23  Evaluating and distinguishing deductive and inductive reasoning

23.1 What the Syllabus Covers

The syllabus head asks the candidate to distinguish deductive from inductive reasoning and evaluate each — that is, judge whether an argument is valid / sound (deductive) or strong / cogent (inductive). PYQ patterns:

  1. Classify a given argument as deductive or inductive.
  2. Judge validity of a deductive argument.
  3. Judge strength of an inductive argument.
  4. Identify the type of induction (enumerative, causal, analogical, statistical, Mill’s methods).
  5. Name the theorist (Aristotle, Bacon, Mill, Hume, Popper, Peirce).

23.2 The Core Distinction

TipDeductive vs Inductive at a Glance
Dimension Deductive Inductive
Direction General → Particular Particular → General
Conclusion Necessary if premises true Probable; never certain
Tested for Validity & soundness Strength & cogency
New information in conclusion No (already implicit) Yes (goes beyond premises)
Falsified by Counter-example to form OR false premise Counter-example to generalisation
Examples Syllogism, mathematical proof Scientific generalisation, polling 

flowchart TB
  R{Reasoning} --> D[Deductive<br/>General → Particular]
  R --> I[Inductive<br/>Particular → General]
  D --> V[Validity<br/>and Soundness]
  I --> S[Strength<br/>and Cogency]
    classDef default fill:#003366,color:#ffffff,stroke:#ffcc00,stroke-width:3px,rx:10px,ry:10px;

23.3 Deductive Reasoning — Evaluating It

A deductive argument is judged on validity and soundness.

TipValidity, Truth, Soundness — Recap
  • Validity — formal: the form guarantees the conclusion if the premises are true.
  • Truth — substantive: the premises are actually true.
  • Soundness — Valid + all premises true.

A valid argument can have false premises (form is right; content is wrong). A sound argument cannot.

23.3.1 Methods to Test Validity

TipThree Ways to Test Validity
  • Truth-table — for propositional arguments (P, Q, →, ¬, ∧, ∨, ↔︎).
  • Counter-example — find a case where premises are true and conclusion false; if exists, invalid.
  • Venn diagram — for categorical syllogisms (Topic 24).

23.3.2 Five Valid Propositional Forms

TipFive Classical Valid Forms
  • Modus Ponens — If P then Q; P; ∴ Q.
  • Modus Tollens — If P then Q; ¬Q; ∴ ¬P.
  • Hypothetical Syllogism — If P then Q; if Q then R; ∴ if P then R.
  • Disjunctive Syllogism — P ∨ Q; ¬P; ∴ Q.
  • Constructive Dilemma — (P→Q) ∧ (R→S); P ∨ R; ∴ Q ∨ S.

23.3.3 Three Classic Formal Fallacies

TipFormal Fallacies
  • Affirming the Consequent — If P then Q; Q; ∴ P. ✗
  • Denying the Antecedent — If P then Q; ¬P; ∴ ¬Q. ✗
  • Undistributed Middle — All A are B; some B are C; ∴ some A are C. ✗

(Topic 21 also lists six syllogistic-rule fallacies: Four Terms, Undistributed Middle, Illicit Major/Minor, Exclusive Premises, Affirmative-from-Negative, Existential.)

23.3.4 Aristotle’s Syllogism — Brief Recap

A categorical syllogism has three terms (S, P, M), two premises, and one conclusion. The four propositional types are A, E, I, O (Topic 21). The 24 valid moods (out of 64) are listed by figure and mood — Barbara (AAA-1), Celarent (EAE-1), Darii (AII-1), Ferio (EIO-1) etc.

23.4 Inductive Reasoning — Evaluating It

An inductive argument is judged on strength and cogency.

TipInductive Strength and Cogency
  • Strong — premises make the conclusion highly probable.
  • Weak — premises make the conclusion only slightly probable.
  • Cogent — strong + all premises true.
  • Uncogent — weak or has false premises.

Inductive strength is a matter of degree, not all-or-nothing.

23.4.1 Five Types of Induction

TipFive Types of Induction
  1. Enumerative / Generalisation — “Every observed X is Y; therefore all X are Y.”
  2. Causal — “X is regularly followed by Y; therefore X causes Y.”
  3. Statistical — Sample → population estimate; uses probability.
  4. Inductive analogy — A and B share many features; A has P; therefore B probably has P.
  5. Predictive — Past patterns will repeat.

