The Logic of Four Alternatives
By K. N. Jayatilleke

Philosophy East and West
V. 17:1-4 (1967)
pp. 69-83

Copyright 1967 by University of Hawaii Press
Hawaii, USA


 

 

p. 69

    I AM WRITING ON the Buddhist logic of the four alternatives (catu.sko.ti), as it appears for the first time in the Pali Nikaayas, for three reasons. First, it appears to be a typical "East-West problem" in philosophy. Secondly, it is evident that if the thesis adduced in this article is correct, the problem has baffled both Indian as well as Western scholars, and among these Indian scholars we have to reckon classical scholars held in great esteem, such as Naagaarjuna. Finally, this system of logic should be of some interest to modern students of logic, not only because it provides a novel method of classifying propositions into logical alternatives, but also because it does so in such a manner that the alternatives are not dependent on the number of truth-values assumed in the system of logic.

    The nature and significance of these reasons would become evident if we give an exact exposition of this logic of four alternatives, which was known in the later Buddhist tradition as the catu.sko.ti (tetralemma). But it must be borne in mind that to call this the "logic of four alternatives" is already to adduce a theory which needs to be proved in the light of all the available evidence. I have previously given some account of this system of logic,[1] but it stands in need of further clarification and exact definition, which I propose to attempt in this article, avoiding as far as possible any repetition of what I have already said on the subject.

    In the Pali Nikaayas the four alternatives are referred to as "these four positions" (imesu catusu .thaanesu)[2] and it is said that religious teachers at the time were trying to state the nature of the existence of the perfect person after death without going "outside these four positions" (a~n~natra imehi catuhi .thaanehi).[3] These references seem to suggest that they were regarded as four logical alternatives and it was believed that the truth with regard to any matter lay, perhaps, in one of these alternatives.


1.    See K. N. Jayatilleke, Early Buddhist Theory of Knowledge (London: George Allen & Unwind Ltd., 1963), pp. 333-351; also, K. N. Jayatilleke, "Some Problems of Translation and Interpretation II," University of Ceylon Review, Vol. VIII, No. I (January, 1950), 45-55.

2.    Sa.myutta Nikaaya, IV, 380. (Reference is to Pali Text Society edition, hereafter P.T.S.)

3.    Ibid.

 

 

p. 70

    Some of the examples given in the texts strongly suggest that the four alternatives present possibilities which are mutually exclusive and together exhaustive, viz.:

(1)    A person is wholly happy
(2)    A person is wholly unhappy
(3)    A person is both happy and unhappy
(4)    A person is neither happy nor unhappy[4]

(1)    X is a person who torments himself
(2)    X is a person who torments others
(3)    X is a person who both torments himself as well as others
(4)    X is a person who neither torments himself nor others[5]

(1)    The universe is finite
(2)    The universe is infinite
(3)    The universe is both finite and infinite
(4)    The universe is neither finite nor infinite[6]

    One example given in the Nikaayas confirms the fact that if one of the four alternatives stated is true, then the others are false.[7] There are, in fact, in the Nikaayas certain examples in which all four alternatives are rejected but these, in my opinion, are not to be explained as violating the exposition we have given below.

    Professor Robinson has also proposed taking the propositions as logical alternatives:

A typical piece of Buddhist dialectical apparatus is the tetralemma (catu.sko.ti). It consists of four members in a relation of exclusive disjunction ("one of, but not more than one of, 'a,' 'b,' 'c,' 'd,' is true"). Buddhist dialecticians, from Gautama onward, have negated each of the alternatives, and thus have negated the entire proposition. As these alternatives were supposedly exhaustive, their exhaustive negation has been termed "pure negation" and has been taken as evidence for the claim that Maadhyamika is negativism. There is thus an extra-logical interest in analyzing the form of the catu.sko.ti.[8]

We agree with this statement of Robinson only insofar as he says that the


4.    Diigha Nikaaya (P.T.S.), I, 31.

5.    Majjhima Nikaaya (P.T.S.), I, 341 ff.

6.    Diigha Nikaaya, I, 22, 23.

7.    Anguttara Nikaaya, II, 25; cf. Jayatilleke, Early Buddhist Theory of Knowledge, pp. 345-346.

8.    See Richard H. Robinson, "Some Logical Aspects of Naagaarjuna's System," Philosophy East and West, Vol. VI, No. 4  (January, 1957), 301-302.

