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Greek science is the classic example, par excellence, of a normal science, whose paradigms, according to Kuhn (TK), have lasted thousands of years. The geometry of Euclid (330 BC) reigned triumphant and undisputed until the introduction of non-Euclidean geometries by Gauss (1777-1855), Lobachewsky (1793-1856), Boylai (1802-60), and Riemann (1826-66). The logic of Aristotle (382-322 BC) is even more resistant to change because it is immersed in the languages which reflect the Western mode of thought.

Immanuel Kant (1724-1804), philosopher and Professor of Logic at Koenigsberg, was fully convinced that "Aristotle did not omit any essential aspect of knowledge; it only remains for us to become more precise, methodical, and orderly."

The research of the Polish thinker J. Lukasiewicz was a sharp departure from the Aristotelian interpretation of logic. Lukasiewicz, a leading member of the Warsaw school of logic, published his paper "0 logice trojwartoscioweJ " ("On Trivalent Logic") in 1920. This publication, the point of departure for non-Aristotelian systems of logic, was not translated into Spanish until 1975 (JL1).

According to J. Ferrater Mora (JFM), there is some evidence that William of Occam (1298-1349) had already suggested the use of three truth-values. Ferrater Mora also indicates that around 1910, the "Russian mathematician N. N. Vasilev of the University of Kazan, published several articles in which he put forward and developed a three-valued logic. Vasilev's fundamental idea consisted in transposing to Logic the rules followed by Lobachewsky in founding his non-Euclidean geometry. Lobachewsky, who had been a Professor at the same University, developed his geometry by eliminating the parallel postulate. Likewise, Vasilev developed his trivalent logic, which he called "non-Aristotelian logic", by eliminating the law of excluded middle. However, the most important and influential contemporary publications on polyvalent logic have been published by Jan Lukasiewicz, Emi1 L. Post, and Alfred Tarski."

In 1930, Lukasiewicz published his paper "Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalkuels" (Philosophical Observations on Polyvalent Systems of Propositional Logic). In this paper the author explains his ideas in great detail, from the point of view of both logic and philosophy. He analyzes the consequences of modal statements which, within the limited framework of bivalent logic, "go against all our intuitions." He also clearly demonstrated the incompatibilities of theorems regarding modal propositions in bivalent propositional calculus" (JL1-69).

In order to give the reader some idea of the insight and courage it took to break with the Aristotelian tradition, I would like to quote the following paragraphs from Lukasiewicz' above-mentioned paper:

"When I became aware of the incompatibility of traditional theorems of modal propositions in 1920, I was in the process of establishing a normal bivalent propositional calculus based on the matrix method. At that time I was convinced that it was possible to demonstrate all the thesis of the ordinary propositional calculus assuming that propositional variables could take on only two values, "0" (false), and "1" (true). This assumption corresponds to the basic theorem that every proposition is either true or false. For brevity's sake, I will refer to it as the law of bivalence. Although it is sometimes referred to as the law of excluded middle, I prefer to restrict this latter term to the well known principle of classical logic which states that two contradictory propositions cannot both be false at the same time."

"Our whole system of logic is based on the law of bivalence, even though it has been fiercely disputed since ancient times. Aristotle knew this law, but he questioned its validity as it referred to future contingent propositions. The law of bivalence was flatly rejected by the Epicureans. Chrysippus and the Stoics were the first ones to develop it fully and incorporate it as a principle of their dialectic, the equivalent of modern day propositional calculus. The arguments regarding the law of bivalence have metaphysical overtones: its supporters are resolute determinists; whereas its opponents generally have an indeterministic Weltanschauung. Thus, we are once again in the area of concepts of possibility and necessity."

In the following paragraph, Lukasiewicz discusses an interesting example of a logical statement using the future tense, and demonstrates that it is not possible to affirm whether it is either true or false, because it is basically uncertain. Next, Lukasiewicz discusses the definitions of his trivalent propositional calculus, which is based on just two three-valued logical propositions, negation and implication, and from which he developed all the remaining propositions necessary for a complete logical system.

