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THE BUDDHIST CONDITIONAL IN SET-THEORETIC TERMS

The Buddhist logical school recognizes two means of valid cognition (two prumd~a): sense perception (prutyak~u) and inference (anumdna) of something not perceived. In the first case we may cognize a fire by seeing its flames, hearing the crackle of the burning fuel, smelling the smoke, or feeling the heat (presumably we will not care to employ the sense of taste); in the second case, being too far from the fire to sense it in any of these ways, we may infer its existence by (a) perceiving smoke, (b) recalling that smoke means fire, and (c) deducing that there is fire in the locus in question. The first of these is a case of direct sense perception, but it is inadequate in itself; it is the fire that interests us, and our perception of smoke is a perception of what is not fire. So we need to proceed. The second, the conditional statement ‘if smoke then fire’ (and for our purposes here we make no distinction between term logic, in which we would say ‘instances of those that possess smoke are instances of those that possess fire’, and propositional logic, in which we would say ‘if x is smoky then x is fiery’) is validated for Dharmakirti by its consonance with two of the three criteria (rilpu) that the antecedent (in this case smoke) possesses. That is, if we consider ‘if smoke then fire’ as valid, then it must be because smoke is present only in cases of fire and because (what is contrapositive to this and equivalent to it) where there is no fire there is no smoke (see Stcherbatsky 1930, pp. 55 ff., and Shastri 1982, pp. 31 ff.). (How it is that we know these things - do we simply observe cases of smoke and fire in the world, so that cognition of the relationship is induced in us, or is there something intrinsic to the nature of smoke or fire or both, by which we can obtain such cognition by deduction? - involves the dispute between the partisans of untartvydpti and the partisans of buhirvyapti, intc which we shall not enter here.)

So thus far we have two perceptions, (a) that of smoke, which is a ‘physical’ perception in a sense (though Buddhists will point out that one person’s perception of the smoke is not another person’s perception, even if it is the same body of smoke - the angle from which it is perceived will differ, if nothing else, hence there is still a subjective and mental aspect to this), and (b) that of the validity of the proposition ‘if smoke then fire’, which may be seen as the mental

Journal of Indian Philosophy 24: 649-658, 1995. @ 1995 Kluwer Academic Publishers. Printed in the Netherlands.

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perception of a mental object.’ We then ‘multiply’ the two, as it were, and obtain by inference the truth of (c) ‘there is fire there’, which again is mental. Skeptics and Can&as may deny the validity of such inference, but historically it appears that most people have accepted it, either intuitionally or by recognition - or belief - in its consistency with the facts of human experience.2

From this point let us renounce consideration of all problems asso- ciated with perception and inference in themselves and look only at the conditional statement that in some sense mediates between them (inasmuch as the conditional or ‘if-then’ statement, applied to a sense- datum such as ‘smoke’, enables us, provided we are not radical skeptics, to infer ‘fire’). For Dharmakirti there are two types, that by essence (svubh&~~) and that by causation (k&ya) (see Shastri 1982, p. 35, pp. 38 ff.; Stcherbatsky 1930, p. 60, pp. 65 ff.) (we omit consideration here of the whole topic of negation; note also throughout that Stcherbatsky’s English terminology and the present writer’s are different). ‘If a pine then a tree’ (the Sanskrit example is that of Si@qa or aiooku tree) is true because a pine has the ‘essence’ of ‘treeness’, but we must add that no implication is made here that the Buddhists regard essences as real entities; indeed, they do not, which is why we now prefer to gloss the whole matter in set-theoretic terms: the set of all pines is a subset of the set of all trees, and it is this fact, this inclusion, this vyapti of treeness (a figure of speech merely, a nomen, Buddhists being radical nominalists) over pineness, that provides the basis for our conditional. The term vyiipti, often and rightly translated as ‘pervasion’ (and we can see this pervasion in such as Figure la, where the domain of that which possesses animalhood pervades the domain of that which possesses catness), can also be translated as ‘inclusion’ (and this too we can see). But set-theoretic inclusion also corresponds to conditional propositions concerning the points within the sets (the sets here being seen strictly as sets of points, bounded by circles, in the Euclidean plane).3 The proposition ‘if a point x is within the “cat” circle then it is also within the “animal” circle’ corresponds exactly to the proposition ‘the set of all cats is included by the set of all animals’. Thus if C(X) means ‘X is a cat’ and A(x) ‘x is an animal’, and if C is the set of all cats and A the set of all animals, we can write, for all x,

C(z) -+ A(z)- C c A

This states the isomorphism between statements of the elementary propositional calculus and statements of elementary set theory.

