Re: [lojban] RE:Trivalent Logics

[py...@aol.com]

I haven't checked xorxes' table to see whether it does give all the cases,
but I trust that it does. The assignments of functions to terms looks about
right and the means of using the functors in different contexts does recall
much of the Aymara approach (so far as i understand it). So we can probably
replicate a plausible three-valued system in lb for them what wants it
(complete with abbreviations for common functors).
I also haven't checked to see whether the system xorxes gives is minimal
(i.e., could we do it with fewer functors), but I suspect it is not -- as I
am sure that the Aymara system is not. xorxes' system lacks one interesting
feature of Aymara, that negation is not a primitive functor, but, since
negation is a given in lb, that would be hard to recreate, in spite of the
interesting thoughts it brings to mind. (In Aymara, negation is something
like "it is certain that it is controversial that," where certainty and
controversiality are primitive functors).
lb does not provide any natural way of upgrading this to a system of binary
connectives unless the gi's that got us into trouble the last time around can
be called to our aid. (I hope they -- or something else -- can be, since
being able to absorb a totally unexpected and odd system would be a nice
demonstration of some property or other than lb is supposed to have.)

[Jorge Llambias]

la pycyn cusku di'e

>The assignments of functions to terms looks about
>right and the means of using the functors in different contexts does recall
>much of the Aymara approach (so far as i understand it).

I realized something else about those assignments after I
posted it. The three assertions:

cai (1,-1,-1) necessarily
sai (1,0,0) probably
ru'e (1,1,-1) possibly

are the three that differ minimally from the simple
assertion (1,0,-1). Maybe that is why they are useful
and the easiest to understand. (The fourth minimal
variation, (0,0,-1) is not an assertion, as it doesn't
start with 1.)

>I also haven't checked to see whether the system xorxes gives is minimal
>(i.e., could we do it with fewer functors), but I suspect it is not -- as I
>am sure that the Aymara system is not.

It is not minimal. For example, {ru'e} is equivalent
to {naicainai}. (Possible = not necessarily not.)

But we don't want a minimal system because some functions
become unusably cumbersome.

>xorxes' system lacks one interesting
>feature of Aymara, that negation is not a primitive functor, but, since
>negation is a given in lb, that would be hard to recreate, in spite of the
>interesting thoughts it brings to mind.

I tried assigning (-1,1,0) to {nai} but it becomes too different
from the binary meaning of {nai}. In any case, {cu'i} = (0,1,-1)
is -1. What would that be? A counter-negation?

>(In Aymara, negation is something
>like "it is certain that it is controversial that," where certainty and
>controversiality are primitive functors).

Isn't (-1,1,0) just plain controversial? I think "necessarily
controversial" is (-1,1,-1).

With my proposal, "controversial", or trivalent negation, comes
out as {naicu'i}, something like "doubtful that not".

(-1,1,-1) is {cu'icai}, "necessarily doubtful".

>lb does not provide any natural way of upgrading this to a system of binary
>connectives unless the gi's that got us into trouble the last time around
>can
>be called to our aid.

Maybe it does: do'egi<f1> ... gi<f2> ... vau<f3>

>(I hope they -- or something else -- can be, since
>being able to absorb a totally unexpected and odd system would be a nice
>demonstration of some property or other than lb is supposed to have.)

Yes, it is working out very nicely.

co'o mi'e xorxes

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[Elrond]

Sorry to disturb here, but although I do not clearly understand what is
this trivalent logical system in the first place, I am very interested in
discovering how this could improve/modify the way a lojbanist (like me)
can build his predications.

In other words, does any document exist on this topic that is both
readable by linguistics newcomers and gives a chance to us mortals down
there to open our mental skills ?

I mean, seriously, such a "tool" (trivalent logic) could lead to very
interesting results if handled by average lojban dudes... Unfortunately
for the moment the topic has only been covered in rather technicals terms.

Any trivalent-logic-for-beginners reading somewhere ?

