Logic तर्कशास्त्र

Quick summary

Logic trains us to reason correctly. This chapter tells how logic began in India, Greece and China, how it is defined, and explains its basic ideas – argument, premises and conclusion, form and content, truth and validity. It ends with the two kinds of argument: deductive and inductive.

Key points

• Logic is traditionally a branch of philosophy and is classified as a formal science (like mathematics).
• Aristotle (384–322 BC) is regarded as the father of logic; his logic is called traditional logic.
• Indian logic grew independently: the Nyaya school, with the Nyaya Sutras of Akshapada Gautama, plus Jain and Buddhist logic (Nagarjuna, Vasubandhu, Dignaga, Dharmakirti). Anviksiki and Tarka are mentioned in the Mahabharata and Kautilya's Arthashastra. Panini's grammar used a logic similar to Boolean logic.
• In China, Mozi (founder of the Mohist school) developed logic.
• Modern symbolic logic grew from the mid-19th century through thinkers like Leibniz, Boole, De Morgan, Frege, Russell and others.
• "Logic" comes from the Greek logos (thought). "Science of the laws of thought" and "science of reasoning" are both too wide.
• I.M. Copi: Logic is the study of the methods and principles used to distinguish good (correct) from bad (incorrect) reasoning. This is the accepted definition.
• Every argument has a form (pattern) and content (subject matter) – like pots of the same shape made of clay or silver.
• Propositions are true or false; arguments are valid or invalid. Never call a proposition valid or an argument true.
• A valid argument can have false premises and a false conclusion; an invalid one can have all true statements.

Important terms

• Argument / inference – a group of propositions in which one (the conclusion) is established on the evidence of the others (the premises).
• Premise – a proposition given as evidence.
• Conclusion – the proposition that is established.
• Deductive argument – claims that the premises give sufficient (conclusive) evidence; the conclusion does not go beyond the premises.
• Inductive argument – premises give only some evidence; the conclusion goes beyond them and is probable.
• Sound argument – a valid deductive argument with true premises (and so a true conclusion).
• Formally valid – validity decided by form alone (deduction). Materially valid – decided by content (induction).

Key facts

Example of deduction: If it rains, roads become wet. It is raining. Therefore roads are wet.
Example of induction: All observed crows are black. Therefore all crows are black.
Bad induction: a cat crossed my way, so something bad will happen today (superstition).

5 important questions

Q1. Who is regarded as the father of logic?

The Greek philosopher Aristotle (384–322 BC). His logic is known as traditional logic.

Q2. Why is the definition "logic is the science of the laws of thought" not accurate?

It is too wide. Thinking includes remembering, imagining and day-dreaming, but logic deals only with reasoning. Also, the process of thinking is studied by psychology, not logic.

Q3. Explain the difference between form and content of an argument.

Content is what the argument is about; form is its pattern. "All men are wise, Rakesh is a man, so Rakesh is wise" and "All doctors are rich, Sunil is a doctor, so Sunil is rich" have different content but the same form: All A is B; X is A; so X is B.

Q4. Distinguish between truth and validity.

Truth and falsity belong to propositions: a proposition is true if it agrees with facts. Validity belongs to arguments: an argument is valid if its conclusion necessarily follows from its premises. A valid argument may have false propositions in it.

Q5. State four differences between deductive and inductive arguments.

(1) Deduction gives sufficient evidence; induction gives some evidence. (2) Deductive conclusion is certain; inductive is probable. (3) Deduction is formally valid; induction is materially valid. (4) Deduction gives no new knowledge; induction gives new knowledge.

Exam tip

Use the exact words "valid/invalid" for arguments and "true/false" for propositions – mixing them is a common mistake in true/false and match-the-pair questions.

Quick summary

A proposition is the basic unit of logic: a sentence that is either true or false. This chapter separates propositions from sentences, classifies propositions as simple and compound, explains truth-functional compound propositions and their five kinds, and teaches how to symbolize propositions using constants, variables, connectives and brackets.

