Syllogism – Advanced Study Notes
Madhya Pradesh TET – Reasoning Ability
(Level: Advanced / University – MIT‑style)
Syllogism is a form of deductive reasoning that uses propositions (premises) to derive a logical conclusion. It is the backbone of many reasoning‑based examinations, including the TET syllabus. Mastery of syllogistic reasoning demands an understanding of formal logic, truth‑functional semantics, and the nuances of categorical inference.
1. From First Principles
| Concept | Formal Definition | Why It Matters |
|---|---|---|
| Proposition | A declarative sentence that is either true or false. | Syllogisms are built from propositions. |
| Premise | A proposition that is assumed to be true within a logical argument. | Provides the premises for inference. |
| Conclusion | A proposition that follows logically from the premises. | The goal of the syllogistic argument. |
| Logical Connective | Operators such as ∧ (and), ∨ (or), ¬ (not), → (implies), ↔ (iff). | They form compound propositions. |
| Validity | An argument is valid if whenever the premises are true, the conclusion is true. | Validity is independent of truth of premises. |
| Soundness | An argument is sound if it is valid and its premises are actually true. | Only sound arguments guarantee a true conclusion. |
The TET syllogisms are always valid. The challenge is to recognize the structure quickly and deduce the conclusion.
2. Classical Syllogistic Logic (CPL)
2.1 Categorical Statements
| Symbol | Ordinary Form | Logical Form | Example |
|---|---|---|---|
| A | All A are B | ∀x (A(x) → B(x)) | All dogs are mammals. |
| E | No A are B | ∀x (A(x) → ¬B(x)) | No cats are reptiles. |
| I | Some A are B | ∃x (A(x) ∧ B(x)) | Some apples are sweet. |
| O | Some A are not B | ∃x (A(x) ∧ ¬B(x)) | Some oranges are not sweet. |
2.2 Mood and Figure
| Figure | Arrangement of Middle Term |
|---|---|
| 1 | P M → S M |
| 2 | M P → M S |
| 3 | P M → S M (but in reverse order) |
| 4 | M P → S M (reversed) |
Mood is the sequence of the three proposition types (e.g., AAA).
Figure indicates where the middle term appears.
2.3 Rules of Validity (V1–V10)
- V1 (Universal Instantiation) – From A or E we can infer an I or O respectively.
- V2 (Existential Generalization) – From an I we infer an E or O.
- V3 (Conversion) – From A we can derive A (reverse).
- V4 (Obverse) – Negate the predicate: A ↔ E, I ↔ O.
- V5 (Contrapositive) – From A derive an A by swapping subject and predicate.
- V6 (Syllepsis) – Syllogistic combination.
- V7 (Disjunctive Syllogism) – From P ∨ Q and ¬P we infer Q.
- V8 (Conditional Syllogism) – From P → Q and Q → R infer P → R.
- V9 (De Morgan) – For compound propositions.
- V10 (Transitivity) – If P → Q and Q → R then P → R.
In TET, the most common moods are AAA 1, EAE 1, AII 3, etc.
3. Truth‑Functional Semantics
3.1 Truth Tables
| Premise 1 | Premise 2 | Conclusion | Valid? |
|---|---|---|---|
| P | Q | P ∧ Q | ✅ |
| P | Q | P ∨ Q | ✅ |
| P | Q | P → Q | ✅ |
| P | Q | P ↔ Q | ✅ |
| P | ¬P | (P ∨ Q) | ❌ (Contradiction) |
- Conjunction (∧) and implication (→) are the core for syllogistic inference.
- A syllogism is a truth‑functional argument iff its truth table contains no row where premises are true and conclusion false.
3.2 Edge Cases
| Edge Case | Description | Handling |
|---|---|---|
| Duality | Statements that are both true and false (e.g., “I am dead” during the day). | Avoided in formal syllogisms; such statements render the argument invalid. |
| Quantifier Ambiguity | “All” vs. “Some” can be misinterpreted. | Use exact logical forms (∀, ∃). |
| Nested Quantifiers | “All X that are Y are Z” → ∀x((X(x) ∧ Y(x)) → Z(x)). | Requires careful parsing. |
| Contradictory Premises | P and ¬P as premises → explosion (any conclusion follows). | Recognize as inconsistent; the syllogism is invalid. |