Thanks for your attention =)

co'o mi'e rafael

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[Jorge Llambias]

Ok, this is what I propose for a trivalent system of
unary operators in lojban:

cai (1,-1,-1)
sai (1,0,0)
ru'e (1,1,-1)
cu'i (0,1,-1)
nai (-1,0,1)

I think {cai}, {ru'e} and {nai} are the easiest to accept.
(-1,0,1) is the most obvious generalization of {nai} from
binary. (1,-1,-1) corresponds to the strongest assertion
(certainty or necessity, depending on what system we use
it on) so it has to be {cai}. (1,1,-1) is possibility or
a weak assertion, so I think {ru'e} fits well.
Now, (1,0,0) is also an assertion, but not as strong
as certainty, something like "this is how it is, but I
give no guarantees". I think {sai} can work for that.
And finally, {cu'i} is for neutral. (0,1,-1) is not
absolutely neutral, it is uncertainty with a bent towards
assertion, but it is the closest to neutral and we do need
it to generate others, so {cu'i} has to be it.

With those 5 it is possible to generate all 27 unary
functors, with at most three of them. For example,
(0,0,1) is {naisai}, (-1,1,-1) is {cu'icai}, (0,0,0)
is {sairu'ecu'i} (among several possibilities), etc.
Only 8 of the 27 need three basic functions, the rest
can be formed from just two.

The nice thing about this system is that it can be used
for different things. For example, for a strictly logical
system we just attach them to {ja'a}, using {ja'acai},
{ja'acu'inai}, etc. And {na}={ja'anai}, so some of them
can be shortened.

But they can also be used for evidentials, attaching them
to {ju'a} for example. Then again there might be some
shortcuts, like {ju'acai} might be {za'a} and {ju'asairu'e}
might be {ca'e}, etc, but we know that we can get all
27 of them from just the simplest, which is always (1,0,-1)
and doesn't take any modifier. We can use {la'a} as the
basis for the probabiliy set, etc.

Would that work?

co'o mi'e xorxes

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[Jorge Llambias]

la pycyn cusku di'e

>Ooops! For functional completeness the system needs min(x,y), too and that
>seems harder to get. Once it is gotten, however, it alone generates all of
>the connectives (binary, unary, more-ary), or rather the Sheffer function,
>min(x,y)+1, does.

Is there a simple :) way to see that this is true?

It seems to me that max(x,y) is the most useful "or", and
min(x,y) is the most useful "and", so it is nice that they
are relatively straightforward to generate.

>I think (disclaimer) that min can be defined with f1=f2 :-1 for -1, 0
>otherwise and f3 as 1for 1, 0 otherwise. But my head is not functioning
>well
>in -1,0,1 arithmetic at the moment.

It does indeed give the minimum. So this is really the same
situation we have in Lojban with respect to 3-way connectives,
right? They can all be generated but not without repeating
some of the arguments in some cases.

Now the question is, do we have anything like a complete
three-value unary system in Lojban? Obviously not a logic
system (we only have na and ja'a there) but maybe with some
set of attitudinals?

{ju'a} or {je'u} (1,0,-1)
{ju'o} (1,-1,-1)
{la'a} (1,0,0)
{ca'e} or {se'o} or {ai} (1,1,1)
{pe'i} (0,0,0) ??

Could we produce some coherent system out of what we have?

co'o mi'e xorxes

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[py...@aol.com]

In a message dated 00-06-25 22:26:50 EDT, xorxes writes:

<< I tried assigning (-1,1,0) to {nai} but it becomes too different
from the binary meaning of {nai}. In any case, {cu'i} = (0,1,-1)
is -1. What would that be? A counter-negation? >>

What you tried for {nai} was a Post or circular negation, that is +1 to each
value. It places an important role in the theory of many-value logics, as
the preferred negation when it comes to looking for minimal connective sets.
The Sheffer function is always this negation of the min function and it is
this negation that gives the analog of double negation (triple negation in
the trivalent case) -- you go round the circle one complete turn by doing n
negations. Your {nai} always gives double negation, even for zillion-valued
logics. Although I can't find anyone who messed with it, your {cu'i} is also
a circular negation, just going the other direction. I wonder if it might
not play a role in defining the other adequate single connective, the Peirce
(amphec) function, NOR. Might that be max(x,y)-1? It is in the 2 case,
which doesn't prove much, and looks at a glance to be in 3 and also to be
adequate.

