Karnataka II PUC Logic (KSEAB) cheat sheet

Logic is the normative science of the formal principles of valid inference; it studies the form of reasoning, not the psychological process of thinking.

An argument = premises (evidence) + conclusion (claim). Only declarative sentences that can be true or false can serve as premises or conclusions.

Deductive validity: it is impossible for all premises to be true and the conclusion false. Validity concerns form; truth concerns content.

Sound argument = valid form + all premises actually true. A sound argument always has a true conclusion; a valid argument may have false premises.

Inductive strength is a matter of degree: premises make the conclusion probable, never certain. Strong induction can still have a false conclusion.

Logic is a normative (prescriptive) discipline; psychology of reasoning is descriptive. Confusing the two is the traditional 'psychologism' error.

Formal logic abstracts content and studies patterns (e.g., All S is P); informal logic studies arguments in natural language, including context and rhetoric.

Logic is distinguished from epistemology (theory of knowledge) and from rhetoric (persuasion): it evaluates inferential correctness, not belief formation or audience effect.

A term is a word or group of words that can be the subject or predicate of a proposition; it stands for a class or an attribute.

Categorematic terms can stand alone as subject/predicate (man, red); syncategorematic terms need other words to function (all, some, not, if, and).

Extension (denotation) = the class of objects the term applies to; intension (connotation) = the set of attributes that determines membership.

Inverse relation for ordinary terms: increasing intension (more attributes) decreases extension (fewer objects), and vice versa.

Univocal term: same meaning in every occurrence; equivocal term: same spelling/sound, different meanings (bank, bat); analogous term: partly same, partly different meaning.

Singular term names one object (Karnataka); general/class term applies to many (student); collective term applies to a group as a unit (team, army).

Contradictories have no middle: term and its negation (living / non-living) exhaust the universe of discourse. Contraries allow a middle (hot / cold).

Definition types: stipulative (assigns meaning), lexical (reports usage), precising (reduces vagueness), theoretical (embeds in a theory), persuasive (carries attitude).

A proposition is what a declarative sentence asserts; it has a truth value. Questions, commands, exclamations and wishes are not propositions.

Simple proposition: one subject–predicate claim. Compound proposition: built from simpler ones using truth-functional connectives.

By quality: affirmative or negative. By quantity: universal, particular, singular. By relation: categorical or conditional. By modality: assertoric, problematic, apodictic.

Truth-functional compound: its truth value is fully determined by the truth values of its components (negation, conjunction, disjunction, conditional, biconditional).

Non-truth-functional compounds include causal and intentional contexts ('because', 'believes that'), where substituting an equal-truth component can change the whole's truth value.

Categorical proposition asserts or denies a predicate of a subject; conditional proposition asserts a relation of dependence between antecedent and consequent.

Two propositions are contradictories if exactly one must be true; contraries cannot both be true but may both be false; subcontraries cannot both be false.

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

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

Traditional square: A–E contrary (not both true), I–O subcontrary (not both false), A–O and E–I contradictory (opposite truth values), A–I and E–O subaltern (A true ⇒ I true; I false ⇒ A false).

Existential import: modern/Boolean interpretation gives A and E no existential import, so A does not imply I; the traditional square assumes the subject class is non-empty.

Conversion (swap S and P): valid for E and I unchanged; A converts only per accidens (to I); O cannot be converted.

Obversion (change quality, replace predicate with its complement): valid for all four forms - A→E, E→A, I→O, O→I.

Contraposition (obvert, convert, obvert): valid for A and O; E yields only a per accidens form; I has no valid contraposition.

Conversion, obversion and contraposition are immediate inferences: conclusion follows from one premise alone, with no middle term.

Categorical syllogism: three categorical propositions, three terms (major, minor, middle), each term occurring in exactly two propositions.

Major premise contains P (predicate of conclusion); minor premise contains S; middle term M appears in both premises and never in the conclusion.

Figure is fixed by the position of M: Fig 1 (M–P, S–M), Fig 2 (P–M, S–M), Fig 3 (M–P, M–S), Fig 4 (P–M, M–S). Mood is the letter sequence of the three propositions.

Rules of validity: (1) middle term distributed at least once; (2) no term distributed in conclusion unless distributed in premise; (3) at least one premise affirmative; (4) a negative premise requires a negative conclusion and vice versa; (5) two particular premises yield no conclusion; (6) two negative premises yield no conclusion.

Fallacy of undistributed middle: 'All dogs are mammals; all cats are mammals; therefore all cats are dogs.'

Fallacy of illicit major/minor: conclusion distributes a term that was undistributed in its premise.

Formal fallacies: four terms, ambiguous middle, illicit process, undistributed middle. Informal fallacies: relevance (ad hominem, appeal to authority/pity/force, straw man), ambiguity (equivocation, amphiboly, accent, composition, division), presumption (begging the question, false cause, false dilemma, hasty generalization).