Key points

• Definition: a proposition is a sentence which is either true or false.
• All propositions are sentences, but not all sentences are propositions – only declarative sentences that can be true or false. Questions, commands, wishes and exclamations are not propositions.
• Simple proposition – contains no other proposition as its component, e.g. "Delhi is the capital of India".
• Compound proposition – contains one or more propositions as components, e.g. "Delhi is the capital of India and it is a crowded city".
• Four kinds of simple proposition: subjectless ("Fire!"), subject–predicate ("Ashok is intelligent"), relational ("Ram is taller than Shyam"), class-membership ("Bhagat Singh was a freedom fighter").
• Compound propositions are truth-functional (truth value decided by the truth values of the components) or non-truth-functional ("I believe that soul exists" – its truth does not depend on whether the soul exists).
• Five truth-functional compounds: negation, conjunction, disjunction, material implication, material equivalence.
• Symbolization removes the defects of ordinary language – ambiguous and vague words, misleading idioms, metaphors – and shows the logical form clearly.
• "Not" works on one proposition, so it is a monadic operator; the other four join two propositions and are binary (dyadic) operators.
• Brackets remove ambiguity: parentheses ( ), box brackets [ ], braces { }.

Important terms

• Propositional constant – a capital letter A–Z standing for one specific proposition, e.g. Y = "Yogasanas unite body and mind".
• Propositional variable – a small letter p, q, r, s… standing for any proposition; a place-holder.
• Propositional connective (logical operator) – the expression that joins or operates on propositions.
• Component – a proposition that appears as a part of a compound proposition.
• Conjuncts / disjuncts – components of a conjunction / disjunction.
• Antecedent / consequent – the "if" part and the "then" part of an implication.

Formulas

Two restrictions while symbolizing:

1. The same constant must be used for a proposition every time it occurs in the same argument.
2. The same constant cannot be used for different propositions in the same argument.

Worked symbolization: "Santosh will take salad or sandwich. Santosh will not take salad. Therefore Santosh will take sandwich." Let S = Santosh will take salad, D = Santosh will take sandwich:

1. S ∨ D
2. ∼S
3. ∴ D

5 important questions

Q1. Distinguish between a sentence and a proposition.

A sentence is a group of words with complete meaning; it may be a question, command or exclamation. A proposition is only that sentence which is true or false. So every proposition is a sentence, but not every sentence is a proposition.

Q2. What is a non-truth-functional compound proposition? Give an example.

A compound proposition whose truth value is not decided by the truth values of its components, e.g. "I believe that soul exists". Whether the soul exists or not, the statement about my belief may still be true.

Q3. Distinguish between a propositional constant and a propositional variable.

A constant (A–Z) stands for one particular proposition and has a definite meaning. A variable (p, q, r…) stands for any proposition at all; it only marks a place where a proposition can be put.

Q4. Name the four kinds of simple proposition with examples.

Subjectless (Bomb!), subject–predicate (Ashok is intelligent), relational (Ram is taller than Shyam), class-membership (Rani Lakshmibai was a great warrior).

Q5. Symbolize: "If it rains then the match is cancelled, and it is not raining."

Let R = it rains, M = the match is cancelled. Symbol: (R ⊃ M) · ∼R. The main connective is the dot.

Exam tip

While symbolizing, first find the main connective, then put brackets around each component. Write the key ("Let R = …") before the symbols – it earns marks.

Quick summary

A decision procedure is a method of deciding whether something belongs to a certain class. In logic it decides whether a truth-functional statement form is a tautology, contradiction or contingency, and whether an argument form is valid or invalid. This chapter teaches the truth table method, the most basic effective decision procedure.

Key points

• An effective decision procedure must be reliable (always correct), mechanical (fixed steps, no imagination needed) and finite (a last step gives the answer).
• Five decision procedures: truth table, shorter truth table, truth tree, conjunctive normal form, disjunctive normal form.
• A truth table is a tabular way of showing the truth values of truth-functional compound statements.
• A truth table has two parts: the matrix (left – all possible values of the variables) and the truth-functional form (right).
• Number of rows = 2ⁿ, where n = number of distinct variables: 1 → 2, 2 → 4, 3 → 8, 4 → 16, 5 → 32.
• Write variables in alphabetical order. For 2 variables the first column is T T F F and the second T F T F.
• Work from the smallest parts outwards; the column under the main connective gives the answer.
• All T under the main connective = tautology; all F = contradiction; some T and some F = contingency.
• To test an argument form: write the premises and conclusion side by side. Look at every row where all premises are true. If the conclusion is true in all such rows, the form is valid; if it is false in even one such row, it is invalid.
• An argument is valid if it is a substitution instance of a valid argument form.

