Logic तर्कशास्त्र

Quick summary

A decision procedure is a method that tells us, in a fixed number of steps, whether an argument is valid or whether a statement form is a tautology. In 11th you used the full truth table. This chapter teaches the shorter truth table (STT) method, which tests a statement form for tautology in a single line. It works by reductio ad absurdum: we assume the opposite and look for a contradiction.

Key points

• I.M. Copi defines logic as the study of the methods and principles used to tell correct reasoning from incorrect reasoning.
• Two main jobs of logic: decide whether an argument is valid, and decide whether a statement form is a tautology, contradiction or contingency.
• A good (effective) decision procedure must be reliable, mechanical and finite.
• A full truth table becomes very long: 4 variables = 16 rows, 5 variables = 32 rows (rows = 2ⁿ). So a shorter method is needed.
• STT is also an effective decision procedure, and it can be done in one row.
• STT is based on reductio ad absurdum – show that the opposite of what is to be proved leads to an absurdity. So it is called an indirect method.
• To test for tautology, we assume the form is NOT a tautology: put F under the main connective and mark it with a star (*).
• Then assign values step by step using the basic truth tables, numbering every step.
• Inconsistency is of two kinds: (i) a rule of a basic truth table is broken, (ii) the same variable gets both T and F. Mark it with "x".
• Inconsistency → assumption wrong → the form is a tautology. No inconsistency → it is not a tautology (it is a contradiction or a contingency).
• When the main connective can be false in more than one way (conjunction: 3 ways; equivalence: 2 ways), check every possibility. The form is a tautology only if every possibility gives an inconsistency.

Important terms

• Decision procedure – a method that decides validity of an argument or the nature of a statement form.
• Tautology – a statement form that is true for all truth values of its components.
• Contradiction – a statement form that is false in every case.
• Contingency – a statement form that is true in some cases and false in others.
• Main connective – the connective that joins the largest parts of the expression.
• Reductio ad absurdum – proving something by showing its denial leads to absurdity.
• Inconsistency – a clash in the assigned truth values.

Formulas

Basic truth tables used in STT:

When the main connective is assumed F:

• p ⊃ q is F in one way only: p = T, q = F.
• p ∨ q is F in one way only: p = F, q = F.
• p · q is F in three ways: (T, F), (F, T), (F, F).
• p ≡ q is F in two ways: (T, F), (F, T).

Worked example 1: (p · q) ⊃ p

Step 1: F under ⊃. Step 2: antecedent T, consequent p = F. Step 3: a true conjunction needs p = T and q = T. Now p is T (step 3) and F (step 2) – inconsistency (x under both p's). So the form is a tautology.

Worked example 2: (p ⊃ q) ⊃ (q ⊃ p)

Step 2: first part T, second part F. Step 3: q ⊃ p is F, so q = T, p = F. Step 4: put the same values in the first part: F ⊃ T = T, which agrees with step 2. No inconsistency, so it is not a tautology.

Worked example 3 (three possibilities): (p ∨ ∼p) · (q ⊃ q)

• (i) first conjunct T, second F: q ⊃ q = F needs q = T and q = F → x.
• (ii) first F, second T: p ∨ ∼p = F needs p = F and ∼p = F (so p = T) → x.
• (iii) both F: the same clashes as above → x.

Every possibility is inconsistent, so it is a tautology.

5 important questions

Q1. What are the conditions of an effective decision procedure?

It must be reliable (always gives the correct answer), mechanical (steps followed by rule, no guessing or imagination needed) and finite (it ends after a limited number of steps).

Q2. Why is the shorter truth table method needed?

A full truth table doubles in size with each new variable (2ⁿ rows), so with 4 or 5 variables it is long, tiring and error-prone. STT decides the same question in one row.

Q3. Why is STT called an indirect method?

It does not prove the result directly. It assumes the opposite (that the form is not a tautology) and shows that this assumption leads to an inconsistency. This is the principle of reductio ad absurdum.

Q4. What are the two types of inconsistency in STT?

(1) Violation of a rule of a basic truth table, e.g. a conjunction marked T with a false conjunct. (2) One propositional variable getting both T and F.

Q5. Test by STT whether p ⊃ (q ⊃ p) is a tautology.

