UPMSP Intermediate Logic cheat sheet

Logic is the normative science of valid inference: it studies the form, not the psychological content, of reasoning.

Deductive validity: an argument is valid iff it is impossible for all premises true and conclusion false; validity concerns form, soundness adds actually true premises.

Inductive strength is a matter of degree: premises make the conclusion probable but never guarantee it; strong arguments can have true premises and a false conclusion.

Formal logic abstracts to logical form (schematic letters for terms or propositions); informal logic studies natural-language argument, context, and fallacy.

Normative vs descriptive: logic prescribes how one ought to infer; psychology describes how people actually reason, so empirical findings do not refute logical norms.

Truth and validity are independent: a valid argument may have false premises; an invalid argument may have all true components.

Scope divisions: traditional (term/syllogistic) logic, modern propositional and predicate calculus, inductive and scientific method, and non-Western traditions such as Nyaya.

Argument vs explanation: an argument offers reasons to establish a claim; an explanation presupposes the claim and shows why it holds.

A term is a word or phrase that can serve as subject or predicate of a proposition; categorical terms denote classes (distribution of a term = reference to all members).

Connotation (intension) = attributes a term implies; denotation (extension) = objects the term applies to; increasing intension normally decreases extension.

Univocal term: same meaning in all occurrences; equivocal: same word, unrelated meanings; analogous: partly same, partly different meanings.

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

Extensional definitions: by enumeration (denotation) or by subclass (genus et differentia listing kinds); intensional: synonymous, operational, genus-and-difference.

Genus and difference: define by placing the term in a wider class (genus) and stating the attribute distinguishing it (differentia).

Rules for good definition: state essential attributes, avoid circularity, avoid obscurity, avoid figurative or emotive language, and neither too broad nor too narrow.

Division differs from definition: it splits a genus into species; rules require a single fundamentum, exhaustive and mutually exclusive members.

Four forms: A (All S are P), E (No S are P), I (Some S are P), O (Some S are not P); quantity universal/particular, quality affirmative/negative.

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

Square of opposition: A-E contraries (not both true); I-O subcontraries (not both false); A-O and E-I contradictories (opposite truth values); A-I and E-O subalterns (truth flows down, falsity flows up).

Existential import: traditional (Aristotelian) interpretation assumes S exists, so subalternation holds; Boolean (modern) interpretation denies it, so universal propositions have no existential import.

Conversion (swap S and P): valid for E and I; for A use per accidens (convert to I) only with existential import; O cannot be converted.

Obversion (change quality, negate predicate): valid for all four forms - A to E, E to A, I to O, O to I.

Contraposition (obvert the converse of the obverse): valid for A and O; E yields only a weakened result; I cannot be contraposed.

Venn diagrams for categoricals: shade for universals (empty regions), place an asterisk for particulars; a two-circle test checks validity of immediate inferences.

Syllogism: three categorical propositions with three terms - major (P, predicate of conclusion), minor (S), middle (M, appears in premises only).

Mood = letter pattern of the three propositions (e.g., AAA, EIO); figure = position of the middle term (4 figures); 256 moods-figures, 24 traditionally valid, 15 unconditionally valid.

Rules of validity: middle term distributed at least once; no term distributed in the conclusion unless distributed in a premise; no two negative premises; a negative premise requires a negative conclusion (and conversely).

Fallacies of distributed terms: illicit major (P distributed in conclusion only) and illicit minor (S distributed in conclusion only).

Undistributed middle: the middle term is never distributed, so the premises cannot link S and P.

Fallacies of negative premises: two negative premises (no conclusion follows) and drawing an affirmative conclusion from a negative premise.

Fallacies of four terms (quaternio terminorum): equivocation makes one word function as two terms, yielding four distinct terms.

Informal fallacies: relevance (ad hominem, appeal to force/pity/popularity, straw man), ambiguity (equivocation, amphiboly, accent, composition, division), and presumption (begging the question, false dilemma, hasty generalization).

Connectives: negation ~, conjunction &, disjunction v, conditional ⊃, biconditional ≡; truth tables define each (conditional false only when antecedent true and consequent false).

Truth-functionality: the value of a compound depends only on the values of its components; number of rows in a table = 2^n for n distinct variables.

Tautology: true on every row; contradiction: false on every row; contingent: mixed; two statements are logically equivalent iff their biconditional is a tautology.

