Indian Logic Forum

Plamen

The Tetralemma

A typical piece of Buddhist dialectical apparatus is the tetralemma (catuskoti). 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,
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(17) This is not quoted in the Karikas.

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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 Madhyamika is negativism. There is thus an extra-logical interest in analyzing the form of the catuskoti.

[46] "Everything is either true, or not true, or
both true and not true, or neither true nor
not true; that is the Buddhas teaching."

(18.8) [47] "He is not to be called empty, nor non-empty,
nor both, nor both-not; but for the sake of
designation he is called...." (22.11)

[48] "It is not said that after final cessation the
Blessed One is, or that he is not, or that
both, or that neither." (25.17)

[49] "If the man is the god, he is thus eternal; the
god would be unborn, since the eternal is
unborn. If the man is other than the god, then
he is non-eternal;
If the man is other than the god, succession
does not obtain.
If one part is divine and one part is human,
then he is both eternal and noneternal; and
that is not possible.
If 'both eternal and non-eternal' were
asserted, then 'neither eternal nor
non-eternal' might be asserted." (27.15-18)

The four alternatives, as expressed before Nagarjuna negates them, are given in example (46). The formula is: [Ax v ?aAx v Ax.?aAx v?a(Ax). ?a?a(Ax)]

It is evident that the first two alternatives ate to be quantified universally for 'x.' Example (49) quantifies the third alternative existentially for 'x. "Some x is A and some x is not A." 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.

Schayer(18) transcribes the fourth alternative [?ap.?a(?ap)l "not-p and not-not-p." This is on the assumption that the four alternatives are propositional functions. However, it is apparent that negations and conjunctions of the basic proposition do not transcribe (49), and,if the terms of the other examples are quantified in the same way, do not transcribe them. "Not-p" is the contradictory of "p," but "some x is A" is not the contradictory of "some x is not A."

Nakamura(19) interprets the tetralemma algebraically as: "a," "-a," "a-a" and "-(a-a)." Since "a-a" equals "0" and "-a-a" equals "O," the third and fourth alternatives are redundant and senseless. However,if the subject
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(18) Aussagenlogik, p. 93. (19) Op. cit., p. 229.

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is not totally distributed in either conjunct in the third alternative, then this form need no longer be considered redundant.(20)

Nagarjuna's reason for negating each of the alternatives is that its terms are null, as defined by his opponents. The fourth alternative is true when one of its terms is null, but not false when the other term is non-null. It is probably for this reason that Nagarjuna rejects the fourth alternative.

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-Schroder notation.

ARISTOTELIAN FORM TETRALEMMA

A ab=0 1 ab=0 E ab=0 2 ab=0 I ab?U0 3 ab?U0.ab?U0(conjunction of I and O forms) O ab?U0 4 ab=0.ab=0 (conjunction of E and A forms)

Dilemmas

It is not surprising to find numerous dilemmas in the works of an author reputedly "bold, baffling and seemingly arrogant" The common form is the "simple constructive" one?upyUqryUq:pvr:yUq?v "r" is generally "not-p," so that "p or r" becomes "p or not-p." The examples are illustrative but not exhaustive.

[50] "If own-being exists, other-being belongs
to nobody; if own-being does not exist,
other-being belongs to nobody." (13.4)

[51] "When nature exists, alteration belongs to
nothing; when nature does not exist,
alteration belongs to nothing." (15.9)

[52] "If the effect is born from the conjunction
of cause and preconditions, and if it exists
in the conjunction, then it cannot be born
from the conjunction; if the effect is born
from the conjunction of cause and
preconditions, and if it does not exist in the
conjunction, then it cannot be born from the
conjunction." (20.1,2)

[53] "If the cause is void of effect, it cannot
produce effect; if the cause is not void of
effect, It cannot produce the effect." (20.16)
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(20) Stcherbastsky The Conception of Buddhist Nirvana, p.90, incidencally quantifies. the third alternative of a tetralemma., but does nor remark on what he has done. "He denies that they are identical (with their causes), that they are different from them, or that they are both (partly identical and partly non-identical).

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[54] "If effect is endowed with real own-being,
cause produces no effect; if effect is not
endowed with real own-being, cause produces no
effect." (20.21)

[55] "Origin and dissolution of the empty do not
take place; origin and dissolution of the
non-empty do not take place." (21.9)

[56] "When the latter is extinguished, the former
entity is not cogent; when the latter is not
extinguished, the former entity is not
cogent." (21.18)

[57] "If the world were finite, there could not
be another world; if the world were infinite,
there could not be another world." (27.21)

Note that this "simple constructive" form consists of two implications, one of whose antecedents must be affirmed. The above examples illustrate Nagarjuna's use of modus ponens. They observe the rule of affirming the antecedent, and so are formally correct. The designation "simple constructive" may seem inappropriate to dilemmas whose consequents are all negations, but "constructive" refers to the propositional structure, and not to the mood of the terms. Nagarjuna eschews affirmation of terms, but he does afffirm propositions.

The next example, though of the same propositional form as the preceding ones, is exceptional in that the first "horn" is the opponents objection, while the second one is Nagarjuna's reply.

[58] (Opponent): "If all this (world) is empty, then
there is no arising and perishing, and no
ones nirvana through abandonment or
cessation is asserted.

(Nagarjuna): "If all this (world) is non-empty,
then there is no arising and perishing,
and no ones nirvana through
abandonment or cessation is asserted."(25.1, 2)

The opponent wishes to deny the antecedent--emptiness of everything--and advances his implication in the expectation that the consequent will be denied by Nagarjuna, who, however, advances a counter-implication with contrary antecedent but identical consequent.

The way of escape from the consequences of these dilemmas is to "take them by the horns," to repudiate the definitions which they presuppose. In so doing, one rejects the whole set of propositions. which Nagarjuna calls "views" (drsti), and arrives at the meaning of emptiness.

[59] "Emptiness was declared by the
Victors as the expeller of all theories;
they declared that those for whom
emptiness is a theory (drsti) are incurable." (13.8)

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Excerpted from the 1957 PEW article of Richard H. Robinson, Some Logical Aspects of Nagarjuna's System.