Propositions, Syllogism & Indian Logic

Take one subject and one predicate and write all four propositions. The square of opposition places A and E on the top corners, I and O on the bottom, and defines four relations. Contradictories (A–O, E–I, the diagonals) always have opposite truth-values. Contraries (A–E, the top side) cannot both be true but can both be false. Subcontraries (I–O, the bottom side) cannot both be false but can both be true. Subalternation (A–I, E–O, the vertical sides) lets truth flow downward from universal to particular and falsity flow upward from particular to universal — never the reverse. NET does not ask for these definitions; it gives you the truth-value of one proposition and asks for the others. The table below is therefore the object to memorise: eight starting points, three consequences each, where "?" means undetermined.

Worked example. Suppose it is given that "Some research scholars are not funded" is false, and the question asks which of the other three propositions can be determined. Step 1: the false proposition is O. Step 2: its contradictory A, "All research scholars are funded", must therefore be true — contradictories never share a value. Step 3: with A true, its contrary E, "No research scholars are funded", must be false — contraries cannot both be true. Step 4: with A true, its subaltern I, "Some research scholars are funded", is true — truth descends. So a single false O fixes all three others: A true, E false, I true. Compare the opposite starting point: if O were true, only A would be fixed (false); E and I would remain undetermined, because a true particular says nothing about its universal, and I–O may both be true.

An immediate inference draws a new proposition from a single given one. Conversion swaps subject and predicate; the result is the converse. Obversion keeps the subject, changes the quality, and replaces the predicate by its complement (P becomes non-P); the result is the obverse. Contraposition replaces the subject by the complement of the predicate and the predicate by the complement of the subject; the result is the contrapositive. Each operation is valid for some forms only, and the pattern is a mirror: conversion is valid for E and I and fails for O (A only by limitation); contraposition is valid for A and O and fails for I (E only by limitation); obversion alone is valid for all four. The contrapositive is not a fresh operation at all — it is obvert, then convert, then obvert again, which is why it inherits conversion's restrictions.

Worked example — deriving the contrapositive of an O proposition step by step. Start: "Some teachers are not researchers" (O). Step 1, obvert: change the quality to affirmative and negate the predicate — "Some teachers are non-researchers" (I). Step 2, convert the I, which converts simply — "Some non-researchers are teachers" (I). Step 3, obvert again — "Some non-researchers are not non-teachers" (O). That is the contrapositive, and it is valid because every step was valid. Now try the same recipe on an I proposition, "Some teachers are researchers": obversion gives "Some teachers are not non-researchers" (O), and the next step is to convert an O — which is impossible. The chain breaks, so I has no contrapositive. The same reasoning shows why O has no converse: to convert O directly you would have to swap the terms of a proposition whose subject is undistributed and whose predicate is distributed, and the new proposition would distribute a term the original did not.

A categorical syllogism has two premises and a conclusion built from exactly three terms. Identify them in a fixed order. The conclusion is found first (look for therefore, hence, so). Its predicate is the major term (P) and its subject the minor term (S); the term appearing in both premises but not in the conclusion is the middle term (M). The premise containing P is the major premise and is written first; the premise containing S is the minor premise. Now read off two labels. The mood is the string of proposition types in the order major–minor–conclusion (AAA, EIO, AOO …). The figure is fixed by where M sits: Figure 1 M is subject of the major and predicate of the minor (M–P, S–M); Figure 2 M is predicate of both (P–M, S–M); Figure 3 M is subject of both (M–P, M–S); Figure 4 M is predicate of the major and subject of the minor (P–M, M–S). Sixty-four moods across four figures give 256 forms. The mnemonic verse Barbara, Celarent, Darii, Ferio … names 19 valid moods; adding the five weakened "subaltern" moods (such as Barbari, AAI-1) gives the traditional 24; on the strict Boolean reading only 15 survive, because nine of them draw a particular conclusion from two universal premises.

