Statement & Conclusion – Advanced Study Notes
(For HTET – Reasoning Ability – “Statement & Conclusion” Section)
Goal – Learn to decide, with absolute certainty, whether a conclusion follows logically from a set of statements (premises).
Why it matters – In competitive exams, the “Statement & Conclusion” (S&C) question tests deductive reasoning, logical inference, and critical thinking. Mastery requires a solid grasp of formal logic, the ability to spot hidden assumptions, and the skill to eliminate fallacious reasoning.
1. Foundations: From Logic to Reasoning
| Concept | Formal Definition | Why It Matters for S&C |
|---|---|---|
| Proposition | A declarative sentence that is either true or false. | S&C premises & conclusions are propositions. |
| Premise | A proposition assumed to be true within an argument. | Determining if a conclusion follows from premises. |
| Conclusion | A proposition that is asserted to follow from the premises. | The question asks whether the conclusion necessarily follows. |
| Inference | A logical transition from premises to conclusion. | Every S&C question is essentially a test of inference. |
| Validity | An inference is valid if there is no possible situation where all premises are true and the conclusion is false. | S&C asks for validity. |
| Soundness | An inference is sound if it is valid and its premises are actually true. | In most S&C questions, truth of premises is not required; only validity matters. |
Key Insight
In S&C, we pretend that premises are true. We check if every truth‑assignment that satisfies all premises also satisfies the conclusion.
If yes → Conclusion necessarily follows.
If not → Conclusion does not necessarily follow.
2. Logical Language & Tools
2.1 Symbols & Operators
| Symbol | Meaning | Example |
|---|---|---|
∧ |
And | P ∧ Q is true iff both P and Q are true. |
∨ |
Or | P ∨ Q is true iff at least one of P, Q is true. |
¬ |
Not | ¬P is true iff P is false. |
→ |
If … then | P → Q is false only when P true & Q false. |
↔ |
If and only if | P ↔ Q true iff P and Q have same truth value. |
∀x |
For all x | ∀x (P(x)). |
∃x |
There exists x | ∃x (P(x)). |
2.2 Inference Rules (Commonly Tested)
| Rule | Example | Conditions | Notes |
|---|---|---|---|
| Modus Ponens (MP) | P → Q, P ⊢ Q |
Premise P true |
Basic “if‑then” rule |
| Modus Tollens (MT) | P → Q, ¬Q ⊢ ¬P |
Conclusion false | Useful for negated conclusions |
| Disjunctive Syllogism (DS) | P ∨ Q, ¬P ⊢ Q |
One disjunct denied | Eliminates one possibility |
| Hypothetical Syllogism (HS) | P → Q, Q → R ⊢ P → R |
Transitive “if‑then” | |
| Constructive Dilemma (CD) | (P → Q) ∧ (R → S), P ∨ R ⊢ Q ∨ S |
Either antecedent holds | |
| De Morgan’s Laws | ¬(P ∧ Q) ≡ ¬P ∨ ¬Q |
Negating conjunctions | |
| Contrapositive | P → Q ≡ ¬Q → ¬P |
Re‑ordering | |
| Biconditional Introduction | P → Q and Q → P ⊢ P ↔ Q |
Mutual implication | |
| Quantifier Rules | ∀x P(x) → Q ≡ ∀x (P(x) → Q) |
Distribution |
2.3 Truth Tables & Truth‑Functional Validity
- Write premises & conclusion in propositional form.
- Construct truth table for all variables.
- Check rows where all premises