Some examples of statements in Spanish may be beneficial for readers unfamiliar with the terminology used in logic. In all languages there are several kinds of sentences. Statements are a kind of sentence. They have both objective and logical meaning, and can be assigned a truth-value. For example:

are statements, because apart from the objective meaning involved in the concepts of "rain" and "muddy", they can also be evaluated as to their truth or falsity. However, sentences like: are not statements because it is not possible to decide whether these sentences are true or false, i.e., they cannot be assigned a truth-value.

Logic is only interested in the truth-value of statement regardless of any conceptual content they may also have. In textbooks of logic, statements are also called "logical propositions."

The main subject of logic is inference, a process by which a conclusion is reached from one or more premises. Premises and conclusions are always statements, i.e., sentences which have specific syntactic characteristics in each language (in which people reason). The consistent use of given syntactic structures develops the operating mechanisms by which the human mind makes inferences. Let's consider the following inferential sohema:

The logical conclusion arrived at does not depend on the conceptual meaning of the statements in the schema. This iriferential schema may be expressed formally by saying:

The concept of implication can be understood using syntactical forms such as: "if x ... then y ...", or forms similar to this. The consistent use of logical meaning in language patterns makes it possible to infer. Any person or computer capable of understanding such syntactic structures can arrive at logical conclusions, though other forms of expression in the language may be primitive.

There is no one universal logic underlying the syntactic structures of all languages. Logic meets the need of man, or of the computer, to manipulate truth-value relations. This need is conditioned, however, by a set of truth-values, adopted at a metalogical level, which are at the basis of the system of logic one wants to operate with. This metalogical level is in turn processed through language, which already has its own logic. The question then arises: how does one switch from one system of logic to another?

The history of logic shows us that it is not possible to switch logic systems while remaining within the conditioning framework of one language. This can only be achieved by resorting to another language. Up to now, recourse has be the formal language of mathematics, whose syntax makes it possible to define structural generalizations. Lukasiewicz used mathematics to generallze bivalent truth-tables and define trivalent truth-values from which he developed a new system of logic which, unlike Aristotle's, can only be understood using formulas.

The mathematization of Aristotelian logic dates back to Boole (1815-1864). Boolean algebra operates with two truthvalues: True ("1"), and False ("0"). Using these binary digits it is possible to unambiguously express any bivalent logical function. Electronic circuits are also binary; thus, computers also "think" according to Aristotelian logic.

To avoid confusion with the notation used in this book for the ternary digits of trivalent logic, "false" will be written "-1" rather than "0" in both the bivalent and trivalent truth-tables. Boolean algebra cannot describe Aymara logic, so it will not be used in this book; therefore, the notation adopted will not lead to any confusion.

A "logical variable" is a symbol, for example "x", which represents the truth-value of a given statement, A "logical function", or "functor" for short, is a relationship p(x) which assigns a truth-value p according to the values of one or several variables. In bivalent logic there are only four one variable p(x) functors which may be represented by the following truth-tables:

x 1 -1 (it is raining) affirmation of x
N(x) -11 (it is not raining)negation of x
T(x) 11 (it is either raining or not)tautology of x
-T(x) -1-1 (it is raining and it isn't) contradiction of x

Also, there are only 16 two-variable functors p(x,y); the most widely used in everyday language are:

x1-11-1 .
y11-1-1 .
x/\y1-1-1-1 conjunction (x and y)
x\/y111-1 alternative (x or y)
x=>y11-11 implication (if x then y)

Truth-tables show the values of a given functor for all possible values of its variables.

The great advantage of truth-tables is that they make it possible to precisely define the logic of sentences in any given inferential schema, regardless of the language used. Moreover, truth-tables are an indispensable tool for reaching conclusions from several complex premises. These conclusions would be very difficult or impossible to arrive at by a process of purely mental inference. Also, this is the only way complicated logical instructions can be given to a computer.

The following example will illustrate how truth-tables are used:


P1 : x=>y"if it has rained, there is no mud"
P2 : N(y)"there is no mud"

Truth- Table

x 1-11 -1
y 11-1 -1
P1 x=>y 11 -11
P2 N(y) -1-11 1

The fourth column is the only one which satisfies both premises; thus, x = -1 (the logical conclusion is, "it has not rained".

