Assam HS Class 11-12 Logic and Philosophy cheat sheet

Logic is the normative science of the formal principles of valid inference; it studies the form of arguments, not their subject matter.

An argument is a group of propositions in which some (premises) are claimed to support another (conclusion); a single proposition is never an argument.

Deductive argument: the conclusion is claimed to follow necessarily; it is valid or invalid, never 'probable'.

Inductive argument: the conclusion is claimed to follow probably; it is strong or weak, cogent or uncogent, never valid.

Validity concerns form alone: an argument can be valid with false premises and a false conclusion (e.g. All A are B; All B are C; therefore All A are C).

Soundness = validity plus all true premises; soundness guarantees a true conclusion in a deductive argument.

Truth applies to individual propositions; validity applies to arguments - the two are independent properties.

Logic is a normative (prescriptive) discipline; psychology of reasoning is descriptive and does not define correctness.

Three basic functions of language in logic: informative (true/false), expressive (emotive), and directive (causing action); ceremonial and performative uses are also noted.

A term is a word or group of words that can serve as subject or predicate of a proposition; terms have both intension (connotation) and extension (denotation).

Intension = set of attributes a term implies; extension = class of objects the term denotes.

Inverse relation: increasing intension generally decreases extension (e.g. 'animal' → 'mammal' → 'dog').

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

Definitions: stipulative (assigns new meaning), lexical (reports standard usage), precising (reduces vagueness), theoretical (embeds a theory), persuasive (influences attitude).

Extensional definition techniques: enumeration, ostensive definition, and definition by subclass.

Intensional definition techniques: synonym, operational, genus-and-difference (definition by genus and specific difference).

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

Categorical proposition: subject and predicate linked by a copula ('is'/'are'), with explicit or implicit quantity and quality.

Four standard forms: A (All S are P), E (No S are P), I (Some S are P), O (Some S are not P).

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

Quality: affirmative (A, I) or negative (E, O). Quantity: universal (A, E) or particular (I, O).

Traditional square of opposition: contradictories (A–O, E–I) always differ in truth value; contraries (A–E) cannot both be true; subcontraries (I–O) cannot both be false; subalterns (A–I, E–O) - truth flows downward, falsity upward.

Existential import: in modern (Boolean) interpretation universal propositions have no existential import, so A and E are not contraries and subalternation fails.

Singular propositions ('Socrates is mortal') are treated as universal (A or E) in traditional syllogistic.

Standard-form categorical proposition requires: quantifier + subject term + copula + predicate term, with the subject class non-empty in traditional logic.

Immediate inference: conclusion drawn from a single premise - conversion, obversion, contraposition.

Conversion: swap subject and predicate; valid for E and I unchanged, A converts only by limitation (to I), O cannot be converted.

Obversion: change quality (affirmative↔negative) and replace predicate with its complement; valid for all four forms (A, E, I, O).

Contraposition: obvert, convert, obvert again; valid for A and O, invalid for E and I (E and I yield only by limitation or not at all).

Complement of a class = all things not in that class; complement of 'P' is 'non-P'.

Distribution rule check: a term distributed in the conclusion must be distributed in the premise for the inference to be valid.

A proposition and its obverse are logically equivalent; conversion of E and I is also equivalence.

Categorical syllogism: three categorical propositions (two premises, one conclusion) containing exactly three terms, each appearing twice.

Major term = predicate of conclusion; minor term = subject of conclusion; middle term appears in both premises and never in the conclusion.

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).

Mood = the three letters of the propositions in order (major premise, minor premise, conclusion), e.g. AAA-1 (Barbara).

Rule 1: the middle term must be distributed at least once (fallacy of undistributed middle).

Rule 2: no term may be distributed in the conclusion unless distributed in the premise (illicit major / illicit minor).

Rule 3: no conclusion follows from two negative premises (fallacy of exclusive premises); a negative premise requires a negative conclusion and vice versa.

Rule 4: from two particular premises nothing follows; if one premise is particular, the conclusion must be particular.

Formal fallacies arise from invalid structure (e.g. affirming the consequent); informal fallacies of relevance include ad hominem, ad baculum, ad populum, ignoratio elenchi; of ambiguity include equivocation and amphiboly.

Logical connectives and symbols: negation ~, conjunction ·, disjunction ∨, conditional ⊃, biconditional ≡.

Truth conditions: ~p is true when p is false; p·q true only when both true; p∨q false only when both false; p⊃q false only when p true and q false; p≡q true when truth values match.

A truth table for n distinct simple propositions has 2^n rows; list values in halves (T T F F / T F T F).

