Goa PSC – Reasoning Ability
Topic: Statement & Conclusion
In “Statement & Conclusion” questions you are given a premise (the Statement) and two propositions (the Conclusions).
Your job is to decide for each Conclusion whether it necessarily follows, could be true, or definitely false from the given Statement.
Why it matters
This format tests deductive reasoning, logical inference, and attention to detail – all skills that are heavily weighted in the Goa PSC and in many competitive exams (IIT‑JEE, JEE‑Advanced, GRE, GMAT, etc.).
1. The Anatomy of a Statement & Conclusion Question
| Component | What it is | Example |
|---|---|---|
| Statement | A fact, observation or scenario that is taken as true. | “All the students present in the hall are wearing blue shirts.” |
| Conclusion 1 | A proposition that may or may not be valid. | “Some of the students in the hall are wearing red shirts.” |
| Conclusion 2 | Another proposition that may or may not be valid. | “All the students in the hall are wearing blue shirts.” |
Answering strategy – For each conclusion you will decide from the set:
| Option | Meaning | Example |
|---|---|---|
| A | Both conclusions are valid. | N/A |
| B | Only Conclusion 1 is valid. | N/A |
| C | Only Conclusion 2 is valid. | N/A |
| D | Neither conclusion is valid. | N/A |
| E | Both conclusions are possible but not necessarily valid. | N/A |
Rule of Thumb:
Validity = “must be true given the Statement.”
Possibility = “could be true, but is not guaranteed.”
Invalidity = “cannot be true regardless of how the Statement is interpreted.”
2. Logical Foundations & Common Inference Rules
| Logical Rule | Formal Representation | Practical Use in Statements |
|---|---|---|
| Modus Ponens | If P → Q and P then Q. | “If it rains, the ground will be wet.” Given P = “it rains”; conclude Q. |
| Modus Tollens | If P → Q and ¬Q then ¬P. | “If it rains, the ground will be wet.” Given Q = “the ground is not wet”; conclude ¬P. |
| Universal Instantiation | If ∀x P(x) then P(a). | “All students are diligent.” Pick any student. |
| Existential Generalization | If P(a) then ∃x P(x). | “John is diligent.” ⇒ “Some student is diligent.” |
| Contrapositive | P → Q is equivalent to ¬Q → ¬P. | Same as Modus Tollens. |
| Disjunctive Syllogism | P ∨ Q, ¬P ⇒ Q. | “Either it is raining or it is sunny.” Given “not raining”, conclude “sunny”. |
| Conjunction & Disjunction | Combine or split conditions. | “A and B” → A ∧ B. “Either A or B” → A ∨ B. |
Key Insight – Most Statement & Conclusion questions reduce to basic propositional logic (P, Q, etc.) or simple set logic (∃, ∀). Recognizing these patterns is the fastest way to answer.
3. Step‑by‑Step Deductive Procedure
- Read the Statement Carefully
- Identify quantifiers: all, none, some, most, at least, exactly.
- Note any conditional phrases: “if”, “only if”, “provided that”.
- Translate into Logical Symbols
- All X are Y → ∀x (X(x) → Y(x))
- No X are Y → ∀x (X(x) → ¬Y(x))
- Some X are Y → ∃x (X(x) ∧ Y(x))
- If X then Y → X → Y
- Derive Immediate Consequences
- Apply the inference rules above to the formal representation.
- Evaluate Each Conclusion
- Must be true? → Valid.
- Could be true? → Possible (but not guaranteed).
- Cannot be true? → Invalid.
- Select the Correct Option
- Map your conclusions to the answer key (A–E).
Tip: If a Conclusion does not appear in the logical consequences list, it is invalid.
4. Common Pitfalls & Tricky Points
| Pitfall | What Happens | How to Avoid |
|---|---|---|
| Assuming “All” means “Exactly All” | “All students are blue” → Some may be also wearing red. | Remember: All = “every member satisfies property”, None = “no member satisfies property”. |
| Treating “Some” as “All” | “Some students are absent” → Concluding “All are absent” is false. | “Some” only guarantees existence, not |