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Shedding some Localic and Linguistic Light
on the Tetralemma Conundrums
F.E.J. Linton
Mathematics Dept., Wesleyan University
Middletown, CT 06459 USA
E-mail: FLinton @ Wesleyan.edu
Abstract. Numerous authors over the centuries have puzzled over what has been called “the Buddhist paradigm of catuskoti.” A classic example: the four statements, considered both mutually exclusive and jointly exhaustive,
(i) the Tathagata exists after death;
(ii) the Tathagata does not exist after death;
(iii) the Tathagata both does and does not exist after death;
(iv) the Tathagata neither does nor does not exist after death.We offer some linguistic gedanken-experiments illustrating everyday situations in which appropriate analogues to these four statement-forms are entirely plausible as mutually exclusive, or jointly exhaustive, alternatives; and we offer a framework, based on the logical paradigms of locale or topos theory, illustrating how forms (iii) and (iv), in particular, need be neither contradictory, nor paradoxical, nor even mutually equivalent.
What the old writings seem to suggest is that, if we write P for the proposition (i) that “the Tathagata exists ...,” the four propositions
(i') P
(ii') ¬P (not-P)
(iii') P & ¬P (P and not-P)
(iv') ¬P & ¬¬P (neither P nor not-P)are mutually exclusive and cover all possibilities. What sort of logic can be at work here?
Classically, of course, where P and ¬P are complementary and ¬¬P = P , the last two formulations are identically trivial and the first two already cover all possibilities (principle of excluded middle, or tertium non datur).
In the intuitionistic logic of a topos, on the other hand, while the first two formulations are at least still mutually exclusive, and the last two, consequently, are still identically trivial, the first two need no longer cover all possibilities, that is, the principle of excluded middle need no longer hold.
If, instead, we envision a logic dual to that of a topos, more like the logic of the lattice of closed subsets of a topological space, we finally reach a situation where both P & ¬P and ¬P & ¬¬P may be non-trivial. But now P , ¬P , P & ¬P , and ¬P & ¬¬P may well no longer be mutually exclusive. Indeed, at least for closed P , we have the order-inclusions
¬P & ¬¬P < P & ¬P < P
and
¬P & ¬¬P < P & ¬P < ¬P ,
so that if P is “regular-closed,” say, that is, if P = ¬¬P , the last two formulations coincide and fall within both P and ¬P (indeed, they constitute the boundary of P ).
Somehow, ¬P must not be getting treated purely as the negation of P . Let us write Q temporarily for this negation of P , and see what we can make of statements (i) through (iv) in such a setting. They become
(i") P ,
(ii") Q ,
(iii") P & Q , and
(iv") ¬P & ¬Q ( = ¬(P ∨ Q)) ,where the last formulation is logically tantamount to the negation of “P or Q (or both),” i.e., to the negation of what the first two alone jointly cover. Certainly the last item here excludes each of the earlier ones, and all are, in general, non-trivial. But if all four are to be mutually exclusive, what (i") and (ii") are implicitly intending to represent must surely be rather
(i''') P & ¬Q
and
(ii''') ¬P & Q ,
respectively. Then, at least classically, we obtain the four mutually exclusive, jointly exhaustive, atomic generators of the free Boolean algebra on the two free generators P and Q , viz.:
(a) P & ¬Q = P − Q ,
(b) ¬P & Q = Q − P ,
(c) P & Q = P & Q , and
(d) ¬P & ¬Q ( = ¬((P − Q) ∨ (Q − P) ∨ (P & Q)) = ¬(P ∨ Q)) .But how is one now to make any sense of the hope that Q may stand for ¬P ? That is, how shall we maintain the mutual exclusivity and individual non-triviality of the four items
(a') P & ¬¬P ,
(b') ¬P & ¬P ,
(c') P & ¬P , and
(d') ¬P & ¬¬P ,obtained from (a)-(d) by putting ¬P in place of Q ?
Let us simplify, for the moment, by assuming that ¬¬P = P , so that the four conjunctions above become
(a") P & P ,
(b") ¬P & ¬P ,
(c") P & ¬P , and
(d") ¬P & P .Now let us imagine that the second occurence of P in each of these four conjunctions is merely a homonym for the P that occurs first. Mostly, in living languages, homonyms are words that sound alike, but are spelled differently and have different meanings, like red, the color, and read, the past participle, or pear, the fruit, pair, the duo or couple, and pare, the verb meaning to peel (and perhaps also cut up) a fruit (perhaps even a pear) or vegetable. But there are homonyms also with both sound and spelling identical, like sucker, which can at once signify a person easily duped or taken advantage of, or a tendril on a vine.
How may we realize the two occurrences of P in (a")-(d") as mere homonyms of each other? It would be enough, for example, were our lattice of propositions somehow spatial, that is, representable as some sort of subsets of some particular space X , to place ourselves in the Cartesian product X × X of the space X with itself. For now, corresponding to P , there arise two clearly distinguishable homonyms of P in X × X : one, the “vertical cylinder” P × X over the P in the first spatial factor X ; the other, the “horizontal cylinder” X × P alongside the P in the second factor X (cf. Figures 1 & 2).
If we now simply treat each first occurrence of P in the forms (a")-(d") as instances of the vertical cylinder P × X , and each second occurrence as the horizontal one X × P , then our four conjunctions correspond to the four rectangles in Figure 3 ( P & P being interpreted, for example, as the intersection, P × P, of P × X with X × P, etc.).
