Logical operators_

Negation

Negation is the only logical operator that acts on a single proposition: it takes a truth value and flips it. When p is true, ¬p is false; when p is false, ¬p is true. It is the building block from which every other connective can be defined.

Conjunction

Conjunction joins two propositions with the word "and". The expression p ∧ q is true only when both p and q are true at the same time; in every other case it is false. It is the strictest connective in propositional logic.

Disjunction

Disjunction joins two propositions with the word "or". The expression p ∨ q is true when at least one of them is true — including when both are. It is false only when p and q are both false. This inclusive "or" is the default meaning in mathematics.

Conditional

The conditional, also called material implication, expresses "if p, then q". The expression p ⇒ q is false in exactly one case: when the antecedent p is true and the consequent q is false. It is the connective students find most confusing, precisely because it is true whenever p is false.

Biconditional

The biconditional, or double implication, expresses "p if and only if q". The expression p ⇔ q is true when p and q share the same truth value (both true or both false) and false when they differ. It is the operator of equivalence.

Exclusive Disjunction

Exclusive disjunction, known as XOR, expresses "either p or q, but not both". The expression p ⊕ q is true when exactly one of the two propositions is true and false when they agree. It is the everyday exclusive "or" made formal.

NAND

NAND is the negation of conjunction: p ⊼ q is equivalent to ¬(p ∧ q). It is false only when p and q are both true and true in every other case. Its importance goes beyond logic: every Boolean function can be built using NAND alone.

NOR

NOR is the negation of disjunction: p ↓ q is equivalent to ¬(p ∨ q) and reads "neither p nor q". It is true only when p and q are both false. Like NAND, it is a universal operator: it alone suffices to build all of propositional logic.

Converse Conditional

The converse conditional, also called replication or reverse implication, is written p ￩ q and means "p if q" or "p is implied by q". It is simply the conditional with the arrow pointing backwards: p ￩ q is equivalent to q ⇒ p. It is false only when q is true and p is false.

Negated Conditional

The negated conditional, written p ⇏ q, asserts that "p does not imply q": p is true and yet q is false. It is the one situation where the conditional p ⇒ q fails, so p ⇏ q ≡ ¬(p ⇒ q) ≡ p ∧ ¬q. Understanding it is understanding what it takes to refute an implication.

Negated Biconditional

The negated biconditional, written p ⇎ q, asserts that p and q are not equivalent: they have different truth values. Its table is identical to exclusive disjunction ⊕, so p ⇎ q ≡ ¬(p ⇔ q) ≡ p ⊕ q. It is the natural tool for spotting the rows where two formulas differ.