flowchart TB
M[Mill's Methods] --> A[Agreement]
M --> D[Difference]
M --> J[Joint Agreement & Difference]
M --> R[Residues]
M --> C[Concomitant Variations]
classDef default fill:#003366,color:#ffffff,stroke:#ffcc00,stroke-width:3px,rx:10px,ry:10px;
22 Deductive and Inductive Reasoning
22.1 Deductive Reasoning
Deductive reasoning moves from general to particular. If the premises are true and the form is valid, the conclusion is necessarily true.
22.1.1 Categorical Syllogism
A categorical syllogism is an argument with exactly three categorical statements — two premises and one conclusion — using exactly three terms.
TipThe Three Terms
- Major term (P) — the predicate of the conclusion.
- Minor term (S) — the subject of the conclusion.
- Middle term (M) — appears in both premises but not in the conclusion.
TipExample with Terms Marked
- Premise 1 (Major): All men (M) are mortal (P).
- Premise 2 (Minor): Socrates (S) is a man (M).
- Conclusion: Therefore, Socrates (S) is mortal (P).
22.1.2 Four Types of Categorical Statements
TipA, E, I, O — the Four Standard Forms
| Letter | Type | Quantity | Quality | Form | Example |
|---|---|---|---|---|---|
| A | Universal Affirmative | Universal | Affirmative | All S are P | All cats are mammals |
| E | Universal Negative | Universal | Negative | No S are P | No fish are mammals |
| I | Particular Affirmative | Particular | Affirmative | Some S are P | Some students are athletes |
| O | Particular Negative | Particular | Negative | Some S are not P | Some students are not athletes |
The letters A, I (affirmative) come from Latin AffIrmo; E, O (negative) from nEgO.
22.1.3 Six Rules of a Valid Categorical Syllogism
TipRules for Validity
| Rule | What it requires |
|---|---|
| 1 | A syllogism must have exactly three terms |
| 2 | The middle term must be distributed in at least one premise |
| 3 | A term distributed in the conclusion must be distributed in its premise |
| 4 | Two negative premises yield no valid conclusion |
| 5 | If one premise is negative, the conclusion must be negative |
| 6 | Two particular premises yield no valid conclusion |
A term is distributed if the statement says something about every member of the class it names.
TipDistribution of Terms
| Statement | Subject | Predicate |
|---|---|---|
| A — All S are P | distributed | not distributed |
| E — No S are P | distributed | distributed |
| I — Some S are P | not distributed | not distributed |
| O — Some S are not P | not distributed | distributed |
22.1.4 Hypothetical and Disjunctive Syllogisms
TipTwo Other Standard Syllogisms
| Type | Pattern | Example |
|---|---|---|
| Hypothetical syllogism | If P then Q; If Q then R; therefore if P then R | If it rains, the road is wet. If the road is wet, accidents rise. So if it rains, accidents rise. |
| Disjunctive syllogism | Either P or Q; not P; therefore Q | Either A is at home or away. A is not at home. So A is away. |
22.2 Inductive Reasoning
Inductive reasoning moves from particular to general. The conclusion is probable given the premises, never certain.
22.2.1 Types of Inductive Inference
TipSix Common Forms of Induction
| Form | Description | Example |
|---|---|---|
| Generalisation | From a sample to a population | “All sampled swans are white → All swans are white” |
| Statistical induction | From % in sample to % in population | “60 % of sampled voters favour A → 60 % of population does” |
| Causal inference | Identifying cause from observed correlation | “Smoking correlates with cancer → Smoking causes cancer” |
| Analogical inference | If A and B are alike in known ways, alike in further ways | “Mars is like Earth → Mars may have life” |
| Inductive prediction | From past pattern to future case | “Sun has risen every day → It will rise tomorrow” |
| Inductive argument from authority | Authority X says P → P is probably true | “Doctor says X is true → X is probably true” |
22.2.2 Strong vs Weak Induction
A strong inductive argument has:
- A large sample (small samples generalise poorly).
- A representative sample (no systematic bias).
- Relevant similarities (in analogies).
