contradicts (S2).

For precise definitions on {PA mei}, we need therefore an explicit definition of {ko'a su'o pa mei} besides (D1).

Once {ko'a su'o pa mei} is defined in some way, (D2) and (D3) are valid for an integer N>=1. (D2) is expanded as follows:

(D2) ko'a N mei 
= ko'a su'o N mei gi'e nai su'o N+1 mei 
= ge ko'a su'o N mei -----(S1)
gi naku ko'a su'o N+1 mei -----(S2)

(S2)
= naku su'oi da poi me ko'a ku'o su'oi de poi me ko'a zo'u 
ge da su'o N mei 
gi de naku me da
= ro'oi da poi me ko'a ku'o ro'oi de poi me ko'a zo'u
naku ge da su'o N mei 
gi de naku me da
= ro'oi da poi me ko'a ku'o ro'oi de poi me ko'a zo'u
ganai da su'o N mei 
gi de me da

Therefore,
ko'a N mei 
= ge (S1) gi (S2) 
= ge ko'a su'o N mei
gi ro'oi da poi me ko'a ku'o ro'oi de poi me ko'a zo'u
ganai da su'o N mei 
gi de me da

Then {ko'a N mei} implies also 
ro'oi de poi me ko'a zo'u de me ko'a

When N=1, 
ko'a pa mei 
= ge ko'a su'o pa mei
gi ro'oi da poi me ko'a ku'o ro'oi de poi me ko'a zo'u
ganai da su'o pa mei 
gi de me da 

In every derivation from (D1) and (D2), {ko'a} may have {ko'e} such that {ko'e me ko'a ijenai ko'a me ko'e}. There seems to be no reason for {ko'a} beeing an individual {ro'oi da me ko'a zo'u ko'a me da} or individuals. If atomicity is implied, that should be caused by the _expression_ {ko'a su'o pa mei}, which is not yet defined.

As a reasonable definition for {ko'a su'o pa mei}, I would suggest as follows:

(D1-1) ko'a su'o pa mei := su'oi da poi me ko'a ku'o ro'oi de poi me ko'a zo'u de me da

When a condition {ije da me de} is also satisfied with (D1-1), {ko'a} is an individual.
Otherwise, {ko'a} of (D1-1) is individuals or non-individual.

(D1-1) says nothing related the number one, but it reflects a property of one-some of non-individual: any non-individual sumti can be one-some. Once non-individual B such that {B me ko'a} is fixed as one-some {B pa mei}, and if C such that {C me ko'a} satisfies conditions (D1) and (D2), C is counted to be an integer, and it is meaningful: at least, an order of cardinality is given to the pair of B and C.


It may be off topic, but if there were a definition for inner fractional quantifier 
{lo piPA broda} =ca'e {zo'e noi ke'a piPA si'e be lo pa broda}
then the language would be richer; this definition would be avaiable both atomist and non-atomist.
Actually, an outer fractional quantifier {piPA sumti} =ca'e {lo piPA si'e be pa me sumti} is available to atomists only.


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