CHSE Odisha Elective Logic cheat sheet

Logic studies inference forms independent of subject matter; its central question is whether a conclusion follows from premises.

Deductive validity is about possibility: no possible world has true premises and a false conclusion.

Inductive strength is a matter of degree: the premises make the conclusion more or less probable.

Logic is descriptive of norms, not of actual reasoning habits; it prescribes correct inference.

Formal logic abstracts content; informal logic evaluates content, context and evidence.

Soundness = validity plus true premises; cogency = strength plus true premises.

Logic underpins proof, definition, consistency testing and the detection of fallacies.

A valid argument can have false premises; validity alone never guarantees a true conclusion.

Categorical propositions relate two classes, subject (S) and predicate (P), by quantity and quality.

A: All S are P (universal affirmative); E: No S are P (universal negative); I: Some S are P; O: Some S are not P.

Distribution: A subject only; E both; I neither; O predicate only.

Traditional reading gives A and E existential import; Boolean reading does not.

Venn shading asserts emptiness; an asterisk asserts existence.

'Only S are P' translates as 'All P are S'; 'Not all S are P' translates as O.

Contradictories differ in both quantity and quality; contraries differ in quality only, both universal.

Singular propositions are treated as universal in traditional syllogistic.

Contradiction: A–O and E–I, exactly one true and one false in every case.

Contrary: A–E, cannot both be true, may both be false.

Subcontrary: I–O, cannot both be false, may both be true.

Subalternation: A→I and E→O, valid only with existential import.

Conversion valid for E and I; A converts per accidens if S exists; O does not convert.

Obversion valid for all forms: change quality, complement the predicate.

Contraposition valid for A and O; partial contraposition for E.

Under the Boolean reading only contradiction, obversion and A/O contraposition remain unconditional.

Three terms and three propositions; middle term appears in both premises and never in the conclusion.

Figure determined by middle-term position: Fig 1 S–M/M–P; Fig 2 P–M/S–M; Fig 3 M–P/M–S; Fig 4 P–M/M–S.

Mood is the letter sequence of major premise, minor premise, conclusion.

Middle term must be distributed at least once; otherwise undistributed middle.

No term may be distributed in the conclusion unless distributed in its premise (illicit major/minor).

Two negative premises give no conclusion; a negative premise forces a negative conclusion.

Two particular premises give no conclusion; a particular premise forces a particular conclusion.

Valid first-figure moods: AAA, EAE, AII, EIO (plus weakened AAI, EAO).

Formal fallacies: undistributed middle, illicit process, affirming the consequent, denying the antecedent.

Ad hominem attacks the person; ad baculum appeals to force; ad misericordiam to pity; ad populum to popularity.

Ad ignorantiam argues from lack of proof; ad verecundiam from unqualified authority.

Hasty generalisation draws a general conclusion from an inadequate sample.

Post hoc ergo propter hoc mistakes sequence for causation; false cause generally misidentifies causal links.

Equivocation shifts word meaning; amphiboly shifts grammatical structure; accent shifts stress.

Composition infers whole from parts; division infers parts from whole.

Begging the question assumes the conclusion; complex question presupposes an answer.

¬P reverses truth value; P∧Q true only when both true; P∨Q false only when both false.

P→Q false only in the row P true, Q false; P↔Q true when truth values agree.

P→Q ≡ ¬P∨Q; contrapositive ¬Q→¬P is equivalent; converse and inverse are not.

Truth table for n letters has 2^n rows; list assignments systematically.

Tautology true on all rows; contradiction false on all rows; contingent mixed.

Argument valid iff no row has all premises true and conclusion false.

De Morgan: ¬(P∧Q) ≡ ¬P∨¬Q; ¬(P∨Q) ≡ ¬P∧¬Q.

'P unless Q' symbolises as P∨Q; exclusive or as (P∨Q)∧¬(P∧Q).

Modus ponens: P→Q, P ⊢ Q; modus tollens: P→Q, ¬Q ⊢ ¬P.

Hypothetical syllogism: P→Q, Q→R ⊢ P→R; disjunctive syllogism: P∨Q, ¬P ⊢ Q.

Simplification: P∧Q ⊢ P; conjunction: P, Q ⊢ P∧Q; addition: P ⊢ P∨Q.

Equivalence rules may be applied in either direction: De Morgan, commutation, association, distribution, double negation, contraposition, implication, exportation.

Conditional proof: assume the antecedent, derive the consequent, discharge and infer the conditional.

Indirect proof: assume the negation of the conclusion, derive a contradiction, infer the conclusion.

Apply elimination rules to premises first; introduce assumptions only when the conclusion is conditional or direct proof stalls.

Every assumption must be discharged before the proof ends.

Inductive conclusions are probable; strength depends on evidence relevance and sufficiency.

Analogy: shared properties support a further shared property; relevance of shared properties and number of disanalogies govern strength.

Generalisation requires a representative, randomly selected, adequately large sample.

Statistical syllogism: X% of F are G; a is F; therefore a is probably G.

Mill's methods: agreement, difference, joint method, concomitant variation, residues.

Correlation does not establish causation; exclude confounders, reverse causation and selection bias.

Hume: induction lacks deductive justification; it rests on habit and the uniformity of nature.

One counter-instance refutes a universal generalisation but only lowers the probability of a statistical one.

Sequence: observation, hypothesis, deduction of consequences, test, confirmation or refutation, revision.

A hypothesis must be testable and falsifiable; unfalsifiable claims fall outside empirical science.

Hypothetico-deductive schema: H→O; O confirms H weakly; ¬O refutes H by modus tollens.

Popper: universal laws are falsifiable but never conclusively verified.

Controlled experiment uses experimental and control groups with random assignment.

Operational definitions specify measurement procedures for each variable.

Duhem–Quine holism: a failed prediction may implicate auxiliary assumptions rather than the central hypothesis.

Hempel's covering-law model: explanation and prediction share the same logical structure of law plus initial conditions.

Symbolise simple statements with letters; combine with ¬, ∧, ∨, →, ↔; use parentheses to fix scope.

WFFs are generated recursively from sentence letters by the connectives.

The main connective determines the form and the final truth-table column.

Logical form abstracts from content; arguments sharing a form share validity status.

∀x(Sx→Px) renders 'All S are P'; ∃x(Sx∧Px) renders 'Some S are P'.

¬∀xFx ≡ ∃x¬Fx, but ∀x¬Fx is a different claim.

'P only if Q' gives P→Q; 'P if Q' gives Q→P; necessary condition is the consequent.

Formalisation permits mechanical validity testing by truth tables, proofs or models.