Venn diagrams and validity
Draw three overlapping circles, shade empty regions and mark occupied ones with X. If the premises already show the conclusion, the syllogism is valid.
Drawing each proposition
| Proposition | On the diagram |
|---|---|
| A: All S are P | Shade the part of S outside P |
| E: No S is P | Shade the overlap of S and P |
| I: Some S are P | X in the overlap of S and P |
| O: Some S are not P | X in the part of S outside P |
Testing a syllogism
- Draw three overlapping circles for S, P and M.
- Diagram the universal premise(s) first, by shading.
- Then diagram the particular premise with an X. If the X could fall in either of two unshaded parts, put it on the line between them.
- Do not draw the conclusion. If it can already be read off the diagram, the argument is valid; if not, it is invalid.
- Venn diagrams use the modern (Boolean) reading: a universal premise alone does not say that anything exists.
Worked example 1: a valid syllogism
All teachers are graduates. Some teachers are poets. Therefore, some poets are graduates. Valid or invalid?
- Terms: S = poets, P = graduates, M = teachers.
- Universal first: 'All teachers are graduates', so shade the part of the teachers circle outside graduates.
- Particular next: 'Some teachers are poets'. The only unshaded part of the teachers-poets overlap lies inside graduates, so the X goes there.
- Conclusion check: 'Some poets are graduates' needs an X in the poets-graduates overlap. It is already there.
Valid (AII in figure 3).
Worked example 2: an invalid syllogism
All lawyers are graduates. All teachers are graduates. Therefore, all teachers are lawyers. Valid or invalid?
- Terms: S = teachers, P = lawyers, M = graduates.
- Shade lawyers outside graduates. Shade teachers outside graduates.
- Conclusion check: 'All teachers are lawyers' needs every part of teachers outside lawyers shaded.
- The region teachers-graduates-not lawyers is still unshaded.
Invalid: undistributed middle ('graduates' is never distributed).
How NET asks this: two premises are given and you choose which conclusion(s) follow, or whether a syllogism is valid. A quick Venn sketch settles it in under a minute.
Typical MCQ: “'No poets are rich. Some teachers are poets.' Which conclusion follows?” 'Some teachers are not rich' (EIO, valid). 'No teachers are rich' does not follow.
Typical MCQ: “Where does the X go when a particular premise could fit two sub-regions?” On the line between them, because the premise does not say which part holds the member.
Shade for all and no, X for some. Universals go first, and never draw the conclusion: just look for it.