4.1 The kimsaku of a functor

As has already been explained in this book, "functor" means the function p(x) which is dependent on the propositional variable x. To facilitate the consistent use of truth-tables for modal and connective statements, let us analyze the following triad of ordered values:

P(1) P(0) P(-1)

The order chosen for x = 1, 0, and -1 (affirmation, symmetrical doubt, and negation) is arbitrary. However, this order must be respected so that the truth-tables in this book may be compared. Thus, the above triads will be called "kimsaku of statement p(x)", and written:

P(x) = (Pl P2 P3)

where P1 = P(1), P2 = P(0) and P3 = P(-1)

If "S" is the Aymara suffix, either simple or compound, which generates the statement having p(x) as a functor, then this will be written:

x.S = (Pl P2 P3)

In the case of two-variable connective statements p(x,y), one does not have a triad, but a 9-element bitriad, which will be called "päkimsaku". The following order will always be respected:

x	=	1 0 -1	1 0 -1	1 0 -1
y	=	1 1 1	0 0 0	-1 -1 -1

päkimsaku p(x,y) = p(1,1) p(0,1) p(-1,1) p(1,0) p(0,0) p(-1,0) p(1,-1) p(0,-1) p(-1,-1)

Note: In Aymara, "kimsa" means "three"; "paya" means "two"; and "pä" can be translated using "a pair (of)" or by the prefix "bi".

Two statements, P(x) and q(x) are said to be logical synonyms whenever,

(Pl P2 P3) = (ql q2 q3)

Two statements, P(x) and q(x), are said to be logical opposites if one negates both the modus and the dictum of the other statement, i.e., whenever,

(P1 P2 P3) = (-ql -q2 -q3)

for example:

"it is possible that it will rain" is the opposite of "it is impossible that it will rain." :

(1 1 -1) is the opposite of (-1 -1 1)

Two statements, p(x) and q(x), are said to be logical "antonyms" when one negates only the dictum of the other statement, in other words when

(Pl P2 P3) = (Q3 Q2 41)

for example:

"it might rain" is the antonym of "it might not rain"
(1 1 0) is the antonym of (0 1 1)

Note: In Aymara there are no homonymous statements, i.e., statements having two or more different logical functions. Thus, a suffix, either simple or compound, can have only one kimsaku. However, because of logical synonymy, a kimsaku can be generated by different suffixes; nonetheless, simple suffixes have no synonyms.

4.2 Method of logical analysis to determine of a statement

The values of the kimsaku (pl p2 p3) corresponding to any modal statement p(x), can be found by following the steps set out below:

1. It is assumed that the modal statement originates in a dialogue between a proponent, who makes an amodal statement x (having no modus, only dictum), and an opponent, who made a modal statement p(x) (having modus and dictum) about the same matter (same dictum), expressed by the Aymara suffix S.

2. The proponent's amodal statement is always assigned the following kimsaku:

• x = (1 0 -1)

• p(x) = (pl p2 p3)

3. To determine the values of pl, p2, and p3, the proponent's and opponent's statements are compared; this results in the following determinations:

determination of pl:

if when the proponent is right (x=1), we have: -the opponent is right (that there is positive certitude about x), then pl = 1; -it cannot be known whether the opponent is right, then pl = 0; -the opponent is not right; then pl = -1.

determination of p2:

if when it cannot be known whether the proponent is right (x = 0), we have: -the opponent is right (that there is doubt about x); then p2 = 1; -it cannot be known whether the opponent is right or wrong, then P2 = 0; -the opponent is not right; then p2 = -1.

determination of p3:

if when the proponent is not right (x = -1), we have: -the opponent is right (that there is negative certitude about x), then p3 = 1. -if it cannot be known whether the opponent is right, then p3 = 0; -the opponent is not right; then, p3 = -1.

Before the general method for determining the kimsakus can be applied, one must consider some aspects of the nature of modal statements which will greatly assist in obtaining their triads.

First, it must be remembered that in a trivalent system of logic the maximum number of simple modal statements is 27, that is to say, 3 ** 3. In a bivalent system there are only 4, i.e., 2 ** 2.

Second, with the exception of the triad (0 0 0), which is its own opposite, the remaining triads form 13 pairs in which each one is the opposite of another (the term "opposite" must not be confused with "antonym.")

It must also be pointed out that truth-values are equidistant from the center, which creates a certain symmetry. For instance, (1 1 0) is symmetric to (0 1 1). When the meanings of statements are analyzed, a certain degree of logical symmetry is also apparent with respect to the concept of "logical antonymy".

4.3 Determination of kimsakus of asseveration

In Aymara, there are approximately twenty modal statements which are used in everyday language. It is difficuIt to tell whether any of the27 triads of trivalent logic are not being used. This book concentrates on the most frequent statements, in order to determine their kimsakus. This material seems to demonstrate that Aymara logic is trivalent, and to show the strictly mathematical manner in which the logical suffixes of this language have been defined.

4.3.1 Amodal kimsaku x = (1 0 -1) (Affirmation)

This amodal kimsaku establishes the order in which comparisons will be made in this book to determine the truth-values pl, p2, and p3 of any modal kimsaku.

Example: x =	"sarta"
	"you went"

4.3.2 Determination of x.wa = (1 0 -1) (Reliability)

The suffix "wa" emphasizes that statement x is reliable but does not involve a logical modality. It is a unit operator, which does not transform the functor.

Examples:
x.wa =	"sartawa"
	"/you/ have gone"
x.wa =	"jumaj roanqtawa"
	"you have eaten"
x.wa =	"jumawa manqtaja"
	"you are (the one who) has eaten"

The meaning added by the suffix "wa" is perfective rather than modal. The examples above show that when wa is added to the pronoun rather than to the verb, emphasis is placed on the actor rather than on the action; however, the fact that "someone has eaten" (the dictum) is not affected modally.

Comments:

(ETA-87): "WA is the most important agglutinator, and is a constituent element in the formation of a great many phrases and sentences." "The functions of WA are equivalent to those of the verbs "to have" and "to be"; this suffix is always used at the end of a sentence."

(MJHB-413) "WA marks the sentence as affirmative and/or personal knowledge." "WA does not occur with the suppositional." (quoted in English by IGR) For a better understanding of the use of the suffix wa and its relationship to the subordinating element ja, the reader is urged to consult existing Aymara grammars, since these are the most widely used and most frequently studied suffixes. However, a few grammatical aspects are unclear, since specialists cannot agree on whether these suffixes have functions equivalent to those of Spanish auxiliary-verbs.

4.3.3 Determination of x.pi = (1 -1 -1) (Certitude)

There is no doubt that this statement corresponds to the modality of logical necessity, or certitude, of classical modal logic.

Examples:

(ETA-288) x.pi	=	"nayapi sarayatja" "yo pues (soy quien) hube ido." "I for sure (it is I (who) had gone"
(ETA-288) x.pi	=	"sarayatpi" "(yo) hube ido pues" "(I) had gone for sure"
(ETA-291) x.pi	=	"çuraninapapi," "(él) tiene que ir a darle pues." "(he) has to go give him for sure"
(HR2-102)	=	"aka çaullt munta?" "este pescado quieres?" "do you want this fish?" x.pi = "jisa, uka çaullpi muntja" "si, ese pescado pues quiero" "yes, I want that fish for sure"
(JEE-82)		Ritisa? quien es? "who is it?" "pedropi" "Pedro (es) pues" "It (is) Peter for sure" x.pi = "akapi" "este(es) pues" "this (is) for sure"

These examples show that besides emphasizing perfection, pi is also a strong modal emphasizer which excludes any possibility of doubt. The suffix pi implies that the verb action is executed "without fail." The suffix "pi" can replace "wa"; when it does it also takes on its, functions as an auxiliary verb (be or have). Thus, the statement x.pi is an asseveration (pl = 1), which excludes the possibility of any uncertainty (p2 = -1) and, of course, excludes the negation (p3 = -1); this makes it easy to determine its kimsaku (1 -1 -1);

Comments:

Ebbing classified pi among the adverbs. Ross said that this suffix is the generator of the "confirmative mood"

(HR-102). "The confirmative mood is used to confirm the truth of something suggested by the listener, or which is evident to him. The primary enclitic pi is always used with this mood."

In the Aymarized Spanish spoken in Bolivia, especially in La Paz, the mood with the suffix pi is so common, that people end their sentences with "pss", for example: "está viniendo pss" /"he is coming PSS"/ (he is coming certainly). pi can also be followed by ni (from the auxiliary verb "have to"), forming the suffix "pini". Example: "sarjepiniwa" (ya se ha tenido que ir siempre /he has already had to leave, already/ (1). In such sentences, "siempre"/always" does not have its usual meaning; it is an idiom indicating logical necessity, certitude. The suffix "puni" is mentioned in grammars as an equivalent; however, "puni" indicates the obligation of having to dosomething. For example:

jutatapuni (you must come pues /for sure/)

4.3.4 Determination of x.su = (1 1 -1) (Possibility)

This is the modality of possibility, i.e., x.su denotes "it is possible that x", or "it may be that x". The truth-table, or kimsaku of x.su corresponds to Tarski's modal function Mx (JL-75).

This modality is generated by the suffix su, often linked to the suffix indicating the listener, but not the speaker. This has had an impact on Aymaranized Spanish sentences: reflexive pronouns are very common, especially in La Paz markets.

Examples:

	x.su =	"alaña munasmawa" 'pueda que te quieras comprarte' (pueda que quieras comprar) /You yourself may want to buy it for yourself/ /You may want to buy it/
(JJE-224)	x.'su =	'jupaj apaspawa uka' (él se podria llevar eso) 'él podria llevar eso' /He himself could carry that/ /He could carry that/
(ETA-126)	x.su =	'nayawa lurirista' (using the actor) 'yo haria' /I would do/
(LBO-34)	x.'su =	'yatiçaspa' '(ojalá) aquel enseñara' '(Would to God) that person taught'
(LBO-35)	x."su =	'yatiçanisu' 'él habria de(ir a) enseñar' /He would have to (go to) teach/

These examples show that this modality can be easily identified. It is an asseveration, so pl = 1. However, there is an element of doubt, since the verb action is only potentially true, then p2 = 1; as the impossibility is excluded, p3 = -1: This results in the well-known triad of possibility (1 1 -1). Aymara scholars do not agree on the number of grammatical moods which can be used for the modality of possibility involving , the suffix su. In the opinion of the famous German linguist Middendorf, there are two moods: the optative and the potential. Ebbing and all those who follow Bertonio's schema of paradigms believe there is only one, which they refer to as either optative or potential. However, they are not fully convinced that the ending s which appears in those moods really corresponds to the suffix su; in some examples they suggest it corresponds to sa. When analyzing the optative mode, Bertonio himself (LBO-35) mentioned vaguely the suffixes su and çu. It should be pointed out that, in Quechua, the suffix çu is related to the concept of possibility, for example "payçusina" (I think it is him). (YSS-121).

