Propositional Logic and Syllogism Cheat Sheet - IIM CAT 2026

Chapter 1: Foundations of Propositional Logic

A proposition is a declarative statement that is deterministically either True (\(T\)) or False (\(F\)), but never both simultaneously. In the IIM CAT DILR section, propositional connective rules govern binary decision trees and conditional arrangement sets.

Core Connectives

• Negation (\( eg P\)): Reverses the truth value of proposition \(P\).
• Conjunction (\(P \land Q\)): Evaluates to True if and only if both statements \(P\) and \(Q\) are True.
• Disjunction (\(P \lor Q\)): Evaluates to True if at least one of the propositions \(P\) or \(Q\) is True.
• Implication (\(P \implies Q\)): Logical conditional where \(P\) is the hypothesis and \(Q\) is the conclusion. Evaluates to False only when \(P\) is True and \(Q\) is False.
• Biconditional (\(P \iff Q\)): Evaluates to True if both \(P\) and \(Q\) share identical truth values.

Truth Table Reference

Chapter 2: Logical Equivalences & Truth Functions

Understanding logical transformation rules allows candidates to simplify complex conditional statements during the exam without constructing full-scale truth tables.

Implication Transformations

• Contrapositive: \(P \implies Q \equiv eg Q \implies eg P\) (Logically Equivalent to original)
• Converse: \(Q \implies P\) (Not logically equivalent)
• Inverse: \( eg P \implies eg Q\) (Not logically equivalent)

De Morgan's Laws

De Morgan's laws dictate how negations distribute over conjunctions and disjunctions:

Distributive Laws

Chapter 3: Syllogisms & Venn Diagrammatic Proofs

A syllogism is a formal deductive argument consisting of a major premise, a minor premise, and a necessary conclusion derived via intersection logic.

Classic Deductive Structure

Premise 1: All humans are mortal.
Premise 2: Socrates is a human.
Conclusion: Socrates is mortal.

Venn Diagram Set Representation

In set-theoretic terms, universal affirmative statements ("All A are B") represent a subset relation \(A \subseteq B\), while existential statements ("Some A are B") signify a non-empty set intersection \(A \cap B eq \emptyset\).

Chapter 4: First-Order Quantifiers

First-order logic introduces quantifiers over domain variables to articulate scope in conditional sets.

• Universal Quantifier (\(\forall\)): Expresses "for all" elements in a domain. Syntax: \(\forall x \, P(x)\).
• Existential Quantifier (\(\exists\)): Expresses "there exists at least one" element in a domain. Syntax: \(\exists x \, P(x)\).

Duality of Quantifier Negation

Chapter 5: Classification of Logical Fallacies

Identifying invalid reasoning patterns is essential for Critical Reasoning questions in CAT VARC and LR sets.

• Ad Hominem: Directing an argument against the opponent's personal traits rather than addressing the substance of their logical claim.
• Strawman: Distorting or oversimplifying a counter-position to facilitate an easy refutation.
• False Dilemma: Erroneously constraining the decision space to two binary choices when plausible alternatives exist.
• Post Hoc Ergo Propter Hoc: Presuming a causal linkage purely on the basis of chronological succession.

CAT Exam Pattern & PYQ Weightage Analysis

In recent iterations of the IIM CAT exam (conducted under NTA/IIM TCS patterns), direct propositional logic and binary logic puzzles account for 1 to 2 complete sets in the Data Interpretation & Logical Reasoning (DILR) section, translating to roughly 12–15% of the total DILR score weightage. Mastery of contrapositives and set intersections directly enhances performance in complex truth-teller vs. liar puzzles and team selection matrices.

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