23.4.2 Six Criteria for Strong Induction

TipSix Criteria for Strong Induction
  1. Sample size — larger is better.
  2. Representativeness — sample mirrors population.
  3. Variety of cases — diverse instances.
  4. No counter-examples ignored.
  5. Logical relevance — premises bear on conclusion.
  6. Consistency with background knowledge.

23.4.3 Bacon, Mill, Hume, Popper — The Four-Person Story

TipFour-Person Story of Induction
  • Francis Bacon (Novum Organum, 1620) — first systematic empirical method; three “tables” of induction.
  • John Stuart Mill (A System of Logic, 1843) — 5 canons / methods of induction.
  • David Hume (1748) — the problem of induction: no logical guarantee from “the future will resemble the past”.
  • Karl Popper (Logik der Forschung, 1934 / Logic of Scientific Discovery, 1959) — replaced inductivism with falsificationism: science doesn’t prove, it disproves.

23.4.4 Mill’s Five Methods — In Detail

TipMill’s Five Methods
Method Idea Worked example
Agreement Common factor across all positive cases is the cause All sick children drank from the same well
Difference Single difference between a positive and a negative case is the cause Two groups identical except for vaccination; only unvaccinated fell ill
Joint Agreement & Difference Combine the two — strongest causal evidence Both above together
Residues Subtract known causes; the residue causes the rest Of three measured effects, two are explained; the residue identifies the third cause
Concomitant Variation When X varies, Y varies → X and Y are causally linked Crop yield rises systematically with fertiliser dose

23.4.5 Common Inductive Fallacies

TipCommon Inductive Fallacies
  • Hasty generalisation — too few cases.
  • Biased sample — non-representative.
  • False cause (post hoc, ergo propter hoc) — temporal sequence ≠ causation.
  • Cherry-picking — only confirming evidence.
  • Survivorship bias — only studying those that “made it”.
  • Ignoring the base rate — base-rate neglect.
  • Confirmation bias.
  • Confusing correlation with causation.

23.5 Abduction — The Third Mode

Charles Sanders Peirce added a third mode: abduction — inference to the best explanation (Topic 18). Sometimes treated as a sub-type of induction; sometimes as a distinct mode.

TipAbduction in One Line

“The lawn is wet. Best explanation: It rained last night.”

23.6 How to Classify a Given Argument

TipThree-Question Classification Test
  1. Does the conclusion claim necessity? If yes → deductive.
  2. Does the conclusion go beyond what the premises strictly contain? If yes → inductive.
  3. Is the premise a sample / observation, with a generalising conclusion? → inductive.

23.6.1 Worked Classifications

TipWorked Classifications
  • “All metals expand on heating. Iron is a metal. Therefore iron expands on heating.” → Deductive (necessary conclusion).
  • “In every observed experiment, this metal has expanded on heating. Therefore this metal always expands on heating.” → Inductive (enumerative).
  • “Every student who attended class scored well. Riya scored well. Therefore Riya attended class.” → Invalid deductive (affirming the consequent).
  • “Smoke is observed near the kitchen. The best explanation is cooking.” → Abductive.

23.7 Common Confusions

TipCommon Confusions to Avoid
  • Validity is not truth. A valid argument may have false premises.
  • Soundness is not certainty about the world. Even a sound argument relies on premises whose truth must be checked.
  • Inductive strength is not validity. Inductive arguments are not valid or invalid; they are strong or weak.
  • Falsifiable does not mean false. A theory is scientific because it could be refuted in principle.
  • Mill’s methods are inductive, not deductive.

23.8 Connections to Research Methods

TipReasoning ↔︎ Research Method
  • Quantitative / experimental research = often deductive (hypothesis testing).
  • Qualitative / grounded-theory research = often inductive (theory emerges from data).
  • Mixed-methods = both.
  • Mill’s methods underlie most causal-comparative and quasi-experimental designs.
  • Popper’s falsificationism shapes modern hypothesis-driven science.
  • Peirce’s abduction appears in hypothesis-generation and machine-learning.

