Connectives

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.