Validity test: an argument is valid iff the conditional (conjunction of premises ⊃ conclusion) is a tautology, equivalently no row has all premises true and conclusion false.

Inference rules: modus ponens (p ⊃ q, p ⊢ q), modus tollens (p ⊃ q, ~q ⊢ ~p), hypothetical syllogism, disjunctive syllogism, constructive dilemma, simplification, conjunction, addition.

Replacement rules: De Morgan (~(p&q) ≡ ~p v ~q), double negation, commutation, association, distribution, contraposition, exportation, material implication (p ⊃ q ≡ ~p v q).

Conditional proof: assume the antecedent, derive the consequent, then discharge the assumption to obtain the conditional; indirect proof: assume the negation of the target and derive a contradiction.

Invalid forms to reject: affirming the consequent (p ⊃ q, q ⊢ p) and denying the antecedent (p ⊃ q, ~p ⊢ ~q).

Predicate logic analyzes inside propositions: predicates (Fx), individual constants (a, b), individual variables (x, y), and quantifiers (∀ universal, ∃ existential).

Quantifier negation: ∀xFx ≡ ∃xFx and ∃xFx ≡ ∀xFx; these are the quantified De Morgan laws and drive most proofs.

Universal instantiation: from ∀xFx infer Fa for any constant; existential generalization: from Fa infer ∃xFx; both are one-way unless other conditions hold.

Existential instantiation requires a new (fresh) constant not previously used; universal generalization requires an arbitrary constant with no special assumptions.

Multiple quantification: order matters - ∀x∃yLxy (everyone loves someone) differs from ∃y∀xLxy (someone is loved by everyone).

Categorical translation: All S are P = ∀x(Sx ⊃ Px); No S are P = ∀x(Sx ⊃ ~Px); Some S are P = ∃x(Sx & Px); Some S are not P = ∃x(Sx & ~Px).

Common errors: translating 'All S are P' as ∀x(Sx & Px) or 'Some S are P' as ∃x(Sx ⊃ Px) - the first wrongly asserts existence, the second is trivially true.

Identity and relations: a=b expresses identity; relational predicates (Lxy, Gxyz) permit formalizing transitive, symmetric, and reflexive properties.

Induction types: enumeration (particular to general), analogy (similar cases), statistical syllogism, causal inference (Mill's methods), and hypothetical-deductive reasoning.

Mill's methods: agreement (only common factor), difference (factor present with effect, absent without), joint method, concomitant variation, and residues.

Necessary condition: cannot occur without it; sufficient condition: guarantees the effect; confusing the two is a standard causal error.

Hypothetico-deductive method: formulate hypothesis, derive testable predictions, test, then confirm, disconfirm, or revise; a single failed prediction can falsify a universal hypothesis.

Confirmation is comparative and gradual; ad hoc rescue of a failing hypothesis weakens it, and auxiliary assumptions complicate disconfirmation (Duhem-Quine problem).

Inductive generalization strength depends on sample size, representativeness, and randomness; hasty generalization and biased sample are the corresponding fallacies.

Analogical argument strength depends on relevance and number of shared properties and on disanalogies; it is never deductively conclusive.

Correlation is not causation: confounds, reverse causation, and coincidence must be excluded before inferring a causal link.

Nyaya treats logic and epistemology together; its core is pramana (means of valid knowledge) and the five-membered inference (pancavayava).

Five members of Nyaya inference: pratijna (thesis), hetu (reason), udaharana (example), upanaya (application), and nigamana (conclusion).

Vyapti (invariable concomitance) between hetu and sadhya is the ground of inference; it is established by observation and causal analysis, not by mere enumeration.

Inference types: purvavat (from cause to effect), sesavat (from effect to cause), and samanyatodrsta (from general observation); anumana is knowledge derived from a hetu.

Fallacies: hetvabhasa (reason that only appears valid) - savyabhichara (deviating), viruddha (contradictory), prakarana-sama (irrelevant), sadhya-sama (unproved), and asiddha (unestablished).

Perception (pratyaksa) is direct and non-erroneous; sabda (testimony) is valid when it comes from a reliable source; upamana is knowledge by analogy or comparison.

Nyaya-Vaisesika categories (padarthas) number six or seven, including dravya (substance), guna (quality), karma (action), samanya (universal), and visesa (particularity).

Buddhist logic (Dignaga, Dharmakirti) uses trairupya - the reason must be present in the subject, in similar cases, and absent in dissimilar cases; Jain logic allows syadvada (conditional predication).