In any case, given the logical consistency of Aymara syntaz, one can chance agreeing with Middendorf that there are really two moods for the modality of possibility: the optative and the potential, and that s never corresponds to sa, because the suffix sa has a well-established logical function, completely different from that of possibility, as will be seen below. It would be the only suffix among over a hundred to have more than one logical function. This would be a remarkable inconsistency, since Aymara suffixes never produce logical homonymy. For this reason, the statement of possibility will be symbolized by x.su regardless of whether, in the translation, the Spanish optative, potential, or Subjunctive moods are required. To reassure the demanding reader who might find the above classification of the suffix su arbitrary, I must say in advance that the mathematical analysis of Aymara logic proves that the suffixes sa and su have different well-defined algebraic characteristics.

Comments:

(E4JM-198) "Vom Optativ: Dieser Modus verdient in der Tat seinen Namen, da seine Formen, ohne weitere Zusätze, wie z.B. in unserer Sprache, 'wenn doch, ich möchte', den Wunsch ausdrücken". (About the Optative: This mood really deserves its name, because its forms, without any additions, express desire, for example, in our language /German/:

"if it is so, I would like to."

(EWM-198) "Der Optativ steht in Konditionalsätzen, wenn die Bedingung wohl als möglich, aber nicht als wahrscheinlich gedacht wird, oder wenn deren Erfüllung überhaupt unmaglich ist, da sie in der vergangenen Zeit häte stattfinden müssen." (The Optative appears in condition al sentences when the condition is considered to be possible, but not probable, or when its fulfilment is totally impossible, because it should have happened in the past.")

(EWM-199) "Der Optativ steht in Konzessivsätzen, in welche etwas Mögliches oder auch Unmögliches eingeräumt wird" (The Optative appears in concessive sentences, in which it is admitted that something can be both possible as well as impossible)

(EWM-199) "Vom Konditional: Dieser Modus kommt in schsatz der Bedingungsätze zur Anwendung, in welchen ausgesagt wird, was geschehen sein würde, wenn die im Vordersatz ausgedrückten möglichen oder unmöglichen Bedingungen erfülli worden wären. (About the Conditional /IGR adds "Potential" in Spanish/: this mood appears in the post-statement in conditional sentences, in which one indicates what would have happened, should the possible or impossible conditions expressed in the ante-statement have occurred."

4.3.5 Determination of x.ki = (1 0 0) (Likelihood)

This modality has no equivalent in other languages. It is typically Aymara, and it is a key element in the psychology of the Altiplano Aymara-thinking people who have altered the meaning of the Spanish word "nomás", to try to express the modality generated by the suffix "ki". "nomás" is used at the end of the sentence , to indicate "probably."

Examples:

(ETA-219)	x.ki=	'sañakí' 'decir nomás' /Say probably/
(ETA-225)	x.ki =	'luranikiwa' 'he de ir a hacer nomás' /I must go do it probably/

x.ki practically means "it is probable that x"; therefore, the best technical name for this modality may be "likelihood."

It is a sentence in which the speaker expresses his good will, but without committing himself, because he is not certain of its impossibility. It does not even indicate doubt. Therefore, since it is an assertion, pl= 1. However, the other two cases are doubtful, thus, p2 = 0, and p3 = 0. So, its kimsaku is (1 0 0).

The suffix "ki" might be called the suffix of "Aymara possibility", which differs from classical or conditional possibility in one respect: the uncertainty concerning doubt is left open (p2 = 0). As was pointed out in Section 4.3.4, the suffix "su" indicates conditional possibility, in which there is some doabt (p2 = 1). It is not by pure chance that the suffix ki appears in the core of the word "waki" (possible). Thus, we have the following verbs:

• wakisiña ("serse" posible) /idiomatic for "to be possible"/
• wakisiyaña (to become possible = can)

For example:

(LB2-148)

sarañama wakisiwa (it is possible that you leave) saraña wakisiyatawa (you may leave)

In his grammar, Bertonio has explained very well the subtle difference between these two closely related verbs: (LBO-106-108) "The first manner (wakisiña) involves possibility or impossibility due to an extrinsic factor, such as illness or some other kind of impediment; the second (wakisiyaña) involves possibility or impossibility, either intrinsic or voluntary."

Although Bertonio explained very clearly the peculiarities of the verb "possum", both in his grammar and in his dictionary, modern grammars do not use it to translate "can", They use only the verb "yatisiña" (to be used to). This is because in Spanish "poder" is a synonym of "force" or "power" and implies certainty that something will be done, whereas the Aymara verb "wakisiyaña' implies the carrying out of a possibility. In the words "wakisiña" and "wakisiyaña", both verbs are made reflexive by the suffix "si". In Spanish the verb "can" is not reflexive; nevertheless, Aymara-thinking people who speak Spanish use it as a reflexive verb, producing sentences such as:

	'te puedes pegarme si quieres, igual no te obedeceré.' /You can yourself hit me; I won't obey all the name/
and:
	'me puedo peinarme' /to myself I can comb my hair/

The verb wakisiyaña (to become possible = can) must not be confused with the verb "wakiçaña" to which the suffix "cha" can be added to express "to make possible" or "to facilitate".

Ludovico Bertonio understood the concept of "limited" affirmation involved in the statement x.ki; thus, in his Spanish translation he used the term "ni otra cosa más" /not another thing/, or "solamente" /only/, which later became "no más" /no more/; the word "no más" did not fully convey the meaning of limitation in the statement, so the idiom "nomás" appeared at the end of sentences, see (LBO-291).

Evidently, the word "only" is close to the sense of likelihood. For example, "estará cansado solamente" /He will only be tired/ is equivalent to "it is likely that He wil l be tired" (and not something else). However, the word "solamente" may have a different connotation in Spanish and imply exclusiveness.

Some translators of religious materials have interpreted the suffix ki this way, and have produced sentences like: "ma Diosaki utjija" when they meant to say "there is only one God"; in fact, they were saying "it is likely there is a God", or, in the popular variety of Spanish spoken in this region: "hay un Dios nomás" /there is a God, probably/

Comments:

(ETA-225) "ki" expresses decision and urgency in an inmediate action; it also denotes avoidance of responsibility, or satisfaction of a desire, or interest. (ETA-219) "ki" is equivalent to the adverb "nomás" (a Bolivian idiom without negative meaning) which generally indicates sufficiency, enough, continuity, or plea."

This quote shows that Tarifa thinks according to a trivalent system of logic, because he states specifically that this suffix has no negative meaning; someone who thought according to a bivalent system of logic would simply have said "affirmation", because if it is not a negation, then it is necessarily an affirmation. However, as we have seen, for a trivalent mind there are many possibilities. Evidently, x.ki is a weak assertion indicating a doubt about its impossibility, but without openly expressing doubt; thus, the value of p2 is not 1, as with the statements of doubt and plausibility.

(HRl-45) "ki" in imperative forms of the verb indicaces encouragement" /quoted in English by IGR/. Ross gives a typical example of this use: 'saraskakim" ('esté yendo nomás' /you may go/ /go ahead/.)

4.3.6 Determination of x.ka = (1 -1 0) (Evidance)

The suffix "/ka" is usually combined with other suffixes. It is used alone in present tense (gerund) and compound statements. Grammar textbooks rarely discuss its logical function when it stands alone.

Examples:

(IIM-G7)	x.ka =	'jumaj sarkam ançiçaw nayaj jiktanim' 'Tú estate yendo, yo ahorita ya te alcanzaré' /You may go; I'll catch up with you/
(IIM-G7)	x.ka =	'niyaw aymar yateqkta.' 'ya estoy aprendiendo aymara' /I am already learning Aymara
(LBO-270)	x.ka =	'mandkaña' 'estar comiendo' 'to be eating'
(LBO-270)	x.ka =	'misskaña' 'estar diciendo misa' /to be saying mass"

In this modal notion there is some evidence that the statement is true. This is the difference between the modal notions of Potential and Evidence. Thus, in compound statements, the modality of evidence is taken as a condition for a second potential statement involving the conditional. Then, p2 = 1. However, the modalit of evidence differs from the modal notion of necessity, in that no clear position is taken as to the impossibility of the statement; some doubt remains about the truthfulness of the statement, because it is in the process of being completed. Thus, p3 = 0. Of course, it is also an asseveration, so pl = 1. The kimsaku for this modal notion is, therefore, (1 -1 0).
The suffix "/ka" can best be translated into Spanish using the gerund ("-ando" and "-iendo" /Eng. "-ing"/). That is why this modality is also called "gerundive". In fact, the truth-value of:

• "it is evident that he comes"

is equivalent to:

• "he is coming"

By this time, the reader will have realized that it is useless to seek analogies between the moods of Spanish grammar and the modalities of trivalent logic. There is no such problem in Aymara, because the meaning of each suffix is precisely defined in the triads. If they are used properly, there are practically as many moods as there are modalities.

In Spanish, there are only three officially recognized moods: the Indicative, the Subjunctive, and the Imperative(2).

The Potential (which in the last few years has also been called Conditional) is not considered a mood. This is due to disagreement between logicians and grammarians, who tend more towards general definitions. To the reader who adhere to the rules of the Spanish Academy, this author would like to apologize for the somewhat arbitrary use of the word "mood" when referring to Aymara modal notions; however, this seems to be the best way to translate the subtleties resulting from this fine modal differentiation.

4.3.7 Determination of x.ska = (1 1 0) (Feasibility)

This is a very interesting modality, because it shows very clearly how well the genial creators of the Aymara language understood the implications of logic for the temporal aspect of statements. The suffix "ska is made up of the suffixes su and /ka (the u is elided); it is used to form the "potential gerund", i.e., it indicates that the verb action would be occurring at this moment.

Examples:

x.ska =	'manqaskatwa' (I) 'would be eating'
x.ska =	'laruskta' '(you) would be laughing'

Both x.ka and x.ska involve a gerund, which affects the logical meaning of the statement.

