IMAT Logical Reasoning · Deductive Logic
Syllogisms: Reasoning with All, Some and No
Two 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.
Why this matters
A 2026 ministry question joined a "some" statement to an "all" statement and asked what follows; a 2012 question asked which statement about three overlapping groups is definitely not true.
Concept 1 of 2: Drawing all, some and no statements as Venn diagrams
Each group is a circle, and each statement tells you something about one region of the picture. "All A are B" says the part of A outside B is empty. "Some A are B" says the overlap has at least one member. Once every statement is drawn, you read conclusions from the regions, not from the words.
Definition
In a Venn diagram each set is a circle; overlapping circles share members.
- All A are B: circle A sits inside circle B (the region "A but not B" is empty).
- No A are B: the circles do not overlap.
- Some A are B: the overlap contains at least one member (mark it with a cross).
- Some A are not B: the part of A outside B contains at least one member.
- "All A are B" says nothing about whether every B is an A: B may be larger.
| 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 |
Draw empty regions first, then place the crosses where they are forced to go.
Practice this conceptself-check · 3 quick reps
"All A are B" does not reverse
All cats are mammals, yet many mammals are not cats. An option that swaps the two groups of an "all" statement is the converse and does not follow.
Concept 2 of 2: Testing a conclusion: must be true, could be true, cannot be true
A conclusion follows only if it holds in every diagram that fits the premises. So try to break it: draw the premises in the least helpful way you can. If one allowed drawing makes the conclusion false, it does not follow. If no allowed drawing makes it true, it cannot be true.
Definition
For each option, ask which of three kinds it is:
- Must be true: true in every diagram that fits the premises.
- Could be true: true in some fitting diagrams, false in others.
- Cannot be true: false in every fitting diagram.
Useful patterns:
- All A are B, all B are C: all A are C.
- Some A are B, all B are C: some A are C.
- All A are B, no B are C: no A are C.
- Two "some" premises never force a conclusion.
Worked example
Premises: "Some cyclists are teachers. All teachers are graduates." Classify: (i) Some cyclists are graduates. (ii) All cyclists are graduates. (iii) Some graduates are not cyclists.
Practice this conceptself-check · 4 quick reps
Two "some" statements prove nothing
"Some A are B" and "some B are C" can be drawn with the A members and the C members in different parts of B. So "some A are C" is only a could-be. Options built from two "some" premises are never "must be true".
Some does not mean not all
"Some students passed" is still true if every student passed. An option that reads "some" as "some but not all" adds a claim that was never made.
Summary — formulas & gotchas at a glance
A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.
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 |
Draw empty regions first, then place the crosses where they are forced to go.
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
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