The rest of the system looks very promising (Hell, very done, except for
usage thoughts -- especially for the binary connectives).

[py...@aol.com]

In a message dated 00-06-24 20:09:46 EDT, xorxes writes:

<< >Ooops! For functional completeness the system needs min(x,y), too and
that
>seems harder to get. Once it is gotten, however, it alone generates all of
>the connectives (binary, unary, more-ary), or rather the Sheffer function,
>min(x,y)+1, does.

Is there a simple :) way to see that this is true? >>

I think the short answer is "No," but, as you note, min is a natural "and"
and, it turns out, +1 is _a _ natural negation, so that we have a kind of
NAND here. If you feel comfy generating all 2-value connectives from NAND (I
have met people who do -- I am not one of them), then this goes just the same
(but it takes three negations to get back to where you started, of course).

<<So this is really the same
situation we have in Lojban with respect to 3-way connectives,
right? They can all be generated but not without repeating
some of the arguments in some cases.>>

Yes. I have not looked at cases that are generated directly to see if there
are any really significant things missing. Everything I could think of (how
many can that be in three-valued logic?) -- various generalizations of binary
conditionals, disjunction, conjunctions, negations, equivalences, fixed-value
functions and the like are easy to get to, though, with the basic formula
(and usually several dozen different ways, given the different ways each of
the unary functions can be presented).

<<Now the question is, do we have anything like a complete
three-value unary system in Lojban? Obviously not a logic
system (we only have na and ja'a there) but maybe with some
set of attitudinals?

{ju'a} or {je'u} (1,0,-1)
{ju'o} (1,-1,-1)
{la'a} (1,0,0)
{ca'e} or {se'o} or {ai} (1,1,1)
{pe'i} (0,0,0) ??

Could we produce some coherent system out of what we have?>>

I suspect we have more than enough to work with. As I noted, there seem to
be several system we could use -- evidentials, confidentials (I forget the
regular name, but you get the idea), probability, necessity-possibility,
concern (there is one unary that seems to mean, "I don't know and I don't
give a damn"). Whether these systems are separated and each complete in
Aymara, I haven't worked out yet (and probably can't given the state of the
paper) but we can fiddle a bit in Lojban. There is not, I think, unary
Sheffer function nor much info about minimal conditions for a complete set
(bivalent systems regularly got to a binary connective for (1,1) and (0,0)
and I think even combinatorics does).
I am not sure that I agree with your assignment of values above, but then I
am not sure I understand what many of these connectives (or these "truth"
values) are meant to mean either. I would, for example, have taken {pe'i} to
be (1,1,-1)-true if true or in doubt, false if false. (0,0,0) is what
Guzman assigns to the "I don't care" marker, maybe comparable to answers to
"Have you stopped beating your wife?"

[py...@aol.com]

I'm not sure whether there is anything readable about three-valued (or any
more-than-two-valued) logic -- I have trouble with most of them and they're
part of my job (but I have a strong bivalent prejudice, too). Most books on
the topic or on Philosophy of Logic or Non-Standard (Deviant) Logics quickly
get off into either constructing a calculus for the notion or trying to make
sense (or show you can't make sense) of the third (and sometimes of the first
two as well) truth values.
In Aymara (as I understand it from the text), the point is less about truth
than about certainty and commitment. So the three positions are roughly
thoroughly committed to, thoroughly committed against, and uncommitted
(leaving the notion of commitment vague). Different functors indicate
different kinds of commitment depending upon the basic situation, the
commitment to the unmarked claim. If I am committed to p, then I am
committed to necessarily p, but if I am not committed to p (either
uncommitted or committed against) the I am committed against necessarily p.
And so on through all the conectives.
As a tool, this could make for finer distinctions in the commitment
categories of lb, starting with making the category itself much clearer.

