Logical Reasoning — Arguments, Venn Diagrams & Indian Logic

Every categorical proposition relates two classes and has exactly four parts: a quantifier (all, no, some), a subject term, a copula (is/are, is not/are not) and a predicate term. Its quality is affirmative or negative; its quantity is universal or particular. Crossing quality with quantity gives exactly four forms, labelled A, E, I, O. The labels come from two Latin words: AffIrmo ("I affirm") donates A and I to the affirmative propositions, and nEgO ("I deny") donates E and O to the negative ones.

Three immediate inferences convert one proposition into another without any second premise. Conversion interchanges subject and predicate: E and I convert simply and validly ("No S is P" gives "No P is S"; "Some S is P" gives "Some P is S"), A converts only by limitation into an I proposition ("All S is P" gives only "Some P is S"), and O cannot be converted at all. Obversion changes the quality and negates the predicate, and it is valid for all four forms: "All S is P" becomes "No S is non-P". Contraposition interchanges the terms and negates both: valid for A and O, and for E only by limitation.

Connotation and denotation are the two dimensions of any term, and NET asks about them directly. The denotation (also called extension) of a term is the collection of all objects to which the term applies: the denotation of "university" is every actual university. The connotation (also called intension) is the set of attributes an object must possess to be named by the term: for "university", attributes such as being an institution, awarding degrees, conducting research. The two vary inversely: as you add attributes to the connotation, fewer objects can qualify, so the denotation shrinks. "Institution" denotes more than "university", which denotes more than "central university", which denotes more than "central agricultural university".

Logic textbooks also distinguish the uses of language, because an argument can only be evaluated once you know what the words are doing. The informative use conveys facts and can be true or false: "The UGC Act was passed in 1956." The expressive use vents or evokes feeling and is neither true nor false: a poem, a slogan, an exclamation. The directive use aims to cause action: a command, a request, a rule. Many real sentences mix uses; ceremonial and performative sentences ("I hereby declare the session open") are the standard mixed examples. Only the informative use can serve as a premise, which is why arguments loaded with expressive language usually turn out to be informal fallacies.

Take the same subject and the same predicate and write all four propositions — A, E, I, O. The square of opposition is the diagram of how the truth of one constrains the truth of the others. Place A at the top left, E at the top right, I at the bottom left and O at the bottom right. The top row holds the universals, the bottom row the particulars, the left column the affirmatives, the right column the negatives. Four relations then run along the sides and diagonals.

One historical caveat is worth knowing because examiners occasionally test it. The classical square assumes existential import — that the subject class is not empty. Modern (Boolean) logic drops that assumption, and with it the relations of contrariety, subcontrariety and subalternation lapse; only the two contradictory diagonals survive. This is why the traditional count of valid syllogisms is 24 while the modern count is 19: the five moods that depend on existential import (Darapti, Felapton, Bramantip, Fesapo and Camestros-type strengthened forms) are rejected in the Boolean reading.

A categorical syllogism is fully identified by two labels. Its mood is the sequence of the three propositions' forms, written in the fixed order major premise, minor premise, conclusion — so AAA, EIO, AOO and so on. Its figure is the position of the middle term in the two premises. Since each of three propositions can be any of four forms, there are 4 × 4 × 4 = 64 possible moods, and with four figures that gives 256 possible syllogisms. Of these only 24 are valid on the traditional reading (19 on the modern Boolean reading).

Rather than memorising 24 names, learn the rules; any syllogism that breaks one is invalid.
(1) There must be exactly three terms, each used in the same sense throughout — otherwise the fallacy of four terms or of equivocation.
(2) The middle term must be distributed at least once, or you commit the undistributed middle.
(3) A term distributed in the conclusion must be distributed in its premise, or you commit the illicit major or illicit minor.
(4) From two negative premises nothing follows — the fallacy of exclusive premises.
(5) If one premise is negative, the conclusion must be negative.
(6) From two particular premises nothing follows.
(7) If one premise is particular, the conclusion must be particular.
(8) Two affirmative premises cannot yield a negative conclusion.

Worked example. "All poets are dreamers. All novelists are dreamers. Therefore all novelists are poets." Identify the parts: the conclusion is "all novelists are poets", so poets is the major term, novelists the minor term, and dreamers the middle term. Both premises are A propositions with dreamers as the predicate, which makes this Figure 2 with mood AAA. Now apply rule 2: an A proposition distributes only its subject, so dreamers is undistributed in both premises. The middle term is never distributed, so the argument commits the undistributed middle and is invalid — even though both premises may well be true. Indeed AAA is valid only in Figure 1.

The single deepest distinction in the unit is between deduction and induction. In a deductive argument the conclusion claims nothing more than what the premises already contain, so if the form is correct the conclusion is necessarily true given true premises. In an inductive argument the conclusion claims more than the premises establish — it is ampliative — so the most it can be is probable. This is why deduction is evaluated as valid or invalid (with no degrees) and induction as strong or weak (in degrees).

An analogical argument is a species of induction. It reasons that because two things share several attributes, they will share one more: "This college and that one have similar faculty, funding and student intake; that one gets excellent NAAC grades; so this one will too." Its strength is judged by six standard criteria: the number of instances compared, the number of respects in which they resemble each other, the relevance of those respects to the attribute inferred, the presence of disanalogies (relevant differences), the variety among the instances, and the modesty of the conclusion — a weaker claim is easier to support. Note the fifth and sixth carefully: greater variety among agreeing instances strengthens the analogy, and a bolder conclusion weakens it.