From a strictly logical standpoint, it is evident that "/He/would be eating" means the same as "/He/ has almost eaten". Obviously, this is an asseveration: pl = 1. However, there is some doubt as to completion of the action, since the person may choke, and the activity could cease without having been terminated; then, p2 = 1. Now, if the person has not eaten, it cannot be deduced whether he has or has not been eating; therefore, p3 = 0. This modality can be called "feasibility" or "gerundive possibility". Thus, we see that x.ska = (1 1 0).

The Greeks also understood the close relationship betwen the time an action occurs and the modal notion of possibility. Diodorus Cronus (died in 307 BC), a disciple of Eucleides of Megara, questioned the concept of possibility, and reformulated it as a temporal variable, saying "it is possible in time t" (JFM). The Qoyas have no such problem; their syntax includes suffixes for both concepts of possibility: one involves the suffix su; and the other a combination of su and /ka, which makes the possibility gerundive.

ska is a compound suffix; this makes it possible to test the validity of the triads given in this book for su and /ka. It stands to reason that x.ska is (x.'su).ka; in other words, it is the result of adding first the suffix su, and then the suffix /ka to the statement x.su. This is one of the many verifications presented in the next chapter; however, we can say in advance that the results are conclusive; the way each and every compound logical suffix is used in Aymara is highly consistent with the logical analysis of each statement.

The notion of feasibility x.ska may become the notion of possibility if the indetermination about that which is uncertain can be eliminated assuming that if the verb action is not completed the statement cannot come true; then, p3 becomes -1, and the triad is that of x.su. This can be done simply by adding the suffix of the modality of necessity:

x.'ska.pi =

'manqaskatapi' 'estarías comiendo pues' /you would be eating for sure/

In a strictly mathematical-operational way, it can also be proven that x.ska.pi s x.su; in other words, both statements are logical synonyms and have the same kimsaku (1 1 -1).

In Aymara:

"it is possible that x" = "it is certainly feasible that x."

4.3.8 Summary of the modalities of asseveration

The determination of the kimsakus of asseverative state ments has led to the identifìcation of 6 different modalities(sic); these notions are all similar in that always pl = l, and p3 is not -1.

The following table is a summary of all these notions. The Spanish forms used to translate the Aymara modalities of asseveration are also listed.

MODAL NOTIONS OF ASSEVERATION IN THE AYMARA LANGUAGE

Asseveration		Opposite		Antonym
x = (1 0 -1)		-x = (-1 0 1)		x = (-1 0 1)
'afirmative that x'		'negative that x'		'afirmative that not -x'
x.wa = (1 0 -1)		x.ka.ti = (-1 0 1)		x.ka.ti = (-1 0 1)
'reliable that x'		'false that x'		'reliable that not -x'
x.pi = (1 -1 -1)		x.pi.ka.ti = (-1 1 1)		x.sa.pi = (-1 -1 1)
'certain that x'		'uncertain that x'		'certain that not -x'
x.su = (1 1 -1)		x.ska.ti = (-1 -1 1)		x.ki.ti = (-1 1 1)
'possible that x'		' impossible that x'		'possible that not -x'
k.ki = (1 0 0)		x.ki.ka.ti = (-1 0 0)		x.ka.ti.ti = (0 0 1)
'likely that x'		'unlikely that x'		likely that not -x'
x.ka = (1 -1 0)		x.ka.ka.ti = (-1 1 0)		x.ti.ti = (0 -1 1)
'evidence that x'		'not evident that '		'evident that not -x'
x.ska = (1 1 0)		x.su.lla = (-1 -1 0)		x.lla.ska = (0 1 1)
'feasible that x'		'not feasible that '		'feasible that not -x'

Modal notions of assertion in Aymara

• x.lla.cha = (1 1 1) Tautology
• x.lla.pi = (-1 -1 -1) Contradiction
• x.lla. la = (0 0 0) 'Abdiction' /the author might mean 'abduction'/ (3)

4.4 Determination of the kimsakus of queries

The negation of a statement is not always its antonym; this is particulaly true of amodal statements. This fact was recognized by classlcal logicians, who introduced the concept of partial negation -the negation of only the modus-, and who established a difference between this and the negation of the whole statement. In modal symbolic logic this depends on the order in which the negation is applied to the modal operator.

For example:

Original Statement	Opposite Statement	Antonymous Statement
x = 'he comes' (1 0 -1)	Nx = 'he does not come' (-1 0 1)	Nx = 'he does not come' (-1 0 1)
Mx = 'he may come' ( 1 1 -1)	NMx = 'it is impossible he may come' (-1 -1 1)	NMx= 'it is possible he may not come' (-1 1 1)
Gx = 'he certainly is coming (1 -1 -1)	NGx = 'it is uncertain whether he may come' (-1 1 1)	GNx= 'he certainly does not come' (-1 -1 1)

This Lable shows that the opposite and the antonym of an amodal statementare identical.

These examples also show a well-known fact of modal logic: the opposite of the notion of possibility is identical to the antonym of the notion of certitude. Using Aymara suffixes:

-x. su = (-x).pi

it is impossible that x s 'it is true that not x"

The adverb "not" plays an ambiguous role; it is used both to form the opposte and the antonym of the statement. To solve this ambiguity, one must use auxiliary verbs such as "ser posible que" /may/, and "ser necesario que" /must/, or their idiomatic equivalents.(4) So, rather than saying "possibly he will not come", and "not possibly he will come", one must say "it is possible that he will not come" and "it is not possible that he will come."

However, when modal statements more subtle than those of certitude and possibility are involved, for example, queries, the adverb "not" is inadequate for a consistent treatment of opposite and antonymous statements. To make the language of symbolic logic compatible with everyday language, the same statements must be generated by resorting to idiomatic forms unsuitable for conversation or for making inferences about practical problems, which is another disadvantage. The difficulties associated with auxiliary verbs are foreign to Aymara, since suffixes, which are easy to apply are used instead. Aymara has a very elegant way of distinguishing an opposite from an antonym: the negation does not involve one suffix, but two. So, it is not necessary for modal statements to consist of two parts (modus and dictum), although both concepts are implicit in the strings which make up the statement. Thus, the formulae of Aymara symbolic logic are totally compatible with the corresponding idiomatic expressions of the language.

At this point, a distinction must be made between Question and query. A question is a sentence which asks for an explanation, or requests some information. One does not answer a question simply by saying yes or no; a complete answer must be given. However, a query is a sentence which demands an answer involving a truth-value.

For example, "Why have you come?" is a question, whereas "By any chance, have you come?" is a query.

The above definitions for the words "question" and "query" are somewhat arbitrary, but useful for our research. It is necessary to distinguish between these words and to determine their specific meanings because a question cannot have a truth-value, and, therefore, is not a logical statement.

Queries are definitely logical statements, because they catn be related to their respective amodal statement, from which its triad of truth-values can be deduced. In Aymara, the suffix ti indicates that a statement is a query; this suffix is never used with questions, as they do not fall within the schema of modal statements. In questions, suffixes such as sa and ja are used, which is a different type of syntactic algorithm.

Examples:

x.ti =	'lurtati?' 'acaso has hecho?' /By any chance, have you done it
x.ti y =	'sarjeti?' 'acaso ya se ha ido (él)?' /By any chance is (he) already gone?/

Each of these modal statements can be compared with its amodal statement, which can be obtained by using wa rather than ti. That is to say, to obtain their kimsakus, the question 'lurtati?' must be compared with its amodal statement 'lurtawa' (you have done).

When a proponent says "he is gone" and the opponent queries "by any chance, is he already gone?", the opponent's statement must be interpreted as an opinion regarding the statement of the proponent (who has used factual language). It can be noticed immediately that this is an assymetric query by comparing it with its antonym, "by any chance, is he not gone yet?". In the first case,we query that he is gone, in the second case, that he is not gone, in other words, when one queries, one is making a statement.

4.4.1 Determination of x.ti = (-1 1 0) (Positive Controversy)

In Aymara, when the suffix ti is used to query the veracity of the amodal statement, the statement is not expected to be true. In the interrogative statement "acaso ya se ha ido?" /by any chance is he already gone?/ an affirmative answer is not expected. If the answer is "yes, he is already gone", the query has been erroneous. So, for the statement x.ti, pl = -1. Now, since the statement involves doubt, p2 must be = 1, because factual incertitude only serves to justify the use of the interrogative. If the factual statement is not true, it cannot be known whether the query was correct or not, because no doubt was involved, nor was it considered, so p3 = 0.

The suffix "ti" implies a controversy and insinuates that the statement may not be true; therefore, it can best be translated into Spanish using the idiom "acasó". This modal notion corresponds to the Spanish interrogative form (5) .

Thus,

x.ti =

'manqtati?' 'has comido acasó?' /have you eaten by any chance(6)/

"Ti" sometimes cannot be translated by "acasó", particularly when combined with other suffixes; then it is translated using the idiom "siempre".(6)

The examples will show the proper way these Bolivian modisms must be used to accurately translate the logical meaning of queries from Aymara into Spanish.

Care should be taken when using this modal notion of controversy to avoid asking impertinent questions; this modality is only appropriate when it is very probable that a negative answer will be given. If it is known that someone has done something, one should not ask "acaso has hecho tú?" / 'by any chance have you done...?' / as this would imply doubt that he has done it, and would be a concealed way of expressing a negative opinion, In Aymara, this kind of question is called "jiskiçukiña" (to ask questions under false pretense); this word is composed of the nucleus "jiski" (question) and the infix of psychological simulation, "çuki".

However, if somebody is accused of having committed an act such as murder, the lawyer should ask "jiwaytati?" ("acaso has matado?") /by any chance, have you killed?/; this indicates a certain degree of trust, because although there is some doubt, an affirmative answer is not expected from the accused.

The formal statement for suffix "ti" is: x.ti = "it is controversial that x".

Negative exhortations: x.ti = (-1 1 0)

Negative exhortation is a type of negative statement which is the opposite not of statements of affirmation, but rather of statements of evidence. An exhortation such as: p = 'jan llullämti' ('you shall not deceive') does not deny the fact that "you have deceived", but implies rather that "you should not deceive in the future". If the exhortation is not effective, i.e., if you do deceive, the exhortation is obviously a failure; so, for x = 1, p = 1. When there is some doubt whether you "have deceived", the exhortation is proven to be valid, since otherwise it would not have been necessary; if one exhorts it is because the possibility exists that you "will deceive"; thus, for x = 0, p= 1. If the exhortation was effective, i.e., if you have not deceived, one can not be sure whether the exhortation was necessary because without it you "might still not have deceived." This last possibility remains open in the modal notion of exhortation; thus, for x = 1, p = 0.