Step 1: F under the main ⊃. Step 2: p = T, (q ⊃ p) = F. Step 3: q ⊃ p is F, so q = T and p = F. Now p is T (step 2) and F (step 3) – inconsistency. Hence p ⊃ (q ⊃ p) is a tautology.

Exam tip

Always write the star under the main connective, number every step, and mark each inconsistency with "x". With conjunction or equivalence as the main connective, show all possibilities – marks are lost if you test only one.

Quick summary

Truth tables decide validity mechanically, but they become clumsy with many variables. A deductive proof shows validity by deriving the conclusion from the premises step by step, using elementary valid argument forms. There are three types: the direct deductive proof, the conditional proof (C.P.) and the indirect proof (I.P.). Each uses the 9 rules of inference and the 10 rules of replacement (19 rules in all).

Key points

• Decision procedures (truth table, shorter truth table, truth tree) decide validity. Deductive proof, conditional proof and indirect proof are not decision procedures; they prove validity.
• Direct deductive proof: the conclusion is deduced directly from the premises by a chain of elementary valid argument forms.
• Rules of inference apply only to whole lines. Rules of replacement can be applied to a part of a line too, because they replace one expression by an equivalent one.
• Conditional proof is used when the conclusion is a conditional (p ⊃ q). Take the antecedent of the conclusion as an extra premise (assumption), deduce its consequent, then close with "C.P.".
• If the conclusion has more than one conditional, the antecedents can be assumed one after another.
• C.P. gives shorter proofs and can prove some arguments the 19 rules alone cannot.
• Indirect proof is based on reductio ad absurdum. Assume the negation of the conclusion (written "I.P."), and derive a contradiction of the form p · ∼p. That contradiction proves the argument valid.
• I.P. is preferred when the conclusion is not a conditional statement.
• Every line in a proof must carry a justification: line numbers used + abbreviation of the rule.

Important terms

• Formal proof of validity – a sequence of statements, each a premise or derived from earlier lines by a rule, ending in the conclusion.
• Rule of inference – an elementary valid argument form (e.g. M.P.).
• Rule of replacement – a pair of logically equivalent forms that may replace each other anywhere.
• Assumption – the extra premise added in C.P. or I.P.
• Contradiction – a statement of the form p · ∼p; it can never be true.

Formulas

Nine rules of inference

Ten rules of replacement

• Double Negation (D.N.): ∼∼p ≡ p
• De Morgan's Law (De M.): ∼(p · q) ≡ (∼p ∨ ∼q); ∼(p ∨ q) ≡ (∼p · ∼q)
• Commutation (Comm.): (p · q) ≡ (q · p); (p ∨ q) ≡ (q ∨ p)
• Association (Assoc.): [(p · q) · r] ≡ [p · (q · r)]; [(p ∨ q) ∨ r] ≡ [p ∨ (q ∨ r)]
• Distribution (Dist.): [p · (q ∨ r)] ≡ [(p · q) ∨ (p · r)]; [p ∨ (q · r)] ≡ [(p ∨ q) · (p ∨ r)]
• Transposition (Trans.): (p ⊃ q) ≡ (∼q ⊃ ∼p)
• Material Implication (M.Imp.): (p ⊃ q) ≡ (∼p ∨ q)
• Material Equivalence (M.Equi.): (p ≡ q) ≡ [(p ⊃ q) · (q ⊃ p)]; (p ≡ q) ≡ [(p · q) ∨ (∼p · ∼q)]
• Exportation (Exp.): [(p · q) ⊃ r] ≡ [p ⊃ (q ⊃ r)]
• Tautology (Taut.): p ≡ (p · p); p ≡ (p ∨ p)

Worked conditional proof

1. ∼P ∨ Q
2. Q ⊃ R / ∴ P ⊃ R
3. P Assumption
4. ∼∼P 3, D.N.
5. Q 1, 4, D.S.
6. R 2, 5, M.P.
7. P ⊃ R 3–6, C.P.

Worked indirect proof

1. P ⊃ Q
2. P ⊃ ∼Q / ∴ ∼P
3. ∼∼P I.P.
4. P 3, D.N.
5. Q 1, 4, M.P.
6. ∼Q 2, 4, M.P.
7. Q · ∼Q 5, 6, Conj.

Line 7 is a contradiction, so the assumption ∼∼P is false and ∼P follows. The argument is valid.

5 important questions

Q1. Name the three types of deductive proof.

(1) Direct deductive proof, (2) conditional proof, (3) indirect proof.

