Hetucakra

Origin. Dignāga (c. 480–c. 540 CE), in three works on reasoning: the Hetucakraḍamaru ("Drum Wheel of Reason"), preserved only in Tibetan translation; the Nyāyamukha, extant in Chinese; and the Pramāṇasamuccaya with his own commentary, the Pramāṇasamuccayavṛtti, four of whose six chapters concern inference and argument. His teacher Vasubandhu's Vādavidhi had already required of a good reason the indispensability relation avinābhāva; the trairūpya consolidates that line into three conditions, and the wheel is the classification Dignāga built on them. Śaṅkarasvāmin (c. 6th century), a pupil of Dignāga, condensed the system into the Nyāyapraveśa, which the Chinese tradition attributes to him and the Tibetan tradition to Dignāga himself. Xuanzang (602–664) translated the Nyāyapraveśa and the Nyāyamukha into Chinese; Yijing's later Chinese translation of the Pramāṇasamuccaya was lost; Kuiji (632–682), Xuanzang's student, fixed the East Asian reading in his Great Commentary on the Nyāyapraveśa.

Models. The validity criterion of Buddhist inference, stated extensionally and decided by exhaustion. The trairūpya lays three conditions on where the reason property occurs — in the subject, among the similar cases, among the dissimilar cases — and the hetucakra crosses the reason's occurrence over the similar cases (all, none, some) with its occurrence over the dissimilar cases, judges each of the nine combinations, and finds exactly two valid. Where Aristotle derives the valid moods from the figures and Nyāya regiments a five-membered demonstration, Dignāga's table is a semantic survey: nothing in it is syntactic, each cell is a relation between the extensions of two properties, and the verdict carries a completeness claim — these nine are all the cases there are. The sibling Navya-Nyāya entry shows the later Indian treatment of the same relation, pervasion, by definition and counterexample; Dignāga decides it by enumerating a finite table.

Formalism.

Schema: One inference form: p is S, because H. p — pakṣa, the subject (sound, in every stock argument). H — hetu, the reason property. S — sādhya, the target property. sapakṣa — the cases similar to p in possessing S. vipakṣa — the cases lacking S.

Trairūpya (three conditions on H):

1. pakṣadharmatva: H present in p — "the reason's definite (eva) existence in the subject".
2. sapakṣe sattvam: H present in the sapakṣa — "existence in subject-like things only (eva)".
3. vipakṣe asattvam: H absent from the vipakṣa — "utter (eva) non-existence in subject-unlike things". Renderings with the eva placements follow Gillon's SEP exposition.

Readings of condition 2: universal: ∀x (sapakṣa x → Hx) existential: ∃x (sapakṣa x ∧ Hx) Condition 3: ¬∃x (vipakṣa x ∧ Hx). sapakṣa inclusive of p, eva restrictive: condition 2 ≡ condition 3. sapakṣa exclusive of p, eva restrictive: the three conditions are jointly inconsistent.

The wheel: condition 1 assumed; H's occurrence over sapakṣa × vipakṣa; cells numbered row by row.

The text states the verdicts geometrically: top and bottom of the middle column valid, the two sides contradictory, the four corners inconclusive as too broad, the centre inconclusive as too narrow.

The nine stock arguments: subject sound throughout; sapakṣa instances, then vipakṣa instances.

1. permanent, because knowable (space; pot) — anaikāntika, sādhāraṇa
2. impermanent, because produced (pot; space) — valid
3. effort-made, because impermanent (pot; lightning, space) — anaikāntika
4. permanent, because produced (space; pot) — viruddha
5. permanent, because audible (space; pot) — anaikāntika, asādhāraṇa
6. permanent, because effort-made (space; lightning, pot) — viruddha
7. not effort-made, because impermanent (lightning, space; pot) — anaikāntika
8. impermanent, because effort-made (lightning, pot; space) — valid
9. permanent, because incorporeal (space, atom; action, pot) — anaikāntika

Verdict names: viruddha (contradictory): cells 4 and 6 — the reason establishes the thesis's negation. anaikāntika (inconclusive): cells 1, 3, 5, 7, 9; the Nyāyapraveśa carries these into its list of pseudo-reasons, naming cell 1 sādhāraṇa ("too broad") and cell 5 asādhāraṇa ("too narrow"). The text closes by noting that the same ninefold combinatorics applies again to doubtful reasons.