Therefore, the kimsaku of statements of negative exhortation is (-1 1 0), which is the kimsaku of the suffix "ti". In fact, in Aymara this kind of negation is generated by adding only the suffix "ti"; the suffix "/ka" is not used at all, which differentiates this kind of statement from the negation of affirmative statements, i.e., x.ka.ti = janiw llullktati' (you have not deceived).

The classical principles of Qoya ethics are good examples of this kind of statement:

• x.ti = 'jan llullämti' (thou shalt not deceive)
• x.ti = 'jan karisimti' (thou shalt not lie /to yourself/)
• x.ti = 'jan jayrämti' (thou shalt not be lazy)

It is interesting to note that in Aymara the verb "to lie" is reflexive (karisiña). The verb "mentirse" /to lie to oneself/ implies that when all is said and done, he who lies (to others) is really lying to himself. Thus, in the popular variety of Spanish spoken in the Altiplano, rather than saying "ha mentido"/he has lied/ people say "se ha mentido"/he has lied to himself/. A "karisiri" (a liar) is a person who is really deceiving himself by not always telling the truth. Other words are used when someone does not tell the truth, but is not "lying" /to oneself/; for example, "kauka" (pranks, pratical jokes, as a child would do.) (NFN-39).

The first scholars who studied Aymara were very interested in the form of negation of negative exhortations, because of its importance for religious preaching. Since in Spanish the adverb "not" plays an ambiguous role, early grammars say that the negation is generated by adding only the suffix "ti"; the suffix "/ka" is not mentioned, although it appears in some illustrative examples. Garcia was the first to notice this remarkable feature of Aymara (JAG).

4.4.2 Determination of x.ka.ti = (-1 0 1) (Negation: it is false that x)

Examples:

x.wa =	'jutapjewa' (they) have come
x.ka.ti =	'janiwa jutapjkiti' (they) have not come
x.wa =	'jutapjayatwa' we (my people) came
x.ka.ti =	'janiwa jutapjakayatti' we ( my people) did not come
x.wa =	'umaj muntwa' /I/ want /some/ water (the ater I want)
x.ka.ti =	'janiw umaj munktti' I want no water 'janiw umaj munktati' /you want no water/
x.wa =	'umaj utji' There is /some/ water
x.ka.ti =	'janiw umaj utjkiti' there is no water

In Aymara, the negation does not play a key role as it does in Spanish. Negation is just one of the modal notions, which might be classified as a query, since formally:

'it is false that x' = 'it is controversial that it is evident that x'

In the third person of the present tense, the suffixes which generate the negation become "kiti" because of elisions; thus, care should be taken to avoid confusion. Apparently, Ludovico Bertonio did not understand this point very well; in his grammar he stated that both the negation and the interrogation [question] are formed simply by adding ti. Fortunately, he realized, especially in his later dictionary (LB1-335), that the suffixes /ka and ti are both involved in any negation of amodal statements. The formation of negative statements in Aymara is well established in modern grammars; however, it is not always clearly explained that the k appearing in negative statements always belongs to the suffix "/ka"; this is because of the lack of a syntactic model, among other things. The determination of the triad for the negation is more than obvious; if the factual statement is true, its denial is a mistake (pl = -1); if the factual statement is uncertain, its denial does not remove the doubt (p2 = 0); and if the factual statement is not true, the negation is correct (p3 =1). Thus, x.ka.ti - x.wa, as was expected.

Comment: (JEE-127) "the following is important: in the affirmative form one says "naya munayata" and "naya laruyata"; the vowels 'a' in muna and 'u' in laru are not dropped because they are part of the root; however, in the negative form one says "munkayati" and "larkayati", because the same ending is used after mun and lar.

4.4.3 Determination of x.ti.ka = (0 1 -1) (Negative Controversy)

This kind of statement also implies controversy, the only difference being that it is symmetric with or the antonym of x.ti.

Examples:

x.ti.ka =	'janiti lurarapkatapaja?' 'acaso no se lo estás haciendo?' /By any chance you are not doing it to him?/
x.ti.ka =	'jumati lurarapkatapaja?' 'acaso tu siempre se lo estás haciendo?' /By any chance are you always doing it to him?/

It is evident that one expects confirmation rather than a negative answer to these questions. If a negative answer is given, one has made a mistake, and is surprised because a negative answer was not expected. That is to say, for this modal notion, p3 = -1. Obviously, the same reasons apply in this case, but only in a symmetrical manner, p2 = 1 (there is doubt), and pl = 0; thus, the kimsaku is (0 1 -1).

A prosecutor would be making a mistake if he asked a defendant "janit jiwaykataja" /By any chance, you would not have killed him?/; he would be assuming the answer would not be negative, when it is factually evident that the defendant "has not killed."

This modal notion shows that suffixes ti and /ka play different roles depending on the order in which they are used. This will be proven mathematically by demonstrating that they are non-commutative operators.

This statement can also be formulated positively if the pronoun rather than the adverb of negation is used to translate the suffix ti:

x.ti.ka =

'jumati lurkataja?" 'acaso tú siempre estás haciendo?' /By any chance, ara yov always doing/it/?/

This example also shows that a negative answer is not expected, because this statement implies a controversy, and leaves open the possibility that what is implied by the question will not prove true.

4.4.4 Determination of x.sa = (0 0 1) (Adversative) (7)

Modern grammars do not distinguish between sa and the possessive suffix sa, which was spelled ssa in early grammars. This is because the phonetic difference is very slight. However, the functions of both suffixes differ widely so that it is advisable to use the letters s and ^s to avoid confusing them when one follows iromediately upon the other, as happens when logical operators in class "I" are linked to the suffixes indicating the recipient, or possession, as the case may be. For example, a distinction should be made between these two suffixes in sentences like 'apirisasa' (ni nuestro portador /not even our porter(8)/) (ETA-82).

Examples:

(ETA-8I) x.sa =	'utamaj jayawa, jaypurusa purkáss' '(la) tu casa es distante, (él) ni a la tarde estará llegando' /(the) your house is far; (he) not even in the afternoon will be arriving/
(ETA-81) x.sa =	'lurañamasa' 'ni en el hacer' not even in the making/
(ETA-83) x.sa =	'alasiñamasa poqatakäniti' 'ni lo que compres ha de estar completo' /not even what you buy will be complete/
(ETA-90) x.sa =	'janisa' 'aunque no' (ni no...) /although not/ /not even not.../

The logical content of statement x.sa shows immediately that this is an adversative modality and indicates negative conjunction; thus, p3 =1, because it affirms non-fulfilment of x. However, values pl and p2 cannot be easily deduced from the statement itself. The statement does not involve any doubt, so p2 can only be 0 or -1. The statement is not symmetric, so pl can not be 1. Thus, there are only four possible triads that can correspond to x.sa, viz, (-1 0 1), (-1 -1 1), (0 0 1), and (0 -1 1). Obviously the first one (negation) and the second one (impossibility) are out of the question, because they do not convey the logical meaning implicit in the statement x.sa. Thus, there are only two triads remaining that may correspond to this statement. Both are very similar; the only difference between them lies in how they treat the doubt. Kimsaku (0 -1 1) is the antonym of (1 -1 0), which corresponds to x.ka, i.e., a "strong adversative" which formally means "it is evident that not-x." Thus, the kimsaku for x.sa must be (0 0 1).

Comments: (LBO-245) "Adversative conjunction are those conjunctions such as "quanque", "uamuis", "licet" (Latin words meaning although, though) and other similar ones which correspond to particles paña and sa, used together, as in the following example: "although you may confess all your sins, if you do not wish to mend your ways, your confession will be worth nothing" = "paña taqe joçanakama confessasisina, jani wanija sasinka, kasikiwa confesasirikta, vel confesasiwuima hani jakurikiti". "Even if the damned weep incessantly, the devil will have no pity" = 'paña manqepaçankkirinaka kunasa wararirikiçeja', supayonaka jani katasa kuya payrikíti'. "Etiam (another example in Latin) si me iccideris in te sperabo, = 'paña jikuyutasma juroaki ullasimama'. So, this particle is not necessary with the nominative gerund, nor is it used when there are different assumptions using the Subjunctive and the Optative; instead of "pana" one can use "sa", for example, "confesasinsa, vel confesasimansa" = "even if you confessed" ; and instead of "sa", one can use "spalla".

Although the logic of these paragraphs by Bertonio is not remarkable, one can at least be certain that the learned linguist gave a great deal of thought to the logical meaning of "sa" and "spalla" and identified them as adversative suffixes, as they were extremely useful in his religious mission. It should be noted that he mentions the compound suffix "spalla",which demonstrates the existence of a logical suffix "lla", which is no longer used, and is not recorded in any modern grammars, with the excception of (FMS).

One of the examples given by Torres Rubio illustrates how the Qoya people have, like their language, remained stubbornly unchanging until now:

(DTR-53) "I shall not say it, even if you kill me." = "pañasa jiwawuasina jani atamasanti."

Neither the suffix "lla" nor the word "paña" are recorded in modern grammars. Tarifa translates the suffix "sa" using the adversative conjunction "ni"/neither/ and the compound suffix "saya" by the adversative conjunction "aunque"/although/; he also sometimes uses the word "siquiera"/ even if or though/, as can be seen in his chapter on conjunctions (ETA-416).

4.4.5 Determination of x.lla = (-1 0 0) (Improbability)

This suffix is very difficult to study because there is a lack of reference material. Sanjines' is the only modern grammar in which it is recorded (FMS-42). Apparently, in present-day Aymara the suffix "lla" is pronounced "ya" (not to be confused with the infix "ya") especially in the compound form "saya" which was previously written "salla". However, there is evidence that it is still pronounced "lla", especially in the Oruro regions.

Examples:

(FMS-42) x.lla =	'uicawalla'	(alli tampoco será) /ìt won't be there either/
(FMS-42) x.lla =	'ukjamalla'	(tampoco será asi) /it won't be so either/
x.lla =	'juttalla'	(tampoco será que viniste) /it isn't likely that you came either/

Of course, the analysis of connective statements will demonstrate that the modality generated by "lla" plays the role of an adversative connective, rendered in the local variety of Spanish as "tampoco será que"/ it is not likely either that./ The formal interpretation of this modality is: 'it is improbable that x'. In fact, this modality is the opposite of x.ki; in other words, it means "it is unlikely that x"; thus, its kimsaku can be equated with the kimsaku for "-x.ki", viz, (-1 0 0).

x.lla is the equivalent of x.pi.lla which, in formal language, means: 'it is improbable that necessarily x." (LBO-328).

