Logical Reasoning

Syllogism & statements

A categorical proposition relates two classes, a subject term and a predicate term, in one of four forms. The A form, All S is P, affirms that every member of S is a member of P. The E form, No S is P, denies that any member of S is a member of P. The I form, Some S is P, affirms that at least one member of S is a member of P. The O form, Some S is not P, denies that every member of S is a member of P. The word some in logic means at least one, and possibly all; it does not mean only some, and this single point decides many NET questions. A statement such as Some students are singers is compatible with all students being singers, so a conclusion that some students are not singers does not follow from it alone.

A syllogism is an argument with two premises and a conclusion, involving three terms: the major term, which appears in the conclusion as predicate, the minor term, which appears as subject, and the middle term, which appears in both premises and links them. The classic valid pattern is: All M is P, All S is M, therefore All S is P. The middle term M disappears in the conclusion, having done its linking work. Validity is tested by asking whether the conclusion must be true whenever the premises are true. If you can imagine, or draw, a situation in which the premises hold and the conclusion fails, the argument is invalid.

Distribution is the technical rule behind many quick decisions. A term is distributed when the statement says something about all of its members. In All S is P, S is distributed but P is not, because the statement says nothing about all P. In No S is P, both terms are distributed. In Some S is P, neither term is distributed. In Some S is not P, P is distributed but S is not. Two rules follow. First, the middle term must be distributed in at least one premise; otherwise the two premises may be talking about different parts of the middle class and prove nothing. Second, a term distributed in the conclusion must be distributed in its premise; otherwise the conclusion claims more than the premises established.

The Venn method

The Venn method turns a syllogism into pictures. Each class becomes a circle, and overlapping circles divide the picture into regions. A universal statement, All S is P or No S is P, is drawn by shading out regions that must be empty: for All S is P, shade the part of S outside P; for No S is P, shade the overlap of S and P. A particular statement, Some S is P or Some S is not P, is drawn by placing a mark in a region that must contain at least one member: for Some S is P, mark the overlap of S and P; for Some S is not P, mark the part of S outside P. When a mark could go in more than one region because an earlier universal premise has not ruled either out, place it on the boundary between them, meaning that at least one of those regions is occupied but you do not know which.

To test an argument, draw the premises in order, universals first so that particular marks respect the shading, and then read off the conclusion. A conclusion follows only if the diagram already shows it: the required region is fully shaded out for a universal conclusion, or already marked for a particular conclusion. If the diagram leaves the conclusion undecided, it does not follow. Work through the standard case: All pens are books, All books are tables. Shade pens outside books, then shade books outside tables. The pens circle now lies entirely inside tables, so All pens are tables follows, and so does Some pens are tables, since a non-empty class with all members inside tables must have some member inside tables, provided the class is taken to have members.

Indian logic: pramanas

Indian logic, as developed in the Nyaya school and debated across schools, asks a prior question to syllogistic form: what counts as a valid source of knowledge, a pramana? A pramana is not merely a belief or a guess; it is a reliable means by which knowledge is produced. The Nyaya school accepts four pramanas: pratyaksha, anumana, upamana, and shabda. The Mimamsa and Vedanta traditions add arthapatti, and some add anupalabdhi as well, while the Carvaka school accepts perception alone. Knowing this table prevents the common error of attributing six pramanas to Nyaya.

Pratyaksha, perception, is knowledge arising from direct contact between a sense organ and its object, such as seeing a pot before you. Anumana, inference, is knowledge of something not directly perceived through a mark invariably connected with it, such as inferring fire from smoke. Upamana, comparison, is knowledge gained by similarity with something already known, such as learning what a wild cow is by being told it is like a cow. Shabda, verbal testimony, is knowledge from the words of a trustworthy speaker, such as learning from a reliable teacher or text. Arthapatti, postulation, is the assumption of something to explain what is otherwise inexplicable, such as postulating that a person who does not eat by day and yet grows fat must eat by night. Anupalabdhi, non-apprehension, is knowledge of absence through the non-perception of something that would have been perceived had it been present, such as knowing there is no pot on a bare table you are looking at.

