Binary Logic (Truth-teller / Liar)

Overview

Binary Logic problems involve statements that are either true or false, with constraints specifying how many (or which) statements from a set can be true simultaneously. On CAT, these are high-difficulty DILR questions where each person makes multiple statements, and you must identify who is lying. On GMAT, basic true/false logic appears in Data Sufficiency and Verbal. The skill is using the mutual constraints to pin down the truth assignment.

Core Setup (CAT format)

Each person makes a set of statements. Constraints specify:

• How many of the statements from each person are true (0, 1, all, etc.)
• Relationships between statements (if statement 1 is true, then statement 2 must be false, etc.)

Goal: Find the truth assignment (which statements are true) that satisfies all constraints.

Types of Binary Logic Problems

Type 1: Exactly one true

Each person makes n statements; exactly one is true. Work out which one by process of elimination.

Type 2: Knight-Knave (Truthers and Liars)

Knights always tell the truth; Knaves always lie. Given a set of statements, determine who is a knight and who is a knave.

Key insight: A knave's statement is false. So if a knave says "X is true," then X is false.

Type 3: Numbered rows/columns with true/false counts

A grid of statements: row i, column j has a statement. The number of true statements in each row and column is given. Solve the binary grid.

Logical Connectives

• AND (∧): True only when both A and B are true
• OR (∨): True when at least one of A, B is true
• NOT (¬): Flips truth value
• IF…THEN (→): A → B is false only when A is true and B is false
• IF AND ONLY IF (↔): True when A and B have the same truth value

Contrapositive: A → B ≡ ¬B → ¬A (logically equivalent)

Inverse: ¬A → ¬B (NOT equivalent to A → B)

Strategy for CAT Binary Logic Sets

1. Identify the constraint type — exactly k true, or knight-knave, or row/column counts
2. Try one assumption — assume person X is a knight (tells truth). Work out the implications. If a contradiction arises, X must be a knave.
3. Use self-referential statements carefully — "I am a knight" is true for a knight and false for a knave (knave would say "I am a knight" → false → knave, consistent). So this statement alone doesn't distinguish.
4. Pin the truth of one statement — once one statement's truth is determined, propagate through connected statements.

Row-Column Grid Example

A grid where each cell is T or F. Column 1 has 2 trues, column 2 has 1 true. Row 1 has 1 true, row 2 has 2 trues. Solve:

Start with row 2 (2 trues) — both cells true: C1R2=T, C2R2=T. Column 2 has 1 true → C2R1 must be false. Column 1 has 2 trues → C1R1=T. Verify row 1: C1R1=T, C2R1=F → 1 true. ✓

Common Mistakes

• Confusing "IF A THEN B" with "IF B THEN A" — these are not equivalent
• In knight-knave: "both are knights" or "both are knaves" — always check consistency of both types
• Forgetting that a false conditional (A → B where A is false) is technically TRUE — the whole "if A then B" statement is true when A is false, regardless of B

Exam Tips

• For CAT: binary logic sets often have a unique solution — if you get two valid assignments, you've missed a constraint
• Try the most constrained person or statement first — the one most directly limited by other constraints
• Use a truth table for small problems (3–4 variables); for larger, use elimination
• Knight-knave shortcut: if two people say "we are the same type," they ARE the same type (both knights or both knaves). If they say "we are different types," they ARE different.