There is an apparent contradiction which must be pointed out: when the suffixes "lla" (improbability) and "sa" (adversative) (9) are combined to form "salla", the meaning of the modal statement generated by the compound suffix is favourable. For example:

x.salla = 'mantapsalla'

(que pase nomás) /he may come in/

However, the effect of this combination can be explained by the following modal theorem:

'favourable that x' = 'unfavourable that it is adversative that x" (9)

Using the algebraic method explained below, it can be demonstraded unequivocally that:

• (x.sa).lla = (0 0 -1)

This kimsaku corresponds closely to the statement "i't is favourable that x." Some linguists (HRl, HR2) have agreed with this interpretation of the compound suffix "salla" and have called those statements concessive mode /or "mood"/.

The following modal theorems are also valid:

• x.salla = -x.sa

that is to say,

• "it is favourable that x" = "it is not adversative that x." (9)

and

• x.salla = (-x).lla

that is to say:

• "it is favoutable that x" = "it is unfavorable that not x"

4.4.6 Determination of x.sti = (1 1 -1) (Positive Eventuality)

Suffix "sti" is made up of sa and ti, in other words, x.sti = x.sa.ti; however, the mere analysis of sentences in which these prefixes are used cannot demonstrate this fact, because the compound suffix is never broken down into its two simple suffixes. This is only natural, since both logical suffixes belong to the syntactic category L. In section 4.7 an algebraic analysis will demonstrate that the suffix sti is indeed composed in this roanner.

The suffix sti is used in statements which can be considered queries. This is a positive manner of asking questions, to which a "yes" is expected rather than a negative reply.

Examples:

x.sti =	'jumawa karistasti?' 'y tú(te) has mentido entonces?' /and you have lied (to yourself) then?/
x.sti =	'jumaraki sarjatasti?' 'y también tú ya te irás?' /and will you also leave?/
x.sti =	'nayasti?' 'y yo?' /and me?/

This last question "and me?" is typical of a child who wants to know whether he can also go to a party. An answer such as "of course!", would confirm his expectations; whereas should the answer "no!" would be disappointed.

The examples show that the person asking the question is right when the factual statement is true, thus, pl = 1. As there is a doubt, p2 = 1. A negative factual situation is not expected in this kind of question, so p3=-1.

In the colloquial variety of Spanish spoken in the La Paz region the conjunction "y" /and/ is used to translate questions of the type x.sti into Spanish, to differentiate them from questions using x.ti. The conjunction "y" /and/ is not really appropriate for simple statements because it is a connective. However, in a certain sense, there is some psychological justification, because the person asking the question expects an affirmative answer and this way of asking demands a second statement in conjunction with(sic) his query to verify the possibilit it implies.

This can be clearly seen in some of Tarifa's examples:

(ETA-307)	'jumaj kunraki luratasti?' (y tú qué has de hacer también?) /and what will you do, too?/
(ETA-308)	'kitimpiraki alayanipjasti?' (y con quién también hemos de ir a hacer compras?) /and with whom will we also go shopping?/

Obviously, these are not queries, but questions, which demand a detailed explanation; however, from the point of view of logic, their aim is to verify the positive truth-value of the factual statement.

It is easy to see that both x.sti and x.su have the same kimsaku. This means that the interrogative statement x.sti and the statement of possibility are logical synonyms. Therefore, the corresponding formal statement is "it is eventual that x."

4.4.7 Determination of x.ti.sti = (-1 1 1) (Negative Eventuality)

This statement is the antonym of x.sti ; accordingly, the triads for both statements are symmetrical. That is to say, the person who asks does not expect an affirmative answer, but does expect a negative one.

Examples:

x.ti.sti =	'jumaj janit jiwaptasti?' 'no siempre has matado?' /you have not always killed/
x.ti.sti =	'jumati jiwaytasti?' 'y tú siempre has matado acaso?' /and you have always killed by any chance?/
(ETA-313) x.ti.sti =	'nayati laruyasti?' 'y yo he de hacer reir acaso?' /and me, roust I make you laugh by any chance?

Note that this query can have both an affirmative and a negative formulation. In any event, doubt is implied, so, p2 = l. An affirmative answer is not expected, therefore, pl = -1. However, since a negative answer is expected, p3 = 1.

To show the logical implication of these interrogative statements in translation, one must again resort to Bolivianisms such as "acasó" /by any chance/; the lengthening of and stress on the final vowel of this word conveys doubt about the "yes". The word "siempre" /always/ is used to translate positive statements. For negative statements, it is sufficient to use "y" /and/, as shown by the innumerable examples. in Tarifa's grammar.

The counsel for the defense, when asking his client whether he has killed or not, must use this kind of interrogative statement, because it conveys his belief that his client has committed no crime, even if there is some doubt.

AYMARA MODALITIES OF QUERY

Query	Opposite	Antonym
x.ka.ti = (-1 0 1) false that x	x.wa = (1 0 -1) reliable that xx = (1 0 -1) false that not -x
x. ti = (-1 1 0) controversial that x	x = ka = (0 1 -1) evident that x	x.ti.ka= (0 1 -1) controversial that not -x
x.sa = (0 0 1) adversative that x (10)	x = sa.lla = (0 0 -1) granted that x	x.la.sa = (1 0 0) adversative that not -x
x.lla = (-1 0 0) unfavorable that x	x.ki = (1 0 0) favorable that x	x.sa.lla = (0 0 -1) unfavorable that not -x
x.sti = (1 1 -1) eventual that x	x.sti.ka.ti = (-1 -1 1) not eventual that x	x.ti.sti = (01 1 1) eventual that not -x

Note:

• modalites
  x.ka.ti
  x.lla
  x.sa
  

Do not demand an answer, thus, they are modalities of impugnation / refutation / rather than queries.

4.5 Determination of kimsakus of conjecture

The modus(sic) of pure and symmetric doubt is governed by the suffix çi. As this suffix has no equivalent in other languages, it is difficult to explain and understand in translation, which can perforce only be an approximation.

Once again, one must resort to Bolivian idioms For the translation of an Aymara suffix. Tarifa's examples throw light on the meaning and uses of the suffix çi; this scholar from Pacajes deftly interpreted these Bolivianisms.

4.5.1 Determination of x.çi = (0 1 0) Contingency (Symmetric doubt)

Examples:

(ETA-274)	x.çi=	'ukakiçiya' 'quízás es pues eso nomás' /perhaps it is PUES that NOMAS/
(ETA-274)	x.çi=	'aka qerunakaj nayatakiçini' 'esas maderas quizás sean pues para mi' /those planks maybe are for me PUES/
(ETA-274)	x.çi=	'janiwa yatkti, uruçi arumaçi' kunapaçapuniya luntataj utaru mantçi, tage qepsu1nkama' 'no sé; habrá sido de día o de noche; cuando siempre se habra entrado el ladrón a la casa, hasta cargar con todo.' /I don't know; it might have been either day or night; when SIEMPRE the thief might have himself broken in the house and stolen everything./
(ETA-275)	x.çi=	'uñjañaçiya' 'habrá que ver pues' /it must be seen PUES/
(ETA-275)	x.çi='luririçi' 'sabrá pues (él) hacer' /he will know how to do it/
(ETA-275)	x.çi=	'sarçiñaniya' 'quizás pues iremos' /Maybe we'll go PUES/
(ETA-277)	x.çi=	'nayaya sarçi, jani jutani ukaj' 'quizás yo pues iré, si no ha de venir' /Maybe I will go PUES, if he is not going to come/
(JEE-123)	x.çi=	'inaj jupa munçi' 'tal vez él quiera' /Maybe he may want to/
(JEE-124)	x.çi=	'inaj juma munçita' 'tal vez tu quisieses' /Maybe you would want to/
(EWM-149)	x.çi=	'istçita' 'vielleicht hast Dugehört' /Maybe you have heard it/
(EWM-79)	x.çi=	'naya çurçi' 'ich werde wohl geben' /Well, I will give/ /Then, I will have to give/
(LBO-275)	x.çi=	'Pedroj aka qollqe luntatirikiçi' 'quizás Pedro hurtaría esta plata' /perhaps Peter stole that silver/
(MJHB-224)	x.çi=	'inas sarçi' 'maybe he went' (I don't know, have no data, and don't care) /quoted in english by IGR/
(MJHB-224)	x.çi=	'ninaçim jiwaraskçi' "I'll bet the fire is going out" /quoted in English by IGR/

The very complex statement x.çi has been illustrated by many examples taken from various grammar textbooks to demonstrate that the suffix çi has been used consistently over the centuries; neither its meaning nor its position in the sentence has changed. Also worth noting are the attempts made by these linguists to translate these sentences into their respective languages. Without a doubt, statements involving çi unequivocally show the difference between those languages which are based on bivalent logic, and Aymara, which is based on three-valued logic. Aymara deals with thesestatements-systematically, and there is no need to invent convoluted sentences to express the various degrees of conjecture.

When Aymara sentences contain the adverb "inasa" the Spanish adverbs of doubt, "perhaps" and "maybe" can be used in the translation: their meaning is close enough. However, in all other cases, these words do not give an accurate translation.