 

 

p. 71

four propositions are "in a relation of exclusive disjunction" and that "these alternatives were supposedly exhaustive." But it is historically incorrect to say that Gautama "negated each of the alternatives." The example cited above shows that when one of the alternatives was taken to be true, the rest were deemed to be false.

    First it is necessary for us to prove (1) that each alternative is mutually exclusive and (2) that all the alternatives are together exhaustive. We should also be able to do so for both quantified and non-quantified propositions.

    Let us take a familiar case. In a two-valued logic of two alternatives, the logical alternatives may be stated as follows :

(1)    p
(2)   
p

Here, p is a proposition of the form "S is P" or "a R b," while p is a proposition of the form "S is not P" or "it is not the case that a R b." The proof that the alternatives are mutually exclusive is that it can be shown that if each of the alternatives is true, the other is necessarily false. Thus,

                    

    The proof that the propositions are together exhaustive consists in showing that it is necessarily false that none of the alternatives is true. Thus the formula, - (p v -p), must be necessarily false. This can be shown to be the case. For if we assume that the formula is true, it leads to a contradiction:

          

Thus, it is logically impossible that the formula could be true and it is, therefore, necessarily false. This means that the alternatives are together exhaustive since at least one of them must necessarily be true.

    If we are to give examples with quantified propositions, we are tempted to select, from among the Aristotelian forms, those which are deemed to be mutually contradictory, e.g.:

(1) O
(2) A

 

 

p. 72

    But there are certain ambiguities in the expressions O and A, which need to be cleared up before examples with a precise meaning can be given. With the common interpretation of O as an existential proposition, there is a certain ambiguity in regard to a false O proposition. In an existential interpretation we mean by the proposition "some dogs are not black" that "there exists an x such that x is a dog and is not-black." In general, an O proposition is interpreted to mean that "there exists an x such that x is an f and a not-g," symbolically written as ($x) (fx•-gx). But when we say that such a proposition is false, we can mean that "there exists an x such that x is an f but it is not the case that such an x which is an f, is a not-g." For example, "it is false that some dogs are not black" means that "there exists an x such that x is a dog but it is not the case that such an x which is a dog is not-black." Or we can mean that "there does not exist an x such that x is an f and a not-g because there is no x which is an f." For example, "it is false that some centaurs are not long-lived" means that "there does not exist an x such that x is a centaur and is not-long-lived because there is no x which is a centaur." These situations may be seen, when we examine the conditions under which an O proposition is false, represented as follows:

($x) (fx -gx)
 T  T T
 T  F F
 F  F T/F

    We can remove the ambiguity with regard to a false O proposition by defining both O and -O existentially. Then we would mean by O: ($x) (fx • -gx) • ($x) fx, assuming and emphasizing the fact that there are x's which are f's. And by -O, we would mean: -($x) (fx • -gx) • ($x) fx. It can be seen that these two propositional functions are mutually contradictory, for when O is true -O is false and when O is false, -O is true:

     

In the proposed sense of an existential O proposition, the proposition implies the existence of at least one x, whether the proposition be true or false. Now an existential -O is equivalent to an existential A proposition, so that the alternatives may be written as:

 

 

p. 73

(1) ($x) O (in the sense just defined)
(2) ($x) A

    However, they are not genuine logical alternatives. ($x) O and ($x) A are mutually exclusive, as we have shown above, but they are not together exhaustive because the following formula may be true:

               

    It may be seen that it is possible for both ($x) O and ($x) A to be false, if there are no fx's:

[($ x) (fx • -gx) ($ x) fx] V [($ x) (fx • -gx) ($ x) fx]
F F F F F

    So ($x) O and ($x) A are logical alternatives only within the framework of existential propositions. For instance, we can truthfully say that either all cats have tails or some cats (i.e., at least one cat) have no tails. Presuming that cats exist, if one of the propositions is true, the other is necessarily false and of the two propositions at least one must be true. But we cannot with the same breath say that either all centaurs are long-lived or some centaurs are not long-lived, for both alternatives are false since there are no centaurs. We have to bear these distinctions in mind when we seek to state the logical alternatives of quantified propositions according to the logic of four alternatives.