For what it’s worth, we exhibit a topos whose internal logic has system of truth values inherently of this form. Indeed, where S is any of the very classical topoi of absolutely standard sets — say, made up of the sets in Gödel’s constructive hierarchy — the topos S × S of ordered pairs of such sets is such a topos. Its truth value object is the ordered pair ( 2, 2 ) consisting of two copies of the usual two-element Boolean algebra formed from the ordinal number 2 = {0, 1}, and this has exactly four global elements: (1, 1) and (0, 0), playing the roles of True and NotTrue , and serving as counterparts of P & P and ¬P & ¬P , respectively; and (1, 0) and (0, 1) , playing the roles of BothTrueAndNotTrue and NeitherTrueNorNotTrue , counterparts in turn of P & ¬P and ¬P & P .
Not every topos whose truth value object has exactly four global elements has them arranged quite in this way, however. For example, if we topologize the ordinal number 3 (whose points are the smaller ordinals 0 , 1 , and 2 ) by declaring open exactly those subsets of 3 that happen themselves to be ordinal numbers (viz., Ø (the empty subset), {0}, {0, 1}, and all of 3 ) , then the topos of sheaves on this space 3 has truth value object whose global elements likewise number four, but correspond exactly, even as to their ordering, to the four open subsets of 3 that make up the topology just described. Here, between True and NotTrue (or 3 and the empty set) we have two intermediate truth values, each neither True nor NotTrue , but one “more true,” as it were, and “less not true,” than the other:
NotTrue = Ø < {0} < {0, 1} < 3 = True .
To within isomorphism, this topos may also be depicted as the topos of double-transitions among sets, that is, as configurations of the form
A ——f→ B ——g→ C
made up of three sets and two functions, as depicted. The truth value object for this topos is the configuration
where 4 , 3 , and 2 are the ordinal numbers 4 = {0, 1, 2, 3}, 3 = {0, 1, 2}, and 2 = {0, 1}, and the functions f+ and f- both carry 0 to 0 and 1 to 1 , but f-(2) = 1 , while f+(2) = f+(3) = 2 , as depicted above.
The four global elements here are simply the four length-two paths, or orbits, seen to emanate from the various members of 4 , the uppermost and lowermost of which it seems plausible to accept as playing the roles of True and NotTrue , respectively, while the remaining two paths, clearly neither True nor NotTrue , somehow represent the values “more True than NotTrue” and “more NotTrue than True.” Or perhaps the catuskotian expressions “both true and yet not true” and “neither true nor yet not true” better convey the sense of these intermediate truth values, though we suspect this is not an illustration of the classical paradigm the catuskoti had in mind.
But in fact, the logic of this topos does realize the way apparent contradictions are commonly used in everyday speech. A daiquiri made with far too much lime juice, for example, and a little too much sugar, may well be called both sweet and not sweet; a coffee prepared with just barely not enough sugar for the taste of a particular coffee-drinker may be disparaged as neither sweet nor not sweet. If the best student to pass through your department in the past ten years has an uncanny knack for getting arrested at student political demonstrations, you will be apt to wonder whether your department should once again post bail for this student who is both really very bright and yet not really very bright. Or, of another student, not quite so bright — generally dealing very well with the more difficult problems and readings, but sometimes inexplicably failing utterly when faced with far simpler ones — and yet having an investment acumen that is simply uncanny, you may well think, somewhat perplexedly, this student is neither really all that bright, nor not really all that bright.
There are, of course, also everyday linguistic settings in which the last two catuskoti options (iii) and (iv), far from being mutually exclusive, coincide competely. A grape-fruit, for example, sour, to some extent, like all its kin, but remarkably less so than most, you might be equally happy to describe as both sour and not sour, or as neither sour nor not sour. Would you like a topos whose truth value object epitomizes just this situation, not envisioned in the catuskoti, of the last two options (iii) and (iv) coinciding? The Sierpiński topos, as it’s known, is a case in point.
The objects of the Sierpiński topos are shortened versions of the configurations shown above: only two sets, B and C , rather than three, and only one function g . The truth value object is the right-hand fragment of the truth value object shown above, and has only three global elements, namely the three one-step paths emanating from the various elements of 3 , which have reason to be thought of as True , NeitherWhollyTrueNorNotWhollyTrue , and NotTrue (taken from top to bottom), respectively, though the middle value may equally well be thought of as TrueInTheLongRunEvenIfNotTrueAtTheOutset . This middle truth value, in other words, is at once BothTrueAndNotTrue and NeitherTrueNorNotTrue , and is the only alternative to the extreme values True and NotTrue .
As a final topic, perhaps not worthy of even this passing mention, let us take up one objection on the part of some commentators to the tetralemma paradigm, namely, that there should by rights be yet a fifth alternative, something like NoneOfTheAbove , to the classical four.
There are indeed topoi, readily described, whose global truth values easily realize the ideal of being five in number. For that matter, that ideal can be realized in three wholly different ways. In all cases, however, the lattice of global truth values must, for purely topos-theoretic reasons (that is, by virtue of what has been called generalized abstract nonsense), be what is known, to those in the lattice trade, as distributive. That requirement rules out the last two lattices depicted in the following Figure 4. The remaining five-element lattices number three: they too appear in Figure 4, on the left: they are all distributive, but are none of them Boolean.
Figure 4
And just which of their intermediate members (between True at the top and NotTrue at the bottom) should be interpreted as BothTrueAndNotTrue , as NeitherTrueNorNotTrue , and which as NoneOfTheAbove , I leave as my parting conundrum to you.
Bibliography
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— 30 November 2003, New Haven, CT —