Statements using çi alone and in combination with other logical suffixes will be discussed below. There is no other way to accurately obtain the corresponding kimsakus. One cannot analyse each and every kind of conjectural Statement, because the translations themselves are arbitrary. After Tarifa, the term "será que.." / it may be that / will be used to translate "ina", and the conjunctions "ni" / neither /; "o" / or /, and "tampoco" / either / will be used to translate the suffixes "sa", "ça" and "lla." As will be explained below, the suffix "lla", sometimes translated by "pues" / perhaps / is very different from the word "pues" / for sure / used to translate the suffix "pi". The Aymara statements are first translated using idioms common in the variety of Spanish spoken in Bolivia. Then, formal expressions used in modal logic, adapted to Aymara, will be used to give a more clear and technical meaning to the translations.

x.çi =	'lurçi' = (0 1 0) 'habrá hecho nomás' /He has done it perhaps/ 'it is feasible that x and not -x' 'it is contingent that x'
x.çi =	'inaj lurçi' 'será que habrá hecho nomás' /He might perhaps have done it/ 'perhaps yes, perhaps not x'
x.sa.çi =	'inasa lurçi' (1 1 0) 'ni será que habrá hecho nomás' /Might it not be that he has done it/ 'perhaps he has done it' 'it is plausible that x'
x.ça.çi =	'inaça lurçi' (0 0 0) 'o será que habrá hecho nomás' /Or perhaps he has done it/ 'it does not matter if he has done it' 'who knows if he has done it' 'it is aoristic that x'
x.lla.ka.çi.ti =	'janilla lurkçiti (-1 1 1) 'acaso tampoco no será que habrá hecho' 'possible, too, ha has not done it at all/ 'It is doubtfull that x (synonym of x.ça)
x.çi.lla =	'jupaçinilla lurirej' (0 -1 0) 'tampoco será que él tendrá que gustar de hacer' 'there is no doubt about doing it, perhaps he likes doing it' 'it is unfavourable that it is contingent that x' 'it is incontingent that x'
x.lla.çi =	'janilla lurçi (0 1 1) 'tampoco no será que habrá hecho' 'it is contingently unfavourable that x' (antonym of x.sa.çi)
x.ti.sa.çi =	'janit inas lurçi (0 1 1) 'acaso, quizás no ha hecho' (synonym of x.la.çi) (antonym of x.sa.çi) 'it is contingent that it is adversative that is questionable that x'

4.5.2 Determination of x.çi.pi = (-1 1 -1) (Total contingency)

• 'jutçipi'
• 'será pues que vendrá'
• 'it is possible that x and that not -x'
• 'it is certainly contingent that /he/ will come'

The convolutions in the Spanish language necessary to translate these statements of conjecture show how difficult this task is. In Spanish, the negation involves only one operator; therefore, modal statements of doubt tend to be either affirmative or negative. There is no room for symmetrical doubt. The syntax of Aymara is based on a three-valued logic; negation involves two independent operators (ka and ti), between which one can intercalate the suffix çi, the operator of the contingency modus(sic); this makes it possible to form completely symmetric statements of doubt which, consequently, are impossible to translate into any language based on a bivalent system of logic.

It would be very interesting to discover whether there are other languages existing today which use two independent suffixes to make negative statements.

It should be pointed out that in Aymara the notion of contingency involved in statements of the type x.çi very different from Aristotle's ("it is not necessary and not at all impossible that x.") The Aristotelian notion corresponds to statements of total contingency in Aymara of the type x.çi.pi.

The modal notions "it is contingent that x" and "it is incontingent that x" differ in this respect: the first statement involves some doubt because it is uncertain whether the statement is true or false. However, the second statement does not involve doubt (p2 = -1); the statement must be either true or false; however, this remains "to be seen" there are no clues in the statement, so p1= p3 = 0. "It is incontingent that x" and "there is no doubt that x" are not identical. For people who think according to a bivalent system of logic, the notion of incontingency is absurd. For someone with a trivalent mind, this is just another logical modality which may be used to reach conclusions.

In a statement of the type x.çi.lla, there is only one certainty: the statement involves some doubt, but it is uncertain whether the statement is true or false. The reason for this is that the statement usually involves two aspects; this can be seen in our example, the amodal counterpart of which is "he will have to do"; there is no doubt that it will have to be done (by somebody), but it is uncertain whether he will be the one to do it. Its triad, therefore, is (0 -1 0).

Both "it is contingent that x" and "it is certainly contingent that x" are clearly conjectural modalities, so in both cases p2 = 1. Both are symmetrical, i.e., pl = p3. They differ in one respect: in the case of x.çi it is not known whether the statement is true or false; however, there is no indication whether it should be affirmative or negative, so p1 = p3 = 0. The statement x.çi.pi, however, implies that it is not necessary and not at all impossible that x; in other words, neither truth nor falsity are expected; thus, p1 = p3 = -1, there is absolute doubt.

There is another statement which is symmetric, and, therefore, impossible to translate. This statement does not involve a doubt per se, but rather the modality "it is aoristic that x." A statement such as x.ça.çi indicates a total lack of interest for any truth-value the statement may have, either because it is impossible to determine or because said truth-value is irrelevant. Thus, the three values in ihe triad are doubtful; the symmetrical kimsaku (0 0 0) is its negation as well as its antonym.

Note: According to Casares' ditctionary, the word "aoristic" means "philosophical doubt", "uncertainty". The word comes from Greek and expresses a grammatical modality(sic) of indefiniteness. I can find no better term in Spanish for a modality in which even doubt itself is doubtful.

The other statements with the suffix, çi are not symmetrical., and, accordingly, can be translated into Spanish. This is particularly true of the statement x.sa.çi, the well-known "perhaps x". There is a doubt, thus, p2 = 1. What is in doubt is the certitude that not-x, so p3 = 0. It is expected that x is true, so pl = 1. Thus, the triad is (1 1 0). The statements x.ti.sa.çi and x.la.çi are the antonym of the statement x.sa.çi; The kimsaku of the antonym is obvious, because it can be obtained by symmetry (0 1 1).

These forms of assymetric doubt will be rendered in Spanish using the following formal expressions:

• x.sa.çi = 'it is plausible that x'
• x.ti.sa.çi = 'it is plausible that not -x'

Here, I should point out that it was precisely after struggling to interpret accurately the modalities expressed by the suffix çi, that I decided to apply the methods of mathematical logic to the study of the logic of the Aymara language. Upon realizing that a statement of the type x.çi was symmetrical, I decided to conduct serious research using mathematical logic to determine the behavior of the different logical suffixes of Aymara, based on the hypothesis that these suffixes generate trivalen statements.

Evidently, to anyone who studies the Qoya language, the suffix çi appears as something very peculiar, and it is precisely the symmetrical doubtful statements (sic) which reveal the non-Aristotelian character of the Aymara system of logic. The following paragraphs quoted (and in some cases translated) from the most significant grammar textbooks illustrate the reactions of several linguists who have analyzed the suffix çi:

(LBO-275) "Regarding the particle chi: this particle is included among the adverbs; in that section it is stated that chi has the meaning of the word "forsitan" ("perhaps" in Latin), and that it is used in sentences involving doubt, as explained therein; it has also been pointed out that chi is used to form conditional sentences in the indicative mood. Here it must be added that the Indians usually insert this particle when communicating information about something."

(LBO-239): "In this language, these adverbs are translated using "inaja" or simply "ja"; the particle "chi" is also added, but it is ornamental, ... ja is added to that which is doubted."

(LBO-239): "These two particles, chi and ; ja, added to the interrogative words mentioned above mean "I do not know", for example, "kitichija" = "I don't know who it is."

(LBO-240): "Finally, the particles ja and chi often serve a purely ornamental purpose when added to the abovementioned interrogative words; however, in those instances they are not interrogative, but "indefinite."

(LB2-174): "ina aro" = "wrong word which need not be taken notice of"; "inaja" = "perhaps, maybe"; "inaki" = "in vain, wrongly".

(LB1-379): "just in case" = "inajaki".

(DTR-159): "inacca, inasa = perhaps" (In the grammar section, there is nothing about chi).

(EWM-149) "CHI": dieser Partikel drueckt Ungewissheit und Zweifel aus und wird in dieser Bedeutung teils zwischen Stamm und Endung eingeschaltet, teils an die Verbalformen eingefuegt. Dass CHI zur Bildung eines zweiten Futurums benutzt wird, ist bereits im Kapitel der Konjugation eingefuehrt worden." (çi: This particle expresses uncertainty and doubt; it has this meaning when it is inserted between the root and the ending; or when it is added to the verbal form. The fact that çi is used to form a second future has already been mentioned in the chapter on conjugation).

(JEE-84) "In Aymara the adverb of doubt is "inaj" = "perhaps, maybe." However, in sentences expressing doubt, the verb has its own conjugation, generally involving the particle chi."

(JEE-314) "ina = in vain"; 'inach = perhaps, maybe; 'inas = perhaps, maybe."

(ETA-275) "The verb tenses in which CHI appears as a verb ending are equivalent to the Future Imperfect and the periphrastic tenses of the Indicative mood or to the present of the Subjunctive mood."

(MJHB-223) "Non-involver: this tense has also been called the guesser, the conjectural, the suppositional, and the lamentor. The use of this tense indicates lack of involvement in the matter, by the speaker primarily, but may invoke subject and/or complement. The nature of the non-involvement is determined by sentence suffixes and/or particles elsewhere in the sentence. Non-involvement may be because there is no information or it may be emotional, or both." /quoted in English by IGR/ (11)

(MJHB-223) "(one common use is) when no data is available, so that the statement constitutes a best guess. However, the implication is usually that the speaker doesn't really care, one way or the other". /quoted in English by IGR/

(IIM2-54)"chi: strong verbal suffix which expresses doubt. It corresponds to the dubitative mood. This suftix cannot be translated when used alone. In the destruction (sic), it is accompanied by "inasa" (perhaps).

4.5.3 Determination of x.ça = (-1 1 1) (Doubt)

The statement x.ça is both conjectural and interrogative meaning; it is also the antonym of statements of possibility. It might more accurately be classified among the modalities of questioning. In compound statements the suffix ça also functions as a connective operator of alternative. Despite its various grammatical functions the suffix ça may have only one kimsaku, as will be determined below:

Examples:

(ETA-271) x.ça =	'jumaça lurta' 'o eres tú el que hizo' /or is it you who did it/ 'it is doubtful that it is you who did it'
(ETA-271) x.ça =	'lurtaça' 'o (tú) has hecho?' /or have (you) done?/
(ETA-272) x.ça =	'luranaça?' 'o hay que hacer?' /or does one have to do it?/
(ETA-272) x.ça =	'yatiritakiça?' 'o es para el que sabe?' /or is it for the one who knows?/
(ETA-273) x.ça =	'aljiriça uka çarkinaka çurtama. janiça ukjamaki?' 'o es el vendedor el que te dio esos tajos (de carne seca), o no es así?' /or is it the salesman who gave you these chunks (of dried meat) or is it not he?/

These examples show that statements in which the suffix ça appears raise an interrogation (sic) (because they can be answered yes or no); however, at the same time they are also questions (because an explanation is required). Also, these statements are all logical equivalents of "it is possible that not p", i.e., they are the antonyms of statements of possibility (this should not be confused with impossibility (which is the antonym of certainty). Yet, these statements containing ça are also the opposite of statements of certainty, because they amount to saying "not necessarily x"; or also,"it is uncertain that x." The x.ça statement involves doubt that x may not happen; so, pl = -1, p2 = 1 and p3 = 1. Thus, the triad for x.ça is the opposite of x.pi, which indicates precisely the uncertainty conveyed by the statement.

Comments: (ETA-271) "CHA, is a disjunctive verbal ending, used to conjugate verbs, much the same as WA". "Such modes of expression reflect doubt or negation." The word CHHA (ça) and the suffix ça also exist, but they are in no way related to CHA."