    Now Robinson has proposed the following alternatives for the catu.sko.ti, which he calls the tetralemma. The examples he gives are in the form of quantified propositions. In his own words:

When the tetralemma is quantified in this way, it is analogous to the four Aristotelian forms in some respects. The similarities and differences tabulate as follows:

Since "No x is not A" equals "All x is A", the fourth lemma is a conjunction of

 

 

p. 74

E and A forms. The third lemma is a conjunction of I and O forms. The fourth lemma is a conjunction of the contradictories of the conjuncts of the third lemma; "No x is A" is the contradictory of "Some x is A", and "No x is not A" is the contradictory of "Some x is not A". Thus there is a reciprocity between the third and fourth lemmas. Negation of the conjuncts of one always produces the other.[9]

We note that he equates "No x is not A" with "All x is A" and says that the fourth lemma is a conjunction of the E and A forms. This is the same as what he has done in his earlier article,[10] where he has stated the formulae using the notation of the Boole-Schroeder logical algebra:

The tetralemma resembles the four Aristotelian forms in some ways. Both sets comprise propositions constructed from two terms and the constants (functors) "all," "some," and "not." However, the third and fourth alternatives of the tetralemma are not simple propositions, but conjunctions. The comparison may be tabulated as follows, using the Boole-Schroeder notation.

    If we re-state Robinson's alternatives using the Aristotelian forms he employs, they would be as follows:

1    A
2    E
3    I • O
4    E • A

It can be shown that if Robinson's analysis is correct, the alternatives are not mutually exclusive or together exhaustive, contradicting his claim (in the same article) that the alternatives are "in a relation of exclusive disjunction"[11] and "were supposedly exhaustive."[12]

    It is generally admitted by modern logicians that the relationships among the Aristotelian forms A, E, I, O in the traditional square of opposition hold


9.    Richard H. Robinson, Early Maadhyamika in India and China (Madison, Milwaukee, and London: The University of Wisconsin Press, 1967), p. 57.

10.     Robinson, "Some Logical Aspects of Naagaarjuna's System," p. 303.

11.    Ibid., p. 301.

12.    Ibid., p. 302.

 

 

p. 75

only if they are given an existential interpretation. Assuming the truths of these relationships, it can be shown that when each of the first three lemmas is true, the rest are false. Thus, when A is true, E being the contrary of A is false. Likewise, O being the contradictory of A is false and, therefore, the conjunction I • O is false since at least one of its conjuncts is false. Similarly, E • A would be false since E is false.

    When E (the second alternative) is true, its contrary A is false and since its contradictory I is false, I • O would be false, while E • A would be false because A is false.

    When I • O (the third alternative) is true, each of the conjuncts, namely I and O, are true. Therefore, A which is the contradictory of O is false and likewise E, the contradictory of I. Since A and E are each false, it would follow that E • A is false.

    But the difficulty is with the fourth lemma, since E and A, being contraries, cannot both be true. Even if we assume its truth, then, since each of the conjuncts E and A would be true, the second and first lemmas must also be concurrently true. If so, the fourth lemma is not a logical alternative and Robinson has not proved his contention that the four members are "in a relation of exclusive disjunction."

    Robinson, however, seems to have suspected this when he made the following statement: "I propose to interpret the fourth alternative as 'No x is A and no x is not A.' This is true when x is null."[13] If this is what he means, then he has to resolve the ambiguity in the E and A forms and distinguish between the existential and non-existential E and A forms. But on the contrary, Robinson has identified the A form of his fourth lemma (E • A) with the A form of the first lemma when he says: "Since 'No x is not A' equals 'All x is A', the fourth lemma is a conjunction of E and A forms."[14] Likewise, he has identified the E form of the second lemma with the E form of the fourth when he writes down both as "No x is A."[15]

    In order to resolve the contradictions, it is necessary to remove the ambiguities of A and E by clearly distinguishing between the existential and non-existential forms. We may do so and state the existential A and E as ($x) A and ($x) E, respectively.