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

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

Modus ponens: p⊃q, p, therefore q. Modus tollens: p⊃q, ~q, therefore ~p. Disjunctive syllogism: p∨q, ~p, therefore q.

Affirming the consequent (p⊃q, q, therefore p) and denying the antecedent (p⊃q, ~p, therefore ~q) are invalid forms.

An argument is valid iff the conditional formed from the conjunction of premises as antecedent and conclusion as consequent is a tautology.

Induction moves from particular observations to a general conclusion; the conclusion goes beyond the premises, so it is probable, not certain.

Types: enumeration by simple induction, induction by analogy, causal inference (Mill's methods), and statistical/generalization reasoning.

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

Method of agreement: find the one circumstance common to all instances where the effect occurs.

Method of difference: compare two instances differing in one circumstance and in the effect; that circumstance is the cause.

Analogy argues that because two things share known similarities, they share a further similarity; strength depends on number of similarities, relevance, and number of disanalogies.

Inductive strength is graded: a strong inductive argument makes the conclusion highly probable; cogency = strength + true premises.

Scientific method stages: observation, hypothesis, deduction of consequences, experimental testing, and confirmation or falsification.

Philosophy: systematic, critical inquiry into fundamental questions about reality, knowledge, value, reason and mind; from Greek 'philosophia', love of wisdom.

Branches: metaphysics (reality), epistemology (knowledge), logic (valid reasoning), ethics (morality), aesthetics (beauty/art), political philosophy.

Metaphysics asks what exists: ontology (being), cosmology (universe), rational psychology (soul/self), theology (God).

Monism holds one ultimate substance; dualism (e.g. Descartes: mind and body) holds two; pluralism holds many.

Epistemology: knowledge is traditionally justified true belief (JTB); Gettier cases challenge its sufficiency.

Rationalism (Descartes, Spinoza, Leibniz): reason is the primary source of knowledge; empiricism (Locke, Berkeley, Hume): experience is.

Kant's synthesis: knowledge requires both sensory content and a priori categories of the understanding; 'concepts without percepts are empty'.

Scepticism (Hume, Pyrrho) questions certainty; idealism holds reality is mind-dependent; realism holds it is mind-independent.

Classification: āstika (orthodox) schools accept the authority of the Vedas - Nyāya, Vaiśeṣika, Sāṃkhya, Yoga, Mīmāṃsā, Vedānta; nāstika (heterodox) - Cārvāka, Jainism, Buddhism.

Cārvāka (Lokāyata): materialism, perception is the only valid means of knowledge, no soul, no afterlife, no karma.

Jainism: anekāntavāda (many-sidedness), syādvāda (seven-fold predication), and ahiṃsā; karma is material and binds the soul (jīva).

Buddhism: Four Noble Truths, pratītyasamutpāda (dependent origination), anātman (no permanent self), and the momentariness of all things.

Nyāya: logic and epistemology; four pramāṇas - perception (pratyakṣa), inference (anumāna), comparison (upamāna), testimony (śabda); five-membered syllogism (pratijñā, hetu, udāharaṇa, upanaya, nigamana).

Vaiśeṣika: atomistic pluralism with six (later seven) padārthas, including dravya, guṇa, karma, sāmānya, viśeṣa, samavāya.

Sāṃkhya: dualism of puruṣa (consciousness) and prakṛti (matter/nature) with three guṇas (sattva, rajas, tamas); Yoga adds disciplined practice and Īśvara.

Advaita Vedānta (Śaṅkara): Brahman alone is real, the world is māyā, and the self (ātman) is identical with Brahman; Viśiṣṭādvaita (Rāmānuja) admits qualified unity.

Logic's core question is not what we think but what follows from what - the relation of implication between propositions.

Propositional logic treats whole propositions as units; predicate (categorical) logic analyses subject–predicate structure.

An argument form is valid if every substitution instance with true premises has a true conclusion - no counterexample exists.

A counterexample to validity is a possible argument of the same form with true premises and a false conclusion.

Cognitive (informative) meaning carries truth value; emotive meaning expresses feeling and does not.

A term's extension may be empty (e.g. 'unicorn'), yet the term still has intension.

Genus-and-difference definition: place the term in a wider class (genus) and state the feature distinguishing it (difference).

Rules for good definition: state essential attributes, avoid circularity, avoid overly broad or narrow wording, avoid ambiguous or figurative language.

Compound propositions are built from simple ones by truth-functional connectives; their truth value depends on components.

Modal propositions qualify assertion: necessary, possible, impossible, contingent.

Categorical propositions can be reduced to standard form by supplying the quantifier and rewriting the predicate as a class term.