A fallacy is an argument that appears sound but is not. Formal fallacies are errors in the structure and can be detected without knowing what the argument is about — the undistributed middle, illicit major, illicit minor, exclusive premises and existential fallacy of the categorical syllogism, plus the two conditional favourites. Affirming the consequent is invalid: "If it rains the ground is wet; the ground is wet; therefore it rained" — a burst pipe would do just as well. Denying the antecedent is equally invalid: "If it rains the ground is wet; it did not rain; therefore the ground is not wet." Contrast these with the valid modus ponens and modus tollens of the first section.

A Venn test converts a logic question into a picture, and the procedure is mechanical. Draw three overlapping circles for the minor term S, the major term P and the middle term M — this creates seven regions inside plus the region outside all three. Now diagram only the premises, never the conclusion. For a universal premise, shade the region asserted to be empty: "No M is P" shades the whole overlap of M and P. For a particular premise, place an X in the region asserted to have at least one member: "Some S is M" puts an X in the S–M overlap. Finally, look: if the diagram of the premises already shows the conclusion, the argument is valid; if you would have to add or move anything, it is invalid.

Worked example — a valid syllogism. "All M are P. All S are M. Therefore all S are P." Diagram the first premise by shading the part of M lying outside P, which forces M inside P. Diagram the second by shading the part of S lying outside M, which forces S inside M. Now S is inside M and M is inside P, so every point of S already lies inside P — three concentric rings. The conclusion is drawn without adding anything, so the syllogism is valid. This is Barbara, mood AAA in Figure 1.

Worked example — an invalid syllogism. "All P are M. All S are M. Therefore all S are P." Both premises push P and S inside M, but they say nothing about how P and S sit relative to each other. You are free to draw P and S as two separate blobs inside M, touching nowhere — a diagram that obeys both premises yet makes "All S are P" false. Because at least one legal diagram refutes the conclusion, the argument is invalid; this is the picture of the undistributed middle. Note the discipline: an argument is valid only if the conclusion holds in every admissible diagram, so a single counter-diagram is enough to destroy it.

The multiple use of Venn diagrams appears when a question gives three or four statements and asks which of several conclusions follow, sometimes including "either–or" options. Two habits protect you. First, treat the given statements as absolutely true however absurd — your knowledge of the world is switched off. Second, when neither conclusion follows on its own, check whether the two conclusions form a complementary pair with the same subject and predicate — Some paired with No, All paired with Some-not, or Some paired with Some-not. If they do, the correct response is "either the first or the second follows", because between them they exhaust every possibility.

Indian logic is examined only in UGC NET, so it deserves separate and careful study. Its central question is not "what follows from what" but "how do we come to know anything at all?" A pramana is a means of valid knowledge — the instrument through which true cognition (prama) arises. The Nyaya school, founded on the Nyaya-Sutra of Gautama (Akshapada Gautama), accepts exactly four pramanas: Pratyaksha (perception), Anumana (inference), Upamana (comparison) and Shabda (verbal testimony). The number four is itself an exam fact, because rival schools count differently.

Anumana has three terms, exactly parallel to the Western syllogism but named from the side of knowledge rather than of form. The paksha is the subject about which something is inferred (the hill); the sadhya is the property to be established (fire); the hetu, also called linga or sadhana, is the mark through which the inference runs (smoke). The engine of the whole process is vyapti — the invariable concordance or universal relation between hetu and sadhya, of the shape "wherever there is smoke there is fire". Without an established vyapti there is no inference at all, only guesswork.

Nyaya divides inference along three axes, and NET can ask about any of them. By purpose: svarthanumana is inference for oneself, an unspoken psychological process needing no formal statement; pararthanumana is inference for another, meant to convince, and only this requires the full five-membered form. By the nature of the vyapti: kevalanvayi (agreement only, where the terms are never absent), kevalavyatireki (difference only, established through negative instances) and anvayavyatireki (both agreement and difference). By the ground of the inference, the classification given in the Nyaya-Sutra itself: purvavat, sheshavat and samanyatodrishta.

A hetvabhasa is literally a hetu-abhasa — a reason that only appears to be a reason. The Nyaya-Sutra lists exactly five, and later Nyaya writers gave four of them alternative names, which is why the same fallacy shows up in NET options under two labels. Learn both columns. The Sutra names are savyabhichara, viruddha, prakaranasama, sadhyasama and kalatita; the later names are savyabhichara (anaikantika), viruddha, satpratipaksha, asiddha and badhita. Note that only viruddha and savyabhichara keep a single name throughout.

It helps to see how neatly the two traditions line up and where they part. The Nyaya hetu is the Western middle term, and savyabhichara — a hetu present both with and without the sadhya — is very nearly the undistributed middle. The five-membered form corresponds to a three-membered Western syllogism with the vyapti spelled out and then applied. But two differences are worth stating in an answer. First, Nyaya makes the example (udaharana) an indispensable member, so its logic is never purely formal: it insists on at least one real instance. Second, Nyaya treats logic as a branch of epistemology — the aim is prama, valid knowledge of what is, not merely a valid form. Western formal logic, from Aristotle onward, deliberately separates validity from truth.