Syllogism

Syllogism

Advanced University‑Level Study Notes
Reasoning Ability – Maharashtra TET (Teacher Eligibility Test)

1. Introduction

A syllogism is a deductive argument that contains two premises (major and minor) and one conclusion. It is the cornerstone of classical categorical logic (Aristotle, Boole, Frege, etc.) and provides a systematic framework for reasoning about categorical propositions.

Example

1. All mammals are warm‑blooded.
2. All dogs are mammals.
3. Therefore, all dogs are warm‑blooded.

In syllogistic reasoning, the form of the argument (its mood and figure) determines whether the argument is valid, regardless of the truth of its premises.

2. Categorical Propositions – First Principles

2.1. Key Principles

1. Distribution – A term is distributed in a proposition if the proposition refers to all members of that class.

   • A: S distributed, P not distributed.
   • E: Both S and P distributed.
   • I: Neither term distributed.
   • O: S distributed, P not distributed.

2. Quality – Affirmative (A, I) vs Negative (E, O).

3. Quantity – Universal (A, E) vs Particular (I, O).

These attributes are the building blocks for determining validity.

3. Syllogistic Figures

A syllogism’s figure is defined by the position of the middle term (M) in the premises:

Illustration:

Figure 1
   (M → P)      (S → M)
   ---------   ---------
      P           S

Figure 4
   (P → M)      (M → S)
   ---------   ---------
      M           S

4. Syllogistic Moods

A mood specifies the types of the three propositions (A, E, I, O).
The classical system identifies 19 valid moods (Aristotle, Boole, etc.).
For example, the syllogism Barbara is AAA-1 (A, A, A in Figure 1).

Why 19?
Each mood must satisfy the Rules of Validity (see §5).
When the premises are true, the conclusion must be true.

5. Rules of Validity (from Aristotle & Boole)

1. Rule of Identity – The conclusion can’t introduce a new term.
2. Rule of Distribution –
   a. The middle term must be distributed at least once in the premises.
   b. If the conclusion distributes a term, that term must be distributed in at least one premise.
3. Rule of Quality – A negative premise cannot lead to a positive conclusion, and vice versa.
4. Rule of Quantity –
   a. A universal premise can’t support a particular conclusion unless the conclusion is universal or the premises are particular (contraposition).
   b. Two particular premises cannot support a universal conclusion (unless both are I).
5. Rule of the Unimprovable – If the premises are true, the conclusion must be true.
6. Rule of the Middle Term – The middle term must appear in both premises; it must not be contradictory or co‑contradictory to the major or minor term.

Illustrative Proof (Barber — AAA-1)

1. All M are P. (A) – M distributed.
2. All S are M. (A) – S distributed.
3. Hence, All S are P. (A) – Conclusion distributes P? No; P not distributed → satisfies rule 2b.
4. All premises satisfy rules of identity, distribution, quality, quantity → Valid.

6. Edge Cases & Common Pitfalls