23.9 Theory Anchors

TipPersons, Years and Key Ideas
Person Year Contribution
Aristotle 4th c. BCE Categorical syllogism; A, E, I, O; “father of deductive logic”
Theophrastus 4th c. BCE Refined syllogistic figures
Francis Bacon 1620 Novum Organum — inductive method; 3 tables
John Stuart Mill 1843 5 methods of induction
David Hume 1748 Problem of induction
C.S. Peirce 19th c. Abduction — inference to best explanation
Karl Popper 1959 Falsificationism replaces inductivism
R.A. Fisher 1925, 1935 Statistical induction; design of experiments
Wesley Salmon 20th c. Statistical relevance theory
Carnap & Hempel 20th c. Inductive logic; confirmation theory

23.10 Practice Questions

Q 01 Distinction Easy

Deductive reasoning moves from:

  • AParticular to general
  • BGeneral to particular
  • CObservation to best explanation
  • DSymbol to meaning
View solution
Correct Option: B
General to particular; conclusion necessarily follows.
Q 02 Validity Medium

Which of the following is NOT a property of inductive arguments?

  • AConclusion is probable, not certain
  • BConclusion goes beyond premises
  • CConclusion is necessarily true if premises are true
  • DStrength can vary in degree
View solution
Correct Option: C
That is the property of deductive arguments; inductive conclusions are only probable.
Q 03 Soundness Medium

A deductive argument is SOUND if and only if it is:

  • AValid
  • BValid AND has all true premises
  • CHas true premises only
  • DConvincing
View solution
Correct Option: B
Sound = Valid + true premises.
Q 04 Classify Medium

"In every observed instance, water boils at 100°C at sea level. Therefore water always boils at 100°C at sea level." This is:

  • ADeductive — valid
  • BInductive — enumerative
  • CAbductive
  • DDeductive — invalid
View solution
Correct Option: B
Generalising from observed instances to a universal claim = inductive (enumerative).
Q 05 Classify Medium

"All birds have wings. A sparrow is a bird. Therefore a sparrow has wings." This is:

  • AInductive
  • BAbductive
  • CDeductive — valid
  • DDeductive — invalid
View solution
Correct Option: C
Classical syllogism, Figure 1 AAA (Barbara). Valid.
Q 06 Fallacy Hard

"If it rains, the road is wet. The road is wet. Therefore it rained." This commits the fallacy of:

  • AAffirming the Consequent
  • BDenying the Antecedent
  • CModus Tollens
  • DHypothetical Syllogism
View solution
Correct Option: A
If P→Q, Q, ∴ P. Invalid. The road could be wet for other reasons.
Q 07 Modus Tollens Medium

"If P then Q. Not Q. Therefore not P." This is:

  • AModus Ponens
  • BModus Tollens
  • CDisjunctive Syllogism
  • DDenying the Antecedent
View solution
Correct Option: B
Modus Tollens — denying the consequent to deny the antecedent. Valid.
Q 08 Mill Hard

In a clinical trial, two groups of patients are identical except that one receives a drug and the other a placebo. Only the placebo group worsens. The drug is concluded to be effective. This uses Mill's:

  • AMethod of Agreement
  • BMethod of Difference
  • CMethod of Residues
  • DMethod of Concomitant Variation
View solution
Correct Option: B
Single difference between positive and negative cases = Method of Difference. Foundational logic of RCTs.
Q 09 Mill Hard

All five villages where cholera broke out drew water from the same well. The well is concluded to be the cause. This uses Mill's:

  • AMethod of Agreement
  • BMethod of Difference
  • CJoint Method
  • DMethod of Residues
View solution
Correct Option: A
Common factor across all positive cases = Agreement. (Classic John Snow / cholera case.)
Q 10 Bacon Medium

*Novum Organum* (1620), the foundational treatise on the inductive method, was authored by:

  • AFrancis Bacon
  • BJ.S. Mill
  • CDavid Hume
  • DKarl Popper
View solution
Correct Option: A
Francis Bacon, Novum Organum, 1620.
Q 11 Hume Hard

The "problem of induction" — that induction cannot be justified by appeal to the uniformity of nature without circularity — is associated with:

  • AAristotle
  • BFrancis Bacon
  • CDavid Hume
  • DJ.S. Mill
View solution
Correct Option: C
David Hume (1748) — the classic critique that induction presupposes what it sets out to prove.
Q 12 Popper Medium

Karl Popper proposed which principle as the criterion of scientific status?

