# Propositional calculus

## Branch of logic / From Wikipedia, the free encyclopedia

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The **propositional calculus**^{[lower-alpha 1]} is a branch of logic.^{[1]} It is also called **propositional logic**,^{[2]} **statement logic**,^{[1]} **sentential calculus**,^{[3]} **sentential logic**,^{[1]} or sometimes **zeroth-order logic**.^{[4]}^{[5]} It deals with propositions^{[1]} (which can be true or false)^{[6]} and relations between propositions,^{[7]} including the construction of arguments based on them.^{[8]} Compound propositions are formed by connecting propositions by logical connectives representing the truth functions of conjunction, disjunction, implication, biconditional, and negation.^{[9]}^{[10]}^{[11]}^{[12]} Some sources include other connectives, as in the table below.

Unlike first-order logic, propositional logic does not deal with non-logical objects, predicates about them, or quantifiers. However, all the machinery of propositional logic is included in first-order logic and higher-order logics. In this sense, propositional logic is the foundation of first-order logic and higher-order logic.

Propositional logic is typically studied with a formal language, in which propositions are represented by letters, which are called *propositional variables*. These are then used, together with symbols for connectives, to make compound propositions. Because of this, the propositional variables are called *atomic formulas* of a formal zeroth-order language.^{[10]}^{[2]} While the atomic propositions are typically represented by letters of the alphabet,^{[10]} there is a variety of notations to represent the logical connectives. The following table shows the main notational variants for each of the connectives in propositional logic.

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Connective | Symbol |
---|---|

AND | $A\land B$, $A\cdot B$, $AB$, $A\&B$, $A\&\&B$ |

equivalent | $A\equiv B$, $A\Leftrightarrow B$, $A\leftrightharpoons B$ |

implies | $A\Rightarrow B$, $A\supset B$, $A\rightarrow B$ |

NAND | $A{\overline {\land }}B$, $A\mid B$, ${\overline {A\cdot B}}$ |

nonequivalent | $A\not \equiv B$, $A\not \Leftrightarrow B$, $A\nleftrightarrow B$ |

NOR | $A{\overline {\lor }}B$, $A\downarrow B$, ${\overline {A+B}}$ |

NOT | $\neg A$, $-A$, ${\overline {A}}$, $\sim A$ |

OR | $A\lor B$, $A+B$, $A\mid B$, $A\parallel B$ |

XNOR | $A\odot B$ |

XOR | $A{\underline {\lor }}B$, $A\oplus B$ |

The most thoroughly researched branch of propositional logic is **classical truth-functional propositional logic**,^{[1]} in which formulas are interpreted as having precisely one of two possible truth values, the truth value of *true* or the truth value of *false*.^{[15]} The principle of bivalence and the law of excluded middle are upheld. By comparison with first-order logic, truth-functional propositional logic is considered to be *zeroth-order logic*.^{[4]}^{[5]}