The other type of conditional, that by causation, may be illustrated by the smoke-fire example. Smoke and fire do not, it seems, share a

THE BUDDHIST CONDITIONAL IN SET-THEORETIC TERMS 651

animals

0 0 cats

(a) Figure 1.

(b>

common essence, even if essences are taken to be purely nominal; nor do ‘smokes’ constitute a subset of the set of fires. But loci on which appears smoke do in fact constitute a subset of the set of all loci of fire; so even here a set-theoretic interpretation is possible, and it will be adopted here.

Let us now see how the conditional works within a syllogistic inference (assuming, as said, for the purposes of this discussion, that we have no theoretical problem with perception as such, on which any inference must at least in part be based, nor with the validity of inference itself, which we here take for granted). We wish to show that Felix is an animal, and by direct perception we ascertain that he is a cat, thus ‘placing’ him within the set of cats (Figure lb). We recall then that the set of all cats is a subset of the set of all animals and infer that Felix must be in this set as well (Figure 2a). By way of a (Western) syllogism, we might say ‘all cats are animals; Felix is a cat; therefore Felix is an animal’. In the more concise (Indian and Buddhist) sv&-thanumiina form we shall have ‘Felix is an animal [the desired conclusion stated first] because of [his] catness’. The thesis that all cats are animals will not be stated explicitly at all, though it is of course implied. In Sanskrit there would be three words only, the thing we are talking about (Felix), called the paksa or ‘locus’ (this is not a literal translation); the probandum (animalness), called the sdhya or ‘thing to be proved’ (the quality, as it were, that is to be proved to be in or on the locus); and the probans (catness), called the hetu or Ziriga ‘reason’ or ‘mark’ by which we infer what is to be inferred (Figure 2b). Things in the outer circle are those that possess (mat) the probandum (stidhya [dharma]), e.g. animals possess animalhood.

Now an inference that is intended to be valid may nevertheless be invalid, and various lists of fallacies have been drawn up. In a sense

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(a> lb)

Figure 2.

they are reducible to two: the relationship between the pak;a and the hetu may be incorrect (Felix may not be in the set of cats after all), or the relationship between the hetu set, the set of all things possessing that quality that is the hetu, and the stihya set may be incorrect. Thus the two basic objections to a proposed inference are the equivalents of asiddha, meaning that the pak;a set is not a subset of the hetu set,4 and avyapta, meaning that the hetu set is ‘not pervaded’ by the stihya (set), i.e. that there is no proper inclusion of the former by the latter. And this can happen in two ways; the hetu set and stihya set may overlap, so that part of the hetu set stands outside, or the two may be entirely disjoint, so that all of the hetu set stands outside. Going back to the relationship between paksa and hetu set, we can see that as long as the pak;a is a single individual, represented by a single point, there is no possibility of overlap; the pak;a is either in or out of the hetu set, and thus the relationship is either correct or one of disjunction. But as we shall see, some pakca are not single individuals but sets, and so here too we can encounter overlap. Thus there are four possible problems with an inference. In looking at an example of each, drawn from Dharmakirti, we shall see what Dharmakirti’s terminology is and discover exactly how overlapping and disjunction render an inference invalid.

Disjunction between pak+a and hetu set is illustrated by ‘sound is impermanent because of its visibility’5 (Figure 3a). In syllogistic form this would be ‘whatever is visible is impermanent; sound is visible; therefore sound is impermanent’. In Sanskrit we have unityah iaabdas’ cllk?usatvdt (see Shastri 1982, p. 89, also Stcherbatsky 1930, p. 173). This is invalid because the pak;a ‘sound possessors’ is in fact disjunct from the hetu ‘visibility’ (Figure 3b). Sound is indeed impermanent, so we continue to place it within the set of nonetemals; but it is not because of visibility (a quality that it does not possess) that it is so;

THE BUDDHIST CONDITIONAL IN SET-THEORETIC TERMS 653

(a) Figure 3.

(b)

sentients

0 sleepers 0 0 trees

sentients

(f-3 trees sleepers

(a> 0)

Figure 4.

another hetu must be sought (and this of course is audibility, and had this been adduced as the reason, the inference would have been correct).