Venn-diagram test: shade what premises assert to be empty, place an X where premises assert existence; the syllogism is valid iff the conclusion's diagram is already contained in the premise diagram.

Connectives: ~ (not), · or ∧ (and), ∨ (or), ⊃ or → (if…then), ≡ or ↔ (if and only if). Precedence: ~ binds tightest, then ∧, then ∨, then →, then ↔.

Truth tables: negation reverses; conjunction true only when both true; disjunction false only when both false; conditional false only when antecedent true and consequent false; biconditional true when truth values match.

A well-formed formula (WFF) is built from sentence letters by finitely many applications of the connectives; 'p ∧ ∨ q' is not a WFF.

Tautology: true on every row; contradiction: false on every row; contingent: true on some, false on other rows.

Material implication oddities: a false antecedent makes p → q true (vacuous truth); 'p → q' is equivalent to '~p ∨ q'.

Equivalences: De Morgan (~(p∧q) ≡ ~p∨~q; ~(p∨q) ≡ ~p∧~q), double negation, contraposition (p→q ≡ ~q→~p), exportation, distribution, material equivalence (p↔q ≡ (p→q)∧(q→p)).

Two formulas are logically equivalent iff they have identical truth values on every row; p → q is not equivalent to its converse q → p or inverse ~p → ~q.

An argument form is valid iff no row makes all premises true and the conclusion false - i.e., premises → conclusion is a tautology.

Decision procedure for propositional arguments: construct a full truth table; valid iff no row has all premises T and conclusion F; a single such row is a counterexample.

Indirect (reductio) truth table: assume premises true and conclusion false, then derive a contradiction in a single row; contradiction ⇒ valid.

Natural deduction rules of inference: modus ponens (p→q, p ⊢ q), modus tollens (p→q, ~q ⊢ ~p), hypothetical syllogism, disjunctive syllogism, constructive dilemma, simplification, conjunction, addition.

Rules of replacement (equivalence) may be applied to any part of a formula; rules of inference apply only to whole lines.

Conditional proof: to prove p → q, assume p as an additional premise, derive q, then discharge the assumption by CP.

Indirect proof (reductio ad absurdum): assume the negation of the desired conclusion, derive a contradiction (r ∧ ~r), then infer the conclusion.

Invalidity proof by truth-value assignment: assign values making the conclusion false, work backward to force premise values, and if a consistent assignment makes all premises true, the form is invalid.

Shorter truth table / consistency test: a set of formulas is consistent iff some row makes all of them true; an argument is valid iff premises plus negated conclusion are inconsistent.

Induction moves from particular observations to general conclusions; it yields probability, not certainty, so it is evaluated as strong or weak, not valid or invalid.

Enumerative (induction by simple enumeration) generalizes from observed instances; reliability rises with sample size, variety and randomness.

Analogical argument: A and B share properties p, q; A also has r; therefore B has r. Strength depends on relevance of shared properties and number of disanalogies.

Mill's methods: agreement (only common antecedent), difference (presence/absence of one factor), joint method, concomitant variation, residues.

Hypothetico-deductive method: formulate hypothesis → deduce testable consequences → test → confirm, disconfirm or revise. Confirmation is never final; one counter-instance can falsify a universal claim.

Abduction (inference to the best explanation) selects the hypothesis with greatest explanatory power, simplicity and fit with background knowledge.

Fallacies of induction: hasty generalization, false cause (post hoc ergo propter hoc), biased sample, slippery slope, false analogy, cherry-picking.

Science requires testability/falsifiability, reproducibility, and control of variables; correlation alone does not establish causation.

Functions of language (Ogden–Richards style classification): informative ( conveying information), expressive (venting or evoking feeling), directive (causing action), and performative/ceremonial.

Cognitive (descriptive) use of language aims at truth; emotive use aims at expressing or arousing attitude; most real discourse mixes both.

A sentence is a grammatical unit; a proposition is a truth-bearing unit; the same proposition can be expressed by different sentences (e.g., translations).

Use vs mention: 'Bengaluru' names a city; ''Bengaluru'' (in quotes) names the word. Confusing them produces the fallacy of equivocation or use–mention error.

Intensional vs extensional context: inside quotation, belief or modal contexts, substituting co-referential terms may change truth value (referential opacity).

Definitions and their purposes: reportive, stipulative, precising, theoretical, persuasive. Good definitions avoid circularity, obscurity and emotive loading.

Ambiguity types: lexical (equivocation), structural/amphiboly, referential, and vagueness (borderline cases); these are semantic sources of fallacy.

Truth and meaning: a declarative sentence's cognitive meaning includes both its sense (intension) and reference (extension); emotive meaning is separate from truth conditions.