Important terms

• Decision procedure – a method that decides whether an object belongs to a given class.
• Matrix – the left part of a truth table listing all truth possibilities.
• Tautology – a form true in all its substitution instances.
• Contradiction – a form false in all its substitution instances.
• Contingency – a form with both true and false substitution instances.
• Main connective – the connective that governs the whole expression.

Formulas

Example 1 – contingency: (p ∨ q) ⊃ (q ⊃ p)

Main column has T and F, so it is a contingency.

Example 2 – tautology: (p · ∼p) ⊃ ∼p

All T, so it is a tautology. (p · ∼p) on its own is all F – a contradiction.

Example 3 – valid argument form: p ⊃ q, ∼q ∴ ∼p

Both premises are true only in row 4, and the conclusion is true there. Valid.

Example 4 – invalid argument form: p ⊃ q, q ∴ p

Both premises are true in rows 1 and 3; in row 3 the conclusion is false. Invalid.

5 important questions

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

It must be reliable (always gives the correct answer), mechanical (steps are followed in order without imagination) and finite (it ends after a limited number of steps).

Q2. How many rows will a truth table for (p ⊃ q) ≡ (r ∨ s) have?

There are 4 distinct variables, so 2⁴ = 16 rows.

Q3. Define tautology, contradiction and contingency.

A tautology is true under all truth possibilities of its components; a contradiction is false under all; a contingency is true under some and false under others.

Q4. How does the truth table test the validity of an argument form?

Construct the table with premises and conclusion in columns. Check every row where all premises are true. If the conclusion is true in each such row, the form is valid; if any such row has a false conclusion, it is invalid.

Q5. Why is the truth table an effective decision procedure?

It is reliable because the answer is always correct, mechanical because it follows fixed steps in a systematic way, and finite because the number of rows (2ⁿ) is limited and there is a last step.

Exam tip

Always fill the matrix in the standard order (T T F F / T F T F) and write the final column under the main connective clearly – the examiner checks that column first.

Quick summary

Besides decision procedures like the truth table, logic uses methods that prove validity. The deductive proof derives the conclusion from the premises step by step, using elementary valid argument forms. It is of three types – direct proof, conditional proof and indirect proof; this chapter teaches the direct proof with the nineteen rules: 9 rules of inference and 10 rules based on the rule of replacement.

Key points

• Methods in logic are of two types: decision procedures (truth table etc.) and methods that are not decision procedures (deductive proof).
• Deductive proof is reliable and finite but not mechanical – it needs intelligence to choose the right rule. So it is not a decision procedure.
• Deductive proof proves validity; it does not decide whether an argument is valid or invalid.
• Direct proof: the conclusion is deduced directly from the premises by a sequence of valid elementary arguments, without any assumption.
• Steps: (1) number the premises; (2) write the conclusion after the last premise, separated by "/ ∴"; (3) derive new lines, each with a justification (line numbers + rule); (4) stop when the conclusion is reached.
• The 9 rules of inference are elementary valid argument forms; they apply only to a whole line.
• The 10 rules of replacement are logical equivalences; they may be applied to a whole line or to any part of it.

Important terms

• Formal proof of validity – a numbered sequence of statements, each a premise or derived from earlier ones by a rule, ending in the conclusion.
• Elementary valid argument form – a basic valid form such as Modus Ponens.
• Justification – the line numbers and rule written on the right of each derived line.
• Substitution instance – an argument got by putting statements in place of the variables of a form.

Formulas

Rules of inference

Rules of replacement

• D.N.: p ≡ ∼∼p
• De M.: ∼(p · q) ≡ (∼p ∨ ∼q); ∼(p ∨ q) ≡ (∼p · ∼q)
• Com.: (p · q) ≡ (q · p); (p ∨ q) ≡ (q ∨ p)
• Assoc.: [p · (q · r)] ≡ [(p · q) · r]; [p ∨ (q ∨ r)] ≡ [(p ∨ q) ∨ r]
• Dist.: [p · (q ∨ r)] ≡ [(p · q) ∨ (p · r)]; [p ∨ (q · r)] ≡ [(p ∨ q) · (p ∨ r)]
• Trans.: (p ⊃ q) ≡ (∼q ⊃ ∼p)
• Impl.: (p ⊃ q) ≡ (∼p ∨ q)
• Equiv.: (p ≡ q) ≡ [(p ⊃ q) · (q ⊃ p)]; (p ≡ q) ≡ [(p · q) ∨ (∼p · ∼q)]
• Exp.: [(p · q) ⊃ r] ≡ [p ⊃ (q ⊃ r)]
• Taut.: p ≡ (p ∨ p); p ≡ (p · p)