Venn diagrams: A shades S outside P; E shades the overlap; I places an X in the overlap; O places an X in S outside P.

Distribution mnemonic: 'All S are P' - subject universal, predicate affirmative; 'No S are P' - both universal and negative, so both distributed.

Obversion of 'All S are P' = 'No S are non-P'; of 'Some S are P' = 'Some S are not non-P'.

Contraposition of 'All S are P' = 'All non-P are non-S'; of 'Some S are not P' = 'Some non-P are not non-S'.

Conversion of 'All S are P' by limitation gives 'Some P are S', which is valid only if S exists (traditional logic assumes existence).

Valid moods by figure: Fig 1 - AAA, EAE, AII, EIO (Barbara, Celarent, Darii, Ferio).

Fig 2 valid moods: EAE, AEE, EIO, AOO (Cesare, Camestres, Festino, Baroco).

Fig 3 valid moods: AAI, IAI, AII, EAO, OAO, EIO; Fig 4: AAI, AEE, IAI, EAO, EIO.

Fallacy of four terms (quaternio terminorum) occurs when the middle term shifts meaning - usually equivocation.

Informal fallacies of presumption: begging the question (petitio principii), complex question, false dichotomy, accident, hasty generalization.

Material implication: p⊃q is equivalent to ~p∨q and to ~(p·~q).

Commutative laws: p·q ≡ q·p and p∨q ≡ q∨p; distributive laws: p·(q∨r) ≡ (p·q)∨(p·r).

Rules of inference: simplification (p·q → p), addition (p → p∨q), conjunction (p, q → p·q), hypothetical syllogism (p⊃q, q⊃r → p⊃r).

Rules of replacement: exportation, absorption, tautology (p ≡ p∨p ≡ p·p), and double negation (~~p ≡ p).

Consistency check: a truth table with no row of all-true premises and false conclusion certifies validity.

Method of residues: subtract from the effect what is known to follow from known causes; the remainder is the effect of the remaining antecedent.

Method of concomitant variation: when one circumstance varies as the effect varies, they are causally connected.

Hasty generalization draws a general conclusion from too few or unrepresentative cases.

Argument from authority is inductive and depends on the reliability of the source in the relevant field.

Hypothetico-deductive method: formulate a hypothesis, deduce testable predictions, and retain or reject it on evidence.

A priori knowledge is independent of experience (e.g. mathematics); a posteriori knowledge depends on experience.

Analytic propositions are true by meaning (e.g. 'All bachelors are unmarried'); synthetic ones add information.

Causation (Hume): we observe constant conjunction, not necessary connection; the idea of cause is a habit of the mind.

Free will debate: determinism holds every event is caused; libertarianism and compatibilism offer rival accounts of responsibility.

Mīmāṃsā (Jaimini): emphasises Vedic ritual and dharma; Kumārila and Prabhākara differ on the theory of error and validity of knowledge.

Nyāya's theory of error is anyathākhyāti (misplacement); Yogācāra Buddhism holds vijñānavāda (consciousness-only).

Advaita's theory of error is anirvacanīyakhyāti - the object is indefinable as real or unreal.

Pramāṇa count varies: Cārvāka accepts only perception; Buddhism accepts perception and inference; Nyāya accepts four.

Vedānta schools: Advaita (Śaṅkara), Viśiṣṭādvaita (Rāmānuja), Dvaita (Madhva) - differing on the relation of self and Brahman.

Identify premise and conclusion markers: 'since', 'because', 'for' introduce premises; 'therefore', 'hence', 'so' introduce conclusions.

Distinguish explanations from arguments: an explanation presupposes the fact it accounts for; an argument seeks to establish it.

An enthymeme is an argument with an unstated premise or conclusion; supply the missing element before testing validity.

Logic's scope includes definition, informal fallacy analysis, formal deduction, and the methodology of science.

Terms can be singular (a named individual) or general (a class); both can function as subject or predicate.

Concrete terms denote objects; abstract terms denote attributes (e.g. 'whiteness').

Collective term refers to a group as a whole ('team'); distributive use refers to members individually.

Ambiguity limits logical analysis; precision in terminology is a precondition of valid inference.

To test form: strip content, keep quantifier and copula - 'All S are P' reveals an A proposition.

Exclamations and imperatives lack truth value and so fall outside propositional logic.

The square of opposition is a diagram of the four standard forms and their truth relations.

Contradictory pairs (A–O, E–I) cannot both be true and cannot both be false.

To find the obverse: keep subject, change quality, and complement the predicate.

To find the contrapositive: obvert, convert, then obvert again - check the form supports it.

Conversion is not generally valid for A propositions unless limited to I.

