Logical Reasoning Cheat Sheet: Propositional & Categorical Logic - IIM CAT 2026

CAT Pattern & Weightage Analysis

In the IIM CAT DILR section, formal propositional logic and syllogisms form the structural backbone of complex arrangement sets, binary logic puzzles, and conditional reasoning questions. Understanding truth tables, implication laws, and valid arguments helps aspirants eliminate incorrect cases rapidly under timed conditions.

• Exam Context: IIM CAT 2026
• Core Focus Areas: Implication, Contrapositive equivalence, Modus Ponens/Tollens, Categorical Propositions (A, E, I, O)
• Application: Binary Logic, Truth/Lie puzzles, Deductive Logic sets, Selection Criteria

Chapter 1: Propositional Logic & Truth Conditions

A proposition is a declarative statement that evaluates strictly to either True (\(T\)) or False (\(F\)). Logical connectives combine propositions into composite compound statements.

Basic Connectives Table

Conditional Variations

For a primary conditional statement \(p \to q\):

• Converse: \(q \to p\)
• Inverse: \( eg p \to eg q\)
• Contrapositive: \( eg q \to eg p\) (Logically equivalent to original statement \(p \to q\))

Chapter 2: Essential Logical Equivalences

Transforming logical expressions into simplified or equivalent forms is critical when analyzing constraints in CAT logic sets.

De Morgan's Laws

\[ eg(p \lor q) \equiv eg p \land eg q\]

\[ eg(p \land q) \equiv eg p \lor eg q\]

Implication Law

\[p \to q \equiv eg p \lor q\]

Distributive Laws

\[p \lor (q \land r) \equiv (p \lor q) \land (p \lor r)\]

\[p \land (q \lor r) \equiv (p \land q) \lor (p \land r)\]

Chapter 3: Rules of Inference & Valid Deductions

Rules of inference establish valid logical steps to derive conclusions from given premises.

Modus Ponens (Law of Detachment)

If \(p \to q\) is true and \(p\) is true, then \(q\) must be true.

\[\frac{p \to q, \quad p}{\therefore q}\]

Modus Tollens (Law of Contraposition)

If \(p \to q\) is true and \(q\) is false, then \(p\) must be false.

\[\frac{p \to q, \quad eg q}{\therefore eg p}\]

Hypothetical Syllogism

Chaining conditional statements together.

\[\frac{p \to q, \quad q \to r}{\therefore p \to r}\]

Disjunctive Syllogism

Eliminating an alternative in a disjunction.

\[\frac{p \lor q, \quad eg p}{\therefore q}\]

Constructive Dilemma

\[\frac{(p \to q) \land (r \to s), \quad p \lor r}{\therefore q \lor s}\]

Chapter 4: Syllogisms & Categorical Logic

Categorical logic deals with relationships between classes using standard quantifiers: All, Some, and No.

Standard Categorical Propositions

• Universal Affirmative (A): All \(S\) are \(P\).
• Universal Negative (E): No \(S\) are \(P\).
• Particular Affirmative (I): Some \(S\) are \(P\).
• Particular Negative (O): Some \(S\) are not \(P\).

Relationships in the Square of Opposition

• Contraries (A and E): Cannot both be true simultaneously, but both can be false.
• Subcontraries (I and O): Cannot both be false simultaneously, but both can be true.
• Subalterns (A to I, E to O): Truth flows downward; falsity flows upward.
• Contradictories (A and O, E and I): Always have opposite truth values.

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