Anumana and its five members

Nyaya analyses inference into five members, and NET questions test both their names and their order. The pratijna is the proposition to be proved, for example, the hill has fire. The hetu is the reason offered, for example, because it has smoke. The udaharana is the example stating the invariable connection together with an instance, for example, wherever there is smoke there is fire, as in a kitchen. The upanaya is the application of the example to the case at hand, for example, this hill has smoke which is invariably accompanied by fire. The nigamana is the conclusion restated as established, for example, therefore the hill has fire. Notice that the hetu must be present in the subject, present in similar cases, and absent in dissimilar cases; a reason failing any of these conditions is a fallacious reason, a hetvabhasa, discussed in the next section.

Two further distinctions sharpen anumana questions. Svarthanumana is inference for oneself, in which the reasoner privately moves from mark to conclusion, while pararthanumana is inference for others, expressed in the five-member form to convince an opponent. What matters for validity is the invariable connection, vyapti, between mark and what is inferred. UGC NET questions often present a short inference and ask which member a given sentence is, or which condition of a good hetu is violated, so practise labelling each sentence of an argument before judging it.

Fallacies

A fallacy is an argument that appears valid or persuasive but fails on inspection. Formal fallacies break the rules of logical form. The fallacy of the undistributed middle occurs when the middle term is distributed in neither premise, so the premises may concern different parts of the middle class. The fallacy of illicit process occurs when a term distributed in the conclusion was not distributed in its premise, so the conclusion claims more than was established. Affirming the consequent argues from If P then Q and Q to P, which fails because Q may have other causes. Denying the antecedent argues from If P then Q and not P to not Q, which fails for the same reason. These forms fail regardless of content, and recognising the shape is enough to reject them.

Informal fallacies fail in context rather than in bare form. The ad hominem fallacy attacks the person making a claim instead of the claim itself. The straw man fallacy misrepresents an opponent position and attacks the misrepresentation. Hasty generalisation draws a general conclusion from too small or unrepresentative a sample. False cause treats mere correlation or sequence as causation. The appeal to authority treats an irrelevant or unqualified authority as decisive, and the appeal to emotion substitutes feeling for evidence. Begging the question, petitio principii, assumes in a premise exactly what is to be proved, so the argument goes in a circle.

Hetvabhasa: fallacies of the reason

Hetvabhasa means the semblance of a reason: a hetu that looks like a genuine reason but fails one of the conditions of a good inference. Nyaya lists five hetvabhasa, and these names recur in NET questions, so learn each with a diagnostic test. Savyabhichara, the irregular or straying reason, is present in dissimilar cases as well as similar ones, so it does not reliably indicate what is to be proved; its mark wanders. Viruddha, the contradictory reason, actually proves the opposite of what is claimed, because it is present where the conclusion is absent. Satpratipaksha, the counterbalanced reason, is opposed by an equally strong reason proving the contrary conclusion, leaving the matter stalemated. Asiddha, the unproved reason, is itself not established: it may be unproved in the subject, unproved in its connection, or based on a doubtful assumption. Badhita, the mistimed or sublated reason, is contradicted by stronger evidence, such as perception, so the conclusion is already known to be false.

A simple diagnostic runs through the five in order. Ask first whether the reason is even established; if not, it is asiddha. Ask whether perception or stronger evidence already contradicts the conclusion; if so, it is badhita. Ask whether the reason proves the opposite; if so, it is viruddha. Ask whether an equal reason proves the contrary; if so, it is satpratipaksha. Ask whether the reason strays into cases where the conclusion is absent; if so, it is savyabhichara. NET questions usually make one of these failures salient: a reason that is present in counterexamples points to savyabhichara, a reason that proves the opposite points to viruddha, and a reason whose very existence is in doubt points to asiddha.

The four categorical propositions

Pramanas compared

The five hetvabhasa

Pramanas accepted by school