The role played by this suffir as a connective will be analyzed in our discussion of compound statements. Bertonio's greatest difficulty was in grasping the logical meaning of some suffixes, such as "ça", which differ radically from European languages in the way they generate connective statements. However, in Bertonio's grammar (p. 244) under the heading "disjunctive conjunctions" there are some examples of the use of "ça" and the connective "miçka". These particles are evidently used to construct etatements which today would te called "alternative." For example: Bertonio records these sentences: "kitiça usu?" (Is anybody sick, by chance?); "Pedroj jutiti janiça?" (Has Peter come or not?)

MODALITIES OF CONJECTURE IN AYMARA

Formal Statement	Antonym
x.sa.çi = (1 1 0) 'it is plausible that x'	x.lla.çi = (0 1 1) 'it is plausible that not -x'
x.ça.çi.ti = (1 1 0) 'it is hesitable that x' (12)	x.lla.çi = (0 1 1) 'it is hesitable that not -x''
x.sa.ça = (1 1 -1) 'it is doubtful that not -x'	x.ça = (-1 1 1) 'it is doubtful that x'

x.çi = (0 1 0) 'it is contingent that x'

x.çi.pi = (-1 1 -1) 'it is certainly contingent that x'

x.çi.lla = (0 -1 0) (opposite of x.çi) 'it is incontingent that x'

x.ça.çi = (0 0 0) 'it is aoristic that x'

The conjecture x.sa.çi is the logical synonym of the feasibility x.ska; the same applies to their respective opposites.

4.6 Double connective statements: päkimsakus.

As used in this book, a double connective statement is a logical proposition consisting of two simple statements linked together by a connective operator.

For example, the proposition:

• p = 'you come and he leaves'

is made up of these two simple state

• x = 'you come'
• y = 'he leaves'

linked by a connective operator "y"; this can be written:

P = x ^ y ('^' represents the conjunction 'and') In Spanish, as in most European languages, double connective statements have the following structure:

p(x,y) = statement 1 + connective + statement 2 Thus, the logical meaning of the statement depends solely on the connective, rather than on the modality of the individual statements. Therefore, in bivalent propositional logic, statements "x" and "y" are considered amodal. In some instances, the connective involves two words, for example, the implication "if x then y": however, these words do not affect the modality of the simple statements involved.

In Aymara, connective statements are expressed in a radically different way; this explains why these kinds of statements have not yet been fully understood and explained by researchers of Aymara grammar, The truth-tables for the connective statements can be analyzed; these statements are also subject to a definition of the function p(x, y). Thus, mathematics can help linguistics towards a better understanding of Aymara syntax.

Connective statements in Aymara differ from those to which we are accustomed(sic) in that in the Qoya language the simple statements which make up the proposition must be modal, i.e, include a modal suffix. The connective is also expressed by modal suffixes. This feature is a complication for anyone trying to translate the modalities and the connective. However, for Aymara-thinking people, it is the simplest way of handling a very large number of connectives; theoretically, the number may reach 3 ** 9 = 19 683 possible combinations. In Aymara, double connective statements have the following form:

p(x,y) = statement 1.M1 + connective A + statement 2.M2 where Ml and M2 are modal suffixes, and A represents the linking suffix.

Modal suffixes M1 and M2 are oftem identical. The connective is generally the word "uka", the pronoun "it", also translated as "that". Thus, far different A suffixes, there are connective expressions such as "ukasti", "ukaska", "ukampisa", etc. Bertonio's dictionary also mentions the connectives "tu" in expressions such as "tullanska" and "mi" as in "miçka" these connectives are apparently no longer used. This singular method of using a connective with its linking modalities has, of course, no similarity with Indo-European languages, giving rise to a strange algebraic etymology of connective words.

As explained in section 4, modern grammars only study the modal function (potential-gerundive) of the compound suffix "ska" (a combination of "su" and "/ka".) However, Bertonio recorded many examples of the connective function of this suffix, not only in the linker "ukaska", but also as a non-verbal suffix. For example: (LB2-148) "masüruska" (yesterday but.)

For this reason, the translation of Aymara functors has always puzzled linguists, who have arrived at the simplistic opinion that Aymara has no precise way of formulatinglogical connective propositions. For example, in his grammar, M. Hartmann B. (MJHB) states that "there are no connective functors in Aymara!"

This is the key to the serious problems of misunderstanding between Aymara- and Spanish-speaking people in Bolivia and Peru. One should probably add to this lack of communication the fact that many of the logical double propositions are not used consistently by Aymaras themselves, because of their low level of education.

Before discussing different kinds of connective statements, it must be pointed out that linguists have not been able to explain precisely the logical meaning of connective words; yet, some of them have grasped the syntactical role of "uka" and its various modal suffixes. Ludovico Bertonio was aware of it; for example, when describing the statement of implication with suffix "çi" he stated:

(LBO-431) "If, when, is (sic) conditional; Chi, for example, "If you are rich, why don't you give alms?" = Ccapacaçitha, cuna supa hani huakhchanaccro ccuyatha? However, the particle chi is used only in the present and past tenses of the Indicative mood, almost never in in the future, or in tenses of other moods. These sentences are conditional because the particle Ucaca, vel Ucasca, or Ucapilla, is used before the second sentence element, for example: "If you die, or died in sin, you will, or would, go to hell" would be rendered = "Hochani hihuahata, hihuasma, vel hiwuiricta, o hiuassina: Ucapilla ucaca vel ucasca infiernoru mirictawa"; the same is true when there are two assumptions, as can be easily seen."

It is also evident that Bertonio needed logic for his missionary work, which forced him to study the form expressing implication. Despite his lack of precision, Bertonio's work is valuable because he recognized that connective statements require modalities and the connective "uka" is followed by modal linking suffixes.

To this day, probably the most comprehensive study of Aymara syntax ever conducted is the research conducted by the scientific team from the University of Florida. Although English aquivalents are not provided, the syntactic function of "uka" and its modal forms are clearly explained in the grammar authored by Hardman-Vazquez and Yapita. The following paragraph serves to illustrate this:

(VY-453): "Uka is about the roost versatile and useful root in the language. Its uses as a demonstrative and a linker have already been discussed, as well as its unique position as head of a noun phrase (jupan uk, 'at her house'). Uka may also act as a resumator of any kind of a grammatical structure, thus reducing this structure to that of a simple noun, and in this way its subordination. Some other features are characteristic of the structure to be subordinated, but subordination is made effective through the resumating action of uka. Indeed, it may act alone, suffixless, as total resumator." / quoted in English by IGR/

Evidently, as Hardman points out, connective words serve as both linkers and resumators. When used a a linker, the connective is placed between the two simple statements. When it acts as a resumator, the connective is added at the end of the two simple statements. Thus, in Aymara propositional logic, a double connective statement may be formally expressed as follows:

• p(x,y) = x.Ml + (xy).A + y.M2
• p(x,y) = x.Ml + y.M2 = (xy).A

In these expressions, the symbols "x" and "y" represent simple statements, followed by their respective modal stands for the suffixes "M1" and "M2". The term "(xy)" stands for the connective between the two simple statements, which in Aymara is almost always "uka". The linkage modality ia conveyed by suffix "A", thus "(xy).A" represents the typical connective word for the logic of the statement, but in relation to the modal suffixes M1 and M2. The same connective may have different logical functions, depending on which simple modal statement it is used with.

The parentheses in the term "(xy).A" indicate that the suffix applies to the "product" of the simple statements, i.e, to both of them. This meaning will be described algebraically below. Suffix "A" always acts upon product xy, therefore, the parentheses can be omitted, and the connective term written simply "xy.A". Sometimes, the connective term contains the adverb of negation "jani" in addition to the linking suffix "A", for example, "jan ukaçti" (but, except). As negation involves a change of sign in our propositional variables, such an example will be written "-xy.çti", or, in general, "-xy.A". This convention can also be algebraically established. As has already been indicated, the logical definition of any simple logical proposition is arrived at by determining its kimsakus since for the three values of x there are three values of p = x.M. However, for a double connective logical proposition, the statement now has nine possible values:

• p = x.M1 + y.M2 + xy.A,

which correspond to the nine combinations of truth-values for "x" and "y". Thus, the logic of a double statement is not specified by a triad, but by a nine-element matrix. It could be referred to as a second-order triad having 3 ** 2 elements. A triple statement, made up of three simple statements, has a third-order triad with 3 ** 3 elements, and so on.

The second-order triad will be displayed as a nine-element row rather than in rows and columns, as matrixes are usually represented. This display will prove very useful for the analysis of truth-tables or matrixes with respect to inferential problems. This kind of second-order triadwill be called "päkimsaku."

As has already been said, whereas in bivalent logic there are only 2 ** 4 = 16 possible combinations (second-order diads), in trivalent logic there are 3 ** 9 = 19683 second-order triads. Thus, it is not practical to give a special name or symbol to each double statement in Aymara, as is usual in bivalent logic. In some instances, one can make logical "analogies", to try to reach an approximate equivalent translation; however, the reader should bear in mind that there is no equivalency between bivalent and trivalent connectives.

The examples used in this book to determine first-order triads, or kimsakus, for modal statements were taken from Aymara grammars printed over the past four centuries. This procedure ensures that the triad assigned to any simple suffix is absolutely correct and originates from the Aymara language itself. They can be, verified algebraically.

It would have been ideal to follow the same procedure of logical analysis to determine second-order triads which correspond to double connective statements. Unfortunately, the grammars give very few examples of connective logical propositions, and the translations of these few examples are not always consistent, and, in some cases, are even contradictory. Therefore, I suggest rather that connective statements be generated by following the logical syntax of Aymara which can be deduced from what has been studied and established up to this point. The logical meaning of these statements can later be verified by checking with Aymara-speaking people. Finally, the trial by fire will be the algebraic method, which obviously must apply also to connective statements, as will demonstrated below.

