Maharashtra HSC Logic cheat sheet

Logic is the normative science of valid reasoning: it studies the form of arguments, not the psychological causes of belief.

Formal logic abstracts content and evaluates structure; material (informal) logic evaluates content-dependent reasoning such as analogies and causal claims.

Deductive logic aims at validity (conclusion follows necessarily); inductive logic aims at strength (conclusion follows with probability).

Logic is normative, not descriptive: it prescribes how one ought to infer, whereas psychology describes how people actually infer.

A sound argument requires both validity and true premises; validity alone never guarantees a true conclusion.

Logic is distinguished from epistemology (theory of knowledge) and from rhetoric (persuasion); truth of premises is a separate empirical question.

Classical Indian and Western traditions both classify inference; in Nyaya, anumana (inference) is one of the pramanas (means of valid knowledge).

Scope includes definition, classification of propositions, inference, proof, scientific method, and detection of fallacies.

A term is a word or phrase that can serve as subject or predicate of a proposition; categorical terms denote classes.

Categorical terms have denotation (the class of objects) and connotation (the attributes implied); increasing connotation generally decreases denotation.

A proposition is a declarative sentence capable of being true or false; questions, commands, and exclamations are not propositions.

Every standard categorical proposition has four parts: quantifier, subject term, copula, predicate term.

An argument is a set of propositions in which some (premises) are offered as support for another (conclusion).

Premise indicators: since, because, for, given that. Conclusion indicators: therefore, hence, so, thus, consequently.

An argument is valid if it is impossible for all premises to be true and the conclusion false; invalidity requires only one possible counterexample.

Explanations, conditional statements, and mere statements of fact are not arguments unless a claim is being supported.

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

Distribution: A distributes S only; E distributes both S and P; I distributes neither; O distributes P only.

Contradictories (A-O, E-I) always differ in truth value; contradictories cannot both be true or both be false.

Contraries (A-E) cannot both be true but may both be false; subcontraries (I-O) cannot both be false but may both be true.

Subalternation: A implies I, and E implies O; truth flows downward, falsity flows upward.

Conversion: E and I convert simply (No S is P = No P is S; Some S is P = Some P is S); A converts only per accidens to I; O does not convert.

Obversion: change quality and replace the predicate with its complement; obversion is valid for all four forms and preserves truth value.

Contraposition is valid for A and O (All S are P implies All non-P are non-S); it is not valid for E and I.

Two terms are complementary when one is the negation of the other; obversion depends on this complement relation.

A categorical syllogism has exactly three terms (major, minor, middle) and three propositions; the middle term appears in both premises but not the conclusion.

Major premise contains the major term (predicate of conclusion); minor premise contains the minor term (subject of conclusion).

Mood names the three proposition types in order (major, minor, conclusion), e.g., Barbara = AAA-1.

Figure is fixed by the position of the middle term: Fig. 1 (M-P, S-M), Fig. 2 (P-M, S-M), Fig. 3 (M-P, M-S), Fig. 4 (P-M, M-S).

Rules of validity: the middle term must be distributed at least once; no term may be distributed in the conclusion unless distributed in a premise.

A valid syllogism cannot have two negative premises, and if either premise is negative the conclusion must be negative.

A valid syllogism cannot have two particular premises; a particular conclusion requires a particular premise.

There are exactly 256 moods-figures; only 24 are traditionally valid (15 unconditionally valid plus 9 weakened by existential import).

Venn-diagram test: shade what premises assert to be empty, place an X where premises assert existence, and check whether the conclusion's content is already depicted.

Five basic connectives: negation (not), conjunction (and), disjunction (or), conditional (if...then), biconditional (if and only if).

Negation reverses truth value; conjunction is true only when both conjuncts are true.

Inclusive disjunction is false only when both disjuncts are false; exclusive disjunction is true when exactly one disjunct is true.

A conditional is false only in the case true antecedent with false consequent; it is true whenever the antecedent is false (vacuous truth).

A biconditional is true when both components have the same truth value.

