Logical reasoning practice

Draw nested groups for 'all' and a shared region for 'some'. Use only overlaps the clues actually guarantee.

Worked example: All owls in a park are birds. No birds there are indoors. An owl in that park cannot be indoors: follow owl → bird → not indoors.

Watch out: 'Some' does not mean 'all', and two groups sharing a larger group do not necessarily overlap each other.

Write one ordered chain. Combine clues through a shared person or object. Leave unrelated pairs undecided.

Worked example: R is before S, and T is before R. Combining the clues gives T → R → S. R must be second among these three.

Watch out: 'Before' does not mean 'immediately before'. Two people both ahead of someone are not ordered relative to each other.

Read 'if A, then B' as a one-way arrow. A guarantees B; missing B rules out A.

Worked example: If a badge is gold, it opens a door. A badge that does not open that door cannot be gold, assuming the rule holds.

Watch out: An open door does not prove a gold badge was used. A rule can describe one sufficient cause without listing all causes.

Follow a known member through each group rule. To test whether a conclusion must hold, try constructing a case where it fails.

Worked example: All kites in a collection are toys. Some toys are red. The red toys could all be balls, so a red kite is not guaranteed.

Watch out: A possible conclusion is weaker than a necessary one. Universal rules alone do not establish that a group has any members.

Number the positions, place fixed items, then treat adjacent items as a block. Test each remaining arrangement against every clue.

Worked example: Four tiles J, K, L, M fill a row. M is first and J-K are adjacent in that order. M-J-K-L and M-L-J-K both work: adjacency alone does not fix a position.

Watch out: Finding one valid arrangement proves it is possible, not that it is the only one. Look for a second arrangement before saying 'must'.

Try to break the rule: find a case where its condition holds but its consequence fails. For a chain, follow each link before drawing a conclusion.

Worked example: To test 'every square token is blue', look for a square token that is not blue. A blue circle does not disprove the rule.

Watch out: 'Only if' introduces a necessary condition. 'If and only if' gives both directions. Keep 'at least one' separate from 'exactly one'.