IMAT Logical Reasoning · Free notes
Deductive Logic: Statements, Conditionals and Puzzles
Deductive Logic has 13 past questions since 2011, and the ministry papers from 2023 on have asked 6 of them. That is close to one in every paper, and the ministry items are short: a statement or two, then one question about what can be deduced. No outside knowledge is needed, only exact reading of small words such as all, some, only if and unless. The difficulty sits in the options, where a statement that is true in real life, or the converse of the given rule, waits beside the one conclusion that actually follows.
The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.
Subtopic notes
Statements, Quantifiers and Negation
0 PYQsA statement is either true or false; words like all, some, none and at least one decide exactly what it claims, and what its opposite claims.
If, Only If and Unless: Conditionals and Valid Inference
0 PYQsAn if-then statement links a condition to a result; only some ways of reasoning from it are valid, and the contrapositive is the one rewording that keeps its meaning.
Syllogisms: Reasoning with All, Some and No
0 PYQsTwo statements about groups can force a third; a Venn diagram drawn in every allowed way shows whether a conclusion must, could or cannot be true.
Certainty, Truth-tellers and Assignment Puzzles
0 PYQsPuzzles give a set of rules and ask what must follow; you solve them by testing cases, building the worst case, and crossing out what the rules forbid.
Formula & revision sheet
2 formulas · 5 reference tables · 14 gotchas across all subtopics — the exam-eve cheat-sheet
Formula & revision sheet
2 formulas · 5 reference tables · 14 gotchas across all subtopics — the exam-eve cheat-sheet
Reference tables (1)
Negating statements with all, some, none and at least one7 rows
| Statement | Its negation | Not the negation |
|---|---|---|
| All A are B | Some A are not B (at least one A is not B) | No A are B |
| No A are B | Some A are B | All A are B |
| Some A are B | No A are B | Some A are not B |
| Some A are not B | All A are B | No A are B |
| At least two came | At most one came (one or none) | Nobody came |
| P and Q | Not P, or not Q, or both | Not P and not Q |
| P or Q | Not P and not Q | Not P, or not Q |
Watch out for (4)
- True is not the same as deducible→ Propositions: statements that are either true or false
- The negation of all is not none→ Negating statements with all, some, none and at least one
- "Not more than" allows equality→ Comparison statements: equivalent wordings and ordering chains
- Items not joined by a chain cannot be compared→ Comparison statements: equivalent wordings and ordering chains
Reference tables (3)
If P then Q: the converse, the inverse and the contrapositive4 rows
| Form | Pattern | Example | Same meaning as the original? |
|---|---|---|---|
| Original | If P then Q | If it is a whale, it is a mammal. | Yes |
| Converse | If Q then P | If it is a mammal, it is a whale. | No |
| Inverse | If not P then not Q | If it is not a whale, it is not a mammal. | No (it matches the converse) |
| Contrapositive | If not Q then not P | If it is not a mammal, it is not a whale. | Yes |
Necessary and sufficient conditions: only if, unless, if and only if6 rows
| Wording | As an arrow | What it tells you |
|---|---|---|
| If P, then Q | P is sufficient for Q; Q is necessary for P | |
| Q if P | Same as above: the word after "if" is the condition | |
| P only if Q | Q is necessary for P | |
| Only Q can be P (only Qs are Ps) | Every P is a Q | |
| P unless Q | Without Q, P happens | |
| P if and only if Q | Each is necessary and sufficient for the other |
Valid inference: modus ponens and modus tollens, and the two fallacies5 rows
| Name | Premises | Conclusion | Valid? |
|---|---|---|---|
| Modus ponens | If P then Q; P | Q | Yes |
| Modus tollens | If P then Q; not Q | Not P | Yes |
| Affirming the consequent | If P then Q; Q | P | No |
| Denying the antecedent | If P then Q; not P | Not Q | No |
| Chain | If P then Q; if Q then R | If P then R | Yes |
Watch out for (4)
- The converse is not equivalent→ If P then Q: the converse, the inverse and the contrapositive
- "Only if" introduces the necessary condition→ Necessary and sufficient conditions: only if, unless, if and only if
- Affirming the consequent looks like modus ponens→ Valid inference: modus ponens and modus tollens, and the two fallacies
- Match the skeleton, not the topic→ Finding the argument with the same logical structure
Reference tables (1)
Drawing all, some and no statements as Venn diagrams5 rows
| Statement | How to draw it | Region you know about |
|---|---|---|
| All A are B | A drawn inside B | A outside B is empty |
| No A are B | A and B drawn apart | The overlap of A and B is empty |
| Some A are B | A and B overlapping, cross in the overlap | The overlap has a member |
| Some A are not B | Cross in A, outside B | A outside B has a member |
| Only As are Bs | B drawn inside A | B outside A is empty |
Watch out for (3)
- "All A are B" does not reverse→ Drawing all, some and no statements as Venn diagrams
- Two "some" statements prove nothing→ Testing a conclusion: must be true, could be true, cannot be true
- Some does not mean not all→ Testing a conclusion: must be true, could be true, cannot be true
Formulas (1)
Watch out for (3)
- "Could happen" is not "must happen"→ What must happen: worst-case reasoning and the pigeonhole principle
- A liar's "both" statement is false, not reversed word by word→ Truth-tellers and liars: assume a case and test it
- Check every clue against your chosen option→ Assignment puzzles: a grid and elimination
PYQ weightage by concept
12 concepts · 0 PYQs — where the marks actually sit, so you know what to drill first
PYQ weightage by concept
12 concepts · 0 PYQs — where the marks actually sit, so you know what to drill first
| Concept | PYQs | Share |
|---|---|---|
| Propositions: statements that are either true or falsefoundation | — | — |
| Negating statements with all, some, none and at least onefoundation | — | — |
| Comparison statements: equivalent wordings and ordering chainsfoundation | — | — |
| Concept | PYQs | Share |
|---|---|---|
| If P then Q: the converse, the inverse and the contrapositivefoundation | — | — |
| Necessary and sufficient conditions: only if, unless, if and only iffoundation | — | — |
| Valid inference: modus ponens and modus tollens, and the two fallaciesfoundation | — | — |
| Finding the argument with the same logical structurefoundation | — | — |
| Concept | PYQs | Share |
|---|---|---|
| Drawing all, some and no statements as Venn diagramsfoundation | — | — |
| Testing a conclusion: must be true, could be true, cannot be truefoundation | — | — |
| Concept | PYQs | Share |
|---|---|---|
| What must happen: worst-case reasoning and the pigeonhole principlefoundation | — | — |
| Truth-tellers and liars: assume a case and test itfoundation | — | — |
| Assignment puzzles: a grid and eliminationfoundation | — | — |