Worked proof 1

1. A ⊃ B
2. B ⊃ C
3. ∼C / ∴ ∼A
4. A ⊃ C 1, 2, H.S.
5. ∼A 4, 3, M.T.

Worked proof 2 (with replacement)

1. ∼(A · B)
2. A / ∴ ∼B
3. ∼A ∨ ∼B 1, De M.
4. ∼∼A 2, D.N.
5. ∼B 3, 4, D.S.

5 important questions

Q1. Why is deductive proof not a decision procedure?

An effective decision procedure must be reliable, mechanical and finite. Deductive proof is reliable and finite but not mechanical, because one needs intelligence to choose which rule to apply.

Q2. Why is the direct proof called "direct"?

Because the conclusion is deduced straight from the premises without making any assumption at any step.

Q3. Construct a formal proof: 1. (P ∨ Q) ⊃ R 2. P / ∴ R

1. P ∨ Q 2, Add.
2. R 1, 3, M.P.

Q4. Construct a formal proof: 1. (P ⊃ Q) · (R ⊃ S) 2. P ∨ R 3. ∼Q / ∴ S

1. Q ∨ S 1, 2, C.D.
2. S 4, 3, D.S.

Q5. Construct a formal proof: 1. M · N / ∴ N ∨ T

1. N · M 1, Com.
2. N 2, Simp.
3. N ∨ T 3, Add.

Exam tip

Simplification gives only the left conjunct – to get the right one, first use Com. Rules of inference cannot be used on a part of a line, so use a replacement rule first when needed.

Quick summary

Deduction gives certainty but no new knowledge; induction gives new knowledge, though its conclusion is only probable. In an inductive inference the conclusion asserts more than the premises. This chapter explains why induction is needed and its four kinds: simple enumeration, analogy, scientific induction and the hypothetico-deductive method.

Key points

• Deduction deals with form, not content; its conclusion is certain but adds nothing new. Induction aims at material truth and adds new knowledge – hence the need for induction.
• Example: gold, silver and iron expand on heating, so all metals expand on heating.
• Simple enumeration and scientific induction establish general propositions; analogy establishes a particular proposition; the hypothetico-deductive method can establish either.
• Simple enumeration – the common man's way of generalizing: what is true of several cases of a kind is true of all cases of that kind. Form: All observed P's are Q; no observed P is non-Q; ∴ all P's are Q.
• Features of simple enumeration: based on uniform and uncontradicted experience; no analysis of why the property is present; unrestricted generality; an inductive leap; low degree of probability.
• Its value depends not only on the number of instances (as Mill and Bacon said) but on variety of experience, wider experience and the nature of resemblances. "All crows are black" is good; "All swans are white" proved bad.
• Analogy – from known resemblances to further resemblance; from particular to particular. Its conclusion is always probable.
• Scientific induction – establishing a generalization on direct and indirect evidence. Mill and Bacon said it always expresses a causal relation, but this is not accepted: "All bats are warm-blooded" is not causal.
• Hypothetico-deductive method (the scientific method) uses both deduction and induction and can establish laws, theories and particular conclusions.

Important terms

• Inductive leap – the jump from observed to unobserved, known to unknown cases.
• Direct evidence – observation of the instances themselves.
• Indirect evidence – support from other established laws (e.g. "all gases expand on heating" supports "all metals expand on heating").
• Restricted generality – a conclusion about a limited group ("all students in this class are smart"); this is not simple enumeration.
• Hypothesis – a tentative solution to a problem.

Key facts

Form of analogy: A has P1, P2…Pn; B has P1, P2…Pn; A also has q; ∴ B probably has q.

• Good analogy (Lowell): Earth and Mars are both planets revolving round the Sun, with water, moderate temperature and atmosphere; there is life on Earth, so there is life on Mars. The resemblances are relevant to life.
• Bad analogy: man and monkey share many body features, man can read and write, so monkeys can – the difference (rationality) is important.
• Value of analogy depends on relevant and important resemblances, important differences, and the conclusion not claiming too much (life on Mars, not human beings on Mars).

