UPSC CSE · CSAT · Reasoning and judgment · TOPIC LESSON
Turn stated rules into valid deductions, and use complete cases to distinguish what must, may and cannot be true.
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BEFORE YOU START
Original educational lesson. Work through the examples before revealing the practice answers. This original common-core lesson supports its declared paper scope. Optional subjects, qualifying language papers and the interview are outside this resource.
LEARN THROUGH ORIGINAL PRACTICE
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01 / UNDERSTAND & APPLY
A proposition is an assertion that is true or false within the model: “The token is in box A” is one. “Put the token in A” is a command, not a truth-valued assertion. A question requests information rather than asserting its answer. Formal deduction works with the supplied assertions and their stated interpretation.
Negation denies the whole proposition. The negation of “Every drawer contains a key” is “At least one drawer contains no key”, not “No drawer contains a key”. The latter is much stronger. Likewise, denying “Both lamps are on” permits one or both lamps to be off.
Treat the stipulated model as your reference. A puzzle may assign unusual roles or perfectly reliable rules; do not reject its premise because real life is less tidy. At the same time, do not import unstated facts to fill gaps.
ORIGINAL PRACTICE · 1
At least one drawer in this cabinet contains no key.
One keyless drawer is enough to make the universal assertion false. Saying every drawer is keyless would exclude additional cases that already falsify the original statement.
02 / UNDERSTAND & APPLY
All A are B means the A set lies inside B. It does not say every B is A or that an A exists. No A is B means the sets do not overlap. Some A are B supplies at least one object in their overlap. Only that existential premise licenses a witness marker.
The negation of some A are B is no A is B. The negation of all A are B is some A are not B. These translations keep the domain fixed. A universal statement can hold in a model with no A objects, so avoid deriving “some” from “all” unless existence is separately supplied.
Two existential statements may concern different witnesses. Some readers are gardeners and some gardeners are singers do not force a reader-singer: the first gardener could differ from the second. Draw separate witnesses unless a premise connects them.
ORIGINAL PRACTICE · 1
No. A model with no gliders satisfies the premise.
The inclusion rule imposes a condition on any glider that exists; it does not supply one. If an additional premise said a glider exists, that object would also be a vehicle.
03 / UNDERSTAND & APPLY
P and Q requires both true. Inclusive P or Q requires at least one true and permits both. Exclusive either P or Q, but not both requires exactly one. The word or is inclusive unless the task clearly restricts it. Neither P nor Q means both are false.
Under the ordinary formal reading, “Unless P, Q” means “If not P, then Q”. It rules out not-P together with not-Q. It does not itself rule out P together with Q. Additional context can impose a stronger rule, but that stronger rule must be stated.
A compact truth check can prevent an intuitive reversal. List every combination of the relevant truth values, then cross out only combinations prohibited by the rule. Do not add exclusions merely because a permitted case sounds unusual.
ORIGINAL WORKED EXAMPLE
A stipulated rule says, “Unless the door is locked, the bell rings.” Let L mean locked and R mean rings. Which combinations are allowed, and does a locked door guarantee silence?
Answer Only unlocked-and-silent is excluded. A locked door may accompany a ringing or silent bell unless another rule says otherwise.
ORIGINAL PRACTICE · 1
No. Q may be true or false.
Both combinations satisfy at least one true. A conclusion that Q is false would require an exclusive-or condition.
ORIGINAL PRACTICE · 2
P is false and Q is false.
Neither excludes each proposition separately; it does not mean only one is false.
04 / UNDERSTAND & APPLY
For an exceptionless model rule P implies Q, P is sufficient for Q and Q is necessary for P. P true gives Q true; Q false gives P false. The latter is contraposition. But Q true leaves P open, and P false leaves Q open.
A biconditional P if and only if Q adds the reverse rule. Without it, converses and inverses are not licensed. For example, every winner may be registered even though many registered people do not win. In that rule registration is necessary, not sufficient.
These formal operations apply to strict premises, not automatically to a tendency or an obligation that can be violated. Keep the model’s wording visible when importing a rule into a deduction chain.