Q2. When is the method of conditional proof used?

When the conclusion of the argument is a conditional (implicative) statement. The antecedent of the conclusion is assumed and its consequent is derived.

Q3. What is the difference between rules of inference and rules of replacement?

Rules of inference are valid argument forms and can be applied only to whole lines. Rules of replacement are logical equivalences; one side may replace the other in a whole line or in any part of a line.

Q4. Construct a formal proof: 1. p ⊃ q 2. q ⊃ r 3. p / ∴ r

1. q 1, 3, M.P.
2. r 2, 4, M.P.

Q5. Construct an indirect proof: 1. A ∨ B 2. ∼B / ∴ A

1. ∼A I.P.
2. B 1, 3, D.S.
3. B · ∼B 4, 2, Conj.

A contradiction is reached, so the argument is valid.

Exam tip

Write the rule abbreviation and the line numbers on every line – a correct step without a justification loses marks. In C.P. the last line must show the range of lines (e.g. 3–6, C.P.).

Quick summary

In propositional logic a simple proposition is taken as one unit, so arguments like "All men are mortal, Socrates is a man, therefore Socrates is mortal" cannot be shown valid there. Predicate logic looks inside a proposition at its subject, predicate and quantity. This chapter covers singular and general propositions, their symbolization, propositional functions, quantifiers, and proving validity by quantificational deduction using the four rules UI, UG, EI and EG.

Key points

• Need: the validity of many arguments depends on the inner structure of simple propositions, which propositional logic ignores.
• Non-compound propositions are of two types: singular (about one individual) and general (about a class).
• Singular proposition "Suraj is wise" = Ws. The predicate letter (A–Z) comes first, then the individual constant (a–w). Negative: "Makarand is not cunning" = ∼Cm.
• Individual variables x, y, z stand for any individual; predicate constants stand for a specific property.
• A propositional function (e.g. Lx – "x is a logician") contains a free variable. It is neither true nor false; it becomes a proposition when the variable is replaced or quantified.
• Two ways of getting propositions from a propositional function: instantiation (gives singular propositions) and quantification / generalization (gives general propositions).
• Universal quantifier (x) = "given any x"; existential quantifier (∃x) = "there is at least one x such that".
• A universal quantification is true only if all its substitution instances are true; an existential quantification is true if at least one is true.
• General propositions: one-class (4 kinds) and two-class (A, E, I, O) – 8 kinds in all.
• In quantificational deduction, UI and EI remove quantifiers; then the 19 rules are applied; UG and EG put quantifiers back.

Important terms

• Singular proposition – states that an individual has or does not have a property.
• General proposition – makes an assertion about all or some members of a class.
• Individual constant – a, b, …, w; stands for the name of an individual.
• Predicate constant – A to Z; stands for a specific property.
• Individual variable – x, y, z; a place-holder for any individual.
• Substitution instance – the singular proposition got by replacing the variable with a constant.
• Quantificational deduction – the method of proving validity of arguments containing singular and general propositions.

Formulas

Symbolizing general propositions

Remember: universal → ⊃ ; existential → · (dot).

Rules of quantification (ν = any individual symbol, y = an arbitrarily selected individual)

• UI: (x) φx ∴ φν – what is true of all is true of each one.
• UG: φy ∴ (x) φx – what is true of any arbitrarily selected individual is true of all.
• EI: (∃x) φx ∴ φν – ν must be a new constant not used earlier in the proof.
• EG: φν ∴ (∃x) φx – what is true of a given individual is true of at least one.
• Tip: when both are needed, apply EI before UI.

Worked proof (UI and UG) – All dogs are animals. All animals are mortal. So all dogs are mortal.

1. (x)(Dx ⊃ Ax)
2. (x)(Ax ⊃ Mx) / ∴ (x)(Dx ⊃ Mx)
3. Dy ⊃ Ay 1, UI
4. Ay ⊃ My 2, UI
5. Dy ⊃ My 3, 4, H.S.
6. (x)(Dx ⊃ Mx) 5, UG

5 important questions

Q1. What is a propositional function?

An expression that contains at least one free individual variable, e.g. "x is a dancer" (Dx). It is neither true nor false, but becomes a true or false proposition when the variable is replaced by a constant or quantified.

Q2. Distinguish between instantiation and quantification.

