There are other logical operators besides ∧, ∨, and ¬. We will consider the conditional operator, →, the biconditional operator, ↔, and the exclusive or operator, ⊕.4 These operators can be completely defined by a truth table that shows their values for the four possible combinations of truth values of p and q.
For any propositions p and q, we define the propositions p → q, p ↔ q, and p ⊕ q according to the truth table:
When these operators are used in expressions, in the absence of parentheses to indicate order of evaluation, we use the following precedence rules: The exclusive or operator, ⊕, has the same precedence as ∨. The conditional operator, →, has lower precedence than ∧, ∨, ¬, and ⊕, and is therefore evaluated after them. Finally, the biconditional operator, ↔, has the lowest precedence and is therefore evaluated last. For example, the expression p →q ∧ r ↔ ¬p ⊕ s is evaluated as if it were written (p → (q ∧ r)) ↔ ((¬p) ⊕ s). But again you should always include the parentheses!
In order to work effectively with the logical operators, you need to know more about their meaning and how they relate to ordinary English expressions. To that end we first consider the conditional operator in more detail in the next section.