RE Trivalent Logic

[pycyn@aol.com]

The "Trivalence" of Aymara
[Disclaimer:  These comments are based on a paper on the
internet at
www.dt.fee.unicamp.br/~arpasi/biblio/igr/igr.html.  This
paper is an English translation of an original in (Bolivian)
Spanish.  The translator as much as admits that he is not
entirely comfortable in Bolivian (i.e., Aymara influenced)
Spanish.  He also clearly does not know logic terminology
very well and is occasionally confused by the original
author's neologisms or peculiar uses of ordinary words for
technical purposes -- or possibly by typos in the original
version (the original text at one point apparently had
"abdiccion," which makes no sense and which the translator
tries to correct to "abduccion," which makes no sense in the
context, though it is a word.  The context suggest
"abdicacion," but that requires seeing what is going on at
that point and I don't think the translator did.). The Aymara
texts are accompanied in most cases by Engish translations
of the original Spanish translations, some of which were
themselves translations from Latin, German, English or
Bolivian.  Finally, the net text is riddled with typos, some of
them at crucial places (the truth tables for connectives),
others even funny ("ro" for "m" countless times).  Under
these conditions, the following is a very tentative set of
comments, not only about Aymara, but also about the real
report on it.]

    The first thing to say is that there clearly is a system
(or several systems) of functors in Aymara and that treating
the functors as for a three-valued logic seems to bring a
order to them that does not appear in earlier treatments,
which tried to reduce them to something in Latin, Spanish,
German or English.  That still leaves the question of
whether that system (those systems) is (are) real in the
language, whether a truth-value sytem is represented, and
how good the representation is from a formal point of view.
    I'll pass over the first question quickly by noting the
author reports that authors for over 400 years have reported
and attempted to deal systematically with the patterns that
comprise the material dealt with, though their reports have
not consistently fit into the patterns here laid out.  the
present patterndoes fit the majority of reports on any given
pattern, however, and deviations ae often explainable in
terms of preconceptions of the earlier author.  
    But both the earlier authors and the present one do
not treat these patterns as truth value patterns  -- or not
exclusively.  Almost all talk about these patterns as much in
terms of  1) confidence levels (certainty, dubiety), 2)
presuppositions (proper since presuppositions met,
improper since not  -- plugs and filters and the like in a
Karttunen system), 3) modality (necessity, possibility,
impossibity), 4) probablility, or 5) plausibility.  And often
several of these undistinguished at once.  Since all of these
have at one time or another been offered as ways to make
sense of many-valued logics for bivalent heads, this may
simply consitute proof that this is a three-valued system
here.  Or it may be taken to mean that it is a three-valued
system but not a truth value system -- epistemic or
metaphysical but not realist.  Or it may be some of each.
    One rason for thinking that there may be more than
one system here is that, formally, the data gives a very
inefficient appearance at first glance.  Some truth functions
do not occur in the data -- and not just strange ones like
Tautology (true for every value of the component sentence),
which actually does occur, but useful ones like Determinate
(true if the component is true or false, false if in doubt).  On
the otehr hand, some functions are repesented several times,
up to four, and including as one of the basic functional
suffixes.  Since an adequate system could surely be done
with three suffixes (and, I seem to recall, actually with one if
you're willing to have moderately long strings -- with three
no string needs be longer than three), the use of nine basic
suffixes suggest that something more is involved.  I have not
done the linguistic work even with the small sample of texts
here to see whether the suffixes do break down into
mutually exclusive combinatory sets or several sets with
some general overlaps, so I don't know that there are
several systems.  I do know that the set of nine suffixes is
complete in the sense that every monadic trivalent function
can be expressed in them, including the ones not found in
the corpus.  But I do not know whether any of the potential
subsystems is complete.  
    The other source of the though that there may be
several systems is that the author talks about the same
function in very different ways when talking about it as
represented by a different suffix (string). The various ways
suggest several of the different things that trivalent values
have been taken to mean and seem to fit into a set of
patterns for about three different systems: some
combination of 1&3, some combination of 4 &5 and then
maybe 2.  Or maybe something entirely different.  Or maybe
just a redundant system.
    The suffixes are directly one-place operators,
mapping a single value to a new value.  But the same system
of suffixes can be used to generate the two-place connective
values -- a real source of efficiency at the next level. 
Unfortunately, at this point either the translation or the
typing goes to pieces comepletely and it is difficult -- maybe
impossible -- to work out the details of the system.  The
idea seems to be that connection p*q can be represented by
a combination of a singular connective on p, a singular
connective on q and singular applied to product of p and q. 
The three values are then summed (both product and sum
are Boolean-like, i.e., for the values1, 0, -1, 1+1 = -1,  
-1+-1 = 1).  This system would give 3^9 different
combinations of functions (27^3), however it does not give
3^9 different final functions, since each function is defined 9
times over: f1(p)+f2(q)+f3(pq), then this with one function
replaced by that one +1 and another by it-1, and then one
with all functions replaced by themselves +1 and another
wiht all replaced by themselves -1.  The text mentioned that
historical records show two connectives (which carry f3)
other than the one used today, so there may once have been
other formula for binary connectives.  But, even if the
system is not complete, 3^7 connectives is surely enough for
everyday use.  
    As for the lbization of all this, Guzman's translations
suggest that the main uses for the unary connectives is in the
area of evidentials or assurances: certainly, necessarily (not
the logical one), (subjective) probably, and so on.  I think
that the use of unary connectives with binaries might work
well for the binaries -- a modifier on each component and
one one the compound (conditionals, conjunctions and
disjunctions are cited).