Instantiation replaces the variable by an individual constant and gives a singular proposition (La). Quantification places a quantifier before the function and gives a general proposition ((x) Lx or (∃x) Lx).

Q3. Symbolize: (a) Some students are not lazy. (b) No snake is a mammal.

(a) (∃x)(Sx · ∼Lx) (b) (x)(Sx ⊃ ∼Mx)

Q4. Prove: All men are mortal. Socrates is a man. ∴ Socrates is mortal.

1. (x)(Hx ⊃ Mx)
2. Hs / ∴ Ms
3. Hs ⊃ Ms 1, UI
4. Ms 3, 2, M.P.

Q5. Prove: 1. (x)(Px ⊃ Dx) 2. (∃x)(Px · Rx) / ∴ (∃x)(Rx · Dx)

1. Pa · Ra 2, EI
2. Pa ⊃ Da 1, UI
3. Pa 3, Simp.
4. Da 4, 5, M.P.
5. Ra · Pa 3, Comm.
6. Ra 7, Simp.
7. Ra · Da 8, 6, Conj.
8. (∃x)(Rx · Dx) 9, EG

Exam tip

Use ⊃ with the universal quantifier and · with the existential quantifier – mixing them is the most common mistake. In proofs, do EI first with a fresh constant.

Quick summary

Traditional logic was founded by the Greek philosopher Aristotle. It studies propositions as made of terms – a subject and a predicate joined by a copula. This chapter covers the traditional classification of propositions (conditional and categorical), the four categorical propositions A, E, I, O, distribution of terms, and the types of inference: immediate inference (square of opposition, conversion, obversion) and mediate inference (three kinds of syllogism).

Key points

• A term is a word or group of words that stands as the subject or predicate of a proposition. A term by itself is neither true nor false; a proposition is.
• A proposition has three elements in the order subject – copula – predicate. In "All apples are red": subject = apples, copula = are, predicate = red.
• Propositions are classified as conditional and categorical.
• Conditional propositions are of two kinds: hypothetical ("If metal is heated, it expands" – antecedent and consequent) and disjunctive ("A line is straight or curved" – alternatives that may be exclusive or inclusive).
• A categorical proposition asserts without any condition. It has quality (affirmative/negative) and quantity (universal/particular). This gives four kinds: A, E, I, O.
• A term is distributed when it refers to the whole class, undistributed when it refers to only a part.
• Inference is inductive (particular to general) or deductive (general to particular). Deductive inference is immediate (from one premise) or mediate (from two or more premises).
• Immediate inference is of two types: opposition of propositions and eduction. Conversion and obversion are the two basic eductions (there are seven eductions in all).
• Mediate inference = syllogism: hypothetical, disjunctive and categorical syllogism.

Important terms

• Copula – the link (is/are, is not/are not) that joins subject and predicate.
• Opposition – the relation between two categorical propositions with the same subject and predicate that differ in quantity, quality or both.
• Conversion – subject and predicate are interchanged (Convertend → Converse).
• Obversion – subject stays the same, quality changes and the predicate is replaced by its complement (Obvertend → Obverse).
• Simple converse – a converse of the same type as the original (E → E, I → I).
• Sub-alternant / sub-alternate – the universal and the particular in a sub-altern pair.

Formulas

Four categorical propositions and distribution

Square of opposition

• Contraries (A–E): cannot both be true, may both be false.
• Contradictories (A–O, E–I): cannot both be true and cannot both be false.
• Sub-contraries (I–O): may both be true, cannot both be false.
• Sub-altern (A–I, E–O): if the universal is true, the particular is true; if the universal is false, the particular is doubtful.

Conversion (rules: quality stays the same; no term distributed in the converse unless distributed in the premise)

Exception: A converts to A when subject and predicate have the same denotation or the predicate is a definition, e.g. "Man is a rational animal" → "A rational animal is man".

Obversion (rules: quality changes; quantity stays; predicate becomes its complement)

5 important questions

Q1. What are the elements of a proposition in traditional logic?

Subject, copula and predicate, in that order. In "No elephants are carnivorous", elephants = subject, are (with "no") = copula, carnivorous = predicate.

Q2. Distinguish between hypothetical and disjunctive propositions.

A hypothetical proposition states a condition and its consequence (If it rains, the ground gets wet). A disjunctive proposition states alternatives (Either Ganesh will sing or dance).

Q3. Why is there no converse of an O proposition?

