Syllogism: Reasoning Ability Notes for Himachal Pradesh TET (Teacher Eligibility)

Syllogism – Advanced Study Notes for TET Reasoning

Audience: Teacher‑Eligibility (TET) aspirants, 12th‑grade level and above
Goal: Build a solid, university‑level understanding of syllogistic reasoning, from first principles to formal logic, and solve a challenging problem.

1. What is a Syllogism?

Term	Definition	Example
Syllogism	A deductive argument that uses two premises to reach a conclusion, all expressed as categorical propositions.	All mammals are warm‑blooded. All whales are mammals. Therefore, all whales are warm‑blooded.
Categorical Proposition	A statement that places all members of one class (subject) into another class (predicate). It can be universal (All/No) or particular (Some/Some not).	All S are P, No S are P, Some S are P, Some S are not P
Major Premise	The premise that contains the predicate of the conclusion.	All mammals are warm‑blooded.
Minor Premise	The premise that contains the subject of the conclusion.	All whales are mammals.
Conclusion	The inference drawn from the two premises.	All whales are warm‑blooded.

2. Classical Syllogistic Structure

Figure	Form of Premises (Major/Minor)	Example	Mood (Form of Propositions)
1	Major: All M are P Minor: All S are M	All mammals are warm‑blooded. All whales are mammals.	AAA (All‑All‑All)
2	Major: All P are M Minor: All S are M	All warm‑blooded animals are mammals. All whales are mammals.	EAE (No‑All‑No)
3	Major: All M are P Minor: All M are S	All mammals are warm‑blooded. All mammals are whales.	AEE (All‑All‑No)
4	Major: All P are M Minor: All S are M	All warm‑blooded animals are mammals. All whales are mammals.	EAE (No‑All‑No)

Notation
A = Universal affirmative (All S are P)
E = Universal negative (No S are P)
I = Particular affirmative (Some S are P)
O = Particular negative (Some S are not P)

3. Logical Foundations

3.1 First‑Order Logic (FOL) Representation

Categorical Proposition	FOL Formula (assuming unary predicates)
All S are P (A)	(\forall x (S(x) \rightarrow P(x)))
No S are P (E)	(\forall x (S(x) \rightarrow \lnot P(x)))
Some S are P (I)	(\exists x (S(x) \land P(x)))
Some S are not P (O)	(\exists x (S(x) \land \lnot P(x)))

3.2 Validity Test (Algebraic)

A syllogism is valid iff the conclusion follows logically from the premises.
In FOL, we check whether

[ \text{Premises} \vdash \text{Conclusion} ]

is provable, or equivalently whether the set ({\text{Premises} \land \lnot \text{Conclusion}}) is unsatisfiable.

3.3 Venn Diagrams & Euler Circles

Venn: 3‑circle diagrams for three terms (M, S, P).
Euler: Simplified circles, useful for proofs of equivalence.

Both allow visual verification of distribution of terms and the truth‑value of the conclusion.

4. Distribution & Quantifier Rules

Term	Distributed in A	Distributed in E	Distributed in I	Distributed in O
Subject	No	Yes	No	No
Predicate	No	Yes	Yes	Yes

Key Rules

Rule of Distribution – A term must be distributed in at least one premise if it is distributed in the conclusion.
Rule of Conversion – A proposition can be converted (subject ↔ predicate) only if the original is I or O.
Rule of Obversion – Converts A ↔ E, E ↔ A, I ↔ O by negating the predicate.
Rule of Contraposition – Applies to A and O; converts All S are P to All non‑P are non‑S.