Le samedi 22 février 2014 22:40:42 UTC+9, xorxes a écrit :
On Sat, Feb 22, 2014 at 5:02 AM, guskant
<gusni...@gmail.com> wrote:
Because {su'o mei} is neither a sequence of logical elements, nor expanded to a sequence of logical elements, a sentence including {su'o mei} itself cannot be a logical axiom or the equivalent. I call a sentence "tautology" only when it is expressed with a sequence of logical elements that is a logical axiom or the equivalent.
But we _are_ defining "su'o mei" (as well as all the other "su'o N mei" and "N mei") as logical elements! That's the whole point of what we're doing, isn't it? Why would you want to give "su'o mei" different meanings in differnet contexts?
"su'o mei" is just the tautological predicate. It has nothing to do with whether or not there are individuals. It is true of anything at all.
{ro'oi da su'o pa mei} alone cannot be expanded to logical elements only, (D1) (D2) neither, because a predicate {N mei} is not a logical element: {N mei} is a predicate that reflects natural number theory, not only predicate logic. They are _distributively_ not tautology.
{ro'oi da su'o pa mei}, (D1) and (D2) give _collectively_ a sequence of logical elements:
ro'oi da poi me ko'a ro'oi de poi me ko'a zo'u de me da
and it is not a logical axiom or the equivalent. {ro'oi da su'o pa mei}, (D1) and (D2) are _collectively_ not tautology.
(D1) and (D2) gives only an order to {su'o N mei} and {N mei}, and they don't give a meaning to the predicate {su'o pa mei}.
The starting point {ro'oi da su'o pa mei} gives indeed a meaning to {su'o pa mei}.
Any of them are fine. (D1-1) is only a "one-shot" definition of a particular ko'a in a particular universe of discourse defined by a speaker. It is not for general use.
But what does D1 even mean if you only know what "su'o mei" means when applied to a particular ko'a? According to D1
ko'a su'o re mei := su'o da poi me ko'a su'o de poi me ko'a zo'u ge da su'o mei gi nai de me da
How is that a complete definition of "ko'a su'o re mei", when there is an undefined term on the right hand side?
In all my definitions "ko'a" was intended as a place holder. They otherwise don't make sense as definitions of the predicates.
It seems that using "ko'a" as a place holder causes a problem.
I use {ko'a} as a plural constant, not as a place holder.
For a place holder, {ke'a} and {ce'u} are suitable, because they are free variables: such usage is not described in CLL, but it is useful at least in the current discussion.
When {ce'u} appears more than two times in a sequence of words, different sumti can be substituted for them, while only a common sumti can be substituted for {ke'a}s. For the current purpose, using {ke'a} is better.
Using {ke'a}, our definitions are described as follows:
(D1-7) ko'a su'o pa mei
(D1) ke'a su'o N mei := su'oi da poi me ke'a ku'o su'oi de poi me ke'a zo'u ge da su'o N-1 mei gi de na me da
(D2) ke'a N mei := ke'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
When (D1) and (D2) are applied to a particular sumti, ke'a are replaced with it. As for (D3), ke'a is in noi-clause, and it is already fixed to zo'e, and is not replaced with another sumti, of course.
Because (D1-7) defines only for {ko'a}, (D1) (D2) (D3) are valid only for sumti that involves a referent of {ko'a} such as {ko'e noi ko'a me ke'a}, {ko'i no'u ko'a jo'u ko'o} etc. (D1) (D2) (D3) are not used for other sumti unless (D1-7) is applied to one of the referents that is involved by the sumti.
Even with (D1-1b), "mi jo'u do su'o pa mei" is true.
(D1-1b) is also a "one-shot" definition defined by a speaker on a particular ko'a that is an individual, and is not applied generally.
It gives a meaning to {su'o pa mei} with a particular ko'a.
For example, suppose a speaker applies (D1-1b) to {mi}:
(D1-1b) mi su'o pa mei := su'oi da poi me mi ku'o ro'oi de poi me mi zo'u de me da ije da me de
Then {mi jo'u do} satisfies (D1) of N=2:
mi jo'u do su'o re mei
From (D1),
ganai ko'a su'o N mei gi ko'a su'o N-1 mei
is always true.
I didn't check your proof in detail, but it seems to me you must be be relying on D1-1, not just on D1. Otherwise both "su'o N mei" and "su'o N-1 mei" are undefined. With D1-1b in effect, the statement is false. From "mi jo'u do su'o re mei" we cannot conclude "mi jo'u do su'o pa mei" if "D1-1b" applies to "mi jo'u do".
I used only (D1) and logical axioms including transitivity of {me}. Any mention of {su'o pa mei} is not necessary for the proof.
As long as D1-1b applies only to "mi", and D1-1 applies to "mi jo'u do", yes. But why would you use different definitions of "su'o mei" in the same context?
If you need {su'o mei} for other sumti that does not involve {mi} in the same context, you must use (D1-7) or (D1-1b) to that sumti. For {ko'a noi naku mi me ke'a}, {ko'a su'o mei} is not defined otherwise.
For example, suppose that a speaker regards {lo nanba} is non-individual:
ro'oi da poi me lo nanba ku'o su'oi de poi me lo nanba zo'u de me da ijenai da me de
That is, the speaker regards a half of {lo nanba} is also {me lo nanba}.
Yes.
Even though there is no individual {lo nanba}, an _expression_ {N mei} is available with (D1-7) (D1) (D2) (D3).
No:
"lo nanba cu su'o pa mei" is true
"lo nanba cu su'o re mei" is true
"lo nanba cu su'o ci mei" is true
I call them {lo nanba xi re} and {lo nanba xi ci} respectively for convenience.
If
(D1-7) lo nanba xi pa cu su'o pa mei
is defined, and if {naku ge lo nanba xi pa cu me lo nanba xi re/ci gi naku lo nanba xi re/ci cu me lo nanba xi pa}, the first sentence is true, and the second and the third are false.
That is to say, if {(D1-7) lo nanba cu su'o pa mei} is defined, and if all the appearances of {lo nanba} have a common referent, the first sentence is true, and the second and the third are false.
The speaker arbitrarily fix a referent to be {lo pa nanba}. If another {lo nanba xi re} is given, {lo pa nanba jo'u lo nanba xi re} is {lo re nanba}.
If "lo pa nanba" satisfies D1-1 and D1 and it also satisfies "ro'oi da poi me lo pa nanba ku'o su'oi de poi me lo pa nanba zo'u de me da ijenai da me de", then it cannot satisfy D2.
(D1) is meaningless for N=1, because (D1)+(D2) for N=1 produces contradiction.
The meaning of {su'o pa mei} is defined by the starting point {ro'oi da su'o pa mei} XOR (D1-7), not by (D1).
(D1) is meaningful only for N>=2, in both procedures of starting with {ro'oi da su'o pa mei} and of using (D1-7).
When
(D1-7) lo nanba cu su'o pa mei
is defined, (D1-7), (D1) for N=2, (D2) and (D3) produce a meaningful {lo pa nanba}. (D1) for N=1 is not used here.
When another {ko'e noi nanba} is given, (D1) can be used for saying {lo nanba (ku) jo'u ko'e noi nanba cu su'o re mei}. Please note that the referent of the first {lo nanba} is different from ko'e.