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|
|\(\vee\)||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.|