Logic is the formal study of reasoning, truth, and inference across mathematics, computer science, philosophy, and artificial intelligence. It provides the foundation for mathematical proof, algorithm design, knowledge representation, and formal verification. At its core, logic distinguishes valid reasoning from fallacious thinking through precise symbolic notation and rigorous rules. A key insight: the validity of an argument depends solely on its structure, not the truth of its content — an argument can be valid with false premises, but a sound argument must have both valid structure and true premises. Modern logic extends far beyond classical two-valued systems to encompass modal, temporal, description, non-monotonic, and type-theoretic frameworks that underpin today's AI, programming languages, and automated reasoning tools.
What This Cheat Sheet Covers
This topic spans 18 focused tables and 158 indexed concepts. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
A jump-to index of every table row in this cheat sheet.
An interactive map of every table and concept in this topic.
Table 1: Logical Connectives and Operators
These are the atoms of formal reasoning—the handful of operators that glue simple propositions into complex statements. Beyond the everyday AND, OR, and NOT, watch the two oddballs at the bottom: the Sheffer stroke (NAND) and Peirce arrow (NOR) are each functionally complete, meaning every other connective can be rebuilt from that single operator alone, a fact that quietly underpins real circuit design.
| Operator | Example | Description |
|---|---|---|
$\neg p$ | Reverses truth value — true becomes false, false becomes true. | |
$p \land q$ | • True only when both operands are true • false otherwise. | |
$p \lor q$ | • True when at least one operand is true • false when both are false. | |
$p \to q$ | • False only when antecedent is true and consequent is false • true otherwise. |