The converse must stay negative, so it must be E or O. In both, the old subject becomes the predicate of a negative proposition and gets distributed, though it was undistributed in the premise. This breaks the rule of distribution.

Q4. Give the converse and obverse of "No umpires are partial".

Converse (E): No partial persons are umpires. Obverse (A): All umpires are non-partial.

Q5. If "All artists are creative" is true, what are the truth values of E, I and O?

E (No artists are creative) – false (contrary). I (Some artists are creative) – true (sub-altern). O (Some artists are not creative) – false (contradictory).

Exam tip

Before converting, first change the sentence into standard form (e.g. "Every mother is anxious" = A; "Not a single stick is straight" = E). Learn the distribution table by heart – it is used again in syllogism rules.

Quick summary

A categorical syllogism is a mediate deductive argument with exactly three categorical propositions (two premises and a conclusion) containing exactly three terms. The theory was given by Aristotle. This chapter explains the structure of a syllogism, its four figures, the rules of validity with the formal fallacies that break them, and a comparison of the Aristotelian syllogism with the Nyaya (Indian) syllogism.

Key points

• Three terms: major term (P) = predicate of the conclusion; minor term (S) = subject of the conclusion; middle term (M) = appears in both premises but not in the conclusion.
• Major premise contains the major term; minor premise contains the minor term. In standard form the major premise is written first.
• The figure of a syllogism depends on the position of the middle term. Aristotle gave three figures; Galen added the fourth.
• A syllogism that breaks any rule is invalid and commits a formal fallacy (a mistake in the form of the argument).
• The middle term links the major and minor terms, so it must be distributed at least once.
• In Indian logic, inference is called anumana. The Nyaya school distinguishes inference for oneself (swartha) from inference for others (parartha); the five-membered syllogism is parartha.
• In Nyaya: minor term = paksha, major term = sadhya, middle term = hetu / linga. The universal relation between hetu and sadhya is vyapti.

Important terms

• Fallacy of four terms – using four terms; often caused by an ambiguous word (fallacy of equivocation: ambiguous major, middle or minor).
• Undistributed middle – the middle term is not distributed even once.
• Illicit minor / illicit major – the minor/major term is distributed in the conclusion but not in its premise.
• Fallacy of negative (exclusive) premises – a conclusion drawn from two negative premises.
• Udaharan – the example in the Nyaya syllogism ("as in the kitchen").

Formulas

Four figures (major premise first, then minor premise)

Rules of categorical syllogism (as in the textbook)

1. Structure: three and only three propositions.
2. Structure: three and only three terms – else fallacy of four terms.
3. Distribution: the middle term must be distributed at least once – else undistributed middle.
4. Distribution: no term may be distributed in the conclusion unless distributed in its premise – else illicit minor / illicit major.
5. Quality: no conclusion from two negative premises – else fallacy of negative premises.
6. Quality: if one premise is negative, the conclusion must be negative, and vice versa.
7. Quality: if both premises are affirmative, the conclusion must be affirmative, and vice versa.

Nyaya five-membered syllogism

1. Pratijna – This hill has fire (thesis).
2. Hetu – Because it has smoke (reason).
3. Udaharan – Wherever there is smoke there is fire, as in the kitchen (vyapti with example).
4. Upanaya – This hill has smoke, which is always with fire (application).
5. Nigaman – Therefore this hill has fire (conclusion).

5 important questions

Q1. What is the figure of a syllogism? Who added the fourth figure?

The figure is the form of a syllogism determined by the position of the middle term in the two premises. Galen added the fourth figure.

Q2. Name the fallacy: All metals are heavy. All stones are heavy. ∴ All stones are metals.

Undistributed middle. "Heavy" is the predicate of an A proposition in both premises, so it is never distributed.

Q3. Name the fallacy: All cricketers are sportsmen. No chess players are cricketers. ∴ No chess players are sportsmen.

Illicit major. "Sportsmen" is undistributed in the major premise (predicate of A) but distributed in the conclusion (predicate of E).

Q4. Name the fallacy: No men are quadrupeds. Some men are tall. ∴ No tall beings are quadrupeds.

Illicit minor. "Tall" is undistributed in the minor premise (predicate of I) but distributed in the conclusion (subject of E).

Q5. Give any three differences between the Aristotelian and the Nyaya syllogism.