A truth table for n distinct statement letters has 2^n rows; convention orders rows T,T,F,F and T,F,T,F.

Tautology: true on every row; contradiction: false on every row; contingent: true on some and false on some.

Two statements are logically equivalent when their columns match on every row; De Morgan's laws: ~(p & q) = ~p v ~q and ~(p v q) = ~p & ~q.

Material implication: p implies q is equivalent to ~p v q and to ~q implies ~p (contraposition).

Natural deduction uses inference rules to derive the conclusion line by line from the premises.

Modus ponens: p implies q, p, therefore q. Modus tollens: p implies q, ~q, therefore ~p.

Hypothetical syllogism: p implies q, q implies r, therefore p implies r. Disjunctive syllogism: p v q, ~p, therefore q.

Simplification: p & q therefore p. Conjunction: p, q therefore p & q. Addition: p therefore p v q.

Conditional proof: assume the antecedent, derive the consequent, then discharge the assumption to obtain the conditional.

Indirect proof (reductio ad absurdum): assume the negation of what is to be proved and derive a contradiction, then conclude the original.

Replacement rules include De Morgan, commutation, association, distribution, exportation, and double negation; they work in both directions.

Formal proof of invalidity: construct one truth-value assignment making all premises true and the conclusion false.

Proof by truth table is decision procedure; natural deduction is a derivation procedure and may require strategy rather than mechanics.

Induction generalizes from observed instances; its conclusion exceeds the premises, so it is never deductively valid.

Enumerative induction is strong when the sample is large, randomly selected, and representative of the population.

Analogical argument: because A and B share properties, and A has F, B probably has F; strength depends on number and relevance of shared properties.

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

Hypothetico-deductive method: formulate hypothesis, deduce observable consequences, test, then confirm or falsify.

A hypothesis is falsifiable when some possible observation would count against it; unfalsifiable claims fall outside empirical science.

Correlation does not establish causation; post hoc ergo propter hoc is the error of inferring cause from temporal succession.

Scientific explanation uses covering-law (deductive-nomological) form: initial conditions plus general laws entail the explanandum.

Inductive strength is a matter of degree; a strong inductive argument can still have a false conclusion.

Formal fallacies occur in structure, e.g., affirming the consequent (p implies q, q, therefore p) and denying the antecedent (p implies q, ~p, therefore ~q).

Fallacies of relevance: ad hominem, ad populum, ad baculum, ad misericordiam, ad verecundiam (appeal to unqualified authority), ignoratio elenchi.

Petitio principii (begging the question) assumes the conclusion among the premises; circulus in probando is circular reasoning.

Equivocation shifts a word's meaning; amphiboly exploits ambiguous grammar; composition and division misapply part-whole properties.

Complex question presupposes an answer, e.g., 'Have you stopped cheating?'

Straw man misrepresents an opponent's position; red herring diverts attention from the issue.

Hasty generalization draws a broad conclusion from too small a sample; false cause and slippery slope project unwarranted consequences.

Critical thinking requires identifying the conclusion, mapping premises, checking validity, and then checking premise truth.

Argumentum ad ignorantiam treats absence of proof as proof of the opposite claim.

Symbols: ~ (negation), & or . (conjunction), v (disjunction), implies arrow (conditional), triple bar (biconditional).

Well-formed formulas are built recursively: statement letters are WFFs, and connectives applied to WFFs yield WFFs.

Scope of a connective is the subformula it governs; the main connective determines the truth value of the whole formula.

Precedence convention: ~ binds tightest, then &, then v, then arrow, then triple bar; parentheses override precedence.

A substitution instance replaces statement letters uniformly with statements or formulas.

A formula is valid (a tautology) when true under every interpretation; it is consistent when true under at least one.

Two formulas are logically equivalent when their biconditional is a tautology.

Quantifier notation: (x) or universal quantifier for 'all', existential quantifier for 'some'; they are interdefinable via negation.

Truth-functional connectives are those whose output truth value depends only on the truth values of their components.