# 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.