    The analysis of the fourth alternative as the conjunction of the denial of each of the conjuncts of the third alternative also appears to be defective. For the fourth alternative is a denial of the first two alternatives as well. In fact, the form of the language seems to suggest this, when we take the non-quantified


13.    Ibid., p. 302.

14.    Early Maadhyamika in India and China, p.57. (Italics mine.)

15.    Ibid.

 

 

p. 76

examples, which are the only examples given in the early Pali texts. The usual linguistic form is "S is neither P nor non-P," which appears to be a denial of a disjunction consisting of "S is P" and "S is non-P," i.e., it is neither the case that "S is P" nor that "S is non-P." But the linguistic form is not perfect and does not manifest the exact logical form, since the proposition is a denial of a disjunction of not only the first two alternatives but of the third alternative as well. We shall therefore symbolize it in that form. When we do so and also remove the ambiguities of A and E, the logical alternatives of the logic of four alternatives, using quantified propositions, may be stated as follows:

(1)    ($x) A
(2)    ($x) E
(3)    ($x) I • O
(4)    -[($x) A v ($x) E v ($x) I • O]

    It can now be shown that each of these is a logical alternative. We have already seen that when (1) is true, (2) and (3) are false. When (1) is true, the fourth alternative is also false, viz.:

               

    Likewise, it can be shown that when the second and third alternatives are each true, the rest are false. With regard to the fourth alternative, it can be shown that when it is true, each of the other alternatives is necessarily false:

               

    The question may be raised as to the conditions under which it is true. It is evident that when the whole expression is true, the disjunction is false. This is so when each of the disjuncts is false. Each of the disjuncts is false when each is non-existential, or, in other words, there are no fx's, viz.:

[($x)-(fx•-gx) •($ x) fx]v[($x)-(fx•gx) •($ x)  fx]v{[($x)(fx•gx) •($ x) fx] •[($ x)(fx•-gx) •($x) fx]}
F F F F F F F F F

 

 

p. 77

    We have now shown that when each alternative is true, the others are necessarily false, and also the conditions under which the fourth alternative is true. This means that the alternatives are mutually exclusive. It now remains to be shown that the alternatives are together exhaustive in the sense that at least one of the alternatives must necessarily be true. In other words, the following expression must be necessarily false, which is seen to be the case:

     

    We may even argue as follows: Assuming that the above expression is true, the whole disjunction within brackets must be false. Therefore, each of the disjuncts, ($x) A, ($x) E, ($x) I • O and -[($x) A v ($x) E v ($x) I • O] must be false. But if ($x) A, ($x) E, and ($x) I • O are each false, then the expression -[($x) A v ($x) E v ($x) I • O] would be necessarily true, viz.:

          

This leads to a contradiction. Therefore, the above expression is necessarily false. It follows that the alternatives are together exhaustive.

    We have thus proved that one and not more than one alternative must necessarily be true. It is, however, necessary to present the alternatives with non-quantified propositions. Robinson's proposal for translating the non-quantified into the quantified is to say that in the above formulae, such as "All x is A," etc., "'x' stands for the attributes of the entity in question."[16] This would result in the Aristotelian error of confusing quantified propositions with the non-quantified.[17] If we do so we would have to translate "Socrates is mortal" as "All the attributes of Socrates are mortal." "Socrates is mortal" would then be logically of the same order as "All men are mortal," when in fact it is not so. When we say "Socrates is mortal" we surely do not mean that "All the attributes of Socrates are mortal," for all the attributes or


16.    Ibid.

17.    See Bertrand Russell, A History of Western Philosophy, Third Impression. (London : George Allen & Unwind Ltd., 1948), p. 219.

 

 

p. 78

constituents of Socrates, taken singly or in isolation from the rest, may very well be immortal, while Socrates is mortal.

    The proposed translation does not make sense where the predicate qualifies the subject as a whole. In a class of cases in which this is not so, it is necessary to select from among the attributes those which belong to the determinable to which the predicate refers. Thus it will be absurd to translate "the universe is finite" as "all the attributes of the universe are finite," but we may very well translate the proposition as "all the spatial or quasi-spatial attributes of the universe are finite."

