In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is an operator that combines or modifies one or more logical variables or formulas, similarly to how arithmetic connectives like + {\displaystyle +} and − {\displaystyle -} combine or negate arithmetic expressions. For instance, in the syntax of propositional logic, the binary connective ∨ {\displaystyle \lor } (meaning "or") can be used to join the two logical formulas P {\displaystyle P} and Q {\displaystyle Q} , producing the complex formula P ∨ Q {\displaystyle P\lor Q} . Unlike in algebra, there are many symbols in use for each logical connective. The table "Logical connectives" shows examples. Common connectives include negation, disjunction, conjunction, implication, and equivalence. In standard systems of classical logic, these connectives are interpreted as truth functions, though they receive a variety of alternative interpretations in nonclassical logics. Their classical interpretations are similar to the meanings of natural language expressions such as English "not", "or", "and", and "if", but not identical. Discrepancies between natural language connectives and those of classical logic have motivated nonclassical approaches to natural language meaning.
Overview In formal languages, truth functions are denoted by fixed symbols, ensuring that well-formed statements have a single interpretation. These symbols are called logical connectives, logical operators, propositional operators, or, in classical logic, truth-functional connectives. For the rules which allow new well-formed formulas to be constructed by joining other well-formed formulas using truth-functional connectives, see well-formed formula. Logical connectives can be used to link zero or more statements, so one can speak about n-ary logical connectives. The boolean constants True and False can be thought of as nullary operators. Negation is a unary connective, and so on.
List of common logical connectives Commonly used logical connectives include the following ones.
Negation (not): ¬ {\displaystyle \neg } , ∼ {\displaystyle \sim } , N {\displaystyle N} (prefix) in which ¬ {\displaystyle \neg } is the most modern and widely used, and ∼ {\displaystyle \sim } is also common; Conjunction (and): ∧ {\displaystyle \wedge } , & {\displaystyle \&} , K {\displaystyle K} (prefix) in which ∧ {\displaystyle \wedge } is the most modern and widely used; Disjunction (or): ∨ {\displaystyle \vee } , A {\displaystyle A} (prefix) in which ∨ {\displaystyle \vee } is the most modern and widely used; Implication (if...then): → {\displaystyle \to } , ⊃ {\displaystyle \supset } , ⇒ {\displaystyle \Rightarrow } , C {\displaystyle C} (prefix) in which → {\displaystyle \to } is the most modern and widely used, and ⊃ {\displaystyle \supset } is also common; Equivalence (if and only if): ↔ {\displaystyle \leftrightarrow } , ⊂ ⊃ {\displaystyle \subset \!\!\!\supset } , ⇔ {\displaystyle \Leftrightarrow } , ≡ {\displaystyle \equiv } , E {\displaystyle E} (prefix) in which ↔ {\displaystyle \leftrightarrow } is the most modern and widely used, and ⊂ ⊃ {\displaystyle \subset \!\!\!\supset } is commonly used where ⊃ {\displaystyle \supset } is also used. For example, the meaning of the statements it is raining (denoted by p {\displaystyle p} ) and I am indoors (denoted by q {\displaystyle q} ) is transformed, when the two are combined with logical connectives:
It is not raining ( ¬ p {\displaystyle \neg p} ); It is raining and I am indoors ( p ∧ q {\displaystyle p\wedge q} ); It is raining or I am indoors ( p ∨ q {\displaystyle p\lor q} ); If it is raining, then I am indoors ( p → q {\displaystyle p\rightarrow q} ); If I am indoors, then it is raining ( q → p {\displaystyle q\rightarrow p} ); I am indoors if and only if it is raining ( p ↔ q {\displaystyle p\leftrightarrow q} ). It is also common to consider the always true formula and the always false formula to be connective (in which case they are nullary).
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