Overlapping between paksa and hetu set is illustrated by ‘trees are sentient beings because they sleep’6 (Figure 4a). In syllogistic form this would be ‘whatever sleeps is a sentient being; trees sleep, therefore tree are sentient beings’. In Sanskrit we have cetanifs taravah svapdt (see Shastri 1982, p. 31, also Stcherbatsky 1930, p. 54). This is invalid because the paksa ‘trees’ overlaps the hetu set ‘sleepers’; that is, not all trees sleep but only some (na hi sarve y-kg rtitrau patrasa~kocabhcrja~ kiqztu kecideva). So the real situation is as in Figure 4b, and we cannot conclude on the basis of ‘sleep’ that trees are sentient beings. (This would prove that some trees are sentient beings, if we accepted that whatever sleeps is a sentient being, but Buddhist logic does not contain the quantifier ‘some’; it is interested only in statements having the implied universal quantifier ‘all’. Or we could conclude that a particular tree ‘possibly’ is a sentient being, but Buddhist logic is not ‘modal’ either and hence does not deal with ‘possible’ conclusions; it desires only certainty.)

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(4 (4 @I

Figure 5.

Both the foregoing are considered by Dharmakirti’s commentator Dharmottara to be cases of asiddhahetu ‘unestablished reason’; the second is ‘unestablished’ in one part (ekadeSundsiddha; there is a lacuna in the Shastri text where (p. 31) the long mark - equivalent to the matron - ought to be), while the first is ‘unestablished’ tout court. Later Dharmakirti will write that in matters relating to the relationship with the paksa (dharmisaqzbandha) the hetu will be asiddha in cases of plain asiddhatva and in cases of sal)zdeha ‘doubtfulness’ (see Shastri 1982, p. 88 and Stcherbatshy 1930, p. 172, where again the examples of sound and trees are discussed). On the basis of his asiddhau sqzdehe ctisiddho hetu we can postulate, then, an casiddhyasiddhahetu and a *saFehasiddhahetu for the two fallacies.

Disjunction between hetu set and sddhya set is illustrated by ‘sound is eternal because a product’ (Figure 5a). In syllogistic form this would be ‘whatever is a product is eternal; sound is a product; therefore sound is eternal’. In Sanskrit we would have nityah subdub krtakutvdt (see Shastri 1982, p. 102, also Stcherbatsky 1930, p. 201). This is invalid because the hetu ‘being a product’ is disjunct from the sadhya ‘etemality’ (Figure 5b); what is produced is not in fact eternal ever. In fact, sound’s being a product shows, not that it is eternal, but the reverse, that it is not; thus the hetu here is called a viruddhahetu, a ‘contradictory reason’, because it proves the reverse of the desired stidhya.

Overlapping between hetu set and stidhya set is illustrated by ‘sound is eternal because it is cognizable’ (Figure 6a). In syllogistic form this would be ‘whatever is cognizable is eternal; sound is cognizable; there- fore sound is eternal’. In Sanskrit we have nityah gabdab prameyatvat (see Shastri 1982, p. 93, also Stcherbatsky 1930, p. 181). This is invalid because while some cognizables are indeed eternal, like space, others are not, like a pot. So we have the situation of Figure 6b, and the hetu

THE! BUDDHIST CONDITIONAL IN SET-THEORETIC TERMS 655

(b)

Figure 6.

Figure 7.

is called anaikcintika ‘not unequivocal’, because it is consistent with both the sadhya desired and its opposite.

We might add that according to some points of view all things are cognizable, so that the real situation is that of Figure 7. The common feature between this and Figure 6b is that a part of the hetu set (cognizables) lies outside the siidhya set; this is the criterion for an anaiktintikahetu, so we may consider that there is still ‘overlapping’ in the required sense. Of course here there is actually pervasion, but it is of the wrong kind; the partisans of the syllogism of Figure 6a desired pervasion between eternality and cognizability, but in Figure 7 it is the other way round, so the thesis is not proved (we might con- sider this ‘false pervasion’ as a special case of ‘overlapping’, although Dharmakirti does not do so).

To sum up, we have four ‘supposedly true’ inferences in Figures 3a, 4a, 5a, and 6a, but the real situations are as in Figures 3b, 4b, 5b, 6b, or 7, in which the faults can be seen. Dharmakirti’s terminology is presented in Table 1.