Five steps of the hypothetico-deductive method (Kon-Tiki example):

1. Observation and feeling of a problem – why are customs of South Sea Islanders and South Americans similar?
2. Formation of an initial hypothesis – people from South America settled on the islands.
3. Collection of additional facts – routes and means of travel.
4. Deductive development – if so, they must have sailed in primitive boats.
5. Verification – a primitive boat actually made the journey, so the hypothesis was accepted.

5 important questions

Q1. Why is there a need for induction?

Deductive conclusions are certain but give no new knowledge. Inductive conclusions are probable but give new knowledge about the world, which science and daily life need.

Q2. Define induction by simple enumeration and give an example.

It establishes a generalization on the basis of uniform and uncontradicted experience of some cases of a kind. Example: one lakh observed crows are black and none is non-black; therefore all crows are black.

Q3. Why does simple enumeration have a low degree of probability?

It rests only on observed cases and direct evidence, with no analysis of why the property is present. A contrary instance may still turn up, as with white swans.

Q4. Explain the factors on which the value of analogy depends.

(1) Resemblances should be relevant and important to the inferred property. (2) There should be no important differences. (3) The conclusion should not claim more than the evidence allows.

Q5. Explain the steps of the hypothetico-deductive method.

Observation and feeling of a problem; formation of an initial hypothesis; collection of additional facts; deductive development of the hypothesis; verification by checking whether the deduced consequences take place. (Illustrate with the Kon-Tiki expedition.)

Exam tip

Remember the pairs: simple enumeration = low probability, scientific induction = high probability; analogy = particular to particular; hypothetico-deductive method = uses both deduction and induction.

Quick summary

A fallacy is an error in reasoning. I.M. Copi says fallacious arguments are those that appear correct but on examination are not. Fallacies are formal (a rule of logic is broken) or non-formal (caused by misleading use of language or irrelevant content). This chapter explains the purpose of studying fallacies and Copi's classification of non-formal fallacies.

Key points

• Any error in reasoning is called a fallacy. The notion of fallacy is psychological, because a fallacious argument looks correct.
• Studying fallacies helps us spot and understand poor reasoning, come closer to the truth, avoid errors in our own arguments, and not be misled by others.
• A complete classification is not possible because errors are countless.
• Formal fallacy – related to the form of a (deductive) argument; committed when a rule of logic is violated, e.g. p ⊃ q, q ∴ p.
• Non-formal fallacy – related to content; committed through misleading use of language: irrelevant conclusion, ambiguous words, wrong use of collective and distributive terms, wrong use of rules in exceptions.
• Six non-formal fallacies (Copi): division, composition, accident, converse accident, ignoratio elenchi, petitio principii.
• A term is used distributively when each member has the quality ("all mangoes in the basket are sweet") and collectively when the members taken together have it ("the mangoes in the basket weigh 5 kg").

Important terms

• Fallacy of division – wrongly moving from the collective use of a term to its distributive use (whole → parts).
• Fallacy of composition – wrongly moving from distributive use to collective use (parts → whole).
• Fallacy of accident – applying a general rule to a special case whose accidental features make the rule unsuitable.
• Converse fallacy of accident – making a general rule from a special case.
• Ignoratio elenchi – irrelevant conclusion: the premises support some other conclusion.
• Petitio principii – begging the question: the conclusion is assumed in the premises.

Key facts

5 important questions

Q1. Define fallacy according to I.M. Copi.

Fallacious arguments are those which appear to be correct but are proved, on examination, not to be so.

Q2. Distinguish between formal and non-formal fallacies.

A formal fallacy concerns the form of an argument and arises when a rule of logic is violated. A non-formal fallacy concerns the content and arises from misleading use of language, such as ambiguity, irrelevance, or wrong use of collective and distributive terms.

Q3. Distinguish between the fallacy of division and the fallacy of composition.

In division one argues from the collective use of a term to its distributive use – what is true of the whole is said to be true of each part. In composition one argues from the distributive to the collective use – what is true of each part is said to be true of the whole.

Q4. What is the fallacy of ignoratio elenchi? Name its kinds.

It is the fallacy of irrelevant conclusion: the premises lead to some other conclusion, but another one is drawn. Its kinds: argumentum ad baculum, ad hominem, ad populum, ad verecundiam, ad misericordiam and ad ignorantiam.