Some biconnective functors of Aymara: p(xy)

	x =	1 0 -1	1 0 -1	1 0 -1
	y =	1 1 1	0 0 0	-1 -1 -1
Conjunctions (x y y):
x.çti + y.çti + xy.ska	=	1 0 -1	0 -1 -1	-1 -1 -1
x.ska + y.ska + xy.ça	=	1 0 -1	0 0 -1	-1 -1 -1
x .mpi + y.mpi + xy.lla.pi	=	1 -1 -1	-1 0 -1	-1 -1 -1
Adjunctions (neither x not y):
x.sa + y.sa + xy.sa.pi	=	-1 -1 -1	-1 -1 0	-1 0 1
x.lla + y.lla + xy.ki	=	-1 -1 -1	-1 0 0	-1 0 1
x.wisa + y.wisa + xy.lla.pi	=	-1 -1 -1-1 0 -1	-1 -1 1
Disjunctions (inclusive or alternative x or y) :
x.ça + y.ça + xy.lla.ska	=	1 1 1	1 0 0	1 0 -1
x.sa.lla + y.sa.lla + xy.sti	=	1 1 1	1 1 0	1 0 -1
Implications (if x then y):
x.ka + y.su + xy.lla.pi	=	1-1 0	1-1 0	-1 0 1
x.ka + y.su + xy.ça	=	1 0 1	1 1 -1	-1 -1 1
x.ska.çi + y.ki + xy.salla	=	1 1 1	0 0 1	-1 0 1
x.lla + y.salla + xy.ça	=	1 1 1	0 1 1	-1 0 1
x.wa + y.çti + xy.ja	=	1 0 1	0 -1 0	-1 -1 1

Examples:

• mantaçti umtaçti ukaska
  y comiste y bebiste 'en ese caso'
  /and you ate and you drank in that case/
• warmipami jilapampi tullanpi jutiwa
  tanto con su mujer como con su hermano 'y todo' ha venido
  /both with his wife and his brother "and all" has come/

4.7 The algebraic method

Following the method of logical analysis outlined in the preceding section, the kimsakus for the functors generated by the various logical suffixes of Aymara were obtained from the logical meaning of the statements to be analyzed.

However, as the reader may already have noticed, this logical meaning is, in the final analysis, a matter of interpretation. In some instances, for example, in negation, it is patently clear that the statement has been assigned the right kimsaku. In other cases, the assigned kimsaku is less obvious, and even arguable. This is particularly so when the Spanish translations are used for the logical analysis of Aymara statements, as the interpretation may lead to serious objections to the methods used.

In this section, the reader will find a rigourous and convincing method for demonstrating that the syntax of Aymara is based on a trivalent system of logic. This is the key finding of the research on Aymara logic presented in this work. The use of mathematical language is unavoidable for this reason; however, given the nature of this monograph, mathematics will be used sparingly and only when absolutely necessary.

Although this may seem surprising, it can be stated without doubt that the logical syntax of Aymara has an algebraic, trinary, ring structure. The trinary digits 1, 0, -1, will be used to define this algebraic ring which shall be called "Aymara siwi" (Aymara ring), in honour of our forefathers, those Qoya scientists who designed the syntax of this language. (sic)

"Aymara siwi" can be defined by the following axioms:

A1: "Aymara siwi" covers the set of trinary digits: 1, 0, and -1. These digits will be called "trits", to draw an analogy between them and the "bits" of Boole's binary algebra.

Trit 1 is equivalent to bit 1; both represent the truth-value "true". Trit 0 represents the third truthvalue (perhaps true and perhaps false.) Obviously, trit 0 has no binary equivalent. Trit -1 is the equivalent of bit 0; both represent the truth-value "false. "

A2: "Aymara siwi" is structured by the operation which we shall call "product of x times y", or simply "x * y", established according to the following table:

x =	1 0 -1	1 0 -1	1 0 -1
y =	1 1 1	0 0 0	-1 -1 -1
x*y =	1 0 -1	0 0 0	-1 0 1

A3: "Aymara siwi" is also structured by an operation which we shall call "addition of x plus y", or simply, "x ± y", established by the following table:

x =	1 0 -1	1 0 -1< td align=middle>1 0 -1
y =	1 1 1	0 0 0	-1 -1 -1
x+y =	-1 1 0	1 0 -1	0 -1 1

These three axioms show that all variables in "Aymara siwi" have the f ollowing properties:

• Pl: There is always a trinary number R such that:
  y = x + R
  number R will be called "y - x" ("y minus x")
• P2: y - x = y + (-x) where (-x) means (-1)*x
• P3: x + x = -x
• P4: x + x + x = 0
• P5: x * x * x = x
• P6: x * y = y * y (product is commutative)
• P7: x t y = y + x (the sum is commutative)
• P8: x * (y * z) = (x * y) * z (product is associative)
• P10: x + (y+z) = (x + y) + z (the sum is associative)
• P11: x * (y+z) = x * y + x * z (distributivity)

It is now evident that the "Aymara siwi" has the properties of a ring. As readers well versed in mathematical logic will surely have noticed, the "Aymara siwi" differs from Boolean algebra in this respect: the algebraic operations of multiplication and addition are not identical to the logical operations of conjunction ("and"), and adjunction (alternative "or"). However, it can be mathematically demonstrated that all functions of Aymara trivalent logic can be expressed using the operators "*" and "+" of the "Aymara siwi."

For example, the following are the polynomial expressions for some of the modal functors mentioned in the preceding sections:

• x.ka = -1 -x
• x..ti = 1 + x
• c.ka.ti = -x
• x.pi = -1 + x + x * x
• x. su = 1 + x - x * x

Using the tables which appear in axioms A1 and A2, the reader will be convinced that the kimsakus which were obtained for the modal statements following the methods of logical analysis are in fact generated by these polynomials. In other words, two different and independent methods lead to identical results, which, accordingly, confirm each other.

The most surprising finding is that the logical suffixes of Aymara are algebraic operators. This can be easily proven as follows:

• let p(x) be the function x.ka
• and let q (x) be the function x.ti
• evidently, the function x.ka.ti becomes q(p(x)) = -x

This same test can be used for more than a hundted logical compound suffixes currently in use in present-day Aymara; and for suffixes recorded in the grammars written four centuries ago. In all cases, without exception, it can be demonstrated that for any compound suffix in a statement determined by suffixes S1 and S2:

it is always true that:

if p(x) = x.Sl and q(x) = x.S2

then q(p(x)) = x.S1.S2

As an exercise, the reader is invited to verify that:

• x.ska = (x.su).ka.

It is further suggested that the reader find the polynomial for x.sa; then use the polynomial for x.ti given above, to prove that:

• x.sti = (x.sa).ti.

Likewise, for the connective functions there are polynomials with several variables; they can be used to calculate the values of any truth-table, for example:

• x y = x.ça + x.ça = xy.lla.ska = -x-y+xy-xx-yy+xxyy
• x y = x.ska + x.ska + xy.ça = -x-y-xy+xx+yy-xxyy
• whera xy = x*y

These results have been presented rather sketchily; however, it will have served to give an idea of the tremendous power of "Aymara siwi": Any modal or connective statement can be represented by its corresponding polynomial, which is formed with just two operators:'*' and '+'.

Now, in "Aymara siwi" any solution to an inferential problem is equal to the solution of a system of (linear) equations. In fact, truth-tables are not necessary, since they are only a representation of the results which can be calculated either manually or with a computer using axioms A1, A2, and A3.

In Boolean algebra, a propositional variable x cannot be (algebraically) related to its negative Nx, which. in "Aymara siwi" is simply -x. Because of this, "Aymara siwi" is much more efficient than Boolean algebra, even for the solution of bivalent inferential problems, or the logic of binary circuits.

Based on these conclusions, I believe we are entitled to speak of Aymara logic as an original theory, which is as complete and consistent as Aristotelian logic. I also believe it is safe to say that Aymara logic is more ancient than Greek logic, as appears to be indicated by the results of archeological excavations at the site of the "taypi qala" ruins, also called Thiawanaku, which was the cradle of Qoya culture.

4.8 Sumary of simple modal suffixes of the Aymara language

: :'it is a synonym of'
-: 'it is an antonym of'

x.pi = ( 1 -1 -1) 'it is certain that x'	: : -x.ça .. : : 'it is indubitable that x'
x. ça = (-1 1 1 ) 'it is doubtful that x'	: : -x.pi : : 'it is uncertain that x'
x."su = ( 1 1 -1) 'it is possible that x'	: : (-x).ça : : 'it is doubtful that not x'
x. ki = ( 1 0 0) 'it is likely that x'	: : -x.lla : : 'it is not unfavorable that x'
x.lla = (-1 0 0) 'it is unfavourable that x'	: : -x.ki : : 'it is unlikely that x'
x.sa = ( 0 0 1) 'it is adversative that x'	: : (-x).ki : : 'it is likely that not x'
x.ka = ( 1 -1 0) 'it is evident that x'	: : -x.ti : : 'it is not controversial that x'
x.ti = (-1 1 0) 'it is controversial that x'	: : -x.ka : : 'it is not evident that x'

x.çi = ( 0 1 0)
'it is contingent that x'

Antonyms:

• x.pi -: x.sa.pi
• x.su -: x.ça
• x.ki -: x.sa
• x.sa.lla -: x.lla
• x.ska -: x.ti.ska
• x.ti.ka -: x.ti

(1) Translator's note: "siempre" means "always"; it is used here in a very peculiar way, which would be difficult to understand in other Spanish-speaking countries

(2) Translator's Note: According to standard Spanish grammars (including the Spanish Academy's) there are five moods: the Indicative, the Subjunctive, the Imperative, the Potential /Conditional/, and the Infinitive.

(3) Translator's Note: IGR uses the word "Abdiccion", which is not recorde in standard Spanish dictionaries. He may have in m 3 the Spanish word "abduccion" (a syllogism in which the Major Premise is evident and the Minor Premise is probable). Webster's Collegiate Dictionary does not give this meaning for the English word "abduction". It must be pointed out that in Charles Sanders Pierce's philosophy, "abduction" is that kind of reasoning that derives an explanatory hypothesis from a given set of facts.

(4) Translator's Note: IGA says the above expressions are auxiliary verbs in Spanish. They are not. However, their English counterparts are. The author might have been thinking of the English forms.

(5) IGR says "interrogative mood" (T's note)

(6) Translator's Note: "acaso" (no accent) normally means "by any chan in these sentences it is used as an adverb of doubt meaning "perhaps, maybe". "siempre" means "always"; it is used here meani "in any case " (app. value)

(7) Translator's "Note: IGR uses the word "adversidad" /adversity, misfortune/

(8) Translator's Note: The word "ni" /neither/ is used in this context to indicate a resounding negative.

(9) Translator's Note: IGR uses "adverse"

(10) Translator's Note: IGR uses "adverse"

(11) Translator's Note: This quote is translated by IGR in the Spanish original. The English word "tense" is translated once by "tiempo" /tense/, but later on the author uses "modo" /mood/ to render it in Spanish. Thus , the reader must be aware of the possibility of any confusion between these two terms in this work.