The following example is less obvious: we want to infer whether there was a full moon last night and whether it is raining today, knowing either that there are or there are not reddish clouds in the sky; the following inferential schema is used:

Statements are:

The premises are: There are three propositional variables, so there are 2^3 = 8 combinations of binary digits; thus, the corresponding truth-table must have 8 colums (or rows, depending on how it is displayed):

1234 5 678
x=1-11 -11-11 -1
y=11 -1-1 1 1-1-1
z=1111 -1-1-1-1
P1:x=>y=11-1 111-11
P2:N(z=>x)=-11-1 1-1-1-1-1
P3: N(x)=-11-1 1-11-11

In this example of inference, the truth-table shows two columns (2 and 4) which contradictorily satisfy the premises; therefore an unequivocal logical conclusion cannot be reached.

The table shows that the set of premises is satisfied both if there was a full moon last night and if there was not, i.e., instead of a logical conclusion we have a contradiction. To solve this contradiction we must reword our premises less categorically using modal expressions which reflect the uncertainty of our knowledge about the implied causal relationships.

As the premises were formulated within the framework of bivalent logic, they cannot represent the concepts of possibility or probability which might have been more appropriate. These statements cannot be formulated with absolute certainty because of their content (uncertainty in the prediction of meteorological phenomena).

Unlike computers, human beings require a more flexible gradation of truth-values than the absolute "either...or." Aristotle himself was aware of this, when he introduced the notions of "possibility" and "contingency" into his modal logic. These concepts of logical modality were first formulated by the American logician Lewis who developed a set of axioms to interpret the concept of "strict implication."

Lukasiewicz used truth-value tables to define his trivalent functors, from which he consistently developed the theorems of modal logic. Lukasiewicz used the symbols 1, 1/2 and 0 to denote the three truth-values of his trivalent logic. Unlike Boole's binary digits, these symbols are numerical, but not algebraic: one canot perform operations with them; they are only used to display truth values: 1 = true; 0 = false; 1/2 = a third truth-value, equidistant from both: "perhaps true and perhaps false."

As will be explained later, although it is also trivalent, Aymara logic is more general than Lukasiewicz's logic, because it is algebraically structured. For this reason, the symbols used for the three truth-values are also ternary digits which can also serve as operators; this makes it possible to handle a larger number of functors than those studied by Lewis and Lukasiewicz.

For those who may wish to consult Lukasiewicz's works, the following table shows the equivalence of symbols used to denote trivalent truth-values:

Representation of trivalent truth-values

Logic true perhaps true and
perhaps false
Boole (binary digits) 1 none0
(it is not an algebraic digit)
Aymara (trinary digit) 10-1
Aymara adverb jisa
(perhaps yes and perhaps no)

Lukasiewicz's approach is new: according to him modal and connective functors are defined according to the following truth-tables which he says "were obtained after detailed consideration and which are more or less plausible": In this work the following symbols are used:
x, y elementary amodal statements
p(x), q(x)elementary modal statements
p(x,y), q(x,y)biconnective statements

Connective statements having more than two variables be considered in this monograph.

Modal Functors p(x) according to Lukasiewicz

x =1 0 -1 amodal statement
N(x)= -10 1 negative statement
G(x)=1 -1-1 statement of certitude ("Gewissheit")
M(x)=11-1statement of possbility ("Mglichkei)

Connective functors p(x,y) according to Lukasiewicz

x=10-1 10-1 10-1 .
y=111 000 -1-1-1.
x=>y=111 011-101 Conditional C(x,y)
x\/y =111 10010-1 Alternative A(x,y)
x/\y=10-1 00-1-1-1-1 Conjunction K(x,y)
x<=>y=10-1 010-1 0-1 Equivalence E(x,y)

In any trivalent system of logic there are 3^3 = 27 modal functors and 3^9 = 19,683 connective (two-variable) functors! As will be demonstrated in the next chapter, these many functors may be dealt with using the logical suffixes of Aymara syntax (just nine modal operators and a subordinative operator). All of them are used in the language as currently spoken: The following examples are useful in illustrating this point:

Some modal functors of Aymara : p(x)

x.wa =10-1 amodal statement of irrefutability
x.ka.pi =-101 =N(x)modal notion of negation
x.pi =1-1-1 =G(x)modal notion of certitude
x.a =-111 modal notion of doubt
x.s =11-1 =M(x)modal notion of possibility modal notion of likelihood
x.i =010 modal notion of contingency =110 modal notion of plausibility (+)
x.ti.i =011 modal notion of plausibility (-)

The notation used here is based on the Aymara suffixes used, to generate logical statements. For example, the sufflxes "ka" and "ti" are used to generate negative statements; thus, the negation of x is symbolized x.ka.ti. For example:

x.wa=juttawa you will come
x.ka.ti=janiw jutktatiyou will not come
x.i=inaj juttayou might come jutta perhaps you will come inas jutitaperhaps you will not come

The next chapter explains the method followed to assign a certain truth-table to any given simple or complex Aymara suffix, be it a modal or a connective functor.

Before concluding this brief introduction to trivalent logic and its relationship to the modal suffixes of the Aymara language, let us use the functors of the language to reword the statements in the inferential sohema given as an example. The examples will now read:

x.wa='qenayrantatanwa' ('it is somewhat cloudy')
y.wa='masarma pajsinwa'('there was a full moon last night')
z.wa='jallunwa' ('it has rained')

Using Aymara modal forms, the premises are:

P1: x.ka + y.s + xy.lla.pi = 1
'qenayrantatkiwa masarma pajsispawa tullanpi; eqawa.'
'having been somewhat cloudy last night, it is possible there was a full moon but; it is correct.'
P2: x.ka + + xy.lla.pi = -1
'jallkiwa qenayrantataspa tullanpi; janiw eqkiti'
'being raining possibly it would be somewhat cloudy but; it is not correct.'
P3: x.ka.ti = 1
'janiw qenayrantatkiti'
'it is not somewhat cloudy'

Using trivalent truth-tables, logical analysis now demands a table of 3^3 = 27 columns, of which only eight (no zeros) correspond to the bivalent table:

x=1 0-1 1 0-1 1 0-11 0-11 0-1 1 0-11 0-11 0-1 1 0-1
y=1 1 10 0 0 -1-1-11 1 10 0 0-1-1-11 1 1 0 0 0-1-1-1
z=1 1 11 1 1 1 1 10 0 00 0 00 0 0-1-1-1 -1-1-1-1-1-1
P1 =1-1 01-1 0 -1 0 11-1 01-1 0-1 0 11-1 01-1 0-1 0 1
P2 =-1-1 1-1-1 1 -1-1 1 1 1 01 1 01 1 00 0-10 0-1 0 0-1
P3 =-1 0 1-1 0 1 -1 0 1-1 0 1-1 0 1-1 0 1-1 0 1-1 0 1-1 0 1

Now the situation has changed radically, because column 9 makes it possible to infer an unequivocal conclusion from the premises. That is to say, one arrives at the logical conclusion that:

The inference is possible because the premises have been toned down. Provided the premises are true, the conclusion reached is also true. Premises which have some elements of "indetermination" make the conclusion less certain compared with an inference based on more "categorical" premises. However, even a clear conclusion with some degree of uncertainty is preferable to a contradiction.

Although the truth-table shows that the inference in our example is perfectly valid, the fact that it is possible to arrive at a very precise conclusion from a set of premises loaded with "uncertainty" must seem absurd, or at least very strange, to minds programmed" according to Spanish bivalent logic. However, as modal suffixes always "program" trivalent truth-tables, people who have been dealing with them since early childhood will be content with conclusions arrived at from modal premises whenever it is not possible to make inferences from true clear-cut premises, because of a lack of information or where aspects of contingency are involved.

It is always more useful to be able to make decisions even if they involve a certain degree of risk, than to do nothing simply because the result will not be certain. This is the practical advantage of using modal logic to make inferences.

Spanish-thinking people feel that "uncertainty is unbearable' and has nothing to do with logic, whereas for an Aymara-thinking person, "ina" is a part of reality, and is as logical as "jisa" or "jani". If Lukasiewicz had been a Qoya, he would probably have considered the bivalent logic of Spanish-speaking people as strange and worthy of study as polyvalent systems of logic.

Chapter index