Q5. Distinguish between hysteron proteron and arguing in circles.

Both are forms of petitio principii. In hysteron proteron the conclusion is directly assumed in the premise (same idea in other words). In arguing in circles the premise is proved by the conclusion and the conclusion by the premise, in a roundabout way.

Exam tip

For "identify the fallacy" questions, first ask: is a rule broken (formal) or is the trick in the words/content (non-formal)? Then match the key clue – force, person, emotion, authority, pity, ignorance, whole/part.

Quick summary

Logic is not only a theory; it is a skill used everywhere. Logic is the study of reasoning, and it trains us to build good arguments and spot bad ones. This chapter shows how logic is applied in everyday life, law, science and computer science.

Key points

• Logical thinking helps us take the right decision at the right time, e.g. choosing a stream after S.S.C. by analysing options, interests and abilities instead of following trends or pressure.
• Logical reasoning is built into the human mind; studying logic makes us better equipped to reason correctly.
• Competitive exams (C.A., Law, UPSC, MPSC) have a reasoning paper to test this ability.
• Everyday life: decision making, clear and precise communication, evaluating arguments of salesmen, advertisements and politicians, avoiding fallacies, and reaching agreement in discussions.
• Law: logic is used in making laws (clear, precise, defined words in laws, contracts and wills), executing laws (fair judgement in trials; lawyers use knowledge of fallacies) and interpreting laws (whether a rule applies to a set of facts).
• "The Rule of Law" – like cases decided alike – rests on inductive reasoning (analogy, simple enumeration); the syllogism helps check whether a legal argument is valid.
• Science: every stage of the scientific (hypothetico-deductive) method rests on logic – suggesting a hypothesis, judging whether it is good, choosing relevant facts and experiments, and deducing consequences for indirect verification.
• Indirect verification commits the fallacy of affirming the consequent, and we cannot know all rival hypotheses, so scientific knowledge is probable, not conclusively proved.
• Relation is one-sided: logic helps science, but science does not help logic. Logical thinking is wider than scientific thinking.
• Computer science: computers use the binary system (0 and 1), like true/false in logic. Logic gates and truth tables, algorithms and flow charts are all logical tools.

Important terms

• Algorithm – a series of steps in a logical order for solving a problem.
• Flow chart – a diagram that shows the steps and decisions of an algorithm; independent of any computer language.
• Program – a sequence of instructions telling a computer how to do a task.
• Machine language – the language a computer understands.
• Logic gates – circuits that handle binary data: AND, OR, NOT, NAND, NOR, EXCLUSIVE-OR.
• Vertical organization – theories bringing order among laws; the highest order in science.

Key facts

Logic gates work like truth tables (1 = true, 0 = false):

Flow chart symbols: oval – start/end; parallelogram – input/output; rectangle – processing; diamond – decision.

Algorithm to add two numbers: 1. Input a and b. 2. Sum = a + b. 3. Print sum. 4. Stop.

Three areas of law where logic works: making laws, execution of laws, interpretation of laws.

5 important questions

Q1. How does logic help in everyday decision making?

It trains us to analyse a situation, list the options, set priorities and judge evidence instead of being swayed by emotions, advertisements or peer pressure – for example, while choosing a career after S.S.C.

Q2. Explain the role of logic in law.

Logic is needed in making laws (precise, clearly defined words), in executing laws (lawyers detect fallacies and argue correctly; judges weigh evidence and reach fair judgements) and in interpreting laws (deciding whether a rule applies to the facts). Like cases being decided alike is based on inductive reasoning.

Q3. Why is scientific knowledge said to be probable?

Most hypotheses are verified indirectly, which logically commits the fallacy of affirming the consequent. To prove a hypothesis we would have to rule out every rival hypothesis, which is not possible. So scientific laws and theories remain probable.

Q4. Explain the use of logic in computer science.

Computers work on the binary system of 0 and 1, like true and false. Logic gates (AND, OR, NOT etc.) process binary data and their working is shown by truth tables. Programs are built from algorithms and flow charts, which are logical sequences of steps.

Q5. "The relation between logic and science is one-sided." Explain.

Logic helps science at every stage of the scientific method and in testing the validity of proofs, but science does not help logic. Logical thinking is wider; scientific thinking is based on it.

Exam tip

For long answers, organise under four headings – everyday life, law, science, computer science – and give one example under each.