  • AVerifiability
  • BFalsifiability
  • CProductivity
  • DAesthetic appeal
View solution
Correct Option: B
Falsifiability. A theory is scientific only if it could in principle be refuted.
Q 13 Peirce Hard

The mode of reasoning called "inference to the best explanation" was named by:

  • AAristotle
  • BJ.S. Mill
  • CC.S. Peirce
  • DKarl Popper
View solution
Correct Option: C
Charles Sanders Peirce — abduction.
Q 14 Strength Medium

Which of the following is NOT a criterion of a strong inductive argument?

  • ALarge sample size
  • BRepresentativeness of sample
  • CValidity of form
  • DVariety of cases
View solution
Correct Option: C
Inductive arguments are not valid/invalid; they are strong/weak. Validity is a deductive concept.
Q 15 Concomitant Medium

Crop yield rises systematically as fertiliser quantity rises. Researcher concludes fertiliser causes yield. This uses Mill's:

  • AMethod of Agreement
  • BMethod of Difference
  • CMethod of Residues
  • DMethod of Concomitant Variation
View solution
Correct Option: D
Variables changing together = Concomitant Variation.
Q 16 Fallacy Hard

"Yesterday I wore a blue shirt and my team won. So I'll wear it again today to make them win." This is the fallacy of:

  • APost hoc ergo propter hoc
  • BAd hominem
  • CBegging the question
  • DEquivocation
View solution
Correct Option: A
Post hoc ergo propter hoc — "after this, therefore because of this". Temporal sequence ≠ causation.
Q 17 Research Medium

A researcher generates theory bottom-up from interview data via open, axial, and selective coding. The dominant logic of inference here is:

  • ADeductive
  • BInductive
  • CAbductive only
  • DFalsificationist
View solution
Correct Option: B
Bottom-up from data to theory = inductive. Grounded theory (Glaser & Strauss 1967) is the classic case.
Q 18 Counter-example Hard

A single counter-example is sufficient to:

  • AProve a strong inductive argument
  • BRefute a universal inductive generalisation
  • CMake a deductive argument sound
  • DEstablish abduction
View solution
Correct Option: B
One black swan refutes "all swans are white" — Popper's famous example.
Q 19 Aristotle Medium

The categorical syllogism is the work of:

  • AAristotle
  • BHume
  • CMill
  • DPopper
View solution
Correct Option: A
Aristotle — father of deductive logic.
Q 20 Match Hard

Match each theorist with the idea most associated:

(i) Aristotle (a) Falsificationism
(ii) J.S. Mill (b) Categorical syllogism
(iii) Hume (c) Five methods of induction
(iv) Popper (d) Problem of induction
  • A(i)-b, (ii)-c, (iii)-d, (iv)-a
  • B(i)-a, (ii)-b, (iii)-c, (iv)-d
  • C(i)-c, (ii)-d, (iii)-a, (iv)-b
  • D(i)-d, (ii)-a, (iii)-b, (iv)-c
View solution
Correct Option: A
Aristotle → syllogism; Mill → 5 methods; Hume → problem of induction; Popper → falsificationism.

23.11 Quick Recall

ImportantQuick recall
  • Deductive: general → particular; necessary; valid/invalid + sound/unsound.
  • Inductive: particular → general; probable; strong/weak + cogent/uncogent.
  • Soundness = valid + true premises.
  • Test for validity: truth-table · counter-example · Venn diagram.
  • 5 valid forms: Modus Ponens · Modus Tollens · Hypothetical Syllogism · Disjunctive Syllogism · Constructive Dilemma.
  • 3 formal fallacies: Affirming Consequent · Denying Antecedent · Undistributed Middle.
  • Aristotle: categorical syllogism; A, E, I, O; 24 valid moods.
  • Inductive types (5): Enumerative · Causal · Statistical · Analogical · Predictive.
  • 6 criteria for strong induction: sample size · representativeness · variety · no ignored counter-examples · relevance · consistency with background knowledge.
  • Bacon (Novum Organum, 1620): 3 tables — agreement, difference, degrees.
  • Mill (System of Logic, 1843) — 5 Methods: Agreement · Difference · Joint · Residues · Concomitant Variation.
  • Hume (1748): problem of induction (uniformity of nature is circular).
  • Popper (1959): falsificationism replaces inductivism. One counter-example refutes a universal claim.
  • Peirce: abduction — inference to best explanation.
  • Inductive fallacies: hasty generalisation · biased sample · post hoc · cherry-picking · survivorship bias · base-rate neglect · confirmation bias · correlation/causation confusion.
  • Quantitative research = often deductive; qualitative = often inductive; mixed = both.