From pycyn@aol.com Wed Jun 21 13:19:07 2000 Return-Path: Received: (qmail 14515 invoked from network); 21 Jun 2000 20:19:05 -0000 Received: from unknown (10.1.10.26) by m4.onelist.org with QMQP; 21 Jun 2000 20:19:05 -0000 Received: from unknown (HELO imo11.mx.aol.com) (152.163.225.1) by mta1 with SMTP; 21 Jun 2000 20:19:05 -0000 Received: from Pycyn@aol.com by imo11.mx.aol.com (mail_out_v27.10.) id a.55.79230d7 (3929) for ; Wed, 21 Jun 2000 16:18:25 -0400 (EDT) Message-ID: <55.79230d7.26827d11@aol.com> Date: Wed, 21 Jun 2000 16:18:25 EDT Subject: RE Trivalent Logic To: lojban@egroups.com MIME-Version: 1.0 Content-Type: text/plain; charset="US-ASCII" Content-Transfer-Encoding: 7bit X-Mailer: AOL 3.0 16-bit for Windows sub 41 From: 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).