In the statement 'the hill has smoke because it has fire', which type of…

In Nyaya logic, a valid inference (anumana) requires vyapti — an invariable, unconditional concomitance between the hetu (the reason, or middle term) and the sadhya (the property to be proved). When the reason offered fails to satisfy this requirement, the argument commits one of the hetvabhasas (fallacies of the reason). Vyāpyatvāsidha is the specific hetvabhasa that arises when the hetu's link to the sadhya is not unconditional but holds only in the presence of an extra condition, called an upadhi — so the seeming universal rule breaks down as soon as an instance lacking that condition is considered.

The standard valid Nyaya inference is "the hill has fire, because it has smoke" — smoke (the hetu) is invariably accompanied by fire (the sadhya) with no known exception, so the vyapti is unconditional. The statement in the question reverses the direction of this inference: it says "the hill has smoke because it has fire", treating fire as the hetu and smoke as the sadhya. But fire does not invariably produce smoke — it only does so when an additional factor, wet or moist fuel, is present; a flame fed by dry fuel, or a red-hot iron ball, has fire without any smoke. Because the fire-to-smoke connection depends on this hidden condition (upadhi) rather than holding in every case, asserting it as an unconditional pervasion is fallacious — this is exactly the vyāpyatvāsidha hetvabhasa.

Checking the reverse case confirms the diagnosis: wherever the standard smoke-to-fire inference is drawn, no counter-example is known (there is no smoke without fire anywhere), so that inference is sound; but the fire-to-smoke direction used in the question fails precisely because of a missing condition, which is the defining signature of vyāpyatvāsidha rather than of the other listed fallacies.

Contrast with the other options:

• Asidha: this concerns whether the hetu itself has genuinely been established for the paksha — a broader question of whether the reason actually holds at all for the subject under discussion. In this statement, fire does genuinely exist on the hill, so there is no failure of that general kind here.

• Sādharna: this applies when the hetu turns up in both sapaksha and vipaksha instances of the sadhya, making it too common to decide the sadhya either way; the issue here is not that fire occurs everywhere regardless of smoke, but that its link to smoke is conditional.

• Asādhārana: this applies when the hetu is found only in the paksha and nowhere else, so no comparison instances exist at all; fire is a hetu that occurs widely elsewhere, so this case does not fit either.

The type of fallacy identified in the statement is vyāpyatvāsidha.