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[Gleki Arxokuna]

doi xorxes xu do pu troci lo nu pilno lo cimei logji xusycmuma'o ca'o lo nu tavla?

[Jorge Llambías]

On Sat, Aug 4, 2012 at 10:43 AM, Gleki Arxokuna
<gleki.is...@gmail.com> wrote:
> doi xorxes xu do pu troci lo nu pilno lo cimei logji xusycmuma'o ca'o lo nu
> tavla?

.i no roi go'i

mu'o mi'e xorxes

[John E Clifford]

Yeah, with the unlikely exception of Aymara, it is hard to imagine what this would look like in practice.  Juggling even the six basics xorxes offers (or the minimal three) probably pushes one's understanding pretty far, unless you have a very concrete notion of what the third value is (and then you may be surprised at what the system offers you in some cases).  We seem to be better with fuzzy (but let's don't get started on that!) than clear-cut triads or more when it comes to dealing with the world.

---

From: Jorge Llambías <jjlla...@gmail.com>
To: loj...@googlegroups.com
Sent: Saturday, August 4, 2012 11:26 AM
Subject: Re: [lojban] RE:Trivalent Logics

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[Gleki Arxokuna]

OK. May be a short tutorial for newbies (non-experts in logic) what this is all about?

May be some kinda table with normal evidentials and your proposals in another?

http://aymara.org/biblio/html/igr/igr3.html  looks perfect but incomprehensible for ordinary people.

I want something that a 5y.o. could understand.

I BELIEVE THAT IT'S ONE OF FEW GREAT IDEAS THAT LOJBAN CURRENTLY LACKS.

So actually I'm quite interested in it.

Although I don't like redefining the meaning of any cmavo including CAI.

[Gleki Arxokuna]

On Saturday, August 4, 2012 8:48:08 PM UTC+4, clifford wrote:

Yeah, with the unlikely exception of Aymara, it is hard to imagine what this would look like in practice.

Why not? Millions of people use it while speaking the language that is highly naturalistic.

Just relexing Aymara to Lojban should be quite easy. It's all the matter of how to teach people to use new but better cmavo.

 

  Juggling even the six basics xorxes offers (or the minimal three) probably pushes one's understanding pretty far, unless you have a very concrete notion of what the third value is (and then you may be surprised at what the system offers you in some cases).  We seem to be better with fuzzy (but let's don't get started on that!) than clear-cut triads or more when it comes to dealing with the world.

---

From: Jorge Llambías <jjlla...@gmail.com>
To: loj...@googlegroups.com
Sent: Saturday, August 4, 2012 11:26 AM
Subject: Re: [lojban] RE:Trivalent Logics

On Sat, Aug 4, 2012 at 10:43 AM, Gleki Arxokuna
<gleki.is...@gmail.com> wrote:
> doi xorxes xu do pu troci lo nu pilno lo cimei logji xusycmuma'o ca'o lo nu
> tavla?

.i no roi go'i

mu'o mi'e xorxes

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[John E. Clifford]

I'm not sure what the Aymara population is, but linguists -- including a number of native speakers of Aymara who have gone on to graduate degrees in linguistics -- agree that Guzman's interpretation of the data (much of which they say he also has wrong) is incorrect.  In short, Aymara does not use trivalent logic, though it does have a number of --to us -- unfamiliar epistemic modalities. 

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[djandus]

I knew nothing of the existence of trivalent logic before this thread, but there is something I can say.

On Sunday, August 5, 2012 3:18:46 AM UTC-5, Gleki Arxokuna wrote:

OK. May be a short tutorial for newbies (non-experts in logic) what this is all about?

May be some kinda table with normal evidentials and your proposals in another?

http://aymara.org/biblio/html/igr/igr3.html  looks perfect but incomprehensible for ordinary ponies. 

I want something that a 5y.o. could understand.

For those interested in diving in, I started getting an idea by instead glancing at wikiponia and then reading over the Aymara document. Also for those diving in, it's useful to know that the author of that other link starts with an introduction, proceeds to some basic description of bivalent logic when the bullets show up, gets into the nitty-gritty description of bivalent logic when the tables show up, and finally gets to the trivalent stuff with "Representation of trivalent truth-values"

I BELIEVE THAT IT'S ONE OF FEW GREAT IDEAS THAT LOJBAN CURRENTLY LACKS.

