Propositional Logic Summary

Mathematical Logic · Axiom Academy

Let's review the fundamental building blocks of logical reasoning: propositions, connectives, truth tables, and the laws that govern symbolic logic. Proposition: A declarative statement that is either true (T) or false (F), but never both and never neither Examples of propositions: "2 + 2 = 4" (true), "The Earth is flat" (false), "It is raining" (true or false depending on context) Not propositions: Questions ("Is it raining?"), commands ("Close the door"), opinions without clear truth value Truth values: Binary nature of logic - every proposition is assigned exactly one value: T or F Variables: We use letters like p, q, r to represent arbitrary propositions in logical formulas NOT ( ¬ ): Negation - flips the truth value. If p is true, ¬p is false, and vice versa AND ( ∧ ): Conjunction - true only when both propositions are true. p ∧ q is true iff p is true AND q is true OR ( ∨ ): Disjunction - true when at least one proposition is true. p ∨ q is false only when both are false IMPLIES ( → ): Implication - false only when premise is true and conclusion is false. p → q means "if p then q" IFF ( ↔ ): Biconditional - true when both propositions have the same truth value. p ↔ q means "p if and only if q" Purpose: Systematic method to determine the truth value of compound propositions for all possible combinations of inputs Construction: List all possible combinations of truth values for variables, then compute the result for each row

This is the written version of the interactive lesson above. See the full Mathematical Logic course.