Study Notes – Statement & Conclusion
(Reasoning Ability – J&K Police SI Exam)
Goal: Master the art of extracting valid conclusions from a set of given statements and recognise when a conclusion is True, False or Undetermined.
Exam‑style: Multiple statements (4–6) + multiple conclusions (5–6). Pick the correct conclusions (single or multiple).
1. What is a Statement & Conclusion?
| Term | Meaning |
|---|---|
| Statement | A premise or a fact that is given (e.g., “All dogs are mammals”). |
| Conclusion | A deduction that follows from the statements (e.g., “If X is a dog, then X is a mammal”). |
| Inference | The logical connection between statements and conclusion. |
Key point:
- A conclusion is valid only if it follows necessarily from all the statements.
- It may be True, False or Undetermined (cannot be decided from the given data).
2. Logical Language & Symbols
| Symbol | Meaning | Example |
|---|---|---|
| ∧ | AND | P ∧ Q → both P and Q are true |
| ∨ | OR | P ∨ Q → at least one of P or Q is true |
| ¬ | NOT | ¬P → P is false |
| → | IF … THEN | P → Q |
| ↔ | IF AND ONLY IF | P ↔ Q |
| ∀ | FOR ALL | ∀x (P(x)) |
| ∃ | THERE EXISTS | ∃x (P(x)) |
Practice tip: Convert every given sentence into symbolic form before drawing conclusions.
3. Basic Truth Tables
| P | Q | P ∧ Q | P ∨ Q | P → Q | ¬P |
|---|---|---|---|---|---|
| T | T | T | T | T | F |
| T | F | F | T | F | F |
| F | T | F | T | T | T |
| F | F | F | F | T | T |
- Derived rules:
- Contrapositive: (P → Q) ≡ (¬Q → ¬P).
- De Morgan: ¬(P ∧ Q) ≡ (¬P ∨ ¬Q); ¬(P ∨ Q) ≡ (¬P ∧ ¬Q).
4. Types of Inference
| Type | Description | Example |
|---|---|---|
| Deduction | From general to specific | All A are B. C is A. → C is B. |
| Induction | From specific to general | 3/4 of sampled people drink tea. → Most people drink tea. |
| Abduction | From effect to probable cause | Stairs are wet → it rained. |
| Transposition | Switching antecedent and consequent (contrapositive) | If it rains, the ground gets wet → If the ground is not wet, it did not rain. |
Exam tip: Most “Statement & Conclusion” questions use deduction.
5. Rules of Inference (Commonly Tested)
| Rule | Symbolic Form | What it Does |
|---|---|---|
| Modus Ponens | P → Q, P ⊢ Q | If P then Q; P is true → Q is true. |
| Modus Tollens | P → Q, ¬Q ⊢ ¬P | If P then Q; Q is false → P is false. |
| Hypothetical Syllogism | P → Q, Q → R ⊢ P → R | Chain of implications. |
| Disjunctive Syllogism | P ∨ Q, ¬P ⊢ Q | One of P or Q; P is false → Q is true. |
| Addition | P ⊢ P ∨ Q | From P, we can add any disjunction. |
| Simplification | P ∧ Q ⊢ P | From conjunction, take any part. |
| Conjunction | P, Q ⊢ P ∧ Q | Combine two true statements. |
| Resolution | (P ∨ Q), (¬P ∨ R) ⊢ Q ∨ R | Classic rule in propositional logic. |