ORIGINAL WORKED EXAMPLE
The model states: if the lamp is powered, the switch is on; if the switch is on, the fuse is intact. The fuse is not intact. What follows about the switch and lamp? What would an intact fuse alone establish?
Answer The switch is not on and the lamp is not powered. An intact fuse alone would not establish switch position or lamp power.
ORIGINAL PRACTICE · 1
No. Uma may be a registered nonwinner.
The proposed conclusion reverses the given inclusion. A model containing Uma as registered but not winning satisfies all supplied facts.
05 / UNDERSTAND & APPLY
Set reasoning becomes reliable when you distinguish boundaries from objects. An inclusion boundary says where any member must lie. A witness marks existence at a particular overlap or exclusion. If a witness is inside A and every A lies in B, that same witness belongs to B.
The diagram here places restorers wholly inside trained people and a volunteer witness inside restorers. Areas are schematic, not population sizes. The volunteer region extends beyond restorers to show that the premises do not force every volunteer to restore things. This extension indicates a permitted possibility, not an asserted census.
For a conclusion involving some, identify the witness that guarantees it. For all or no, identify the rule chain that excludes every counterexample. An attractive drawing is not a substitute for checking those logical commitments.
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ORIGINAL WORKED EXAMPLE
All restorers are trained. Some volunteers are restorers. Which conclusions follow: some volunteers are trained; all volunteers are trained; all trained people are restorers?
Answer Only “some volunteers are trained” is guaranteed among the three proposed conclusions.
ORIGINAL WORKED EXAMPLE
All roses are plants. No plants are metal. Some displayed objects are roses. What follows, and could a different displayed object be metal?
Answer Some displayed objects are not metal, and no roses are metal. A different displayed object could be metal; the premises do not forbid it.
ORIGINAL PRACTICE · 1
Yes.
The poet-teacher witness must be a reader under the all-teachers rule. This uses the same object through the chain; it does not require every poet to teach.
06 / UNDERSTAND & APPLY
In a finite puzzle, a global condition such as exactly one statement true must be checked across the whole set. It does not mean every speaker independently tells one truth, and it does not identify which statement is true in advance.
Create a row for each possible underlying state. Evaluate every statement within that row, count the true statements and retain only rows satisfying the global rule. A statement may refer to the same location as another in opposite wording, which can simplify but should not replace the full check.
ORIGINAL WORKED EXAMPLE
Exactly one token is in one of boxes A, B and C. The labels state: A says “The token is in B”; B says “The token is not in B”; C says “The token is in A”. Exactly one of the three labels is true. Where can the token be?
Answer The token can be in B or C; it cannot be in A. The location is not uniquely determined, but A’s exclusion is certain.
ORIGINAL PRACTICE · 1
Q is true.
At least one must be true, and P is not. The exactly-one restriction is satisfied with Q true and P false.
07 / UNDERSTAND & APPLY
A possible conclusion holds in at least one valid model. A necessary conclusion holds in every valid model. An impossible conclusion holds in none. In the token example, “token in B” is possible; “token not in A” is necessary; “token in A” is impossible.
An unresolved target does not make every conclusion unresolved. You can know an invariant while several complete arrangements remain. Conversely, finding one satisfying arrangement proves possibility but does not prove necessity. To disprove necessity, present another valid arrangement where the proposed conclusion fails.
ORIGINAL PRACTICE · 1
B must be on; A can be on but need not be.
The possible on-pairs are AB, AC and BC before applying the implication. AC violates A implies B. AB and BC remain. Both contain B, only one contains A, and the list is exhaustive.
08 / UNDERSTAND & APPLY
For each proposed conclusion, state whether you are proving necessity, showing possibility or ruling out a case. Use the right evidence: a chain applying to all models for necessity, one complete valid model for possibility, and a contradiction for exclusion.
Check every premise against a countermodel, including existence, exactly-one conditions and the domain. A counterexample that quietly changes “all” to “usually” or inserts an unlisted box is not a counterexample to the given puzzle.
Finally distinguish contradiction from missing information. Contradictory premises admit no valid case and should be reported as inconsistent in an applied puzzle. Multiple valid cases indicate underdetermination. Neither should be disguised as a uniquely solved arrangement.
KEEP THE BIG IDEAS
READ WITH CONTEXT
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