>[!abstract] >In [[Propositional logic|propositional logic]], a logical connective is a logical constant used to connect logical formulas. Common connectives include negation, disjunction, conjunction, implication, and equivalence. In standard systems of classical logic, these connectives are interpreted as truth functions. ("Logical connectives", 2025). | Connective | Boolean | English | Infix | Same as | | -------------------------- | ------- | --------------------------------------- | ------------------------- | -------------------------------------------------------------------------------------------------------------------------------------------- | | Conjunction | AND | A and B<br>(both) | $ A \land B $ | | | Material biconditional | XNOR | A if and only if B<br>(both or neither) | $ A \leftrightarrow B $ | $ (A \rightarrow B) \land (B \rightarrow A) $<br>$ (A \land B) \lor (\neg A \land \neg B) $ | | Material conditional | IMPLY | If A then B | $ A \rightarrow B $ | $ \neg(A \land \neg B) $<br>$ \neg A \lor B $ | | Non-conjunction | NAND | Not both A and B | $ A \uparrow B $ | | | Non-disjunction | NOR | Neither A nor B | $ A \downarrow B $ | $ \neg(A \lor B) $<br>$ \neg A \land \neg B $ | | Negation | NOT | Not A | $ \neg A $ | $ A \rightarrow \bot $ | | Disjunction<br>(inclusive) | OR | A and/or B | $ A \lor B $ | $ \neg((\neg A)\land(\neg B)) $<br>$ (\neg A) \rightarrow B $<br>$ (A \rightarrow B) \rightarrow B $ | | Disjunction<br>(exclusive) | XOR | Either A or B | $ A \oplus B $ | $ (A \lor B) \land \neg(A \land B) $<br>$ (A \land \neg B) \lor (\neg A \land B) $<br>$ \neg((A \land B) \lor (\neg A \land \neg B)) $ | ## References - Logical connectives. (2025, January 13). In *Wikipedia*. https://en.wikipedia.org/w/index.php?title=Logical_connective&oldid=1263757771 ## Related - [[Propositional logic|Propositional calculus]]