Worked example 1 — illicit major. "All dogs are mammals. No cats are dogs. Therefore no cats are mammals." The conclusion is "No cats are mammals", so P = mammals, S = cats, M = dogs. Major premise: "All dogs are mammals" (A, M–P). Minor premise: "No cats are dogs" (E, S–M). Middle term subject of the major and predicate of the minor: Figure 1, mood AEE. Now the rules. Rule 2: M (dogs) is the subject of an A, hence distributed — satisfied. Rule 3a: the conclusion is E, which distributes both terms, so P (mammals) is distributed in the conclusion; but in the major premise mammals is the predicate of an A, which is undistributed. A term distributed in the conclusion but not in its premise — illicit major; the syllogism is invalid, and the true premises with the false conclusion prove it. Confirm by mood: AEE is valid only in Figures 2 and 4 (Camestres and Camenes), never in Figure 1.

Worked example 2 — illicit minor. "All philosophers are thinkers. All philosophers are human. Therefore all humans are thinkers." Conclusion: "All humans are thinkers" — P = thinkers, S = humans, M = philosophers. Major: "All philosophers are thinkers" (A, M–P). Minor: "All philosophers are human" (A, M–S). M is the subject of both premises: Figure 3, mood AAA. Rule 2 is satisfied (M is subject of an A, distributed). Rule 3b: the conclusion is A, which distributes its subject S (humans); in the minor premise humans is the predicate of an A and so undistributed — illicit minor. The premises only license the weaker conclusion "Some humans are thinkers" (AAI-3, Darapti — valid traditionally, an existential fallacy on the Boolean reading). Worked example 3 — exclusive premises. "No reptiles are mammals. Some birds are not reptiles. Therefore some birds are not mammals." Mood EOO: both premises negative, so rule 4 fails at once and nothing follows — reptiles being excluded from mammals tells us nothing about where the non-reptile birds sit.

The Venn test on Worked example 1. Draw three overlapping circles labelled S (cats), P (mammals) and M (dogs); they create seven regions. Diagram only the premises. Premise 1, "All dogs are mammals": shade every part of M that lies outside P — two regions vanish. Premise 2, "No cats are dogs": shade the entire overlap of S and M — two more regions vanish (one of them already shaded). Now inspect the conclusion, "No cats are mammals", which would require the whole S–P overlap to be shaded. Look at the region that is inside both S and P but outside M: it is unshaded, and nothing forbids a cat from being there. Because at least one region admitted by the premises would make the conclusion false, the argument is invalid — the same verdict the rules gave, reached by a picture. The discipline is always the same: universals shade, particulars place an X, universals before particulars, and the conclusion is valid only if it is already drawn when the premises are finished.

A formal fallacy can be exposed by replacing the words with letters; an informal fallacy survives only because of what the words mean or how they are used. Logicians since Copi group informal fallacies into four families. Fallacies of relevance: the premises are psychologically persuasive but logically beside the point — ad hominem, straw man, red herring, appeals to force, pity, popularity and inappropriate authority. Fallacies of weak (defective) induction: the premises are relevant but far too weak — hasty generalisation, post hoc, slippery slope, false analogy, appeal to ignorance. Fallacies of presumption: an unproved assumption is smuggled in — begging the question, false dilemma, complex question, accident. Fallacies of ambiguity: a shift in meaning does the work — equivocation, amphiboly, accent, composition, division. Knowing the family lets you eliminate two options at once.

Worked identification. Read: "The committee proposes a dress code. Its chairman, mind you, was once a fashion model, so his motives are obvious. Besides, once you regulate clothing you will regulate hairstyles, then speech, and finally thought itself. And everyone on campus opposes it — a poll of my three room-mates was unanimous." Three separate fallacies are stacked here. "Was once a fashion model, so his motives are obvious" attacks the proposer's circumstances rather than the proposal: circumstantial ad hominem. "Regulate clothing … then thought itself" is an unsupported chain of consequences: slippery slope. "A poll of my three room-mates" generalises from a tiny, unrepresentative sample to "everyone on campus": hasty generalisation (with a dash of ad populum, since even a real majority would not settle the merits). Notice that none of the three tells us anything about whether the dress code is a good idea — which is the working definition of irrelevance.