Dharmakīrtian reworking: Three admissible reason types: svabhāvahetu ("this is a tree, because it is a śiṃśapā"), kāryahetu ("there is fire, because there is smoke"), anupalabdhihetu ("there is no pot here, because none is perceived"). Pervasion (vyāpti) restated as anvayavyāpti (co-presence) and vyatirekavyāpti (co-absence), grounded in svabhāvapratibandha, connection by the natures of the relata. Dharmakīrti's pervasion quantifies over actual and non-actual items alike, which no survey of the wheel's two classes can establish.

Symbols.

Metatheory. As a combinatorial object the wheel is an exhaustive classification of the extensional relations between the reason property and the target property over the induction domain: the domain is partitioned by the target into sapakṣa and vipakṣa, the reason's trace on each side takes one of three values, and the nine combinations are judged case by case, so the verdict that exactly cells 2 and 8 yield knowledge comes with its completeness argument. R.S.Y. Chi, whose "Buddhist Formal Logic" (Royal Asiatic Society, 1969) is the first book-length formal study, reads the whole construction as an extensional study of the possible relations between two properties. What the conditions quantify over is the contested part. Gillon's SEP exposition locates the ambiguity in the construal of sapakṣa and the force of the particle eva: with sapakṣa inclusive of the subject and eva restrictive, the second and third conditions are logically equivalent; with sapakṣa exclusive, the three conditions are jointly inconsistent — so the wording underdetermines the logic. Hayes ("Diṅnāga's Views on Reasoning", 1980; "Dignāga on the Interpretation of Signs", 1988) reads Dignāga as developing an inductive form of reasoning; Oetke ("Studies on the Doctrine of Trairūpya", 1994; "Ancient Indian Logic as a Theory of Non-monotonic Reasoning", 1996) as articulating defeasible reasoning; Katsura ("Dignāga on Trairūpya Reconsidered: A Reply to Prof. Oetke", 2000) answered Oetke; the class terms themselves have a literature — Staal ("The Concept of pakṣa in Indian Logic", 1973), Tillemans ("On sapakṣa", 1990), Katsura ("Pakṣa, sapakṣa and asapakṣa in Dignāga's Logic", 2004). He and van der Kuijp return to the text itself in "Once Again on the Hetucakraḍamaru: Rotating the Wheels" (Journal of Indian Philosophy 44, 2016).

Applies to. The inference theory of Buddhist pramāṇavāda; the epistemological doctrine the apparatus serves — the two pramāṇas, apoha semantics — lives in the Epistemologies collection (Buddhist Pramāṇavāda; Dharmakīrti's Epistemology). East Asian Buddhist logic (yinming), which received the system through Xuanzang's translations and Kuiji's commentary and spread to Korea and Japan. The Nyāya counter-theories, beginning with Uddyotakara's reply to the Pramāṇasamuccaya. Comparative logic, where the wheel is the standard test case for reconstructing Indian inference in modern notation. The prehistory of the pervasion problem that Navya-Nyāya made its central question.

Limitations. The trairūpya governs a single schema — one subject, one reason, one target — and the wheel classifies that schema's instances; there is no consequence relation among arbitrary propositions, no connectives, no derivations, and so nothing about which soundness or completeness in the modern sense could be proved. The classification is relative to an agreed stock of examples: the stock verdicts presuppose the disputants' ontologies (the atom counted permanent, space counted real), so a cell assignment can be contested by contesting an instance rather than the criterion. A reason true only of the subject falls in cell 5 by construction — audibility is absent from both classes because only sound is audible — so the wheel cannot evaluate a reason that lacks instances outside the pakṣa. The textual base is thin: the Hetucakraḍamaru survives only in Tibetan, and Chi's complete paraphrase of it runs to eleven verses, so reconstruction leans on the Nyāyamukha, on the Nyāyapraveśa — whose authorship the Chinese and Tibetan traditions already disputed — and on Kuiji's commentary. And the quantificational dispute is open: deductive, inductive, and defeasible readings of the second condition each have defenders, so no single modern formalization is canonical.

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