    In view of the difficulty of such translation without loss of meaning, it is necessary to examine the classification of such propositions in accordance with the logic of four alternatives on their own merits. Let us examine one of the historical examples given in the texts cited at the beginning of this article, which is typical of many of the examples in the Pali Nikaayas (i.e., in early Buddhism), namely:

(1)    A person is wholly happy
(2)    A person is wholly unhappy
(3)    A person is both happy and unhappy
(4)    A person is neither happy nor unhappy

In general these propositions are of the form:

(1)    S is P
(2)    S is non
-P
(3)    S is P and non
-P
(4)    S is neither P nor non
-P

On a relational analysis of the proposition, the alternatives would be:

(1)    a R b
(2)    a non
-R b
(3)    a R and non
-R b
(4)    a neither R nor non
-R b

An example of the latter would read as follows:

(1)    a is to the east of b
(2)    a is to the west of b
(3)    a is to the east and the west of b (e.g., if "a" is a straight road running east
-west through "b")
(4)    a is neither to the east nor the west of b

    It is evident from the historical examples that the second alternative is to be taken as the contrary of the first. We designate this by non-P instead of

 

 

p. 79

not-P and non-R instead of not-R. Let us symbolize (1) as p, and the contrary of p as p. It is evident that the third alternative is not a conjunction of the first two alternatives or of the contradictories of the first two alternatives (on the analogy of the quantified propositions). What, then, does it mean? Certain historical examples make the sense clear. When the statement "the universe is both finite and infinite" is made, it is explained as "the universe is finite in one dimension and infinite in another."[18]

    In general, the third alternative is, therefore, of the form: "S is partly P and partly non-P." Instead of partly we may substitute, and in historical examples we often meet with, synonymous expressions such as "in some respects," etc. This would have to be contrasted with the first alternative, where the sense and sometimes the wording make it clear that the meaning is "S is wholly P," while the second alternative means "S is wholly non-P." It is, therefore, necessary to define the conventional logical relations between "S is wholly P" and "S is partly P." We have represented "S is wholly P" as just p. We may then symbolize "S is partly P" as: Δp. The four alternatives would thus be symbolized as follows:

(1)    p
(2)    p
(3)    Δ p •Δ p
(4)    - [p v p v (Δ p •Δp)]

    When p is true, we may choose to adopt the convention that Δ p is false. This is, however, arguable in the light of conventional usage, for we may as well say that if p is true, then Δ p is also true. But there is no such uncertainty in usage with regard to Δ p, for if p is true, then Δ p is false. It follows from these conventions that if the first alternative is true, then the second which is the contrary of p is false, while the third and fourth are also false, viz.:

     

    Similarly, if p is true, the rest are false:

     


18. Uddham-adho anta-sa~n~nii lokasmi.m viharati, tiriya.m ananta-sa~n~nii, Diigha Nikaaya (P.T.S.), p. 23.

 

 

p. 80

    Likewise, if (Δ pΔ p) is true, the rest are false:

     

    In the same way, if -[p v p v (Δ p •Δ→ p) ] is true, the rest are false

     

    As in the case of quantified propositions, the question may be raised as to the conditions under which this proposition can be true. Since the second alternative is the contrary and not the contradictory of P and the third asserts that the subject has a combination of some of the contrary characteristics, there is left a part of the determinable constituting the universe of discourse which is referred to by the fourth alternative.

    We may illustrate this by considering the following four alternatives:

(1)    A person is wholly happy
(2)    A person is wholly unhappy
(3)    A person is both happy and unhappy
(4)    A person is neither happy nor unhappy

Happiness (sukha) in this context is a determinate quality characterizing a person's hedonic tone (vedanaa). When we remove the qualities of "happiness," "unhappiness" (dukkha), or a mixture of the two, we are left with "neutral hedonic tone" (adukkha-m-asukha vedanaa). So a person who is "neither happy nor unhappy" comprises the class of people experiencing a neutral hedonic tone. Such a class need not necessarily be a null class, although it could be so sometimes. This is another reason why Robinson's proposal to translate non-quantified propositions into quantified ones, by taking the predication as being made of the attributes of the subject, is unsatisfactory. Robinson's translation for the above example may read somewhat as follows: "The experiences of a person are neither happy nor not happy." This can be true only when there are no experiences, since it would mean "there is nothing that is both an experience and is happy and nothing that is both an experience and is not happy."

