Le vendredi 21 février 2014 06:43:48 UTC+9, xorxes a écrit :
On Thu, Feb 20, 2014 at 1:50 AM, guskant
<gusni...@gmail.com> wrote:
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
(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
For precise definitions on {PA mei}, we need therefore an explicit definition of {ko'a su'o pa mei} besides (D1).
That's why I started by saying "ro'oi da su'o pa mei", which is to say that "su'o pa mei" is a tautological predicate, always true of anything.
Yes, and in order to say "ro'oi da su'o pa mei", an axiom that is not an logical axiom should be given. That's why an explicit definition for {ko'a su'o pa mei} is necessary especially for the case that ko'a is an individual.
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:
[...]
Then {ko'a N mei} implies also
ro'oi de poi me ko'a zo'u de me ko'a
"ro'oi de poi me ko'a zo'u de me ko'a" is true independently of whether "ko'a N mei" is true or not. It's just a case of the general "ro'oi de poi broda zo'u de broda".
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
Yes, and since "su'o pa mei" is a tautology, that reduces to:
ko'a pa mei
= ro'oi da poi me ko'a ku'o ro'oi de poi me ko'a zo'u de me da
which says that "ko'a" is an individual. (Which is to be expected, what else would a one-some be if not an individual?)
Because "ro'oi da su'o pa mei" is based on a non-logical axiom, it cannot be called "tautology" in normal meaning. With this axiom, {ko'a pa mei} says that "ko'a" is an individual, of course.
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}.
I don't think that can happen if "ko'a pa mei" is true.
You are right under the condition that "ro'oi da su'o pa mei" is true. However, it is a non-logical axiom or the equivalent. I discussed that (D1) (D2) (D3) without any non-logical axioms are meaningful even in the case that ko'a is non-individual in the point that they give an order of cardinality.
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
Since that is also a tautology ("ko'a" itself will instantiate "su'oi da poi me ko'a"), it works, but it's more complicated that it needs to be. We can just as well define it as:
ko'a su'o pa mei := ko'a me ko'a
or:
ko'a su'o pa mei := ko'a du ko'a
or any other tautology. Or just state that "su'o pa mei" is the tautological predicate.
The complicated form of (D1-1) is intended to add a non-logical part {ije da me de} in order to say explicitly the case that ko'a is an 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.
If by "one-some" you mean "pa mei", then only indiciduals can satisfy it. If you mean "su'o pa mei", then yes, anything satisfies it, it's a tautology. Or am I missing something?
I mean "pa mei" by "one-some". As I mentioned above, In order to say {pa mei} is an individual, a non-logical part {ije da me de} is necessary to be added to (D1-1). This addition is equivalent to a non-logical axiom "ro'oi da su'o pa mei", but explicitly mentions the condition for ko'a being an individual. Because (D1) (D2) (D3) give only an order of cardinality, they alone can be used both cases of individuals and non-individual. Starting with a non-logical axiom "ro'oi da su'o pa mei" is available only to the case that ko'a is an individual or individuals, but (D1) (D2) (D3) themselves are more generally available without non-logical axioms.
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.
I assume "lo piPA broda" will have some such meaning , but it's a different system. And it relies on a previous definition of "si'e", which we don't have from basics like the ones we're discussing here for "mei".
I agree. I just want to suggest it on my personal gadri page for symmetry of definitions of quantifiers.