COLOUR THEME
The four forms A, E, I and O, the traditional square of opposition, existential import, and immediate inference by conversion, obversion and contraposition.
5 sub-topics, in the order to study them. Tap one to jump to it.
Revision mode hides the full explanations and keeps each section's key terms, lists and tables. Use it from your second read onward.
In brief
A standard-form categorical proposition has four parts: quantifier (All / No / Some), subject term (S) , copula (is / are) and predicate term (P) .
A standard-form categorical proposition has four parts: quantifier (All / No / Some), subject term (S), copula (is / are) and predicate term (P). It states a relation between two classes.
In brief
The letters come from the Latin AffIrmo (I affirm) and nEgO (I deny).
The letters come from the Latin AffIrmo (I affirm) and nEgO (I deny).
| Form | Statement | Quantity | Quality | Subject distributed? | Predicate distributed? |
|---|---|---|---|---|---|
| A | All S is P | Universal | Affirmative | Yes | No |
| E | No S is P | Universal | Negative | Yes | Yes |
| I | Some S is P | Particular | Affirmative | No | No |
| O | Some S is not P | Particular | Negative | No | Yes |
Memory rule for distribution: Universals distribute the subject; negatives distribute the predicate. So A distributes S, E distributes both, I distributes neither, O distributes P.
A term is distributed when the proposition says something about every member of the class the term denotes.
In brief
The square arranges A, E, I, O at its corners (A top-left, E top-right, I bottom-left, O bottom-right).
The square arranges A, E, I, O at its corners (A top-left, E top-right, I bottom-left, O bottom-right).
| Relation | Between | Rule |
|---|---|---|
| Contradictory | A–O and E–I (diagonals) | Cannot both be true and cannot both be false. Exactly one is true. |
| Contrary | A–E (top) | Cannot both be true; can both be false |
| Subcontrary | I–O (bottom) | Can both be true; cannot both be false |
| Subalternation | A to I, and E to O (sides) | Truth passes down (universal to particular); falsity passes up (particular to universal) |
Inference table (traditional view):
| If this is... | A | E | I | O |
|---|---|---|---|---|
| A is true | T | F | T | F |
| A is false | F | ? | ? | T |
| E is true | F | T | F | T |
| E is false | ? | F | T | ? |
| I is true | ? | F | T | ? |
| I is false | F | T | F | T |
| O is true | F | ? | ? | T |
| O is false | T | F | T | F |
(? means undetermined.) The shortcut: contradictories always flip; contraries cannot both be true; subcontraries cannot both be false; "true travels down, false travels up".
In brief
The traditional (Aristotelian) view assumes every class named in a proposition has at least one member, so universal propositions imply the existence of their subject.
The traditional (Aristotelian) view assumes every class named in a proposition has at least one member, so universal propositions imply the existence of their subject. This makes subalternation, and the contrary and subcontrary relations, valid.
George Boole rejected this. On the Boolean (modern) view, universal propositions (A and E) make no existence claim, while particular propositions (I and O) do. Consequences: only the contradictory relation survives on the Boolean square; contrary, subcontrary and subalternation fail; and "All unicorns are white" does not imply "Some unicorns are white". The existential fallacy is committed when a syllogism with two universal premises draws a particular conclusion, which is valid only on the Aristotelian view.
In brief
Immediate inference draws a conclusion from one premise .
Immediate inference draws a conclusion from one premise. The three classical operations are conversion, obversion and contraposition.
Conversion (interchange S and P, keep quality):
| Original | Converse | Type |
|---|---|---|
| A: All S is P | Some P is S | By limitation (per accidens) |
| E: No S is P | No P is S | Simple |
| I: Some S is P | Some P is S | Simple |
| O: Some S is not P | Not valid | O cannot be converted |
Obversion (change the quality, replace the predicate with its complement; valid for all four):
| Original | Obverse |
|---|---|
| A: All S is P | No S is non-P |
| E: No S is P | All S is non-P |
| I: Some S is P | Some S is not non-P |
| O: Some S is not P | Some S is non-P |
Contraposition (replace S with non-P and P with non-S; equals obvert, convert, obvert):
| Original | Contrapositive |
|---|---|
| A: All S is P | All non-P is non-S |
| E: No S is P | Some non-P is not non-S (by limitation) |
| I: Some S is P | Not valid |
| O: Some S is not P | Some non-P is not non-S |
Memory: O cannot be converted; I cannot be contraposed; A and O are the only forms that contrapose simply; obversion works for everything.
Practise with free MCQs and answer keys, then return here to fix what you got wrong.
↑ Module 1: Western Logic: Arguments and Categorical Reasoning · All 19 topics in Logical Reasoning · Unit overview and sample MCQs
Last updated: . Notes are a revision aid; check official UGC sources for any rule or scheme that changes over time. Spotted a mistake? Write to netlawverse@gmail.com.