Quantifiers, Negation and Square of Opposition: All, Some, No Statements, Conversion, Obversion and Contraposition for SSC

Statements with the words all, some and no form the base of syllogism, Venn diagram and inference questions. This guide on quantifiers, negation and the square of opposition for SSC explains the four types of categorical propositions, the relations between them, the immediate inferences (conversion, obversion and contraposition) and the correct negation of statements, with solved examples, practice questions and FAQs.

Four Categorical Propositions

In logic, “some” means at least one and does not rule out “all”.

Square of Opposition

• If A (All S are P) is true, then I (Some S are P) is true, E is false and O is false.
• If E (No S is P) is true, then O is true, A is false and I is false.
• If I is true, then E is false; A and O are doubtful (unknown).
• If O is true, then A is false; E and I are doubtful.
• If A is false, then O is true; E and I are doubtful.
• If I is false, then E is true, and O is true; A is false.

Negation (Contradiction) of Statements

The negation of “All students passed” is “Some students did not pass” and not “No student passed” (that is the contrary).

Immediate Inferences

Conversion

Interchange the subject and the predicate.

Obversion

Change the quality (affirmative to negative or the reverse) and replace the predicate by its complement. It is valid for all four propositions.

Contraposition

Replace the subject with the complement of the predicate and the predicate with the complement of the subject.

Solved Examples

Example 1. If “All books are pens” is true, what is the status of “Some books are pens” and “No book is a pen”?
“Some books are pens” is true (subalternation) and “No book is a pen” is false.

Example 2. If “Some cats are dogs” is false, what is the status of “No cat is a dog”?
I false means E is true.

Example 3. Write the converse of “No teacher is a thief”.
No thief is a teacher.

Example 4. Write the negation of “All the members are present”.
Some members are not present.

Example 5. Write the obverse of “All roses are flowers”.
No rose is a non-flower.

Example 6. Write the contrapositive of “All mammals are warm-blooded”.
All non-warm-blooded animals are non-mammals.

Tips for the Exam

• In a possibility (Either–Or) question, the pairs of the contradictory statements (A–O, E–I) make the Either–Or conclusion.
• Use the converse of E and I freely; do not take the converse of A or O as a full statement.
• “Some” does not mean “some but not all” in logic unless the question says so.
• Remember that A and E are contraries, not contradictories.

Quick Tips

• Memorise the contradictory pairs: A–O and E–I.
• Remember that E and I convert simply.
• Remember that A and E are contrary, and I and O are subcontrary.
• Write the negation as the contradictory and not the contrary.

Quick Revision Points

• A: All S are P; E: No S is P; I: Some S are P; O: Some S are not P.
• A and O are contradictory; E and I are contradictory.
• A and E are contrary; I and O are subcontrary.
• E and I convert simply; A converts to I.
• The negation of All is Some not.

Practice Questions With Answers

1. What is the negation of No S is P?
   Answer: Some S are P
2. If All S are P is true, is Some S are P true?
   Answer: Yes
3. What is the converse of Some S are P?
   Answer: Some P are S
4. What is the converse of All S are P?
   Answer: Some P are S
5. Which pair is contradictory with A?
   Answer: O
6. If No S is P is true, what is the status of Some S are P?
   Answer: False
7. Can both A and E be true at the same time?
   Answer: No (contrary), unless S is empty
8. Can both I and O be false at the same time?
   Answer: No (subcontrary)
9. What is the obverse of No S is P?
   Answer: All S are non-P
10. What is the contrapositive of All S are P?
    Answer: All non-P are non-S

How to Practise This Chapter

Practise 10 questions every day and write the steps on paper. Then attempt timed sets from our practice papers and the previous year papers page. See all topics on the Logical Reasoning notes page, read the previous topic Critical Reasoning: Paradox, Flaw and Evaluate.

Frequently Asked Questions

What are A, E, I and O propositions?

The four types: All S are P, No S is P, Some S are P and Some S are not P.

What does some mean in logic?

At least one, possibly all.

What are contradictory statements?

Statements that cannot be both true or both false, such as A and O.

What are contrary statements?

Statements that cannot both be true but can both be false, such as A and E.

What are subcontrary statements?

Statements that cannot both be false but can both be true, such as I and O.

What is conversion?

Interchanging the subject and the predicate.

What is obversion?

Changing the quality and taking the complement of the predicate.

What is the negation of All S are P?

Some S are not P.

Does the O proposition convert?

No.

Is this topic asked in SSC?

Yes. It helps in syllogism, Venn diagrams and either-or questions.

What is the contrapositive of an A proposition?

All non-P are non-S.

What is subalternation?

The relation from the universal to the particular of the same quality.