How to judge an argument: sound or fallacy

What is a sound or fallacy puzzle?

Each argument has five parts: a claim ("The hill has fire"), a reason ("because it has smoke"), a rule with an example ("Wherever there is smoke, there is fire, as in the kitchen"), the reason applied ("The hill has smoke") and the conclusion ("So the hill has fire"). Above the arguments is a list headed "What you're told".

Take what you're told as true, even where it sounds odd. The question is never whether hills really burn; it is whether the reason, given what you were told, carries to the claim. Each argument either holds, or one of five things breaks it.

Where does it come from?

From Nyaya, "method", India's classical school of logic. Its founding text, the Nyaya Sutras, is attributed to Akshapada Gautama; scholars date it anywhere from the 6th century BCE to the 2nd century CE. It sets out an argument in the same five parts: claim (pratijna), reason (hetu), example (udaharana), application (upanaya) and conclusion (nigamana). Its standard illustration is the hill with smoke on it: like a kitchen hearth, and unlike a lake.

The Nyaya Sutras (1.2.4) also name five hetvabhasa, reasons that only look like reasons. Yukti's five answers besides Holds are a simplified, checkable reading of them: Reason not shown (sadhyasama), Has an exception (savyabhicara), Proves the opposite (viruddha), Evenly matched (prakaranasama) and Contradicted (kalatita, in its later reading).

How do you judge one?

Ask these questions in order. The first one that settles it gives the answer:

1. Is the reason true here? Look for it in what you're told about this place. If nothing says so: Reason not shown.
2. Do the rules carry the reason to the claim? Follow the "wherever" lines, one after another if you need to. If they reach the claim: Holds. If they reach the claim being false: Proves the opposite.
3. If no rule settles it, were you told the claim is false here? Then: Contradicted.
4. Does somewhere have the reason without the claim? Then the argument's own "wherever" isn't true: Has an exception.
5. Does another sign here point the other way? If something else true of this place goes only with the claim being false: Evenly matched.

A worked example: smoke and fire

What you're told: Wherever there is smoke, there is fire. The hill has smoke. The kitchen has smoke and fire. The valley has mist. The forest has mist and rain. The cave has mist, and no rain.

1. "The hill has fire, because it has smoke." You were told the hill has smoke, and the first rule carries smoke to fire: Holds.
2. "The lake has fire, because it has smoke." The rule is fine, but nobody said the lake has smoke: Reason not shown.
3. "The valley has rain, because it has mist, as in the forest." The valley does have mist, but no rule links mist to rain, and the cave has mist with no rain: Has an exception.

These are the three answers on Gentle days. Steady days add two more, and Tough days all six.

Steady and Tough days: three more answers

Add to the list: Wherever there is water, there is no fire. The lake has water. The meadow has clouds and frost. The well has frost, and no rain.

• "The lake has fire, because it has water." The rules lead from water to no fire: Proves the opposite.
• "The cave has rain, because it has mist." No rule settles it, and you were told the cave has no rain: Contradicted.
• "The meadow has rain, because it has clouds." Nothing says clouds go with no rain, but the meadow also has frost, and the well, the only other place with frost, has no rain. One sign points each way: Evenly matched (Tough days only).

How does Yukti help when you are stuck?

An answer that isn't certain is stopped, and the board says what doesn't hold, with the lines that show it. It still counts as a step. A nudge asks you a question, such as "Were you told the lake has smoke?". The day after, yesterday's route gives each argument's answer with its reason.

Sources

Read on 26 October 2026: Nyaya Sutras (Wikipedia); Nyaya (M. R. Dasti, Internet Encyclopedia of Philosophy).