So actually I'm quite interested in it.

Although I don't like redefining the meaning of any cmavo including CAI.

I would like to mention here that rather than redefining CAI, you could:

• define an experimental cmavo as "a trivalent logic marker" and use CAI on it as one was proposing to use CAI on ja'a
  
• define a couple of base experimental cmavo to represent a few of the base trivalent logics, as was done in lojban with the bivalent logical connectives, and then define how CAI alter them.

Past that, I actually have a fundamental question that, when answered, should probably help a newbie jump into discussion. What are the examples (in lojban) of instances what these terms refer to: modal functor, connective (two-variable) functor.

Honestly, I find it something that will not be of particular use to me, but it sounds pretty cool if {rodo} manage to get it working for yourselves. Thus, make an orthogonal usage of Lojban, please.

[Gleki Arxokuna]

On Sunday, August 5, 2012 5:49:49 PM UTC+4, clifford wrote:

I'm not sure what the Aymara population is, but linguists -- including a number of native speakers of Aymara who have gone on to graduate degrees in linguistics -- agree that Guzman's interpretation of the data (much of which they say he also has wrong) is incorrect.  In short, Aymara does not use trivalent logic, though it does have a number of --to us -- unfamiliar epistemic modalities. 

If so the task to implement trivalent logic is no longer of utter importance for me. I thought that the inventors of Aymara were much wiser.

uinai they were not. 

ta'o Why implementing this logic won't solve any problems? What can solve them? Quaternary logic? Do you have any thoughts of how to implement it?

[Gleki Arxokuna]

Imagine or not but the whole Lojban is of no use for me. And for most inhabitants of the Earth. 

but it sounds pretty cool if {rodo} manage to get it working for yourselves. Thus, make an orthogonal usage of Lojban, please.

If I only knew how to explain all that stuff in plain language. I understand it in formulae but when it comes to choosing words in real conversation  it'll be a nightmare for me.

I guess we just need to produce a bunch of sentences and learn them by rot in order to use these new cmavo IRL.

[John E Clifford]

Guzman's claim (not in the monograph so much) that Aymara is a language invented by van Daniken gods is also at least suspect (there are other related languages in the area, with all the expected diachronic connections).

What problems a trivalent or other polyvalent logic will solve, if any, is pretty much to the imagination of the person setting up the system and typically guides his choices for which of 19,682 binary functions he picks for OR, AND, and IF (and which of the 27 unary is NOT).  I suppose the favorite is future contingents, followed by other epistemic/causal puzzles.  The record of  accepted success is limited (nil, I think), though some issues have been clarified a bit.  As the cases suggest, various kinds of modal logics seem to offer better hope (though without much better success so far).

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From: Gleki Arxokuna <gleki.is...@gmail.com>
To: loj...@googlegroups.com
Sent: Monday, August 6, 2012 10:20 AM

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[John E Clifford]

Happily, you don't need to know much logic to use a logical language (I cite Lojbab as my classic case).  It is enough to have the tools needed and know how to use them; what they mean in some formal sense is not important except as an occasional guide about usage.  We have, for example, gotten along fairly well with the bivalent IF (ganai gi) for 50 years, with only occasional worries in ordinary conversation.  In more technical discussions, we have worked out some modal solutions, which are not commonly used nor well understood.  If the need for these become acute, a quick lesson or two should suffice to get the usage right, whatever the technical completeness is.  That is, Lojban is buying symbols and terminology, not complete theories (the properties of time -- finite, infinite, circular, dense, discrete, linear, branching, and so on -- don't affect how tense work or even how they are interpreted). 

---

From: Gleki Arxokuna <gleki.is...@gmail.com>
To: loj...@googlegroups.com

Cc: jjlla...@gmail.com
Sent: Monday, August 6, 2012 10:24 AM

Subject: Re: [lojban] RE:Trivalent Logics

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