ConnectiveExampleAlternative Symbols
Negation
Disjunction (or)
Conjunction (and)
Conditional (implies)
Biconditional (equivalent)
Exclusive Or

Order of Precedence and Associativity

Both conjunction and disjunction are associative, so we can write something like without parentheses and don’t need to worry about ambiguity.

To further reduce the number of necessary parentheses, some sources introduce precedence rules for connectives. For the big 5 connectives in propositional logic, we can define precedence and associativity as follows, with operators at the top having higher precedence

Operator

Some sources also define and as right associative. So, for example, is the same as .

Truth Functional Connectives

A connective is truth functional if the truth-value of the compound proposition is entirely determined by the truth values of component propositions.

All the above connectives are truth functional. But we can have non-truth functional connectives if we use casual English. For example, unlike , the English sentence “if then ” does not have a true/false value when is false.

See Also