Logical Connectives

Mathematical Logic · Axiom Academy

The building blocks for constructing complex propositions in mathematical logic In propositional logic, we combine simple propositions into complex ones using logical connectives . These five fundamental operators allow us to express sophisticated logical relationships. 1. The Five Logical Connectives Mathematical logic uses five primary connectives to build compound propositions from simpler ones. Each connective transforms the truth values of its inputs in a specific way. Each connective takes one or more propositions and produces a new proposition with its own truth value. Negation is the simplest connective—it reverses the truth value of a proposition. If p is true, then ¬p is false, and vice versa. The negation symbol ¬ is read as "not" Conjunction is true only when both propositions are true. It represents the logical "and" operation. Disjunction is true when at least one proposition is true. In logic, "or" is inclusive —both can be true simultaneously. Note: In mathematical logic, ∨ always means inclusive OR unless otherwise specified. Implication p → q is read "if p, then q" or "p implies q". It is false only when the hypothesis is true but the conclusion is false. When the hypothesis is false, the implication is automatically true—this is called "vacuously true". The biconditional p ↔ q is read "p if and only if q" (abbreviated "iff"). It is true when both propositions have the same truth value . 7. Order of Operations (Precedence)

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