There is in fact a good Sanskrit name for the sadhya set: sapaksa or ‘set of all similar cases’; they are similar to the pak;a in possessing the sadhyadharma ‘quality to be proved’. When we proved Felix to be an

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TABLE 1

Overlapping Disjunction

Paba (set) - hetu set saFdeh&iddhahetu asiddhyasiddhahetu relationship (Figure 4b) (Figure 3b)

Hetu set - sadhya set anailuintikahetu viruddhahetu relationship (Figure 6b) (Figure Sb)

animal we proved him to have the quality ‘animalhood’ in company with his ‘similars’ the animals. The pa@z itself is often thought of as a set (e.g. trees); like any set it may have only one member, as in the paksa that is Felix. The basic task of proof then is to show that the paksa (set) is a subset of the sapaksa (set). This is done by showing that the paksa has that quality known as the hetu (in Felix’s case, cathood) and that this quality is not found outside the sapaksa (set of animals). There seems to be no good Sanskrit word for hetu set, the set of all things possessing the quality.

It should be mentioned that although a term like ‘quality’ may seem to imply a ‘substance’ to which a quality is attached, on a deeper level Buddhists are highly critical of notions of substance and are capable of arguing that there are really no substances, only qualities or phenomena. Likewise the ‘essences’ or ‘universals’ such as cathood and animalhood are not considered as reals (as they generally are in the Western tradition, Platonic and Christian, up to the time of O&ham) but as abstractions, valid of course on their level. Thus Buddhist logic does not imply any metaphysic (which exists of course elsewhere in the Buddhist tradition) and in this sense is ‘mathematical’ and formal.

NOTES

’ Buddhist logic has not been elaborated in terms of the Abhidharma; indeed that most basic of terms, ‘perception’, is rendered by one Sanskrit word (v@7na) in the Abhidharma and by quite another (pratyaka) in logic, or rather pramnnavcida (epistemology; theory of knowledge). But here we recall that in the Abhidharma a sense-perception is ‘registered’ in the mind in the ‘following moment’ by the manovij%ina, thus becoming a part of our mental experience; and that purely mental experience, not following on sense-perception, is also recognized. It is seen as ‘perception’ of a nonphysical kind, in which the manas (that function of the mind that perceives dharma) is considered a perceiving organ analogous to the eye, ear, etc. Thus the mentally perceived dharma are considered as objects; various lists of them are given, and they include such experiences as faith (Sraddhn) and (for

THE BUDDHIST CONDITIONAL IN SET-THEORETIC TERMS 657

negative example) anger (krodha) as well as neutral ones. Our experience of being angry is thus a perception on the part of our munas of the dharma anger. But mental perception of the validity of a conditional statement is not included in the traditional lists; hence the virtual separation of pramdy theory from Abhidharma, though they are not necessarily inconsistent. ’ If the validity of inference is intuitional, then from ‘smoke on the hill’ and ‘if smoke then fire’ we reach the conclusion ‘fire on the hill’ directly and inescapably and can do no other; it is a question of a fundamental law of thought or even of existence itself. If, on the other hand, we deem such a conclusion merely consistent with experience, this means that we encounter no contradictions in reasoning in such fashion. There is a third and still weaker mode of acceptance of the validity of inference: we observe that others accept it, and we do so also so that we shall be able to debate with them, for the motive of establishing, if not the truth of our own theses, at least their probability. (And this in mm would be motivated, for Buddhists, by the desire to help sentient beings by producing in them correct cognitions concerning things transcendent over the world, if any, and even things of the world.)

To prove the validity of inference, i.e. of proof itself, would seem to be impossible because of the circularity involved; the proof of the validity of proof would itself be a proof, and it could only be valid itself if proof were first valid, but this is what requires to be proved. On the other hand, proof that proof is not valid would be equally impossible, because if proof is not valid (in the general case), then the proof that proof is not valid would be, or at least might be, itself invalid. We may imagine a Buddhist and a Carvaka wrangling over this issue. Buddhist: ‘You say that “proofs are not valid”; are we to accept this on faith, or do you claim to prove this? If you claim to prove it, then you contradict yourself; your “proof” that proofs are invalid will itself be invalid ipso facto.’ Carvaka: ‘You say that “proofs are valid”; are we to accept this on faith, or do you claim to prove this? If you claim to prove it, your reasoning is circular; your “proof” that proofs are valid has validity itself only if proofs are already valid-but this is just the point of contention.’ It is a perfect stalemate. In the end we probably do have just to accept one point of view or the other on faith. 3 Such diagrams are of course not new. They are usually attributed to L. Euler (1701-1783) as they are used in his Lettres h we princesse d’dllemagne of 1768 (see Bochenski 1961, p. 260). But as Bochenski also shows, they were used earlier by Leibniz and possibly invented by him: ‘. . . new, or newly developed features, such as . . . the so-called “Eulerian” diagrams’ (p. 258). The photographic reproduction given between pp. 260 and 261 of a Leibniz autograph leaves no doubt: in the first lines on the page we find ‘omne B est C’, ‘Null. D est C’, and ‘Erg. Null. D est B’, all of which corresponds exactly to the diagrams on the right; in particular, ‘omne B est C’ (all B is C) corresponds to the circles marked B and C (the former inside the latter).

