Deductive Logic/Categorical Sentence Schemata/Example Modus Bocardo (OAO-3) syllogisms
Understanding Modus Bocardo (OAO–3)[1]
| Role | Quantity–Quality | Logical Form |
| Major Premise (O) | Particular Negative | Some M are not P |
| Minor Premise (A) | Universal Affirmative | All M are S |
| Conclusion (O) | Particular Negative | ∴ Some S are not P |
Figure 3 rule:
The middle term (M) is subject of both premises.
The reasoning holds because some M’s (that are not P) are included in S (since all M are S), so those S must not be P.
Ten Valid Modus Bocardo (OAO–3) Syllogisms
| # | Major Premise (O) | Minor Premise (A) | Conclusion (O) |
| 1 | Some mammals are not carnivores. | All mammals are animals. | ∴ Some animals are not carnivores. |
| 2 | Some metals are not magnetic. | All metals are elements. | ∴ Some elements are not magnetic. |
| 3 | Some poets are not philosophers. | All poets are writers. | ∴ Some writers are not philosophers. |
| 4 | Some triangles are not equilateral. | All triangles are polygons. | ∴ Some polygons are not equilateral. |
| 5 | Some students are not athletes. | All students are people. | ∴ Some people are not athletes. |
| 6 | Some birds are not flyers. | All birds are animals. | ∴ Some animals are not flyers. |
| 7 | Some laws are not just. | All laws are human creations. | ∴ Some human creations are not just. |
| 8 | Some medicines are not safe. | All medicines are chemical compounds. | ∴ Some chemical compounds are not safe. |
| 9 | Some books are not novels. | All books are works of literature. | ∴ Some works of literature are not novels. |
| 10 | Some humans are not kind. | All humans are animals. | ∴ Some animals are not kind. |
All follow: Some M are not P; All M are S; ∴ Some S are not P.
Valid vs. Invalid Bocardo Syllogisms Comparison
| # | Valid Example | Invalid Example | Why Invalid |
| 1 | Some mammals are not carnivores. All mammals are animals. ∴ Some animals are not carnivores. | All mammals are not carnivores. All mammals are animals. ∴ Some animals are not carnivores. | Major premise made universal (A→O error). Universal negative (A?) invalid mood (A A O). |
| 2 | Some metals are not magnetic. All metals are elements. ∴ Some elements are not magnetic. | Some elements are not metals. All metals are elements. ∴ Some elements are not magnetic. | Middle term misplaced — wrong figure (M not subject in both). |
| 3 | Some poets are not philosophers. All poets are writers. ∴ Some writers are not philosophers. | Some writers are not poets. All poets are writers. ∴ Some writers are not philosophers. | Middle term reversed in minor → figure violation. |
| 4 | Some triangles are not equilateral. All triangles are polygons. ∴ Some polygons are not equilateral. | No triangles are equilateral. All triangles are polygons. ∴ Some polygons are not equilateral. | Major premise universal negative (E) instead of particular (O) → wrong mood (E A O). |
| 5 | Some students are not athletes. All students are people. ∴ Some people are not athletes. | Some athletes are not students. All students are people. ∴ Some people are not athletes. | Middle term predicate once, subject once → not figure 3. |
| 6 | Some birds are not flyers. All birds are animals. ∴ Some animals are not flyers. | Some animals are not birds. All birds are animals. ∴ Some animals are not flyers. | Middle term mispositioned → violates 3rd figure structure. |
| 7 | Some laws are not just. All laws are human creations. ∴ Some human creations are not just. | Some laws are unjust. All human creations are laws. ∴ Some human creations are not just. | Reverses subject/predicate roles → figure 1 pattern. |
| 8 | Some medicines are not safe. All medicines are chemical compounds. ∴ Some chemical compounds are not safe. | Some unsafe things are medicines. All medicines are chemical compounds. ∴ Some chemical compounds are not safe. | Major premise not in proper form (O) → “unsafe things” breaks middle term role. |
| 9 | Some books are not novels. All books are works of literature. ∴ Some works of literature are not novels. | No books are novels. All books are works of literature. ∴ Some works of literature are not novels. | Major premise universal negative (E), not particular (O). |
| 10 | Some humans are not kind. All humans are animals. ∴ Some animals are not kind. | Some animals are not humans. All humans are animals. ∴ Some animals are not kind. | Middle term reversed in minor → wrong figure (becomes figure 1). |
Common Fallacies in Invalid Bocardo Forms
| Fallacy | Description |
| Wrong Mood | Major premise must be particular negative (O). If it’s universal negative (E) or affirmative (A), invalid. |
| Wrong Figure | In figure 3, middle term must be subject in both premises; reversing one breaks the structure. |
| Illicit Middle Term | M not distributed or positioned properly. |
| Illicit Major Term | A term distributed in the conclusion but not in its premise. |
| Existential Error | If both premises are universal, a particular conclusion (O) is unjustified. |
| Reversed Roles | Switching subject and predicate destroys figure structure. |
Summary of Modus Bocardo (OAO–3)
| Property | Description |
| Mood | O A O (Particular Negative + Universal Affirmative → Particular Negative) |
| Figure | 3 (M subject in both premises) |
| Form | Some M are not P; All M are S; ∴ Some S are not P |
| Conclusion Type | Particular Negative (O) |
| Essence of Validity | Some of the M that are not P must also be S (since all M are S); therefore, some S are not P. |