(1) Aristotelian has three propositions; Nyaya has five. (2) Aristotelian is deductive and formal; Nyaya is deductive-inductive and both formal and material (the udaharan shows the induction behind the major premise). (3) Aristotelian is analytical and better for testing validity; Nyaya is expository and rhetorical, the method of actual debate. Also, in Nyaya all three terms come together in the upanaya.

Exam tip

To name a fallacy, first mark S, P and M, then write the type (A/E/I/O) of each proposition and check distribution. Check the rules in order: terms → middle → illicit process → quality.

Quick summary

Science establishes laws by induction, moving from observed cases to all cases. This jump from "some" to "all" is called the inductive leap, and it needs justification. It is justified by the formal grounds (principle of uniformity of nature and principle of causation) and the material grounds (observation and experiment). This chapter explains observation, its characteristics, conditions and fallacies, and experiment with its nature and characteristics.

Key points

• Inductive leap: the evidence covers some cases, the conclusion covers all cases, so the conclusion is only probable.
• Principle of uniformity of nature: there is order in nature; what happens once will happen again under similar circumstances.
• Principle of causation: some events are causally connected, and the same cause always produces the same effect.
• Formal grounds alone are not enough; the material truth of a law needs facts collected by observation and experiment.
• "Observation" comes from Greek – ob (before) and servare (to keep): keeping something before the mind.
• Observation is selective perception of facts with a definite purpose. Every observation is perception, but every perception is not observation.
• Characteristics of observation: purposive (e.g. discovery of Neptune), selection of significant facts, selection of significant aspects of a fact (a doctor notes BP and pulse), neglect of illusory aspects (a stick looks bent in water), use of instruments (telescope, microscope, X-ray).
• Conditions of good observation: right mental set and intellectual condition (alert, unbiased, patient; public verifiability), awareness of limits of sense organs and instruments, external conditions (e.g. fog), training in observation techniques.
• Experiment is observation under conditions controlled by the investigator.
• Variables in an experiment: independent (factor changed, e.g. type of soap), dependent (its effect, e.g. dirt left), relevant/controlled (kept constant, e.g. water amount, temperature, time).
• Characteristics of experiment: deliberately undertaken, sets up an artificial situation (coin and feather fall together in a vacuum), systematic variation of conditions, can be repeated.

Important terms

• Perception – becoming aware of whatever comes to our notice, with no purpose or selection.
• Fallacy of non-observation – overlooking relevant facts. Two forms: neglect of instances, neglect of operative conditions.
• Neglect of instances – missing relevant cases, due to a narrow range of experience (we cannot hear below about 20 Hz), unfavourable physical conditions (not seeing the Sun in an eclipse) or bias.
• Neglect of operative conditions – missing the real cause; e.g. Digby's sympathetic powder was credited for healing when the real cause was keeping the wound clean.
• Fallacy of mal-observation – wrongly interpreting what is sensed, e.g. rope seen as a snake, mirage seen as water.

Key facts

Causes of mal-observation: unfavourable physical conditions, the observer's peculiar mental state (a white shirt seen as a ghost after a horror film), lack of experience (a baby thinks its mirror image is another baby).

5 important questions

Q1. What is the inductive leap?

In induction we pass from observed to unobserved cases, from "some" to "all". This passage is the inductive leap. It makes the conclusion probable, so it has to be justified by the formal and material grounds of induction.

Q2. Distinguish between observation and perception.

Observation has a definite purpose and involves selection of facts; perception has no purpose and no selection. Everything observed is perceived, but not everything perceived is observed.

Q3. Explain the fallacy of neglect of operative conditions with an example.

It means ignoring the essential conditions that really produce an effect and treating some other factor as the cause. In the 17th century people believed Digby's sympathetic powder, rubbed on the weapon, healed wounds. The real cause was that the wound was kept clean.

Q4. Explain the three types of variables in an experiment.

Independent variable – the factor the experimenter changes (type of soap). Dependent variable – the effect measured (amount of dirt removed). Relevant/controlled variables – other factors that could affect the result and are kept constant (water, temperature, washing time).

Q5. Why is the experimenter called the master and the observer the slave of nature?

The observer must wait for events to happen in nature and cannot control them. The experimenter creates and changes the conditions as he wishes and can repeat the experiment.

