Keep a condition and its reverse separate
“Every accepted application is signed” means accepted → signed. A signature is necessary for acceptance under this rule. It does not say that a signature alone guarantees acceptance.
| Statement | Does it follow? |
|---|---|
| X is accepted, so X is signed. | Yes: this uses the given rule. |
| X is unsigned, so X is not accepted. | Yes: this is the contrapositive. |
| X is signed, so X is accepted. | No: a signature may not be sufficient. |
| X is not accepted, so X is unsigned. | No: a signed application might be rejected for another reason. |
Use a counterexample
To challenge “Every signed application is accepted,” imagine a signed application that misses a deadline and is rejected. It still obeys the original rule: every application that was accepted was signed. One such consistent counterexample shows that the reversed conclusion is not forced.
You are not claiming that this rejection actually happened. You are checking whether the proposed conclusion is logically necessary.
Translate all, some and none
- All A are B: the A set lies inside B.
- Some A are B: at least one member belongs to both sets.
- No A are B: the two sets do not overlap.
“All poets in a club are readers. Some club members are poets.” The second statement establishes that those poets exist, so some club members are readers. It does not establish that all readers are poets or that every member is a poet.
In formal reasoning, an “all” statement alone does not establish that the named set contains a member. Use the item’s stated assumptions and any explicit existence claim.
Worked selection rule
A team chooses exactly two people from A, B and C. If A is chosen, B must also be chosen. Which pair is forbidden?
Show the valid and invalid pairs
A–B is valid. B–C is valid because the rule does not require A whenever B appears. A–C is forbidden: it includes A without the required B. Listing the three pairs makes the distinction visible.
Check an argument’s missing link
Separate the conclusion from its evidence. If an argument says that students using a study tool scored higher, that association alone does not establish that the tool caused the difference. Starting ability or study time might also differ. A relevant assumption or weakening option should address the link used by that particular argument.
Try the rule in a new setting
Only registered members may enter a reading room. Sara entered the room in accordance with the rule. What follows?
Answer and explanation
Sara is a registered member. “Only registered members may enter” makes registration necessary for permitted entry. It does not say that every registered member entered or that registration is the only condition the room may impose.
Examples on this page are original teaching exercises. For exam dates, eligibility, required stream and official instructions, consult the current HEC notice and ETC portal.
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