- Reproducibility (in causal inferences).
22.3 Mill’s Methods of Inductive Inference
The British philosopher John Stuart Mill (1843) systematised five methods of inductive reasoning, especially for identifying causes.
TipMill’s Five Methods
| Method | Principle | Working idea |
|---|---|---|
| 1. Method of Agreement | If two or more cases of the phenomenon share only one circumstance in common, that circumstance is the cause | Two food poisoning cases — only common food is the cause |
| 2. Method of Difference | If a case where the phenomenon occurs and a case where it does not occur differ in only one circumstance, that circumstance is the cause | Same field, two plots; only one fertilised; only that plot grows differently |
| 3. Joint Method of Agreement and Difference | Combines (1) and (2) | More reliable than either alone |
| 4. Method of Residues | After known causes are accounted for, the residual phenomenon is due to the residual circumstance | Discovery of Neptune from Uranus’s orbital residual |
| 5. Method of Concomitant Variations | When two phenomena vary together in a regular way, one causes the other (or shares a cause) | Heart rate rises with exercise intensity |
22.4 Comparing Deduction and Induction
TipDeduction vs Induction at a Glance
| Dimension | Deduction | Induction |
|---|---|---|
| Direction | General → Particular | Particular → General |
| Conclusion | Necessary if premises true | Probable; never certain |
| Strength term | Valid / Invalid | Strong / Weak |
| Quality term | Sound / Unsound | Cogent / Uncogent |
| Information | No new content beyond premises | Conclusion goes beyond premises |
| Truth-preserving | Yes | No |
| Defeasible | No (within given premises) | Yes — overturned by new evidence |
22.5 Practice Questions
Q 01
Categorical Syllogism
Easy
A categorical syllogism contains exactly:
View solution
Correct Option: B
A categorical syllogism has 3 terms (major, minor, middle) and 3 statements (two premises and one conclusion).
Q 02
Categorical Statement Types
Medium
"Some students are not athletes." This is a categorical statement of type:
View solution
Correct Option: D
"Some S are not P" is the O form — Particular Negative.
Q 03
Middle Term
Medium
In a valid categorical syllogism, the middle term must be:
View solution
Correct Option: A
The middle term must be distributed in at least one premise; if not, the syllogism commits the fallacy of "undistributed middle".
Q 04
Distribution
Hard
In the statement "All cats are mammals", which terms are distributed?
View solution
Correct Option: B
In an A statement (All S are P), only the subject is distributed. The predicate is not — we don't claim "all mammals are cats".
Q 05
Mill's Methods
Hard
Two students fall ill after the same picnic. The investigators find that the only food common to both is a particular salad. They conclude the salad caused the illness. This is an example of:
View solution
Correct Option: A
Method of Agreement — when two or more instances share only one circumstance, that circumstance is the cause.
Q 06
Concomitant Variations
Medium
"As exercise intensity increases, heart rate increases." This causal inference uses Mill's:
View solution
Correct Option: C
Concomitant variations — when two phenomena vary together in a regular pattern, one causes the other or shares a cause.
Q 07
Two Negative Premises
Medium
According to the rules of categorical syllogism, two negative premises:
View solution
Correct Option: B
Rule 4 of categorical syllogism: two negative premises yield no valid conclusion.
Q 08
Deduction vs Induction
Easy
Which of the following is a feature of inductive reasoning that distinguishes it from deductive reasoning?
View solution
Correct Option: C
In induction, the conclusion goes beyond the premises (with corresponding loss of certainty). Deduction is truth-preserving but adds no new content.
ImportantQuick recall
- Categorical syllogism: 3 terms (major, minor, middle), 3 statements.
- Four categorical types: A (All S are P), E (No S are P), I (Some S are P), O (Some S are not P).
- Distribution: A → S; E → S, P; I → none; O → P.
- Rules: 3 terms; middle term distributed once; no two negatives; if one premise negative, conclusion negative; no two particulars.
- Mill’s methods: Agreement, Difference, Joint A&D, Residues, Concomitant variations.
- Deduction = certain, truth-preserving; Induction = probable, content-extending.