Which of the following statements are correct as per Classical Indian School of logic (Nyaya)?

A. Nyaya syllogism has three propositions.
B. Nyaya syllogism has five propositions.
C. Nyaya syllogism has three terms.
D. Nyaya syllogism has five terms.
Choose the correct answer from the options given below:

The correct answer is B and C only — meaning statements B and C are correct: the Nyaya syllogism has five propositions and three terms.

Let us carefully understand the structure of the Nyaya syllogism to see why this is the case.

Background: What is the Nyaya School?
Nyāya is one of the six classical schools (Darshanas) of Hindu philosophy, founded by the sage Gautama (also called Akshapada). It is primarily a school of logic and epistemology, and its theory of inference (Anumāna) is one of its most celebrated contributions. The Nyaya system is deeply concerned with valid reasoning and correct knowledge (Pramāṇa).

Why Statement B is Correct: Nyaya Syllogism has FIVE Propositions (Avayavas)
Unlike the Aristotelian syllogism of Western logic — which has only three propositions (Major Premise, Minor Premise, and Conclusion) — the Nyaya syllogism is structured around five members (Pañcāvayava). These five propositions are:

1. Pratijñā (Proposition / Thesis) — The statement of what is to be proved.
    Example: "The hill is fiery."

2. Hetu (Reason / Ground) — The reason or ground that supports the proposition.
    Example: "Because there is smoke on it."

3. Udāharaṇa (Example / Universal Proposition) — A universal rule illustrated by a familiar example.
    Example: "Wherever there is smoke, there is fire — as in a kitchen."

4. Upanaya (Application) — The application of the universal rule to the present case.
    Example: "The hill has smoke."

5. Nigamana (Conclusion) — The conclusion that follows from the above.
    Example: "Therefore, the hill is fiery."

This five-membered structure is the hallmark of Nyaya inference. Each step serves a pedagogical and logical purpose, making the argument complete, publicly verifiable, and immune to doubt. Therefore, Statement B is correct.

Why Statement C is Correct: Nyaya Syllogism has THREE Terms
Even though the Nyaya syllogism has five propositions, it revolves around only three logical terms:

1. Paksha (Minor Term / Subject) — The subject about which the conclusion is drawn.
    Example: "The hill."

2. Sādhya (Major Term / Predicate) — The property that is to be proved about the subject.
    Example: "Fire / Fiery."

3. Hetu (Middle Term / Reason) — The mark or characteristic that provides the logical link between the subject and the predicate.
    Example: "Smoke."

The relationship among these three terms drives the entire inference. The Hetu (smoke) is invariably associated with Sādhya (fire), and since Paksha (the hill) possesses the Hetu, we can conclude it also possesses the Sādhya. This is essentially the same three-term structure as in Aristotelian logic, but the manner of presentation is expanded to five steps. Therefore, Statement C is correct.

Why Statements A and D are Incorrect:
— Statement A says Nyaya syllogism has three propositions. This is incorrect; three propositions belong to the Aristotelian (Western) syllogism, not Nyaya. Nyaya has five propositions.
— Statement D says Nyaya syllogism has five terms. This is incorrect; the Nyaya syllogism has only three terms (Paksha, Sādhya, Hetu), regardless of the number of its propositions.

Key Takeaway:
The Nyaya system is unique because it separates the number of terms (three) from the number of propositions/steps (five) in a syllogism. The five steps serve a communicative and didactic function, ensuring that every member of a debate audience can follow and verify the reasoning — while the logical core still rests on three fundamental terms. This is what makes the Nyaya system both philosophically rigorous and practically oriented toward public discourse and debate (Vāda).

Hence, B and C only is the correct answer.