Indian epistemology asks how valid knowledge (prama) arises, and names the instrument of valid knowledge a pramana; the knower is the pramata and the object known the prameya. The Nyaya school, founded on Gautama (Akshapada) and his Nyaya-Sutra with Vatsyayana's Nyaya-bhashya as the classic commentary, lists sixteen categories (padarthas) of which pramana is the first, and admits exactly four pramanas. Other schools admit fewer or more, and the differences are a fixed NET favourite: Charvaka 1 (pratyaksha only), Buddhists and Vaisheshika 2 (add anumana), Sankhya and Yoga 3 (add shabda), Nyaya 4 (add upamana), Prabhakara Mimamsa 5 (add arthapatti), Bhatta (Kumarila) Mimamsa and Advaita Vedanta 6 (add anupalabdhi). The order in which pramanas join the list never changes, so the whole table can be rebuilt from the sequence pratyaksha, anumana, shabda, upamana, arthapatti, anupalabdhi.

Nyaya analyses anumana into named parts that map onto, but are richer than, the Western syllogism. The paksha is the locus about which something is inferred (the hill); the sadhya is the property to be proved (fire); the hetu, also called linga or sadhana, is the mark (smoke). Two further classes matter: the sapaksha is any locus known to possess the sadhya (the kitchen, where smoke and fire are seen together) and the vipaksha any locus known to lack it (the lake, where neither is found). The vyapti is the invariable, unconditional concomitance "wherever smoke, there fire"; it is anvaya (positive: smoke → fire) and vyatireka (negative: no fire → no smoke). Vyapti is grasped by repeated observation (bhuyodarshana) together with the non-observation of exceptions, and is secured against doubt by tarka (hypothetical reasoning: "if smoke could exist without fire, it would not be an effect of fire"). The pervaded term (smoke) is the vyapya, the pervader (fire) the vyapaka: fire is wider than smoke, which is why one may infer fire from smoke but never smoke from fire.

Inference for another (pararthanumana) is stated in five members — the panchavayava — because a hearer must be led through it; inference for oneself (svarthanumana) is a silent mental act needing none. Using a second classic example: Pratijna (thesis) "Sound is non-eternal"; Hetu (reason) "because it is produced"; Udaharana (universal with instance) "whatever is produced is non-eternal, as a pot"; Upanaya (application) "sound is produced, like the pot"; Nigamana (conclusion) "therefore sound is non-eternal". The immediate cause of the inferential cognition is the third step, linga-paramarsha — the knowledge "the paksha has a hetu that is pervaded by the sadhya". Anumana is further classified by the kind of vyapti it rests on: kevalanvayi, where only positive concomitance is available because the sadhya is present everywhere ("this is nameable because it is knowable" — nothing exists that is not knowable, so no vipaksha exists); kevalavyatireki, where only negative concomitance is available because the sadhya is found nowhere else ("earth differs from the other elements because it has smell" — no sapaksha exists); and anvaya-vyatireki, the ordinary case with both (smoke and fire).

• Purvavat — from perceived cause to unperceived effect: heavy dark clouds, therefore rain will come.
• Sheshavat — from perceived effect to unperceived cause: the river is swollen and muddy, therefore it rained upstream.
• Samanyatodrishta — from a general uniformity that is not causal: whatever changes position moves, so the sun, seen in different places, moves.
• By purpose: svartha (for oneself, no formal statement) versus parartha (for another, five members); the Buddhist logician Dignaga required only three marks (trairupya) of the hetu, while Nyaya requires the five listed below.