 

 

p. 81

    We have now shown that the four alternatives of these non-quantified propositions are mutually exclusive, since when one is true the rest are false. It remains to be shown that they are together exhaustive. This means that the following expression must be necessarily false, which is seen to be the case:

     

    In using the above two-valued logic of four alternatives, early Buddhism did not discard the common-sense two-valued logic of two alternatives. Both logics are used side by side and utilized for purposes of classification according to the subject-matter dealt with. Thus they are treated as complementary logics.[19]

    It would, of course, be self-contradictory to negate all the alternatives. The apparent instances in which this is done in early Buddhism are those in which, for some reason or other, each of the alternatives is misleading, being based on false assumptions, and therefore is to be rejected.[20] This rejection is not the same as negation. In a two-valued system of two alternatives, we can pose the following as logical alternatives:

(1)    He stopped smoking
(2)    He did not stop smoking

According to the laws of this system of logic, we may argue that one of the alternatives must necessarily be true, but if we are talking of a person who has never smoked, each of the two alternatives is misleading and, therefore, both have to be rejected. It was for similar reasons that the Buddha rejected each of the four alternatives about the existence of the one-who-has-attained-the-Transcendent (Tathaagata) after death because none of the alternatives "fit the case" (na upeti),[21] just as much as "he stopped smoking" or "he did not stop smoking" do not "fit the case" of the person who has never smoked, however arbitrarily we may like to define the words so as to affirm one of them as true and the other false, on the basis of our Procrustean interpretations.

    It is not the intention of this article to examine in detail what Naagaarjuna does with this early Buddhist logic of four alternatives. Suffice it to say that


19.    See Jayatilleke, Early Buddhist Theory of Knowledge, pp. 301-304.

20.    Ibid., p. 346.

21.    Ibid., p. 475.

 

 

p. 82

there is little evidence that Naagaarjuna understood the logic of the four alternatives as formulated and utilized in early Buddhism. Early Buddhism regarded each of the logical alternatives as being either true or false. It also utilized, as we have observed, the two-valued logic of two alternatives side by side with the above two-valued logic of four alternatives. Naagaarjuna, however, says that according to the Buddha, propositions could be true, false, both true and false, and neither true nor false.[22] But such an extension was not made in the early Buddhist texts, however legitimate it may have been.

    Naagaarjuna dismisses the third and fourth alternatives as being logically impossible: "If one part were to be divine and another part human, he would be both eternal and non-eternal and this is not possible (na yujyate)." "If it is declared to be the case that he is both eternal and non-eternal, then one might as well say that he is neither eternal nor not eternal." Despite his statement that according to the Buddha "everything could be both true and false (sarva.m . . . tathya.m caatathya.m)," he questions how "the true and the false (saccaasacca) can be together for this is mutually contradictory (paraspara-viruddha.m hi)."[23] Likewise, as Robinson has shown,[24] he invokes the law of excluded middle, dismissing altogether a third alternative (t.rtiiya.h) to p and -p.[25]

    The proposal of some commentators[26] to regard the four alternatives as representing progressive degrees of truth is to deny that they are in a relation of exclusive disjunction. If this were so, each would be true from some standpoint. This would be the case according to the relativistic logic proposed by the Jains, to which Buddhism was opposed.[27]

    It is evident that Naagaarjuna and some of his commentators, ancient and modern, refer to this logic with little understanding of its real nature and significance. I said that this appears to be a typical "East-West problem" because it is the kind of problem which, owing to the historical development of the Aristotelian tradition in the West (leaving aside the modern developments in the field of logic), one would expect the Western scholar to find it difficult to comprehend. To some extent this is true. When Poussin referred to this logic and observed that Indians "never clearly recognised the principle of contradiction,"[28] what was at fault was his own failure to understand the


22.     Muulamaadhyamikakaarikaa, 18, 8.

23.    Ibid., 27, 17-18; 8, 7.

24.     "Some Logical Aspects of Naagaarjuna's System," pp. 295-296.

25.    This cannot be understood as a complementary use of the logic of two alternatives.

26.     Robinson, Early Maadhyamika in India and China, p. 57.

27.     Jayatilleke, op. cit., pp. 348 ff.

28.    Louis de La Vallee-Poussin, The Way to Nirvana (Cambridge: Cambridge University Press, 1917), p. 111.

 

 

p. 83

structure of this logic, probably because he was convinced that there could be only one system of logic, namely, the Aristotelian. But on the other hand, one could not wholly blame the Western scholar for this intellectual blindness, for although they have written about it with seeming authority, few of the traditional classical Indian scholars and hardly any of the modern Indian and Japanese writers appear to have comprehended this logic.