Exam tip

For "explain with example" questions, use the textbook's standard examples: Neptune (purposive), stick in water (illusory aspect), Digby's powder (operative conditions), rope-snake and mirage (mal-observation), coin and feather (experiment).

Quick summary

Scientific inquiry starts with a problem and a guess about how the facts are connected. That guess is a hypothesis – a tentative solution that must be tested. This chapter covers the meaning and characteristics of a hypothesis, how hypotheses originate, the conditions of a good hypothesis, and how a hypothesis is verified directly or indirectly, along with the limits of verification.

Key points

• "Hypothesis" comes from Greek hypo (under) and tithenai (to place).
• Coffey: a hypothesis is an attempt at explanation, a provisional supposition made in order to explain scientifically some facts or phenomenon.
• Simply put, it is a tentative solution to a problem. Once verified, it becomes the explanation.
• Example: when the orbit of Uranus did not match calculations (Bouvard, 1820), astronomers supposed an unknown planet beyond it. The Berlin telescope later found it – Neptune.
• Characteristics: an important stage of scientific investigation; an attempt at explanation (Jenner's idea that cowpox pus protected milkmaids from smallpox); provisional; an organising principle (Newton's gravitation linked falling bodies, planets and tides); result of rational activity; result of keen and creative imagination (Appert's canning).
• Origin (sources): keen and creative imagination; painstaking work (Kepler tried 19 wrong hypotheses before elliptical orbits); adequate and wide knowledge (Pasteur cured the silkworm disease); chance/accident (Fleming's penicillin, 1928); insight (Archimedes in the bath); induction by simple enumeration and analogy (Lowell's analogy of Earth and Mars about life on Mars).
• Conditions of a good hypothesis: relevance, self-consistency, testability (verifiable at least in principle), compatibility with established knowledge, explanatory power, predictive power, simplicity.
• An irrelevant hypothesis: a follower of Galen said thigh bones are curved in nature and narrow trousers made them straight.
• Simplicity: the Copernican (Sun-centred) hypothesis replaced the Ptolemaic (Earth-centred) one because it made fewer independent assumptions.

Important terms

• Verification – finding out whether a hypothesis agrees with facts; if yes it is confirmed, if not it is rejected or modified.
• Direct verification – observing the very facts the hypothesis refers to, by observation or experiment. Such hypotheses are instantial / empirical.
• Indirect verification – deducing consequences from the hypothesis and testing them against facts. Such hypotheses are non-instantial / theoretical.
• Insight – a sudden flash that gives the solution unexpectedly.
• Analogy – an inference based on observed resemblances.

Key facts

Steps of indirect verification: (1) deductive development – assume H true and deduce consequences C1, C2, C3; (2) check whether these consequences actually happen.

Limits of verification: "If H, then C; C happens; ∴ H" commits the fallacy of affirming the consequent. Verification shows a hypothesis can explain the facts, not that it is the only explanation. Proof requires showing "if and only if H, then C".

5 important questions

Q1. Define hypothesis.

A hypothesis is a provisional supposition or tentative solution put forward to explain facts that cannot be understood without it (Coffey: an attempt at explanation made to explain scientifically some facts).

Q2. What is meant by instantial and non-instantial hypotheses?

Hypotheses that can be verified directly by observation or experiment are instantial (empirical). Those that can be verified only indirectly, by testing their consequences, are non-instantial (theoretical).

Q3. Explain chance and insight as sources of hypothesis.

Chance: Fleming noticed that mould on a culture plate stopped germs growing nearby, which led to penicillin; but only his knowledge let him use the chance. Insight: the solution flashes suddenly, as when Archimedes saw water overflow from his bath and framed the principle of loss of weight in liquids.

Q4. Explain any four conditions of a good hypothesis.

Relevance – it must explain the facts in question. Self-consistency – it must not contradict itself. Testability – it must be verifiable at least in principle. Simplicity – it should make the minimum number of independent assumptions (Copernicus over Ptolemy).

Q5. Explain indirect verification with an example.

Consequences are deduced from the hypothesis and tested. The hypothesis that South Sea Islanders came from South America implied they could have travelled by sea in primitive boats. The Kon-Tiki crew sailed such a raft and the currents took them to the islands, confirming the hypothesis indirectly.

Exam tip

Pair every point with its textbook example (Neptune, Rayleigh-Argon, Kon-Tiki, Fleming, Archimedes, Kepler, Copernicus). Examiners look for the example as much as the definition.