Bochenski also hints (p. 260) that there may be earlier precedents still. The diagrams of John Venn (1860) are later (p. 262); the Venn equivalent of the Leibniz-Euler conditional is (in the current adaptation of Venn’s own practice)

m B C

in which the hatching indicates an ‘impermissible’ region; for the permissible regions a point in B is a point in C as well, thus ‘if B then C’ analogous to our Figure la ‘if cat then animal’.

658 BRIAN GALLOWAY

4 There must be the presence (sat&a) of the quality of catness in Felix, i.e. in set-theoretic terms, Felix must be in the set of cats. Thus in general we require the presence of the quality of the hetu in the pa&r (but not all of the hetu in the pa&z): lirigusytinumeye suttvum evu (Zirigu means hetu; unumeyu means pukqu); see Shastti (1982) p. 3 1, Stcherbatsky (1930) p. 5 1. Not all camess need be in Felix. 5 Or ‘the spoken word is impermanent . . . ’ For the Brahmanical theory that words are permanent and that any instance of the speaking of a word is merely a manifestation of the eternal word, see Stcherbatsky (1930) p. 127 n. 2. Dharmakirti of course as 2 Buddhist does not accept this theory.

Some trees ‘sleep’ in the sense that they fold their leaves at night (r&r-uu puttrusuqzkocu). Dharmakirti here does not challenge the idea that plants sleep in the same sense that animals do; that is, he accepts for the sake of argument that there is a correct relationship between hetu and sti&yu, in that ‘whatever sleeps is a sentient being’ and even that ‘sleep’ is properly applied to some trees. This of course lends the impression that he would regard some trees as sentient beings. This as a Buddhist he cannot do, because the Buddha himself denied sentience to plants, according to the RutnurGi satra as quoted by Santideva in his &X$isumuccuyu (p. 111; Bendall & Rouse p. 195). But none of this is of concern in the present case, because Dharmakirti wishes only to make a point about the relationship of p&u and hetu; the former must be entirely within the latter and the latter must apply to all of the former in a valid inference.

That Dharmakirti is capable of distinguishing between the sense that a predicate might have with regard to a plant and the sense that it might have with regard to an animal is shown in another example. Some Jains offer the inference ‘trees are sentient beings because they die’; Dharmakirti replies that they do not die in the sense that animals, who really are sentient beings, do, because in sentient beings death means the extinguishing of the sense-perceptions, and trees in the Buddhist theory have no sense-perceptions to be extinguished. The Jain speaks (see Shastri 1982, p. 89 and Stcherbatsky 1930, pp. 173-174) ‘without making a difference between a death concomitant with the predicate (sentient being) or not so concomitant’. That is, ‘What dies is a sentient being’ is attacked on the ground that ‘die’ has two senses, one associated with ‘sentient being’ and one not. The same could have been said for ‘sleep’ in ‘what sleeps is a sentient being’.

REFERENCES

Bochenski, I. M. 1961. A History of Formal Logic (rev. tr. by Ivo Thomas of Formule Log&). Notre Dame, Indiana: Univ. of Notre Dame Press.

&ntideva. Sik$isumuccuya. P. L. Vaidya, ed. Darbhanga: Mithila Institute,l961. Eng. tr. C. Bendall and W. Rouse. Delhi: Motilal Banarsidass, 1971.

Sh%&i, C. S. 1982, ed. Nyciyu Bind@ by Dharma Kirti with a Commentary of Sr~dharmotturtichh~u. Varanasi: Chaukhambha Sanskrit Sansthan.

Stcherbatsky, T. 1930. Buddhist Logic. Leningrad: Academy of Sciences of the USSR (Bibliotheca Buddhica 26). Reprinted New York, Dover, 1962. All references are to the second volume, which contains Stcherbatsky’s translation of Dharmakirti’s Nyiiyabindu.

Berkeley, California, USA