A term is distributed if the proposition says something about every member of the class it denotes.

Identify major and minor premises by the terms of the conclusion, not by order of appearance.

Test validity by the five rules; a single rule violation suffices to declare the syllogism invalid.

A syllogism with a particular conclusion from universal premises is valid (e.g. AAI-3), but a universal conclusion from a particular premise is not.

Fallacies of ambiguity arise from shifting meaning; fallacies of relevance arise from irrelevant appeals; fallacies of presumption hide the premise.

Translate 'unless' as 'if not': 'p unless q' = ~q⊃p = p∨q.

'Only if' marks a necessary condition: 'p only if q' = p⊃q.

'If and only if' translates to the biconditional ≡.

Test equivalence by comparing truth-table columns row by row.

Inductive arguments are assessed for strength, not validity; cogent arguments are strong with true premises.

Analogy strength increases with relevant similarities and decreases with relevant dissimilarities.

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

Mill's methods are eliminative: they seek the necessary and sufficient circumstance among observed cases.

The problem of induction (Hume): past regularities do not logically guarantee future ones.

The mind–body problem: interactionism, epiphenomenalism, and identity theory are competing positions.

Theories of truth: correspondence, coherence, and pragmatic.

Sources of knowledge (pramāṇa in Indian terms): perception, reason, testimony, and intuition.

Nyāya five-membered inference: pratijñā (thesis), hetu (reason), udāharaṇa (example), upanaya (application), nigamana (conclusion).

Buddhist logic (Dignāga, Dharmakīrti) accepts two pramāṇas and develops the theory of apoha (exclusion) for meaning.

Jain syādvāda: every judgment is qualified by 'syāt' (in some respect).

Cārvāka rejects inference as a pramāṇa because universal propositions cannot be established from perception alone.

Formal logic abstracts from content; material logic concerns the truth of premises.

A valid argument with false premises may still have a true conclusion - validity does not guarantee truth.

An invalid argument may have all true premises and a true conclusion - truth does not guarantee validity.

Logic is sometimes called the science of the laws of thought, but its subject is inference, not psychology.

Definitions by genus and difference are the most rigorous intensional definitions.

Persuasive definitions smuggle evaluation into description and are common in rhetoric.

Operational definitions specify a test procedure (e.g. 'soluble' = dissolves in water).

Vagueness and ambiguity are different defects: vagueness is borderline application, ambiguity is multiple meaning.

The copula 'is' may express class inclusion ('all S are P') or identity ('Scott is the author of Waverley').

A proposition's truth value is independent of whether anyone knows or asserts it.

Universal affirmative A: subject distributed; predicate undistributed.

Particular negative O: subject undistributed; predicate distributed.

Obversion preserves truth value for all four forms; conversion and contraposition do not.

The complement of a class is relative to a universe of discourse.

Contraposition of an E proposition is invalid; of an I proposition, invalid.

Immediate inference has one premise; mediate inference (syllogism) has two or more.

A syllogism with two affirmative premises cannot have a negative conclusion.

A syllogism with two universal premises and a particular conclusion is valid only in Figures 3 and 4 (AAI, EAO).

The fallacy of illicit process occurs when a term is distributed in the conclusion but not in the premise.

The fallacy of the undistributed middle is the most common formal fallacy in categorical reasoning.

p⊃q is not equivalent to q⊃p (converse); the converse is a fallacy when used as inference.

p⊃q is equivalent to its contrapositive ~q⊃~p.

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

The number of rows in a truth table doubles with each additional simple proposition.

Inductive generalization: from a sample to a population; reliability depends on sample size and representativeness.

A strong inductive argument can have a false conclusion; strength is about probability, not guarantee.

Mill's method of agreement is limited because it cannot distinguish necessary from merely common circumstances.

Falsification (Popper): a hypothesis is scientific if it is testable and could be refuted by observation.

Descartes' cogito: 'I think, therefore I am' - the one certainty surviving methodical doubt.

Locke: the mind is a tabula rasa; all ideas derive from sensation and reflection.

Hume's fork: relations of ideas (a priori) versus matters of fact (a posteriori).

Kant's Copernican revolution: objects conform to our forms of intuition and categories, not vice versa.

Sāṃkhya's three guṇas: sattva (lightness/clarity), rajas (activity), tamas (inertia).

Yoga's eight limbs (aṣṭāṅga): yama, niyama, āsana, prāṇāyāma, pratyāhāra, dhāraṇā, dhyāna, samādhi.

Buddha's Four Noble Truths: suffering, its cause, its cessation, and the path.

Advaita's māyā explains the appearance of plurality without denying Brahman's unity.