# Connectives

### From Learning Logic for Computer Science

In an arithmetic expression like \(x + 3 * y\) the second symbol, \(+\), is identified with addition—a function which takes two numbers as arguments and maps them onto a new number. A **connective** is the same in the context of logical formulas: it is a symbol that is used to combine formulas; at the same time it has a meaning in the sense that it operates on truth values.

There are the following aspects to each connective:

- What does the actual symbol look like?
- What is its meaning?—The boolean function associated with it.
- How many formulas does it connect?–Its Arity (or arities).
- What is the name of a formula with that symbol at its root.–Its name.
- How is the symbol read as part of a formula?
- How are the formulas it combines named?
- How strong does it bind when parentheses are ommitted?–Precedence.

# List of connectives

Symbol | Arity | Precedence | Name | How to read | Arguments | Boolean function \(f_C\) | Remarks |
---|---|---|---|---|---|---|---|

\(\neg\) | 1 | 1 | negation | not ... | neg | ||

\(\vee\) | 2 | 2 | disjunction | ... or ... | disjunct | or | |

\(\bigvee\) | 0, 1, 2, ... | 2 | disjunction | ... or ... or ... | disjunct | Or | |

\(\wedge\) | 2 | 2 | conjunction | ... and ... | conjunct | and | |

\(\bigwedge\) | 0, 1, 2, ... | 2 | conjunction | ... and ... and ... | conjunct | And | |

\(\bar\wedge\) | 2 | 2 | exclusion | not both ... and ... | nand | Also known as Sheffer stroke \( \mid \). | |

\(\bar\vee\) | 2 | 2 | neither ... nor ... | nor | Also known as Peirce arrow \(\downarrow\) and Quine's dagger \(\dagger\). | ||

\(\dot\vee\), \(\oplus\) | 2 | 2 | exclusive disjunction | ... either, but not both ... | xor | ||

\(\not\leftrightarrow\) | 2 | 3 | exclusive disjunction | ... either, but not both ... | xor | ||

\(\dot\bigvee\) | 0, 1, 2, ... | 2 | exclusive disjunction | Xor | |||

\(\bigoplus\) | 0, 1, 2, ... | 2 | parity | Parity | Not to be confused with \(\dot\bigvee\). | ||

\(\rightarrow\) | 2 | 3 | conditional | if ..., then ... | antecedent, consequent | cond | |

\(\leftrightarrow\) | 2 | 3 | biconditional | ... if, and only if, ... | bicond | Not to be confused with \(\equiv\), which is a relation between formulas. |